Gravity measurement analysis techniques

Abstract

Techniques and systems are provided for determining and generating a mass density distribution using gravity measurements acquired by an array of gravity sensors. The gravity sensor may generate a respective signal based on a mass positioned near the gravity sensors. The respective signals indicate a respective gravitational force associated with the mass. Further, techniques provided herein may include generating an image using the mass density distribution.

Description

GRAVITY MEASUREMENT ANALYSIS TECHNIQUES BACKGROUND

[0001] The present disclosure generally relates to techniques for generating or determining a mass density distribution using gravity measurements acquired by multiple gravity sensors.SUMMARY

[0002] It should be understood that these aspects are presented merely to provide the reader with a brief summary of these certain embodiments and that these aspects are not intended to limit the scope of this disclosure. Indeed, this disclosure may encompass a variety of aspects that may not be set forth below.

[0003] The brief summary presented below is intended only to familiarize the reader with certain aspects and contexts of embodiments of the present disclosure without limitation to the claimed subject matter.

[0004] Conventional techniques for detecting or imaging an object involves transmitting EM or acoustic waves through a media and detecting or imaging the object based on attenuation of the waves due to any materials that are present. The attenuation results from interactions (e.g., reflection, absorption, and so on) between the EM or acoustic waves and materials. Certain materials can completely block or shield certain frequencies of EM or acoustic waves, making it difficult to use a single type of EM or acoustic wave source for different applications.

[0005] The problem of imaging and detecting is fundamentally important for a vast number of application domains including satellite imaging, biomedical imaging, ocean surface imaging, the Oil and Gas sector and various others. This application relates to a revolutionary technology to solve the fundamental problem of imaging for these application domains. For example, in the Oil and Gas sector, sub-surface exploration relies on accurate imaging techniques to determine the position and shape of sub-surface hydrocarbon reservoirs. Currently, electromagnetic (EM) or acoustic waves are used for all imagingapplications. These waves cannot penetrate complex environments (such as sea water or salt formations), and EM waves especially cannot penetrate conducting surfaces and sub-surface soil. High intensity radiations of these waves are also harmful to nature and organisms. These challenges together make imaging in unconventional environments costly and invasive.BRIEF DESCRIPTION OF THE DRAWINGS

[0006] These and other features, aspects, and advantages of the present disclosure will become better understood when the following detailed description is read with reference to the accompanying drawings in which like characters represent like parts throughout the drawings, wherein:

[0007] FIG. 1 shows a gravity measurement system that includes one or more measurement collection systems and a mass density analysis system, in accordance with the present disclosure;

[0008] FIG. 2 shows a schematic diagram of a gravity sensor array that may be used in a gravity measurement system of FIG. 1 , in accordance with present disclosure;

[0009] FIG. 3 is a flow diagram of a method for generating a mass density distribution based on gravity measurements acquired by the measurement collection system of FIG. 1, in accordance with the present disclosure;

[0010] FIG. 4 shows a schematic diagram of a point mass measured by a measurement collection system having two sensors, in accordance with the present disclosure;

[0011] FIG. 5 shows a schematic diagram of a point mass measured by a measurement collection system having two sensors, in accordance with the present disclosure;

[0012] FIG. 6 shows a schematic diagram of a target mass measured by a measurement collection system using a discretized domain, in accordance with the present disclosure;

[0013] FIG. 7 shows the schematic diagram of the measurement collection system with sensors imaging the object of FIG. 2 using a discretized domain, in accordance with the present disclosure;

[0014] FIG. 8 shows a graph illustrating example factors for determining a number of gravity sensors and / or arrangement of gravity sensors for obtaining an image with a desired accuracy, in accordance with the present disclosure;

[0015] FIG. 9 shows a schematic diagram of a measurement collection system with gravity sensors imaging an object using a discretized domain with pixels, in accordance with the present disclosure;

[0016] FIG. 10 shows the measurement collection system with gravity sensors imaging the object using a discretized domain with voxels, in accordance with the present disclosure;

[0017] FIG. 11 shows multiple measurement collection systems disposed around a volume of a container, in accordance with the present disclosure;

[0018] FIG. 12A shows a simulated mass density distribution image generated a first number of gravity sensors, in accordance with the present disclosure;

[0019] FIG. 12B shows a simulated mass density distribution image generated a second number of gravity sensors, in accordance with the present disclosure;

[0020] FIG. 12C shows a true image associated with the simulated mass density distribution images of FIGS. 12A and 12B, in accordance with the present disclosure;

[0021] FIG. 13 shows a flow diagram for generating a mass density distribution output, in accordance with the present disclosure;

[0022] FIG. 14 shows an example of a method for pre-processing gravity measurements that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0023] FIG. 15 shows a flow diagram of a method for obtaining matrix information using metadata that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0024] FIG. 16 shows a flow diagram of a method for generating matrix information using a sensor mapping that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0025] FIG. 17 shows a flow diagram of a method for determining a pixel allocation that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0026] FIG. 18 shows a flow diagram of a method for modifying a pixel allocation based on a mass density variation between locations that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0027] FIG. 19 shows a flow diagram of a method for modifying a pixel allocation based on the gravitational potential field that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0028] FIG. 20 shows a flow diagram of a method for solving a matrix by generating block submatrices that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0029] FIG. 21 shows a flow diagram of a method for solving a matrix to determine the mass density distribution using one or more regularization parameters that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0030] FIG. 22 shows a flow diagram of a method for generating a clear image using the mass density distribution output of FIG. 13, in accordance with the present disclosure;

[0031] FIG. 23 shows a flow diagram of a method for generating a clear image using a trained neural network (NN) model architecture that may be utilized in the method of FIG.13, in accordance with the present disclosure;

[0032] FIG. 24 shows a flow diagram of a method for training a neural network model architecture that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0033] FIG. 25 shows a flow diagram of a method for training an autoencoder architecture that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0034] FIG. 26 shows a flow diagram of a method for training a U-net architecture that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0035] FIG. 27 shows a flow diagram of a method for determining an L2 norm regularization parameter that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0036] FIG. 28 shows a flow diagram of a method for adjusting gravity measurements that may be utilized in the method of FIG. 13, in accordance with the present disclosure;

[0037] FIG. 29 shows a graph of mass versus sensor distance, in accordance with the present disclosure;

[0038] FIG. 30 shows a first example of a pixel allocation that is a uniform grid pixel allocation, in accordance with the present disclosure;

[0039] FIG. 31 shows a second example of a pixel allocation that is a boundary concentrated pixel allocation, in accordance with the present disclosure;

[0040] FIG. 32 shows a third example of a pixel allocation that is an adaptive grid, in accordance with the present disclosure; and

[0041] FIG. 33 shows a technique for computing a mass density distribution over a relatively large volume, in accordance with the present disclosure.DETAILED DESCRIPTION

[0042] Certain embodiments commensurate in scope with the present disclosure are summarized below. These embodiments are not intended to limit the scope of the disclosure, but rather these embodiments are intended only to provide a brief summary of certain disclosed embodiments. Indeed, the present disclosure may encompass a variety of forms that may be similar to or different from the embodiments set forth below.

[0043] As used herein, the term “coupled” or “coupled to” may indicate establishing either a direct or indirect connection (e.g., where the connection may not include or include intermediate or intervening components between those coupled), and is not limited to either unless expressly referenced as such. The term “set” may refer to one or more items. Wherever possible, like or identical reference numerals are used in the figures to identify common or the same elements. The figures are not necessarily to scale and certain features and certain views of the figures may be shown exaggerated in scale for purposes of clarification.

[0044] As used herein, the terms “inner” and “outer”; “up” and “down”; “upper” and “lower”; “upward” and “downward”; “above” and “below”; “inward” and “outward”; and other like terms as used herein refer to relative positions to one another and are not intended to denote a particular direction or spatial orientation. The terms “couple,” “coupled,” “connect,” “connection,” “connected,” “in connection with,” and “connecting” refer to “in direct connection with” or “in connection with via one or more intermediate elements or members.”

[0045] Furthermore, when introducing elements of various embodiments of the present disclosure, the articles “a,” “an,” and “the” are intended to mean that there are one or more of the elements. The terms “comprising,” “including,” and “having” are intended to be inclusive and mean that there may be additional elements other than the listed elements. Additionally, it should be understood that references to “one embodiment,” “an embodiment,” or “some embodiments” of the present disclosure are not intended to be interpreted as excluding the existence of additional embodiments that also incorporate the recited features. Furthermore, the phrase A “based on” B is intended to mean that A is at least partially based on B. Moreover, unless expressly stated otherwise, the term “or” isintended to be inclusive (e.g., logical OR) and not exclusive (e.g., logical XOR). In other words, the phrase A “or” B is intended to mean A, B, or both A and B.

[0046] This disclosure relates to systems and methods for determining a mass density distribution using gravity measurements. As referred to herein, the “mass density distribution” refers to data indicating masses in different locations within a physical location (e.g., an area, a volume) of interest where an object may be present. For example, the mass density distribution may be a numerical representation of the masses, such as a matrix with elements that correspond to different physical locations where the object may exist. In particular, the physical location is represented as a discretized domain that approximates the physical location. The discretized domain is subdivided into locations (e.g., pixels, voxels) that approximate portions of the physical space. As such, each element of the matrix for a mass density distribution may include a mass value for a respective location with a discretized domain. The mass density distribution may indicate a presence or absence of an object based on the mass values across the discretized domain (e.g., at each location). Thus, the mass density distribution may be useful for applications that utilize object detection or imaging.

[0047] The gravity measurements (e.g., measurements of gravitational acceleration) used to determine the mass density distribution are measured or acquired by an array of gravity sensors. In general, a gravity sensor is a type of sensor with a sensing component that produces a measurable signal due to presence of masses near the sensing component that subject gravitational forces to the sensing component. The relationship between the gravitational force subjected to the sensing component of the gravity sensor (e.g., the gravity sensor, for simplicity) by an object near the gravity sensor may be expressed by Newton’s Law of gravitation FG— (Gm1m2) / r2, where G is the gravitational force constant, mi is a first object (e.g., the gravity sensor), m2 is a second object (e.g., an object positioned near the gravity sensor), and r is the distance between the object and the sensor. Accordingly, the gravity measurements have a linear relationship with the masses and an inverse square relationship with the distance.

[0048] The gravity sensors may include accelerometers, gravimeters (e.g., absolute gravimeters, relative gravimeters, spring gravimeters, magnetic field gravimeters) or anycombination thereof. As further non-limiting examples, the gravity sensor may be a microelectromechanical systems (MEMS) sensors, levitated opto-mechanical systems (LOMS) sensor, or a combination thereof. An example LOMS sensor is described in Phys. Rev. A 98, 053831, (2018), |https: / / doi.orq / 10.1103 / PhysRevA.98.05383l|, which is incorporated herein by reference for all purposes. For a LOMS sensor, the sensing component is a rotating nanometer sized material (e.g., a silicon and / or silica nanoparticle) that is suspended or levitating in a chamber using focus lasers (e.g., optical tweezers). When a mass is placed near the LOMS, the angle of rotation of the nanometer sized material changes. The change in the angle of rotation produces a detectable change in an output Newtonian force, which is an example of a gravity measurement that can be used to determine the mass and distance of the object. Although this example of a gravity measurement relates to operation of a LOMS sensor, it should be noted that other types of gravity measurements may be used, such as measurements obtained by gravimeters, MEMS sensors, and other sensors capable of producing a measurement based on gravitational forces subjected to the sensor. Other sensing components of gravity sensors may include spring-based sensing components, falling corner cube gravimeters, and so on. The gravity measurements obtained by these gravity sensors may each be useful for determining or generating the mass density distribution.

[0049] The gravity sensors may be arranged or implemented as a virtual and / or physical array of gravity sensors. In a virtual array, a single gravity sensor is used to obtain multiple gravitational acceleration measurements, such as by moving the single gravity sensor to each position of the array to obtain the gravitational acceleration measurement. Multiple sensors could similarly be moved to different positions to obtain such measurements. In a physical array, a plurality of gravity sensors is positioned at different locations, and thus the array is occupied by physical sensors. As described in more detail in this disclosure, tailoring the shape of the array, the number of sensors, dimensions, and spacing of the array may adjust the resolution and / or processing times of the gravitational acceleration measurements to produce the mass density distribution and / or image. The measurements are then assembled using a processor (e.g., an onboard processor of the measurement collection system) to generate or determine the mass density distribution of the discretized domain, which may be an area (e.g., 2D) or a volume (e.g., 3D) of a mass density distribution, or 4D- including time.

[0050] The mass density distribution may be used to produce certain outputs, such as an alert, a control signal, a visualization (e.g., a two-dimensional (2D) image, a three-dimensional (3D) image, or a four-dimensional (4D) image varying over time) indicating the mass density distribution of the discretized domain. In any case, the output may be used in conjunction with other devices or systems, such as monitoring systems, maintenance systems, imaging systems, or alert systems, as discussed in more detail herein. As one non-limiting example, the disclosed techniques include an imaging device (e.g., camera) that creates a 2D or 3D image using an array of gravity sensors. Detecting objects or imaging using gravitational acceleration measurements provides an advantage over certain existing imaging techniques, such as optical imaging (e.g., imaging using particular wavelengths of light, such as infrared, visible, and so on) and electrostatic imaging. In particular, the gravitational waves can pass through objects or are otherwise not blocked or shielded by materials. As such, the sensors for measuring gravitational acceleration are capable of detecting objects that are behind other objects (e.g., objects behind walls, objects underground), as well as identifying empty space behind other objects (e.g., tunnels, empty containers, and so on). Accordingly, it is presently recognized that it may be advantageous to utilize the disclosed measurement collection system in a variety of applications, such as cargo shipping container monitoring, occupancy detection (e.g., for controlling heating, ventilation, and air condition (HVAC) units and systems), inventory monitoring, monitoring of large masses underwater, and other applications described herein.

[0051] With the foregoing in mind, FIG. 1 shows a diagram of a gravity measurement system 10 that includes a measurement collection system 12 and a mass density analysis system 14. As discussed in further detail herein, the measurement collection system 12 includes multiple gravity sensors 16 that each may detect, measure, or otherwise acquire a gravity measurement that are used to determine a mass density distribution. The mass density distribution indicates the presence of one or more masses from the subjected gravitational force onto the gravity sensors 16. In some embodiments, the mass density distribution may correspond to a predetermined area or volume bounded by and / or surrounding the gravity sensors 16. As discussed herein, the gravity measurement is a signal indicating or related to the gravitational force imparted by masses on gravity sensors 16. The mass density analysissystem 14 obtains or receives the gravity measurements from the gravity sensors 16 (e.g., the measurement collection system 12) and processes the measurements to generate or determine a mass density distribution. In some embodiments, the mass density analysis system 14 may generate an output based on the mass density distribution. For example, the output may include an image of objects within a threshold range of the gravity sensors 16. In some embodiments, the output is a three-dimensional volume of the mass surrounding the gravity sensors 16. In some embodiments, the output indicates the presence of a target mass or target density within the area or volume surrounding the measurement collection system 12.

[0052] To perform the operations described herein, each measurement collection system 12 may include a processor 18, a memory 20, communications circuitry 22, and one or more gravity sensors 16. The processor 18 may be any type of computer processor or microprocessor capable of executing computer-executable code. The processor 18 may also include multiple processors that may perform the operations described below. The memory 20 may be any suitable articles of manufacture that can serve as media to store processorexecutable code, data, or the like. These articles of manufacture may represent computer-readable media (e.g., any suitable form of memory or storage) that may store the processorexecutable code used by the processor 18 to perform the presently disclosed techniques. Generally, the processor 18 may execute software applications that include programs that process gravity measurements according to the embodiments described herein. It should be noted that, although the illustrated embodiment shows the measurement collection system 12 including a processor 18, in some embodiments, the measurement collection system 12 may not include a processor 18. Instead, the measurement collection system 12 may include one or more gravity sensors 16 and a memory 20 that stores the gravity measurements. As described in more detail below, the mass density analysis system 14 may be capable of retrieving or otherwise obtaining the gravity measurements 64 from the memory 20. In some embodiments, the measurement collection system 12 may communicate the gravity measurements 64 to the mass density analysis system 14 using the communications circuitry 22.

[0053] The memory 20 may also be used to store the data, analysis of the data, the software applications, and the like. The memory 20 may represent non-transitory computer-readable media (e.g., any suitable form of memory or storage) that may store the processorexecutable code used by the processor 18 to perform various techniques described herein. It should be noted that non-transitory merely indicates that the media is tangible and not a signal. The communication circuitry 22 may be a wireless or wired communication component that may facilitate communication between the measurement collection systems 12 and the mass density analysis system 14.

[0054] To perform the operations described herein, the mass density analysis system 14 may include a processor 26, a memory 28, communication circuitry 30, an input / output device 32, and a display 34. In general, the processor 26, the memory 28, the communication circuitry 30 may include generally similar features as described above with respect to the processor 18, the memory 20, and the communication circuitry 22. However, the processor 26 of the mass density analysis 14 may have more processing capabilities (e.g., more memory bandwidth) than the measurement collection system 12. As such, it may be advantageous for the mass density analysis system 14 to perform operations described herein such as solving systems of equations or matrices based on the gravity measurements, generating a mass density distribution output, generating an image, or deblurring an image.

[0055] The display 34 may depict visualizations associated with software or executable code being processed by the processor 26 of the mass density analysis system 14. In one embodiment, the display 34 may be a touch display capable of receiving inputs from a user accessing a computing system associated with the mass density analysis system 14. The display 34 may also be used to view a numerical representation of a mass density distribution, a 2D or 3D image generated using the mass density distribution, and the like. The display 34 may be any suitable type of display, such as a liquid crystal display (LCD), plasma display, or an organic light emitting diode (OLED) display, for example.

[0056] As shown, the measurement collection system 12 is separate from the mass density analysis system 14. For example, the gravity measurement system 10 may be communicatively coupled to the mass density analysis system 14 (e.g., via a network).However, it should note that this example is meant to be non-limiting. In some embodiments, the mass density analysis system 14 may include each of the gravity sensors 16 or may otherwise be communicatively coupled to the gravity sensors 16.

[0057] In some embodiments, the measurement collection system 12 may include at least one housing that holds or encapsulates gravity sensors 16. For example, the measurement collection system 12 may include gravity sensors 16 in a housing that maintains the gravity sensors 16 in a relatively fixed position with respect to other gravity sensors 16 in the housing. It should be noted that maintaining gravity sensors 16 in a relatively fixed position may aid the mass density analysis system 14 in determining or generating the mass density distribution by reducing unknown variables corresponding to the gravity measurements acquired by the gravity sensors 16. In some embodiments, the measurement collection system 12 may include multiple housings that each hold a respective set of gravity sensors (e.g., a first housing holding a first set of gravity sensors 16, a second housing holding a second set of gravity sensors 16, a third housing holding a third set of gravity sensors 16, and so on). It should be noted that utilizing multiple sets of gravity sensors 16 may improve the accuracy of the mass density distribution determined by the mass density analysis system 14 by aiding acquisition of multiple gravity measurements at different locations relative to a domain or volume.

[0058] In some embodiments, one or more gravity sensors 16 may be separated from other gravity sensors 16 of the measurement collection system 12 rather than being contained in a housing or a single unit. For example, each gravity sensor 16 may be disposed in different locations within an area and separated by about 1 meter (m) or less, about 5 m or less, about 10 m or less, about 100 m or less, about 1000 m or less. As one specific, non-limiting example, individual gravity sensors 16 may be distributed across space (e.g. a building, site, town, country, etc.) and send information to a central location (e.g., that includes the mass density analysis system 14) for processing.

[0059] As described above, each measurement collection system 12 may be capable of acquiring a gravity measurement of a physical domain (e.g., the physical location) surrounding the measurement collection system 12. To illustrate this, FIG. 2 shows a diagramof a measurement collection system 12 that includes a gravity sensor array 40. As shown, the gravity sensor array 40 includes multiple gravity sensors 16 arranged as a two-dimensional (2D) array of gravity sensors 16 extending along an x-axis 42 (e.g., lateral axis) and a y-axis 44 (e.g., longitudinal axis). However, it should be noted that the gravity sensor array 40 may alternatively be a three-dimensional (3D) array of gravity sensors 16. In embodiments where multiple measurement collection systems 12 are utilized, a portion of the measurement collection systems 12 (e.g., 1, 2, 3, 4, 5, or more than 5) may include a 2D array of gravity sensors 16, while a remaining portion of the measurement collection systems 12 may include a 3D array of gravity sensors 16. At least in some instances, one or more of the measurement collection systems 12 may include a linear array, or even a random array, of gravity sensors 16.

[0060] As shown, the gravity sensor array 40 includes a 4-by-4 array of gravity sensors 16. The 4-by-4 array is shown for simplicity, but the disclosed techniques may be utilized in larger arrays of gravity sensors 16. Although only one gravity sensor array 40 is shown, it should be noted that the measurement collection system 12 may include one or multiple gravity sensor arrays 40. In such embodiments, at least one gravity sensor array 40 may include a different number or arrangement of gravity sensors 16. Additional gravity sensor arrays 40 may be placed such that they surround two or more sides of an object or a physical domain where an object is positioned.

[0061] As described herein, the gravity sensor 16 is a sensor that detects an object based on a change in gravitational forces subjected to the sensing component of the gravity sensor 16. The gravity measurements obtained by the gravity sensors 16 may be a series of numerical values indicating the gravitational force subjected to the respective sensing components of the gravity sensors 16. The gravity measurements may include time stamps indicating when a gravity measurement was obtained and / or metadata indicating the particular gravity sensor 16 and / or gravity sensor array 40 that obtained the gravity measurements. The gravity sensors 16 may include micro-electromechanical systems (MEMS) sensors, levitated opto-mechanical systems (LOMS) sensor, gravimeters, or other devices capable of acquiring gravity measurements. The choice of type of gravity sensors 16may depend on different considerations, such as the sensitivity of the gravity sensors 16, the mass of the object intended to be detected, the speed the object is moving, cost of gravity sensors. For example, it may be advantageous to utilize certain MEMS sensors, which are relatively easier to fabricate as compared to LOMS sensors. However, LOMS sensors may have a higher sensitivity, and thus may provide more granular or higher resolution information about an object and / or may provide more accurate detection of objects having a mass within certain mass threshold ranges. Such considerations of the gravity sensors 16 of the measurement collection system 12 are described in more detail with respect to FIGS. 8 and 30. Depending on the sensitivity, the gravity measurements obtained by the gravity sensors 16 may have 9 or more decimal points. Since gravitational forces of certain detected objects may be relatively low in comparison to other nearby objects, the portion of the gravity measurement that actually correspond to the target mass 52 may be decimal places on the order of 10’4, 10’5, 10’6, 10’7, and so on. In any case, the mass density distribution may be generated by gravity measurements obtained by one or more types of gravity sensors 16.

[0062] In the illustrated embodiment of FIG. 2, the gravity sensors 16 are separated from adjacent gravity sensors 16 by a distance 46 along the x-axis 42. Further, the gravity sensors 16 are separated from adjacent gravity sensors 16 by a distance 48 along the y-axis 44. The distance 46 and / or distance 48 may be the same or different between adjacent gravity sensors 16 of the gravity sensors array 40. As described in more detail with respect to FIG. 8, it may be advantageous to arrange the gravity sensors 16 based on the mass density of an imaging target, the relative location of the imaging target, and so on.

[0063] Each gravity sensor 16 has an alignment 50 that relates to the orientation of the sensing component. As used herein, the “sensing component” of the gravity sensor 16 refers to a mass of the gravity sensor 16 that undergoes a change in a property (e.g., a rotational angle) when subjected to a gravitational force by a mass proximate to the gravity sensor 16 (e.g., the target mass 52). For example, in an embodiment where the gravity sensor 16 is a LOMS device, the sensing component is the levitating particle. The alignment 50 of the LOMS device may refer to the direction of precession of the levitating particle. In an embodiment where the gravity sensor 16 is a MEMS device, the alignment of a MEMS deviceis a vector that is substantially parallel to a direction of oscillation of the mass coupled to the capacitor of the MEMS device.

[0064] The alignment 50 is represented as a vector having a component that extends along the x-axis 42 and / or the y-axis 44, but it should be noted that the alignment 50 may also have a component that extends along a third axis (e.g., a z-axis that is perpendicular to the x-axis 42 and the y-axis 44). The alignment 50 may correspond to an asymmetry (e.g., a material composition asymmetry, mass asymmetry, or shape asymmetry) in the particle of the gravity sensor 16. In some embodiments, the sensing component may be an anisotropic particle (e.g., an ellipsoidal particle, a rod-like particle, or otherwise a particle having at least one dimension longer than another dimension). As such, the alignment 50 corresponds to direction the longest dimension of the anisotropic particle is extending.

[0065] As shown, the alignment 50 of the gravity sensors 16 do not all point the same direction (e.g., the gravity sensors 16 are “randomly” aligned). It should be noted that there are advantages to both randomly aligned gravity sensors 16 and pointing all of the alignments 50 in the same direction. Arranging the gravity sensors 16 such that their respective alignments 50 point towards the target mass 52 may maximize the signal acquired by the gravity sensors 16 since the sensing component of the gravity sensors 16 will experience the greatest gravitational force. However, it is presently recognized that it may be difficult to arrange all of the gravity sensors 16 in this manner. For example, it may be difficult to determine the alignment 50 of a particular gravity sensor 16 with a direct measurement due to the size of the gravity sensor 16. Instead, it may be easier and less time-consuming to determine the alignment 50 of all of the gravity sensors 16 analytically. As a non-limiting example, the alignments 50 of all the gravity sensors 16 may be solved analytically using a system of equations (e.g., Cx — b) used to solve for the masses at each location 55, which is described in more detail in FIG. 3.

[0066] Having a “random” or “semi -random” alignment 50 for the gravity sensors 16 may also be useful in certain applications. For example, the location of the target mass 52 may not be known, and thus, it may not advantageous to arrange all of the gravity sensors 16 such that their respective alignments 50 point in a particular direction. Instead, when thealignments 50 are “random” or “semi-random, the gravity sensor array 40 is likely to have at least one alignment 50 pointing towards a location 55, and thus an improved signal for different directions. This increases the likelihood of obtaining more accurate data of the target masses 52 in different locations of the physical domain. Moreover, having a gravity sensor 16 that does not obtain a maximum signal (e.g., its alignment 50 is orthogonal to the gravitational force vector at a location) may be useful as negative information since it may confirm that a mass is not present in a particular location 55.

[0067] In operation, the gravity sensors 16 each acquire gravity measurements that may be used to detect a target mass 52 within physical domain. To facilitate calculating the mass density distribution, the physical domain is represented as a discretized domain 54 which include locations 55 that are discretized to provide an approximation of the physical domain. To simplify discussion, detecting the target mass 52 is discussed with respect to the locations 55 of the discretized domain 54 rather than the physical location where the target mass 52 exists. For simplicity, the target mass 52 is represented as a 2D shape (i.e., a circle), but it should be noted that the target mass 52 may include more complex or irregular shapes, such as 3D shapes, concave shapes, irregular shapes, and so on. In some embodiments, the target mass 52 is an inanimate object. In some embodiments, the target mass 52 (e.g., object) may be a living thing, such as a person or animal.

[0068] The discretized domain 54 is subdivided into the locations 55 (e.g., pixels, voxels) that represent a portion of the total area or volume of the discretized domain 54. The discretized domain 54 has a pixel allocation 56, which refers to the distribution of the locations 55 within the discretized domain 54. For example, the pixel allocation 56 may refer to the size of each location 55, the shape of each location 55, the arrangement of each location 55, the number of locations 55 within the discretized domain 54, or a combination thereof.

[0069] The pixel allocation 56 of the discretized domain 54 in FIG. 2 is a uniform grid. Other types of pixel allocations 56 that may be used are a boundary concentrated grid where there are more locations 55 (e.g., smaller pixels or voxels) at a boundary of the discretized domain 54 that is nearest to the gravity sensors 16 and the number of locations 55 gradually decreases moving away from the gravity sensors 16. An example boundary concentrated gridpixel allocation 56 is shown in FIG. 31 and described in more detail herein. The pixel allocation 56 may change during the determination or generation of the mass density distribution. For example, the pixel allocation 56 may be an adaptive grid where the mass density analysis system 14 may iteratively adjust (e.g., increase or decrease) a subset of the locations 55 (e.g., increased or decreased) based on a likelihood that the mass values at the subset of locations indicate the target mass 52 is present or absent. An example adaptive grid pixel allocation 56 is shown in FIGS. 32 and 33 and described in more detail herein.

[0070] Dividing the discretized domain 54 into the locations 55 may provide several advantages for processing the gravity measurements to determine or generate the mass density distribution. Processing gravity measurements obtained by gravity sensors 16 is difficult because the gravity sensors 16 cannot be focused on a particular area to, for example raster scan a room to find a target of interest. Instead, gravity sensors 16 may measure most or all of the masses surrounding the gravity sensors 16. Forming the discretized domain 54 allows the mass density distribution to be calculated using a matrix that approximates the physical location. In general, the problem for solving for the mass density distribution is described herein as solving Cx — b.

[0071] By creating a discretized domain 54, each location 55 may be represented as an element in a matrix with a respective index, such as location 55tj (e.g., 55 ij, 551,2, 551,3, and so on). The masses at each location 55ij may be represented as the matrix x, where the order or size of the matrix x match the dimensions of the discretized domain 54. The mass density distribution described herein may be the matrix x or otherwise a numerical or visual representation of the mass values of each element in x. The gravity measurement (e.g., the measured signal) by each gravity sensor 16 may be represented as b. The contribution of each measured signal by a respective gravity sensor 16 at each location 55 may be represented as C, and includes information such as gravity sensor sensitivity, gravity sensor alignment, gravity sensor location relative to a location within the volume, and so on. Each location 55, j is at a known distance from each gravity sensor 16, and thus the problem is to solve x for Cx — b, where the unknowns are the masses. In practical applications, this may be a very difficult problem to solve due the large size of x. However, the pixel allocation 56 of thediscretized domain 54 can be modified, and thus provides an adjustable resolution depending on the desired degree of precision for the mass density distribution. Moreover, having the unknowns as the masses may further aid solving Cx — b. For example, Cx — b may be an ill-posed problem where an exact solution may not be possible. However, depending on the application, a user may provide constraints indicating masses or energies of the physical system that can greatly reduce the parameter space, and provide a stable solution for Cx — b.

[0072] The measurement collection system 12 may transmit the gravity measurements obtained by the gravity sensors 16 to the mass density analysis system 14. The mass density analysis system 14 may have more processing capabilities than the measurement collection system 12, and thus, may be useful for solving the system of equations or matrices to determine or generate a mass density distribution using the gravity measurements. FIG. 3 shows an example flow diagram of a process 60 for determining or generating the mass density distribution.

[0073] At block 62, the mass density analysis system 14 obtains gravity measurements 64. For example, the measurement collection system 12 may send a control signal to the gravity sensors 16 causing the gravity sensors 16 of a gravity sensor array 40 to acquire one or more gravity measurements 64. In some embodiments, the gravity measurements 64 may be obtained by one or more multiple measurement collection systems 12 or gravity sensor arrays 40.

[0074] At block 66, the mass density analysis system 14 generates and solves one or more equations that represent a distribution of the locations 55 within the discretized domain 54, as described with respect to FIG. 2. The one or more equations may be a system of equations or matrices represented as Cx — b for the discretized domain 54. The size of the matrices in Cx — b may be determined by an input such as the matrix information 68. In general, the matrix information 68 includes physics-related information for one or more target masses 52 within the discretized domain 54. The matrix information 68 may include the equations, matrices, and constants that provide the dimensions for Cx — b. For example, the matrix information 68 may include the size of the matrix where each element corresponds to alocation 55 within the discretized domain 54. As such, the matrix information 68 is a mathematical representation of the pixel allocation. Further, the matrix information 68 may include a mass of the gravity sensors 16, an arrangement of the gravity sensors 16, a relative location of the gravity sensors 16 within the measurement collection system 12, a number of gravity sensors 16, a distance from each of the gravity sensors 16 and pixels of a discretized domain (e.g., as described in more detail with respect to FIG. 7), and other information that provides inputs (e.g., other than the unknown masses in x) to Cx — b.

[0075] Cx — b may be an ill-posed problem that cannot be solved analytically when the number of unknowns within x, is larger than the number of independent equations (e.g., the rank of C). Accordingly, solving the system of equations or matrix may utilize numerical methods. In some instances, solving for x may involve applying an inverse optimization technique. While solving Cx — b (e.g., solving an inverse optimization problem) may be difficult due to the large size of these matrices, it is presently recognized that one or more inputs may simplify solving Cx — b. For example, the mass density analysis system 14 may receive inputs, such as pre-process inputs 70, constraints 72, or any combination thereof, that reduce the parameter space of possible solutions for Cx — b.

[0076] The pre-process inputs 70 generally include information about the accuracy of and / or strength of data acquired by the gravity sensors 16. For example, the pre-process inputs 70 may include an accuracy, sensitivity, an alignment 50, a drift factor (e.g., based on sensor temperature and / or age), a sensor mapping, or a combination thereof, of the gravity sensors 16. In some embodiments, the pre-process inputs 70 may include a background gravity factor that may be used to remove noise related to masses outside of the discretized domain 54. As referred to herein, the “background gravity factor” may be a gravity measurement obtained by gravity sensors in an area with a known mass density distribution. For example, it should be noted that the gravity sensors 16 are subject to gravitational waves from any mass, such a nearby gravity sensors 16 within the measurement collection system 12, objects proximate to the discretized domain 54, and even the Earth, moon and other planets. Moreover, the gravity sensors 16 may also have variation in signal due to randomness in orientation, sensor drift, and angular variations (e.g., randomness oforientation), as described in more detail with respect to FIG. 13. Accordingly, it may be advantageous to utilize pre-process inputs 70, including the background gravity factor, to remove and / or adjust signals and / or noise that may inadvertently obscure or adversely affect a signal related to the imaging target.

[0077] The constraints 72 generally include physics-based information to ease solving one or more equations or matrices, such as by reducing the solution space of x to eliminate invalid solutions, limit ranges of possible values, add mathematical relationships between variables, and other techniques for solving ill-posed systems of equations. In some embodiments, the constraints 72 may indicate a mass range or a density range expected to be within the discretized domain 54. For example, the constraint 72 may be a mass range or density range corresponding to steel when the discretized domain 54 includes a shipping container, water when the discretized domain 54 includes a person or animal, and other types of matter expected to be within a particular physical domain represented by the discretized domain 54. As such, the mass density analysis system 14 may solve the one or more equations such that the solution is guided towards a solution that includes masses or densities within the mass ranges or density ranges that represent the physical system within the discretized domain 54 by eliminating potential values for the elements of x that do not correspond to what might actually exist within an area being measured with gravity sensors 16. In some embodiments, the constraints 72 may include one or more regularization parameters used to guide the solution of Cx — b. Accordingly, the constraints 72 may prevent the mass density analysis system 14 from utilizing computational resources to provide solutions with masses that are not realistic or otherwise not expected to be within the discretized domain 54.

[0078] At block 74, the mass density analysis system 14 determines or generates a mass density distribution output 76 (e.g., the mass density distribution within a discretized domain 54) related to the discretized domain 54 using the solved equations or matrix, which is described in more detail with respect to FIG. 13. In some embodiments, the mass density analysis system 14 may utilize the mass density distribution output 76 to determine a location of the target mass 52 within the discretized domain 54, or to determine that the target mass52 is no longer in the discretized domain 54. In some embodiments, the mass density analysis system 14 may utilize the mass density distribution output 76 to determine an indication of a change in density distribution of the target mass 52 within the discretized domain 54 based on a previous determination of mass density distribution output 76.

[0079] Multiple gravity sensors 16 are utilized to solve the equations or matrix of equations (e.g., Cx — b) to produce a mass density distribution. To explain how a matrix may be used to represent the discretized domain 54, FIGS. 4 and 5 show gravity sensors 16 measuring a point mass 80 within a domain 100 (e.g., a non-discretized domain). For example, FIG. 4 shows a block diagram of a measurement collection system 12 that includes two gravity sensors 16 (e.g., a first gravity sensor 16a and a second gravity sensor 16b). The first gravity sensor 16a is separated from the second gravity sensor 16b by a distance 82 (e.g.,rSA,SB) along the x-axis 44. The point mass 80 is separated from the first gravity sensor 16a by a distance 84. The point mass 80 is separated from the second gravity sensor 16b by a distance 86.

[0080] As described herein, the gravity sensors 16 detect objects based on the gravitational force applied to the gravity sensor 16 by the objects. The first gravity sensor 16a is subject to the following force, FSA:Where O / SA is the mass of the first gravity sensor 16a, m\ is the mass of the point mass 80, TOSB is the mass of the second gravity sensor 16b, rsA,i is the distance 86, and / 'SA.SB is the distance 82. The term yk relates to the sensor accuracy or sensitivity and may be related to one or more of the pre-process inputs 70.

[0081] It should be noted that the diagram shown in FIG. 4 is meant to be a simplified example. For example, the only masses included in the equation (1) are the mass of the first gravity sensor 16a, the second gravity sensor 16b, and the point mass 80. The masses of the housing of the gravity sensor 16, for example, and any additional masses that may be present are not included in the equation (1). In some embodiments, a background gravitymeasurement may be utilized by the mass density analysis system 14 (e.g., described in FIG.3) to account for any additional masses that may be present.

[0082] The second gravity sensor 16b is subject to the following force, FSB:Where rsB,i is the distance 86. To facilitate discussion of a matrix used to solve for the mass density distribution of the point mass 80, the equations (1) and (2) may be rewritten as:>

[0083] By rewriting rsA in terms of rsB,i and rsA,SB, equations (3) and (4) may be used to solve two unknowns, such as the mass mi and the distance rsB.i- Equations (3) and (4) may provide the inputs to the terms (e.g., C, b, and x) of the matrix Cx — b. For example, the term b may include the measurement of force subjected to sensor A (e.g., FSA / G), the variables in the termmsBms^as well as the measurement of force subjected to sensor B (e.g., FSA / G).rSA, SB-SAThe term - — corresponds to the gravitational force resulting from the interactionrSA,SBbetween the first gravity sensor 16a and the second gravity sensor 16b. As such, this may be used as a correction to remove background signals for sensor to sensor interactions. The term C may include yk. The term x may include the masses and distances, such as mi, rSA 1, andrSB,l ■

[0084] Adding another point mass 80 introduces more unknowns and increases the likelihood that Cx — b is ill-posed. For example, FIG. 5 shows a block diagram of a measurement collection system 12 that includes two gravity sensors 16 (e.g., a first gravity sensor 16a and a second gravity sensor 16b). The illustrated embodiment includes two points masses 80 (e.g., a first point mass 80a and a second point mass 80b). The first gravity sensor 16a is separated from the second gravity sensor 16b by a distance 82 (e.g., Y A.SB along thex-axis 44. The first point mass 80a is separated from the first gravity sensor 16a by a distance 84. The first point mass 80a is separated from the second gravity sensor 16b by a distance 86. The second point mass 80b is separated from the first gravity sensor 16a by a distance 88. The second point mass 80b is separated from the second gravity sensor 16b by a distance 90.

[0085] In a similar manner as described above with respect to FIG. 4, the first gravity sensor 16a is subject to the following force, FSAWhere ms.A is the mass of the first gravity sensor 16a, m\ is the mass of the first point mass 80a, TOSB is the mass of the second gravity sensor 16b, m2 is the mass of the second point mass 80b, FSA,SB is the distance 82, rsA,i is the distance 84, FSA,2 is the distance 88, and y is an sensitivity term that factors in the alignment 50 for each gravitational force term. The term k may also factor in the sensor accuracy and other weight factors that may affect the gravitational force measurements by a gravity sensor 16 (e.g., factors other than the mass of the object in a particular location 55 and the distance between the location 55 and the gravity sensor 16). The second gravity sensor 16b is subject to the following force, FSB.'Where rsB.i is the distance 86, rsB,2 is the distance 90.

[0086] Equations (5) and (6) may be rewritten as a system of linear equations in the form:

[0087] There are several unknowns in equations (7) and (8), such as mi, m2, / 'SA.2, rsA,i, rsB,2, and rsB,i. Unless there are enough independent equations to solve for these unknowns,numerical methods may be employed to solve for the unknowns. As it should be appreciated, detecting the location of the point masses 80a and 80b may be relatively complex.

[0088] However, practical applications will not measure a point mass 80, but a mass having a volume, which further complicates solving for any unknowns in Cx — b . It is presently recognized that using the discretized domain 54 may simplify solving the equations or matrix of equations used to detect objects by turning the problem into a simpler approximation (e.g., based on the number of locations 55). That is, by using the discretized domain 54, unknowns related to the distances between sensors and masses may be eliminated since the respective distance between the gravity sensors 16 and the locations 55 are known (e.g., the locations 55 are created or picked).

[0089] To that end, FIG. 6 shows a schematic diagram for measuring the target mass 52 using the measurement collection system 12 with a discretized domain 54. To form the discretized domain 54, the mass density analysis system 14 virtually discretizes (e.g., in pixels, voxels etc.) a physical domain into the discretized domain 54 with unknown mass density distribution (in 2D, 3D, ...) for mass density distribution determination and possibly image reconstruction. As such, the discretized domain 54 includes an array 102 (e.g., a pixel array, a voxel array) of locations 55 that corresponds to an observation area or volume where a target mass 52 may be present (e.g., the physical domain). The mass density analysis system 14 may generate the discretized domain 54 using, for example, finite difference discretization methods (FDDs), finite element techniques, and other techniques understood by one of ordinary skill in the art.

[0090] The discretization provides an adjustable resolution within the discretized domain 54 that may simplify solving for the mass density distribution output 76. For example, the mass density analysis system 14 may add one or more locations 55 within certain areas of the discretized domain 54. Adding more locations 55 within the array 102 may be useful for increasing the resolution of the mass density distribution output 76 in areas where the target mass 52 is likely located or where the sensitivity of the gravity sensors 16 is higher, and thus more accurate. It should be noted that although the illustrated embodiment of the measurement collection system 12 only includes one gravity sensor 16 for purposes ofdiscussing this technique, the measurement collection system 12 may include any number of gravity sensors, so a set of equations as discussed below and in reference to FIG. 5 may be generated for each location 55 to determine a mass associated with each respective location 55 within the domain 100.

[0091] As generally mentioned above, the discretized domain 54 includes a pixel or voxel array 102 of locations 55 where the target mass 52, or additional objects, may exist. In general, each location 55iJ (e.g., location 551,1, location 551,2, and location 551,3) of the array 102 is at a different distance 106 from each gravity sensor 16 of the measurement collection system 12. By forming the discretized domain 54, each respective distance between a location and gravity sensors 16 is known. In the illustrated embodiment, the array 102 is a two-dimensional 3x10 array, but it should be noted that the array 102 may be any suitable size pixel or voxel array, such as 100x100, 256x256, 512x512, 100x100x100, 512x512x512, and so on. The gravitational force subjected to the gravity sensor(s) 16 in the measurement collection system 12 of FIG. 6 may be represented as:

[0092] Where i is the index for a row of the array 102, and j is the index for a column of the array 102. Each force value, FSA, shown in equation (9) may generally be rewritten in the form of Cx — b in a generally similar manner as described with respect to FIGS. 4 and 5. In this case, the unknown x is the mass at each location 55 of the array 102 since the discretized domain 54 is discretized into locations 55 with known or measurable distances from each gravity sensor 16. As the target mass 52 is not evenly distributed amongst each location 55 of the array 102, the value of the mass at each location may vary, which is generally shown in the mass density distribution output 76, which may be a visualization 108 of the mass density distribution output 76. In general, the pixel shading of the visualization 108 are darker for areas with a higher mass. Equation (9) may be solved as generally described with respect to FIG. 3, thereby providing a density distribution output 76 that indicates a respective mass within each location 55 of the discretized domain 54. Further, the mass density distribution output 76 may be used to determine whether the target mass 52 is present orabsent within the discretized domain 54, as well as its location (e.g., subset of locations 55) within the discretized domain.

[0093] As described above, the alignment 50 of the gravity sensors 16 may be useful even if the alignment 50 is orthogonal to the gravitational force imparted by a mass. FIG. 7 illustrates the alignment 50 of gravity sensors 16 with respect to the discretized domain 54, which in this example is an array 102 that is a 6 x 6 two-dimensional pixel array. To simplify discussion, the relative distance of the gravity sensors 16 are discussed with respect to the gravitational force imparted by a mass corresponding to the location 554,3 of the discretized domain 54.

[0094] The gravitational force imparted by any mass in the location 55 onto the first gravity sensor 16a may be represented by a vector 103. A gravitational field due to the mass at location 554,3 that is subjected to the first gravity sensor 16a is represented by the vector 103a. The gravitational field due to the mass at location 554,3 that is subjected to the second gravity sensor 16b is represented by a vector 103b. Although the first gravity sensor 16a is further away from the location 554,3 than the second gravity sensor 16b, the first gravity sensor 16a may experience a larger gravitational force from the mass at the location 554,3. In particular, the alignment 50 of the first gravity sensor 16a is parallel with the vector 103a, while the alignment 50b of the second gravity sensor 16b is perpendicular with the vector 103b. As such, in the matrix Cx — b, the element in b for the first gravity sensor 16a may be larger than the element in b for the second gravity sensor 16b.

[0095] The gravity sensor array 40 includes a first row 110, a second row 112, a third row 114, and a fourth row 116 arranged along the axis 42. The first row 110 is separated from the second row 112 by a first distance 118. The second row 112 is separated from the third row 114 by a second distance 120. The third row 114 is separated from the fourth row 116 by a third distance 122. In some embodiments, the first distance 118, the second distance 120, and the third distance 122 may be equal. Alternatively, at least one of the first distance 118, the second distance 120, and the third distance 122 may be different. Moreover, although each gravity sensor 16 is shown as being aligned with the gravity sensor 16 of another row (e.g., the first row 110, the second row 112, the third row 114, or the fourth row116), it should be noted that the gravity sensors 16 in adjacent rows may be offset (e.g., the centers of gravity sensors 16 in adjacent rows are not aligned) from one another. Positioning the gravity sensors 16 offset from one another may improve the efficiency of stacking or packing gravity sensors 16, thereby increasing the number of gravity sensors 16 that are disposed near the boundary of the discretized domain 54. That is, gravity follows an inverse square law, it is advantageous to provide gravity sensors 16 closer to the area represented by the discretized domain 54 to maximize the signal of the gravity measurements 64 obtained by the gravity sensors 16. In any case, it is presently recognized that it may be advantageous to include multiple rows of gravity sensors 16. Without wishing to be bound by theory, it is believed that each row of gravity sensors 16 provides additional accuracy with respect to the measured gravitational force at a particular location (e.g., a pixel or a voxel).

[0096] Utilizing multiple rows (e.g., the first row 110, the second row 112, the third row 114, and the fourth row 116) may improve the accuracy of the mass density distribution by providing gradiometry. Gradiometry of gravitational fields is a measure of gravitational field gradients, which indicates how the gravitational fields are changing over distance. Gradiometry measurements may provide advantages such as more information related to local variations in each location 55, improved resolution, and a reduction in noise.

[0097] For simplicity, gradiometry from a gravity measurement in accordance with the gravity sensor array 40 shown in FIG. 7 is discussed with respect to the location 556,6 along the axis 44. As shown, the third gravity sensor 16c, in the first row 110, is a first distance 150 from the location 55e,6. The fourth gravity sensor 16d, in the second row 112, is a second distance 152 from the location 55e,6. The fifth gravity sensor 16e, in the third row 114, is a third distance from the location 55e,6. The sixth gravity sensor 16f, in the fourth row 116, is a fourth distance from the location 55e,6. As such, gravity measurements from each of the gravity sensors 16c, 16d, 16e, and 16f may provide a gravity measurement corresponding to a higher order derivative of the gravity measurement. As such, the combination of the gravity measurements more accurately represents the gravitational fields at each location 55.

[0098] It may be advantageous to arrange the gravity sensors 16 based on the mass density of an imaging target, the relative location of the imaging target, and so on. Withoutwishing to be bound by theory, FIG. 8 shows a graph 130 indicating a non-limiting example of factors that may be useful for determining an arrangement of sensors suitable for measuring a particular target mass 52. The graph 130 includes a first axis 132 (e.g., x-axis) corresponding to the distance from a sensor to boundaries or changes in mass density within the domain. In general, it is presently recognized that objects having a varying mass density due to boundaries between multiple objects, end surfaces of objects, porosity, and so on, may have a mass density that is relatively difficult to represent mathematically. For example, it may be easier for the mass density analysis system 14 to detect an object having a uniform mass density as compared to an object that is porous or is composed of a variety of materials having different mass densities. The graph 130 also includes a second axis 134 (e.g., y-axis) corresponding to the total number of variations in mass density to be imaged in a domain (e.g., the discretized domain 54). Further, the graph 130 includes a third axis 136 (e.g., z-axis) corresponding to the number of folds and / or the accuracy of the folds that are produced in the image. As referred to herein, “folds” refers to the variation in the curvature of gravitational fields (e.g., higher order effects in the gravitational field) resulting from masses at different locations. The graph 130 includes a distribution function 138 (e.g., a three-dimensional Gaussian function) that indicates a particular combination of values corresponding to the first axis 132 and the second axis 134 for obtaining a degree of accuracy (e.g., the third axis 136). Accordingly, it is presently recognized that there may be a predetermined arrangement of gravity sensors 16 for measuring gravitational forces of a target mass 52 that facilitates detection of the target mass 52, as opposed to leading to false positives, false negatives, or otherwise not identifying the target mass 52.

[0099] Although the examples of measurement collection systems 12 in FIGS. 6 and 7 generally show gravity sensors 16 disposed on one-side of a target mass 52, it should be noted that it may be advantageous to provide gravity sensors 16 on multiple sides of the target mass 52. For example, measurements obtained from gravity sensor arrays 40 positioned on multiple sides of the target mass 52 may be combined to provide a more accurate measurement of the mass density distribution output 76. In particular, providing gravity sensors 16 that surround the discretized domain 54 may result in more gravity sensors 16 being closer to the target mass 52 as opposed to providing multiple layers of gravity sensors16 along one side of the target mass 52. Since the sensitivity of the gravity sensor 16 follows an inverse square relationship, surrounding the discretized domain 54 with gravity sensor arrays 50 may improve the signal obtained by the gravity sensors 16. FIG. 9 illustrates the gravity measurement system 10 with multiple gravity sensor arrays 40a, 40b, 40c, and 40d (e.g., collectively, 40). Although four gravity sensor arrays 40 are shown, it should be noted that the system may include any number of gravity sensor arrays 40.

[0100] As shown, each gravity sensor array 40 includes a 4-by-4 array of gravity sensors 16. The 4-by-4 arrays are illustrated for simplicity, but it should be noted that denser arrays and / or 3D arrays may be used to more accurately determine the mass density distribution output 76 of the target mass 52 in an actual implementation. In the illustrated embodiment of the gravity measurement system 10, four gravity sensor arrays 40a, 40b, 40c, and 40d are positioned on different sides of the imaging target 52. As noted above, positioning gravity sensor arrays 40 on multiple sides of the imaging target 52 (e.g., or a discretized domain 54 where the imaging target 52 may reside) improves the accuracy of gravity measurements since more gravity sensors 16 are positioned closer to the locations 55. As such, the gravity measurements obtained by the gravity sensor arrays 40 will more accurate since they have a higher signal.

[0101] Gravity sensor arrays 40 may also be implemented as 3D arrays, which may further provide for imaging over more locations 55. A simplified example of gravity sensor arrays 40 arranged about a 3D location 55 (e.g., a voxel) is shown in FIG. 10. Although the gravity sensor arrays 40a, 40b, 40c, and 40d are shown as 4-by-4-by-4 arrays, it should be noted that denser arrays may be used. Similar, the discretized domain 54 may have more locations than the 7-by-4-by-6 array shown. Each location 55 may be represented with an index “z, j, k”. The gravity measurements 64 obtained from a 3D array may be processed together such that x is a tensor (e.g., having a size i-by-j-by-k). However, it may be easier for the mass density analysis system 14 to process multiple matrices that each represent a cross-section of the discretized domain 54 shown in FIG. 10 (e.g., the discretized domain 54 may be divided into four matrices for each j). In any case, providing 3D gravity sensor arrays40 may improve the accuracy of the mass density distribution by improving signal-to-noise ratio (e.g., with more sensors) and improving gradiometry measurements.

[0102] Although the discussion above generally relates to determining a target mass 52 within a non-specific discretized domain 54, it is presently recognized that the ability of the gravity measurement system 10 to detect unknown objects within a space may be applied to a variety of specific applications. As one specific, non-limiting example, FIG. 11 shows the gravity measurement system 10 used to determine the presence of target masses 52a and 52b within a shipping container 180, although it should be understood that this example would be similarly applicable to a room, building, or any other specified three-dimensional space and thus could be useful in locating and assessing stock, produce, employees etc. In the illustrated embodiments, measurement collection system 12 includes multiple gravity sensor arrays 40a, 40b, 40c, and 40d disposed on or near multiple surfaces of the shipping container 180. However, it should be noted that in some instances, the gravity sensor arrays 40 may be disposed on or near any number of surfaces of the shipping container 180, such as one surface, two surfaces, or more than two surfaces.

[0103] In operation, each gravity sensor 16 of the measurement collection systems 12a, 12b, 12c, and 12d may obtain measurements reflecting the mass within the shipping container 180. To detect the presence of the imaging targets 52a and / or 52b, the mass density analysis system 14 may receive the sensitivity term, y, that indicates the parameters of the gravity sensors 16 and the matrix information 68. The matrix information 68 may indicate a number of locations 55 within the discretized domain 54, the spatial dimensions corresponding to each location 55, and other information as described herein. To further facilitate detecting the presence of the target masses 52a and 52b, the mass density analysis 14 may also receive constraints 72 that indicate a range of mass densities likely to be within the interior and / or walls of the shipping container 180. In this way, the mass density analysis system 14 may be more likely to detect the presence of the imaging targets 52a and / or 52b within the shipping container 180 by reducing the parameter space of the unknowns in the equations (e.g., Cx — b)~ as described with respect to block 66 of FIG. 3. It should be noted that in this example, the target masses 52a and 52b are illustrated as being people working in the shippingcontainer 180, but any mass within the shipping container 180 can be detected. Indeed, in other examples, the target masses could include illegal contraband, items for inventory determinations, items that may have broken during shipping, hazardous materials, etc.

[0104] The number of sensors utilized in the measurement collection system 12 affects the accuracy of the mass density distribution output 76. The accuracy of the mass density distribution output 76 may, in turn, affect the resolution of any image generated based on the mass density distribution output 76. It is presently recognized that is may be desirable or sufficient to generate a mass density distribution output 76 that generally corresponds to the mass and / or position of the target mass 52 as opposed to fully resolving all details that may be useful for imaging the target mass 52. That is, in an application as described in FIG. 11, it may be sufficient to determine that there is an object within the shipping container 180 that has a mass density that corresponds to human physiology, for example, as compared to accurately resolving a body. With this in mind, FIGS. 12A and 12B show images 200 and 202 that correspond to the domain 100 shown in FIG. 11. FIG. 12C shows the actual image 204 of the target masses 52a and 52b. More specifically, FIG. 12A shows the image 200 that is based on the mass density distribution output 76 obtained by a first number of gravity sensors 16. FIG. 12B shows the image 202 that is based on the mass density distribution output 76 obtained by a second number of gravity sensors 16, and the second number is greater than the first number. As generally shown, the image 202 includes two features 206 and 208 at locations and having a mass density that generally corresponds to the target masses 52a and 52b. The gravity sensors 16 that obtained the gravity measurements 64 used to generate the images 200 and 202 may be a subset of available gravity sensors 16. For example, the measurement collection system 12 may receive input indicating a desired resolution for an image. As such, the measurement collection system 12 may determine a subset of the gravity sensors 16 to utilize to obtain the gravity measurements 64 based on the desired resolution. Alternatively this determination (of the subset of gravity sensors 16 to utilize) may be made elsewhere in the system and communicated to the measurement collection system 12. Accordingly, while more gravity sensors 16 may generally provide high resolution images, fewer sensors may provide sufficient information for detecting imaging targets 52. Selectively using a subset of available gravity sensors when lowerresolution is acceptable can save on computational load when determining the mass density distribution.

[0105] The discussion of FIG. 3 provides one example of a technique for determining or generating a mass density distribution output 76 using gravity measurements 64. FIG. 13 shows an example flow diagram of a process 300 illustrating one technique for generating the mass density distribution output 76 using matrix information 68, pre-process inputs 70, and constraints.

[0106] At block 302, the mass density analysis system obtains gravity measurements 64 for an object (e.g., the target mass 52) in a volume. As described herein, the gravity measurements 64 may be acquired by gravity sensors 16 within a measurement collection system 12. For example, the gravity measurements 64 may be obtained by one or more measurement collection systems 12 arranged about a volume (e.g., corresponding to the discretized domain 54). In some embodiments, the measurement collection systems 12 may transmit the gravity measurements 64 to the mass density analysis system 14. In some embodiments, the mass density analysis system 14 may be capable of remotely controlling one or more measurement collection systems 12. For example, the mass density analysis system 14 may transmit a control signal (e.g., based on user input) to the one or more measurement collection systems 12 that cause the one or more measurement collection systems 12 to measure the gravitational force subjected to the gravity sensors 16 of the measurement collection system 12. In turn, the one or more measurement collection systems 12 may transmit the gravity measurements 64 to the mass density analysis system 14.

[0107] At block 304, the mass density analysis system 14 pre-processes the gravity measurements 64 to obtain pre-processed gravity measurements (e.g., pre-processed data signals). In general, pre-processing the gravity measurements 64 may include filtering and / or adjusting the gravity measurements 64 to account for noise and / or other variations in the signals of the sensor measurements unrelated to the gravitational force subjected to the gravity sensors 16 by the target mass 52 as described further below, such as sensor drift and sensor accuracy. In some embodiments, the mass density analysis system 14 may receive one or more pre-process inputs 70 to facilitate pre-processing the gravity measurements 64.In general, the pre-process inputs 70 may include information relative the accuracy and strength of data acquired by the gravity sensors 16, such as a sensor accuracy, a sensor weight factor, an alignment 50 of a sensor, a sensor drift factor, a denoising factor, a sensor mapping, a sensor noise, or any combination thereof, of the gravity sensors 16.

[0108] As mentioned above, the pre-process inputs 70 may include a sensor drift factor. As referred to herein, the “sensor drift factor” is an adjustment that may be applied by the mass density analysis system 14 to the gravity measurements 64 to account for variations in the gravity measurement 64 due to age of the gravity sensor 16, weather conditions (e.g., temperature and / or humidity) where the gravity sensors 16 are operating, and the like. For example, the sensor drift factor may indicate how the signal measured by the gravity sensors 16 varies with weather conditions. Additionally or alternatively, the sensor drift factor may indicate how the distances between the gravity sensors 16 changes with weather conditions (e.g., due to expansion or contraction of the housing that houses the gravity sensors 16). The sensor drift factor may be stored in a storage component (e.g., the memory 28 or memory 20) and obtained by the mass density analysis system 14 based on conditions (e.g., the age of the gravity sensors and / or the weather conditions) to adjust the gravity measurements 64 based on the conditions. In some instances, the sensor drift factor may be implemented as a lookup table. In some embodiments, the sensor drift factor may include a drift vector that indicates how a sensor measurement changes over time. As such, pre-processing the gravity measurements may include the mass density analysis system 14 removing or adjusting the gravity measurements for sensor drift. In such embodiments, it may be advantageous to perform a calibration process to determine a sensor drift factor for one or more of the gravity sensors 16 so that the sensor drift factor may be applied to subsequent gravity measurements 64 obtained by the gravity sensors 16.

[0109] As one example, to determine the sensor drift factor, the mass density analysis system 14, or another suitable processor, may obtain gravity measurements from the gravity sensors 16 acquired while the gravity sensors 16 are disposed in a location with known masses within the location. The mass density analysis system 14 may identify a trend or set of changes associated with the gravity measurements over the time. Based on the trend orthe set of changes, the mass density analysis system 14 may determine the drift vector, which may be used to correct gravity measurements obtained by the gravity sensors 16 in subsequent measurements. For example, the drift vector may indicate a how gravity measurements 64 obtained by the gravity sensors 16 may change over a time period (e.g., about 1 minute, about 5 minutes, about 10 minutes, about 30 minutes, about 1 hour, about 2 hours, and so on). Accordingly, the mass density analysis system 14 may utilize the drift vector to correct for potential changes in gravity measurements 64 that are obtained over the time period.

[0110] In some embodiments, the pre-process inputs 70 may include a denoising factor. As such, pre-processing the gravity measurements may include removing or adjusting the gravity measurements to remove noise. As referred to herein, the “noise” may include fluctuations in the gravity measurements 64, such as background noise, phase noise during the rotation of a sensing component, and other types of noise that may obscure signals indicative of masses within the discretized domain 54. As one non-limiting example, the mass density analysis system 14, or another suitable computing device having a processor, may obtain gravity measurements from a first gravity sensor 16 of the measurement collection system 12. Then, the mass density analysis system 14 may determine a variance of the gravity measurements obtained by the first gravity sensor 16. For example, the mass density analysis system 14 may determine that the first gravity sensor 16 has a noise that may be represented by a Gaussian. Then, the mass density analysis system 14 may determine a number of gravity measurements (e.g., about 100 measurements, about 1,000 measurements, about 10,000 measurements, and so on) or a time period (e.g., about 1 minute, about 5 minutes, about 10 minutes, about 30 minutes, about 1 hour) to obtain by the first gravity sensor 16 a value within a particular degree of certainty that may be provided as an input by an operator. The mass density analysis system 14 may obtain the gravity measurements using the first gravity sensor 16 in accordance with the determined number of gravity measurements and / or time period. In some embodiments, the mass density analysis system 14 may average multiple gravity measurements 64 to increase the signal-to-noise ratio.

[0111] At block 306, the mass density analysis system 14 receives, acquires, or otherwise obtains a matrix (e.g., one or more matrices, one or more equations) for a given set of sensors(n pixels / voxels) and provide inputs to the indices of the matrix based on the pre-processed sensor measurements. In general, the matrix represents a combination of knowns and unknowns within a discretized domain 54. For example, the knowns of the matrix may include the sensor mass, the sensor mapping, the number of sensors, the number of sensors per side, the distance between the gravity sensors 16 and the locations 55, the number of locations 55, and so on. The unknowns of the matrix may include the mass at each location 55. In some embodiments, the mass density analysis system 14 may receive, acquire, or otherwise obtain the matrix using constraints 72. For example, as described in more detail in FIG. 18, the mass density analysis system 14 may modify a pixel allocation 56 based on the mass density analysis system 14 identifying locations 55 having a mass value that falls within a range of mass densities corresponding to a target mass 52 or composition of matter (e.g., air, water, a metal frame of a shipping crate, and so on) expected to be near the target mass 52.

[0112] In some embodiments, obtaining the matrix may include the mass density analysis system 14 utilizing reference data to determine the matrix to use for a particular measurement collection system 12. In general, the reference information is data that facilitates the identification of a particular set of sensors. The reference information could be a particular identifier of the sensors (e.g., sensorOOl and sensor002), location information (e.g., a geographic location of the sensors), a MAC address of the sensors, an IP address of one or more of the sensors, and so on. In some embodiments, the reference information is from one sensor that is associated with a cluster of sensors. In this way, the reference information from the sensor may be representative for the cluster of sensors. In any case, the mass density analysis system 14 may determine a matrix that corresponds to the measurement collection system 12 based on the reference information.

[0113] In some embodiments, obtaining the matrix may include the mass density analysis system 14 generating the matrix that may be stored for multiple subsequent uses. To do so, mass density analysis system 14 may receive, acquire, or otherwise obtain or generate a pixel arrangement for a domain as described in more detail with reference to FIGS. 17-19. For example, the mass density analysis system 14 may receive sensor arrangement information,such as a total number of sensors disposed about the domain, the sensor spacing, the number of sides of the discretized domain 54 that are occupied by the gravity sensor arrays 40, a number of gravity sensors 16 disposed around each side of the domain, and the like.

[0114] At block 308, the processor 26 solves the one or more equations or matrices that correspond to the discretized domain 54, as described with respect to FIG. 2. For the purposes of this discussion, “solving” the equations or the matrices does not necessarily mean obtaining an actual solution, since the equations can sometimes be difficult to solve (e.g., the equations or the matrices may represent an ill-posed problem). Hence, “solving” also includes various techniques to iteratively, convergently, or otherwise drive the equations toward an estimated solution. For example, in some embodiments, solving the matrix may include utilizing a cost function that guides convergence of the matrix towards a solution. Accordingly, the processor 26 may iterate through unknown values that correspond to the mass at each location 55 and determine whether a solution provides a minimum for a cost function. If a minimum is achieved, then the processor 26 outputs a mass density distribution (e.g., the mass density distribution output 76 described with reference to FIG. 3).

[0115] Due to the complexity of solving the mass density distribution output 76, it may be advantageous to utilize additional techniques that may improve the accuracy of the mass density distribution output 76. Example techniques that may be utilized to perform the functions described in the blocks of the method 300 are described below with reference to FIGS. 14-21.Pre-Processing Sensor Measurements

[0116] The mass density analysis system 14 may pre-process the gravity measurements 64 to remove noise that may make it difficult for the mass density analysis system 14 to solve the equations or matrices to generate the mass density distribution output 76. To illustrate an example of this, FIG. 14 shows a flow diagram of a method 350 for generating pre-processed gravity measurements that may be utilized in the method of FIG. 13.

[0117] At block 352, the mass density analysis system 14 obtains the gravity measurements 64. In general, the mass density analysis system 14 may obtain the gravity measurements 64 in a generally similar manner as described with respect to block 302 of FIG. 13. At block 354, the mass density analysis system 14 determines a time period corresponding to the gravity measurements. In general, the time period corresponding to the gravity measurements may be a numerical value that indicates one or more readings over a time horizon. For example, the system 12 may acquire gravity measurements 64 over a time period and process each gravity measurement 64 to determine a respective mass density distribution output 76. In some instances, multiple gravity measurements 64 can be averaged or processed together to improve the signal -to-noise ratio, and thus providing a more accurate mass density distribution output 76. Additionally, the mass density analysis system 14 may process the gravity measurements 64 together to determine that the object is moving and correct the gravity measurements 64 based on the movement (e.g., generate a gravity measurement 64 representative of a stationary object as opposed to a moving object).

[0118] At block 356, the mass density analysis system 14 generates adjusted gravity measurements by applying one or more pre-process inputs 70 to the gravity measurements, where the one or more pre-process inputs 70 that are used are relevant to the time period. In some embodiments, the pre-process inputs 70 may include a sensor drift factor that indicates how a sensor measurement changes over time. As such, the mass density analysis system 14 may apply the sensor drift factor by increasing or decreasing the gravity measurements based on the magnitude of the time period corresponding to the gravity measurements. For example, the sensor drift factor may be a mathematical function that defines how measurements of the gravity sensor 16 change over time. Accordingly, the mass density analysis system 14 may extrapolate an adjusted gravity measurement using the sensor drift factor. In some embodiments, the mass density analysis system 14 may utilize the pre-process inputs 70 by applying a reference or look-up table. For example, the mass density analysis system 14 may obtain a gravity measurement 64 and determine a temperature corresponding to the time period the measurement was taken. Then, the mass density analysis system 14 may determine a correction or other adjustment to the gravity measurement 64, thereby removing noise or artifacts from the gravity measurement 64 that may make itdifficult for the mass density analysis system 14 to determine or generate the mass density distribution.

[0119] At block 358, the mass density analysis system 14 outputs the pre-processed gravity measurements 360 based on the adjusted gravity measurements. In general, the pre-processed gravity measurements 360 are the output of one or more adjustments to the gravity measurements 64. For example, the mass density analysis system 14 may perform block 356 iteratively for each pre-process input 70.

[0120] As discussed above, with reference to block 306, the mass density analysis system 14 may obtain or generate matrix information 68 that is used to determine the mass density distribution output 76. The matrix information 68 may be generated beforehand, such as by performing a calibration experiment where the mass density analysis system 14 determines a discretized domain 54 (e.g., matrix information 68 for the discretized domain 54) for a particular gravity sensor array 40. Then, the matrix information 68 may be stored in the memory 28 or other suitable storage component, until the mass density analysis system 14 receives additional gravity measurements for processing. Alternatively, the mass density analysis system 14 may generate the matrix information 68 “on-the-fly” or otherwise “ -situ”. To illustrate one example, FIG. 15 shows a flow diagram of a method 362 for obtaining previously generated matrix information 68 using metadata associated with the gravity sensors 16 that may be utilized in the method of FIG. 13.

[0121] At block 364, the mass density analysis system 14 obtains the pre-processed gravity measurements 360. In general, the pre-processed gravity measurements 360 are the output of the mass density analysis system 14, or other suitable processor, performing the method 362. Accordingly, in some instances, the method 362, which represents an example of the block 306, may be performed in parallel with block 304. For example, the mass density analysis system 14 may obtain or generate the matrix information 68 while pre-processing the gravity measurements 64.

[0122] At block 366, the mass density analysis system 14 identifies metadata associated with the pre-processed gravity measurements 360 (e.g., pre-processed data signals). Ingeneral, the metadata may indicate reference information, such as a particular identifier of the sensors (e.g., sensorOOl and sensor002), location information (e.g., a geographic location of the sensors), a MAC address of the sensors, an IP address of one or more of the sensors, and so on. The reference data may indicate a particular gravity sensor array 40 that obtained the gravity measurements 64, and thus, the mass density analysis system 14 may query memory (e.g., the memory 28) to determine whether matrix information 68 exists for the gravity sensor array 40. In any case, the mass density analysis system 14 may determine a matrix that corresponds to the measurement collection system 12 based on the reference information.

[0123] At block 368, the mass density analysis system 14 determines one or more matrices based at least in part on the metadata. Accordingly, the mass density analysis system 14 may obtain matrix information 68 for a particular gravity sensor array 40 that is identified using metadata. Retrieving the matrix information 68 may include the mass density analysis system 14 accessing a storage component, such as the memory 28 of the mass density analysis system 14, the memory 20 of the measurement collection system 12, or other suitable memory, and retrieving the matrix information 68 from the storage component.Determining Matrix Information Using a Pixel Allocation

[0124] The discussion below relates to techniques for generating and / or adjusting pixel allocations 56 to simplify solving for the mass density distribution. The pixel allocation 56 generally refers to the distribution of location 55 and / or the dimensions of the locations within a discretized domain 54. Referring to the simplified equation Cx — b , the pixel allocation 56 affects the number of elements in the matrix x. The mass density analysis system 14 may generate matrix information 68 based on the equation that includes information related to the pixel allocation. That is, each element of the matrix x includes a number that corresponds to the mass at a location 55 within the discretized domain 54.

[0125] FIG. 16 shows a flow diagram of an example of a method 380 for generating matrix information 68 using a pixel allocation that may be utilized in the method of FIG. 13 (e.g., in blocks 304 and 306). This example method 380 may be performed by the massdensity analysis system 14 to generate matrix information 68 in situ, such as while the gravity sensors 16 are obtaining the gravity measurements.

[0126] At block 382, the mass density analysis system 14 receives an indication of gravity measurements 383 that were measured or being measured by one or more gravity sensors 16. For example, the mass density analysis system 14 may receive the indication based on a control signal that causes one or more gravity sensors 16 to obtain gravity measurements 64. As another non-limiting example, the mass density analysis system 14 may receive the indication based on the gravity measurements 64 being received by the mass density analysis system 14.

[0127] At block 384, the mass density analysis system 14 receives a sensor mapping associated with the measurement collection system 12. In general, the sensor mapping indicates the arrangement of the gravity sensors 16 for a measurement collection system 12 and / or the gravity measurement system 10. For example, the sensor mapping indicates a first location corresponding to a first gravity sensor 16, a second location correspond to a second gravity sensor 16, or distances between gravity sensors 16, and so on. Additional nonlimiting examples of sensor mapping may include the distances (e.g., the first distance 150, the second distance 152, the third distance 154, or the fourth distance 156) and / or number of rows (e.g., the first row 110, the second row 112, the third row 114, and the fourth row 116) as described in FIG. 7. Further, the sensor mapping may include number of sides surrounding a discretized domain 54.

[0128] At block 386, the mass density analysis system 14 determines a discretized domain 54 based on the pre-process inputs 70, such as the sensor mapping. That is, the mass density analysis system 14 determines a pixel allocation for a discretized domain 54 that represents an area or volume of interest. As one non-limiting example, the sensor mapping may indicate an area that the gravity sensors 16 surround. Accordingly, the processor may determine the discretized domain 54 such that it encompasses the area and includes locations 55 within the discretized domain 54 that are sized based on the accuracy of the gravity sensors 16 at the different locations. Accordingly, the mass density analysis system 14 maydetermine a discretized domain 54 that includes an area with the distance range where the sensitivity of the gravity sensors 16 is above the threshold.

[0129] At block 388, the mass density analysis system 14 determines a pixel allocation based on the pre-process inputs 70 (e.g., the sensor mapping) and / or the constraint 72. To determine the pixel allocation based on the sensor mapping, the mass density analysis system 14 may determine an area with dimensions such that the area is enclosed by the gravity sensors.

[0130] In some embodiments, the mass density analysis system 14 may utilize constraints to determine the pixel allocation. As one non-limiting example, the constraints 72 may indicate physical dimensions or a shape of a target mass 52. The constraints 72 may be provided by a user based on prior knowledge of the target mass 52 or the environment surrounding the target mass 52. As such, the mass density analysis system 14 may determine a pixel allocation such that enough pixels are included within the discretized domain 54 to accurately represent the shape and / or size of the target mass 52.

[0131] At block 390, the mass density analysis system 14 generates matrix information 68 based on the pixel allocation. In general, the matrix information 68 may include information that relates each element of the matrix x to a location 55 within the discretized domain 54. Further, the matrix information 68 may include data indicating the distance between each index and the gravity sensors 16 of the measurement collection system 12.

[0132] In some embodiments, generating the matrix information 68 may include updating the matrix information 68. For example, the mass density analysis system 14 may solve the system of equations or matrices in an iterative manner and update the matrix information at each step in the iteration. For example, the mass density analysis system 14 may solve the systems of equations or matrices to arrive at a first solution where the solution is the matrix x and each element of x represents a potential mass at each location 55. The mass density analysis system 14 may then determine whether to generate a new matrix that corresponds to a discretized domain 54 with more locations 55 in certain areas based on the mass values or changes between mass values between the elements of the matrix x. For example, the massdensity analysis system 14 may perform an adaptive discretization and add more elements (e.g., more locations 55 to the discretized domain) where a change in mass occurs. As described herein, the change in mass in a matrix solution may indicate that the target mass 52 is present. As such, it may be advantageous to include more elements to ensure the solution (e.g., the mass density distribution) more accurately represents the target mass 52. This is described in more detail below.Generating a Pixel Allocation for a Discretized Domain

[0133] Since gravity follows an inverse square law, the gravitational potential field applied by an object to the gravity sensors 16 decays at an exponential rate as the object is further from or moves away from the gravity sensors 16. As such, numerical errors may become more frequent at certain distances from the gravity sensors 16, which may reduce the accuracy of the mass density distribution output 76. Accordingly, it may be advantageous to determine a pixel allocation such that the discretized domain 54 includes higher resolution (e.g., more locations 55 with smaller dimensions) when the locations 55 are closer to the gravity sensors 16. To illustrate this, FIG. 17 shows an example of a method 400 for determining a pixel allocation (e.g., a grid or array of pixels) using certain pre-process inputs 70 that may be utilized in the method of FIG. 16. Accordingly, the method 400 may be used to adapt a pixel allocation such that smaller pixels are present where the gravity measurements 64 may have relatively less error.

[0134] At block 402, the mass density analysis system 14 receives sensor mapping 404 (e.g., pre-process inputs 70). In general, the domain data 406 is the discretized domain 54 generated by the mass density analysis system 14 as described with respect to block 386 of FIG. 16. As described herein, the sensor mapping 404 indicates the arrangement of the gravity sensors 16 for a measurement collection system 12 disposed about a discretized domain 54. For example, the sensor mapping 404 indicates a first location corresponding to a first gravity sensor 16, a second location correspond to a second gravity sensor 16, and so on.

[0135] At block 408, the mass density analysis system 14 identifies a boundary of the discretized domain 54 using the domain data 406. For example, the domain data 406 may indicate the physical dimensions of the discretized domain 406. As such, the processor 406 may determine which location 55 are closest to the gravity sensors 16, and thus, may provide gravity measurements 64 that are within a threshold accuracy. At block 410, the mass density analysis system 14 identifies one or more sensors located near the boundary of the discretized domain 54. For example, the mass density analysis system 14 may determine the sensitivity of the gravity sensors 16 closest to the boundary of the discretized domain 54 to determine minimum dimensions for the locations 55 nearest to the gravity sensors 16.

[0136] At block 411, the mass density analysis system 14 determines a pixel allocation 412 based on the gravity sensors 16 located near the boundary. In general, the larger the grid size, e.g., the greater the number of pixels, the larger the numerical system of equations that needs to be solved. As such, improving grid resolution may make the optimization problem (e.g., block 308 of FIG. 13) numerically more challenging and time consuming to reconstruct the mass density distribution output 76. Accordingly, to determine the pixel allocation 412, the mass density analysis system 14 may allocate relatively more pixels where the error associated with the pixel is below a threshold. Since gravitational force follows an inverse square law with respect to distance between two objects, the error for a pixel increases exponentially as the distance between the pixel and the gravity sensor 16 increases. Accordingly, the mass density analysis system 14 may allocate more pixels at the boundary and decrease the number of pixels at locations further from the gravity sensors 16. In this way, the process 400 may reduce the computational complexity of generating the mass density distribution output 76 by focusing computation (e.g., solving for masses at pixels) on regions where data has less error.Iterative Adjustment of a Pixel Allocation

[0137] Increasing the number of locations 55 (e.g., the pixel allocation) within the discretized domain 54 increases the computational complexity for solving the system of equations to generate the mass density distribution output 76. Rather than using a pregenerated or predetermined pixel allocation, it may be advantageous to start with a simplerpixel allocation for a discretized domain 54 that includes large locations 55 and iteratively adapt the pixel allocation by adding more locations 55 when there is a change of mass between locations 55. In particular, a change in mass between two location 55 may indicate the presence of an object. Accordingly, increasing the number of locations 55 where an object may be located may increase the accuracy of the mass density distribution output 76. Similarly, it may be advantageous to not add or remove locations 55 to the discretized domain 54 where there is likely not an object. In any case, it may be more efficient to solve the equations or matrices (e.g., block 308) multiple times, and each time modify the matrix information 68 based on the results from the previous solution. FIG. 18 shows a flow diagram of a method 420 for adjusting a pixel allocation based on a mass density variation between pixels. The method 420 may be utilized in block 388 of the method of FIG. 16.

[0138] At block 422, the mass density analysis system 14 determines a first pixel allocation. In some embodiments, the first pixel allocation may be a predetermined allocation of pixels that is specific to a sensor mapping 68 or universal to all gravity sensors 16. For example, the mass density analysis system 14 may use a default pixel allocation 56 to provide a starting point for solving the equations or matrices, and subsequently modify the pixel allocation 56 to a different pixel allocation (e.g., a denser pixel allocation, a boundary concentrated pixel allocation, an adaptive grid pixel allocation, and the like).

[0139] At block 424, the mass density analysis system 14 identifies a mass variation between the locations 55 corresponding to the first pixel allocation. To do this, the mass density analysis system 14 may solve the equations or matrices (e.g., perform block 308) using a first matrix that corresponds to the first pixel allocation to generate a first mass density distribution. The first mass density distribution represents a solution of the equations or matrices that includes mass values at locations 55 corresponding to the first pixel allocation. Then, the mass density analysis system 14 may compare the mass values between locations 55 to determine mass density variations between the locations 55.

[0140] At block 426, the mass density analysis system 14 determines a subset of the locations 55 for the first pixel allocation having a mass density variation that exceeds a threshold, is below a threshold, or outside of a threshold range. For example, the mass densityanalysis system 14 may determine whether one or more of the mass density variations between the locations 55 exceeds a threshold, which may indicate that the target mass 52 is present. That is, and for simplicity, if a discretized domain 54 does not include the target mass 52 but is instead, for example, empty, the first mass density distribution may indicate that the mass at each location 55 is the same. However, if an object is present within the discretized domain 54, it is expected that the first mass density distribution will indicate that the locations 55 that include the target mass 52 will have different masses than surrounding areas that are empty (e.g., do not include the target mass 52). If the mass density variation exceeds a threshold, this may indicate the target mass 52 is present, and thus it may be advantageous to allocate more locations 55 where the object may be present. If the mass density variation is below the threshold, this may indicate the target mass 52 is not present, and thus it may be advantageous to reduce the number pixels in a subset of pixels where the object may be absent.

[0141] At block 428, the mass density analysis system 14 determines a second pixel allocation based on the subset of locations having a mass density variation that exceeds the threshold. The second pixel allocation may include a denser distribution of locations 55 in areas of the discretized domain 54 where the mass density variation exceeded the threshold. Further, the second pixel allocation may include a less dense distribution of location 55 where the mass density variation is below a threshold.

[0142] Accordingly, the mass density analysis system 14 may utilize the method 420 to search for potential locations where a mass of interest is present, which is indicated by the mass density variation between pixels. In turn, the mass density analysis system 14 may increase the number of pixels in locations where the mass of interest may be present, while decreasing the number of pixels in locations where the mass of interest is likely not present. This may reduce the computational complexity of solving the systems of equations by preventing pixels from being allocated to locations where the mass of interest is not present.Three-dimensional Space Discretization into Volumetric Pixels

[0143] Forming a discretized domain 54 such that the locations 55 represent 3D locations (e.g., voxels) is a computationally difficult task. It would be useful to reduce the number of locations 55 without compromising the quality of the mass density distribution output 76 since solving for a 3D discretized domain 54 increases the computational complexity for solving the systems of equations. One technique to efficiently solve for the discretized domain 54 is to utilize the integration of the scalar gravitational potential field over voxels. To do so, each location 55 (e.g., voxel) is treated as a point mass and the mass density analysis system 14 calculates the gravitational potential for each point mass. Accordingly, rather than treating the gravitational potential field as a continuous field across a volume, it is discretized into gravitational potential fields at specific locations 55, which simplifies the calculation.

[0144] FIG. 19 shows a flow diagram of an example of a method 440 for determining a pixel allocation that may be utilized in the method 380 of FIG. 16. Although the process 440 is described as being performed by the mass density analysis system 14 of the mass density analysis system 14, it should be noted that the process 440 may be performed by any suitable processor.

[0145] At block 442, the mass density analysis system 14 determines a pixel allocation (e.g., a pixel allocation 56). In general, the mass density analysis system 14 may determine the pixel allocation based on the sensor mapping 68 and / or the domain data 406 as described with respect to FIGS. 16 and 17. At block 444, the mass density analysis system 14 integrates the scalar gravitational potential field over a subset of locations 55 of the pixel allocation. For example, the mass density analysis system 14 calculates a gravitational potential field for a location 55 by calculating gravitational potential field due to other locations 55 that surround the location 55. Then, the mass density analysis system 14 may sum all gravitational potential fields due to the surrounding pixels to determine the gravitational potential field at the location 55. The mass density analysis system 14 may continue to do this calculation for each location 55, thereby providing an approximation of the gravity potential field within the discretized domain 54 that may serve as a reference for a minimum gravity potential energy for the discretized domain 54.

[0146] At block 446, the mass density analysis system 14 identifies a subset of the locations 55 having a mass density distribution within one or more threshold ranges. In some embodiments, the one or more threshold ranges may correspond to certain materials, elements, or compositions expected to be within an area sensed by the gravity sensors. For example, the one or more threshold ranges may correspond to water (e.g., which may indicate a human body), iron (e.g., which may indicate metals and / or metal alloys including iron), air (e.g., which may indicate areas where a mass is absent), and other compositions or materials. The subset of pixels that include pixel values within these threshold ranges may indicate realistic solutions. As such, the mass density analysis system 14 may retain these locations 55 while adjusting other locations 55 and iterate through solving for the mass density distribution again. At block 448, the mass density analysis system 14 may determine a second pixel allocation 450 that includes the subset having the mass density distribution within the threshold range, while other pixels are modified. In some embodiments, determining the second pixel allocation 450 may include increasing the density of locations when the mass values correspond to masses within the one or more threshold ranges. For example, the mass density analysis system 14, at block 446, may determine that a first subset of pixels correspond to air and / or an object of interest, and that a second subset of pixels correspond to mass densities outside of an expected threshold range. As such, rather than solving or determining an entirely new pixel allocation, the mass density analysis system 14 may only modify or adjust the locations 55 within the second subset. In this way, the process 440 may reduce computational resources utilized for solving for the mass density distribution output 76 by maintaining locations 55 that provide realistic solutions, while adjusting other locations 55 until they provide a realistic solution. This may be computationally more efficient than solving for the mass density distribution with the same pixel allocation 56 repeatedly until the calculation arrives at a realistic solution. In some embodiments, the mass density analysis system 14 may iterate through FIG. 19 multiple times to further improve the second pixel allocation 450.Solving for the Mass Density Distribution by Subdividing a Matrix into Subblocks

[0147] Solving the equations or matrices to generate the mass density distribution output 76 may be computationally difficult due to the large size of the matrix. In particular, while Cx — b may be solved using an inverse optimization technique to generate the mass density distribution output 76, inverting relatively large matrices is computationally expensive and not readily parallelizable. Further, for very large sized operator matrices (e.g., such as floating point 32-bit square matrix of size 50,000x50,000 and above), it may be difficult to fit the whole matrix in a standard computer memory. One technique to ease solving the equations is to utilize iterative inversion processes, such as Fast Non-Negative Least Squares (FNNLS) approach. In this approach, the matrix is subdivided into blocks, and the inversion for each matrix is performed using a cluster of CPU / GPU machines. FIG. 20 shows a flow diagram of a method 460 for solving a matrix to determine the mass density distribution output 76 that may be utilized in the method of FIG. 13.

[0148] At block 462, the mass density analysis system 14 generates multiple block submatrices using the matrix information 68 and the pre-processed gravity measurements 360. In general, the mass density analysis system 14 may divide the matrix x into multiple block submatrices using an iterative approach such as FNNLS. In particular, it is presently recognized that FNNLS may be useful for solving matrices that include gravitational force values since these values are not negative and FNNLS solves the matrix x in ||Cx- b||2subject to x>0. In some embodiments, the mass density analysis system 14 may determine a size for each block submatrices based on an amount of memory available to ultimately solve the matrices.

[0149] At block 466, the mass density analysis system 14 determines or computes an inverse of the multiple block submatrices using a distributed computing technique, such as DASK. Once the inverse of each block submatrix is determined, the mass density analysis system 14 may determine the inverse for the larger matrix. At block 468, the mass density analysis system 14 solves Cx — b using the inverse. Accordingly, the method 460 provides a technique for solving for the inverse in a distributed manner. Thus, performing the process 460 may utilize less computational resources (e.g., memory and computing time) as compared to using a single computing device.A Regularization Parameter Based on an Approximation of the Lowest Energy of Mass Density Distributions

[0150] A regularization parameter may facilitate solving the systems of equations or matrices (e.g., at block 308 of FIG. 13). One example regularization parameter that may be used may be based on the principle of energy minimization. One such regularization parameter is the L2 norm, which utilizes entropy and energy minimization principles. FIG.21 shows a flow diagram of a method 470 for solving a matrix to determine a mass density distribution using one or more regularization parameters, such as the L2 norm. At block 472, the mass density analysis system 14 obtains one or more L2 regularization parameters 473 (e.g., constraints 72). In general, it is presently recognized that it may be advantageous to utilize L2 norm regularization when a problem is ill-posed. In general, L2 norm regularization utilizes entropy and energy minimization principles to solve for the matrix x. In some embodiments, the L2 regularization parameters may include one or more of a minimum mass, a maximum mass, a minimum energy, a maximum energy, and so on. As a few non-limiting examples, the mass density analysis system 14 may utilize a Tikhonov regularization parameter, a Lasso regularization parameter, or both.

[0151] At block 474, the mass density analysis system 14 solves for the matrix x using the one or more L2 regularization parameter (e.g., using the matrix information 68 and pre-processed gravity measurements 460 or gravity measurements 64), thereby obtaining the mass density distribution output 76. Solving the matrix using the one or more regularization parameters may include minimizing ||C% — b||2+oc 7?(%), subject to conditions such as x >0, where oc / ?(x) is the one or more regularization parameters. Solving this problem includes iterating multiple times to fine tune the one or more regularization parameters for a given problem (e.g., a particular target mass 52). Accordingly, the method 470 provides a technique for solving for the mass density distribution output 76 using regularization parameters, which help provide a stable solution for ill-posed problems.Generating an Image Using the Mass Density Distribution

[0152] The mass density distribution may be used to generate an image that provides a visualization of the target mass 52 within the physical location (e.g., represented by the discretized domain 54). The term x in the simplified equation Cx — b represents the mass at each location 55 (e.g., area or volume) within the discretized domain 54. An image may be generated using the solved x, where the image is formed of pixels that have a size proportional to the dimensions of the location 55. Further, the image may have a color or pixel value that corresponds to the mass in each element of the matrix x. To illustrate, FIG. 22 shows a flow diagram of a method 480 for generating a clear image 482 using the mass density distribution output 76.

[0153] At block 484, the mass density analysis system 14 generates a blurred image based on the mass density distribution output 76. To do so, the mass density analysis system 14 may generate pixels for each element of the matrix x. The dimensions of each pixel may be equal to dimensions of the location 55. Further, the mass density analysis system 14 may determine a color based on the range of values in the elements of the matrix x, and assign each pixel a pixel value that corresponds to the mass at each location 55.

[0154] At block 486, the mass density analysis system 14 de-blurs the blurred image using deblurring techniques. As described in further detail below, the deblurring techniques may include using a suitable neural network model architecture. The suitable neural network model architectures may be trained using blurred images and corresponding clear or smooth images. Accordingly, the neural network model architecture may store relationships between subsets of pixels of the blurred images and subsets of pixels of the smooth images. Further, the neural network model architecture may be capable of transforming a blurred image into a smooth image based on the relationships.

[0155] At block 488, the mass density analysis system 14 generates a clear image 482. To generate the clear image 482, the mass density analysis system 14 may iterate through the deblurring process multiple times until the image has a suitable resolution. In this way, FIG.22 shows an example technique whereby the mass density analysis system 14 generates an output such as a clear image 482 that corresponds to a mass density distribution. In turn, the clear image may be further utilized to determine whether certain objects are present. Forexample, the mass density analysis system 14 may be trained using certain object-based image analysis techniques, and thus, be capable of determining whether a subset of pixels of the clear image 482 corresponds to a particular object of interest.Deblurring Images Using a Trained Neural Network Model Architecture

[0156] The initial image generated (e.g., at block 484) using the mass density distribution output 76 may be blurred due to, for example, the number of gravity sensors 16 used to generate the mass density distribution output 76, the arrangement of the gravity sensors 16, the regularization parameters used to solve the systems of equations or matrices (e.g., at block 308 of FIG. 13), and so on. In any case, it may be advantageous to deblur the image using a trained neural network (NN) model architecture. To illustrate this, FIG. 23 shows a flow diagram of a method 490 for generating a clear image using a trained neural network (NN) model architecture that may be utilized in the method of FIG. 13.

[0157] At block 492, the mass density analysis system 14 obtains a trained model architecture. In general, the model architecture may include any suitable NN model architectures that may be utilized for removing blur in an image. For example, the trained model architecture may include a U-net architecture, an auto-encoder architecture, a generative adversarial network (GAN) NN architecture, and so on. The trained model architecture may be trained using blurred and deblurred training images such that the trained model architecture is capable of providing a deblurred image output based on a blurred image input. As referred to herein, a “blurred image” refers to an image having a resolution below a resolution threshold. In contrast, a “deblurred image” refers to an image having a resolution above a resolution threshold. In some embodiments, obtaining the trained model architecture may include retrieving the trained model architecture from a storage component, such as from the memory 28, a cloud database, or any other suitable storage component. In some embodiments, obtaining the trained model architecture may include the mass density analysis system 14 training or generating the trained model architecture.

[0158] In any case, at block 494, the mass density analysis system 14 provides the mass density distribution output 76 as an input to the trained model architecture. Accordingly, atblock 496, the mass density analysis system 14 receives a deblurred image (e.g., a mass density distribution output 76) as an output of the trained model architecture. As generally mentioned above, the deblurred image may have a high image resolution than the blurred image. For example, the deblurred image may have a first resolution that is 5% or greater, 10% or greater, 15% or greater, 20% or greater, or 25% or greater than a second resolution of the blurred image. At least in some instances, the mass density analysis system 14 may repeat the process 490 one or more times, using the same trained model architecture or a different trained model architecture. For example, the mass density analysis system 14 may repeat the process 490 until a resolution or smoothness of the clear image 482 coverages.Training an NN model Architecture

[0159] The trained NN model architecture is trained or retrained (e.g., refined) using blurred training images and corresponding clear training images. This training process is generally illustrated in the method 500 of FIG. 24.

[0160] At block 502, the mass density analysis system 14 obtains the training data (e.g., blurred training images 504 and deblurred training images 506). In general, one blurred image may correspond to one or more deblurred images. Thus, the training data may include a first image that is the blurred image (e.g., having a resolution below a resolution threshold). The deblurred images may include one or more deblurred images of the blurred image with varying degrees of blurring. By training on such images, the NN model architecture may be capable of correlating varying levels of blurring to a true image.

[0161] At block 508, the mass density analysis system 14 generates or trains a neural network model architecture 510 using the training data (i.e., the blurred training images 504 and the deblurred training images 506). Accordingly, the neural network model architecture may be trained to draw inferences between certain patterns (i.e., a first subset of pixels and / or voxels) in the blurred images that correspond to patterns (i.e., a second subset of pixels and / or voxels) in the deblurred images. In some embodiments, the mass density analysis system 14 may train the neural network model architecture using GPUs.Autoencoder NN model Architecture

[0162] One type of NN model architecture that may be used to deblur the blurred image generated by the mass density distribution output 76 is an autoencoder architectures. The auto-encoder architecture consists of an encoder block and a decoder block. The encoder block has multiple convolution layers and pooling layers. The decoder has multiple layers of transpose-convolution layers. The neural network (NN) learns to deblur the reconstructed mass density distribution images. The NN may be trained on a dataset of ten thousand true and blurred images. In some embodiments, the blurred images for the dataset may be obtained from a gravity camera simulator. As referred to herein, a “gravity camera simulator” is a software application, model, or code that is capable of generating a mass density distribution and / or an image of the mass density distribution using inputs from the gravity sensors and the information of the surroundings (e.g., volumetric and discretization information of the space of interest, objects that may impart a gravitational force to the gravity sensors 16 but are outside of the discretized domain 54). The gravity camera simulator may generate the image by calculating the scalar potential field measured by an array of simulated gravity sensors 16. The number of gravity sensors 16 in the array and / or the number of sides of the object that the gravity sensors 16 surround may be adjustable to increase or decrease the resolution of the image. It may be useful to simulate multiple images to provide a large enough training set to increase the accuracy and robustness of the gravity camera simulator. FIG. 25 shows a flow diagram of a method 520 for generating an autoencoder architecture that may be utilized in the method of FIG. 13.

[0163] At block 526, the mass density analysis system 14 provides training images to an encoder. As discussed above with reference to block 502 of FIG. 24, the training images may include blurred training images and deblurred training images. At block 528, the mass density analysis system 14 obtains a latent space representation. In general, the latent space representation transforms the training images from a first space to a second space that is a lower dimensional space. At block 530, the mass density analysis system 14 provides the latent space representation to the decoder. The decoder attempts to reconstruct the input (e.g., a true image) using the latent space representation. The output of the decoder (e.g., thereconstructed image) is then compared to the deblurred training image input to determine whether the decoder is capable of producing a match, which provides feedback for refining the model. This process may then be iterated through multiple images until providing a trained decoder that is capable of producing a reconstructed image that matches the deblurred training image, thereby providing an autoencoder architecture 532 that may be used to deblur images.Generative Adversarial Network Model Architecture

[0164] Another type of NN model architectures that may be used to generate the clear image 482 is a U-net architecture. A U-net architecture is composed of a generator 542 and a discriminator 544. The generator 542 creates images starting from random noise, and the discriminator 544 determines whether the image created is true or fake. During the model training process, the generator 542 learns how to generate real-looking images such that it may be capable of deceiving the discriminator 544. FIG. 26 shows a flow diagram of a method 540 for training a U-net architecture that may be utilized in the method of FIG. 13.

[0165] As shown, blurred training images 504 are provided to the generator 542. The generator 542 attempts to generate a deblurred image or “true image”. The deblurred training image 506 and / or the true image (e.g., the output of the generator 542) are provided to a discriminator. The discriminator 544 determines whether the output of the generator 542 is an accurate representation of, or otherwise matches the deblurred training image 506. If the discriminator determines that the output of the generator 542 matches the deblurred training image 506, the discriminator 544 may output an indication that the deblurred image and / or the true-like image is accurate or otherwise “real”. Alternatively, if the discriminator determines that the output of the generator 542 does not match the deblurred training image 506, the discriminator 544 may output an indication that the output of the generator 542is not accurate or otherwise “fake”. As shown, the “real” or “fake” indication may be provided back to the discriminator 544 as feedback, thereby refining the ability of the discriminator to distinguish between deblurred images and fake images (e.g., images generated by the generator 542). Similarly, the “real” or “fake” indication may be provided back to thegenerator 542 as feedback, thereby refining the ability of the generator to generate deblurred images using the blurred training images 504.Regularization Parameter Selection for Mass Density Image Reconstruction

[0166] Because the equation Cx — b is ill-posed, solving the equation may include solving an inverse optimization problem. An inverse optimization problem utilizes one or more regularization parameters to recover optimized unique solution. The choice of the regularization parameter depends on the physics of the problem. Since gravity is a fundamental force of nature that follows energy minimization and entropy principles, it may be advantageous for the regularization parameter to include the L2 norm of the scalar gravitational potential field, which corresponds to the energy of the system. Additionally or alternatively, the regularization parameter may include the scalar hyperparameter, X, along with the L2 norm. The hyperparameter may be determined using binary search over a large logarithmic range (e.g., 10’5to 10+5). The mass density analysis system 14 may perform an iterative search within different subranges to identify the lowest error associated with the hyperparameter. The hyperparameter is unknown beforehand, but once known, it provides insight into what the minimum energy may be, thereby facilitating the optimization process. The choice of L2 energy for regularization and a scalar tunable hyperparameter make the solution to inverse problem better fit to the true mass density distribution output 76. An example process 550 for determining or selecting regularization parameters is shown in FIG.27.

[0167] At block 554, the mass density analysis system 14 integrates the scalar gravitational potential field over locations 55 using training images 552. To do so, the mass density analysis system 14 calculates the scalar gravitational potential field at each location 55 by summing up the contributions of the gravitational potential field from each surrounding location 55. The training images 552 may include a mass that resembles (e.g., has similar dimensions, has similar composition) one or more target masses 52 to be detected by a gravity sensor array 44. For example, the training images 502 may include training images of one or more people occupying different locations within the shipping container 180. As such, the calculated scalar gravitational potential field may be similar to what may be detected by thegravity sensors 16. As such, the resulting L2 norm may more accurately represent the physical system.

[0168] At block 556, the mass density analysis system 14 determines an L2 norm 558 based on the integration over the scalar gravitational potential field over the pixel allocation. That is, the mass density analysis system 14 may calculate the L2 norm for all of the gravitational potential fields of the locations 55. This L2 norm (e.g., the L2 norm of the scalar gravitational potential field) may be used as a regularization parameter and it represents the minimum energy in the physical system. The mass density analysis system 14 also identifies a hyperparameter by selecting different hyperparameters within a range and determines which hyperparameter provides a realistic solution for solving for the mass density distribution output 76. This may include using the L2 norm of the gravitational potential field with a selected hyperparameter as a regularization parameter to solve for the mass density distribution output 76 (e.g., the mass density distribution within a discretized domain 54). As solving for the mass density distribution output 76 is performed multiple times, the mass density analysis system 14 may vary the hyperparameter. By varying the mass density analysis system 14, the mass density analysis system 14 may identify a particular hyperparameter that increases the likelihood that solving the mass density distribution output 76 provides a realistic solution. Once identified, the mass density analysis system 14 may utilize this hyperparameter for subsequent calculations, rather than having to recalculate the hyperparameter for each calculation. In this way, the process 550 involves using the gravitational potential field which corresponds to the energy of the system as a regularization parameter. Using such an energy parameter may indicate what the minimum energy of the system may be, and thus, guide the optimization process to provide a more accurate mass density distribution output 76.Adjusting Precision of Gravity Measurements

[0169] As described with reference to block 304 of FIG. 13, the mass density analysis system 14 may modify the gravity measurements 64 based on a degree of precision or sensitivity associated with the gravity sensors 16 that obtained the gravity measurements 64.To illustrate this, FIG. 28 shows a flow diagram of a method 560 for adjusting gravity measurements 64.

[0170] At block 562, the mass density analysis system 14 receives gravity measurements 64. At block 564, the mass density analysis system 14 determines a degree of precision for the gravity measurements 64. For example, the mass density analysis system 14 may determine a type of gravity sensor 16 that obtained the gravity measurements 64. As generally mentioned herein, different types of gravity sensors 16 may have different degrees of precision. At block 566, the mass density analysis system 14 adjusts the gravity measurements 64 based on the degree of precision. For example, the mass density analysis system 14 may truncate or augment the gravity measurements 64 based on decimal places of the data not changing within a threshold range. It should be noted that certain techniques may be performed using floating point (FP) 32-bit. FP 32-bit may be sensitive to small perturbations in data, due to noise. To avoid these errors, it may be advantageous to use 64-bit or 128-bit, which may provide higher precision, and thus, more insight into forces. At least in some instances, to avoid losing numbers or decimals during multiplication, the mass density analysis system 14 may separate integer and floating point, treat as logs, and utilize other techniques known to one of ordinary skill in the art. At block 568, the mass density analysis system 14 outputs the adjusted gravity measurements 570, which may be further utilized in block 306 described with reference to FIG. 13.

[0171] An optimal placement of the gravity sensors 16 (e.g., a distance between the gravity sensor 16 and the target mass 52 where the target mass 52 may be detected) depends on the sensitivity of the sensors. FIG. 29 shows a graph depicting mass (e.g., y-axis) of an object and distance (e.g., x-axis) from gravity sensors 16, which may be useful for selecting gravity sensors 16 for a particular imaging or detection application. The graph in FIG. 29 includes a box 602 that indicates a sample range of sensitivities for certain gravity sensors 16. Further, the graph includes a first point 604, a second point 606, a third point 608, a fourth point 610, and a fifth point 612, which indicate different examples of objects. The first point 604 corresponds to a person in a room, the second point 606 corresponds to a plane at an airport, the third point 608 corresponds to a large vessel (e.g., having a weight betweenabout 100 to 1000 tons), the fourth point 610 corresponds to relatively large boat vessel, and the fifth point 612 corresponds to an asteroid. In general, the graph shows that gravity sensors 16 with higher sensitives may be capable of detecting larger objects and at further distances from the objects.Example Pixel Allocations

[0172] The mass density analysis system 14 may utilize one or multiple pixel allocations 56 to solve for the mass density distribution that accurately represents the discretized domain 54. To that end, FIGS. 30-32 show different examples of pixel allocations 56 that may be utilized in the techniques described herein. For example, FIG. 30 shows an image 650 with a pixel allocation 56 that is a uniform grid (e.g., a uniform pixel allocation). The image 204 shows a true image (e.g., as shown in FIG .12C) and the image 650 shows a simulated image using an array of gravity sensors 16 positioned on one side of the target masses 52a and 52b. In the uniform grid, each location 55 has substantially the same dimensions (e.g., across the x-axis 42 and y-axis 44). That is, the locations 55 are allocated substantially uniformly. A uniform grid may be easier to utilize because it may not utilize additional processing (e.g., by the processors 26) to determine a pixel allocation. However, it may be advantageous to vary the size and / or shape of locations 55 to improve the likelihood of detecting an object and / or reduce computational complexity.

[0173] FIG. 31 shows an image 660 generated based on a pixel allocation 56 that is a boundary concentrated grid (e.g., stretched grid, boundary concentrated pixel allocation, stretched pixel allocation). The image 204 shows a true image and the image 660 shows a simulated image using an array of gravity sensors 16 on the left side of the target masses 52a and 52b. In the boundary concentrated grid, there are relatively more locations 55 (e.g., pixels) at the boundaries of the image 660 and the dimensions of the locations 55 generally increase moving away from the left boundary, which corresponds to where the gravity sensors 16 obtained measurements. Since gravity follows inverse square law, the error for a detected mass at a location 55 increases exponentially as the distance between the locations 55 and the gravity sensor 16 increases. Accordingly, the mass density analysis system 14 may allocate more locations 55 at the boundary such that the result mass density distribution output 76 hasmore pixels where there is higher gravity sensor resolution. In a similar manner, the mass density analysis system 14 may decrease the number of locations 55 further from the gravity sensors 16.

[0174] FIG. 32 shows an image 670 generated based on a pixel allocation 56 that is an adaptive grid. The image 204 shows a true image and the image 670 shows a simulated image using an array of gravity sensors 16 on the left side of the target masses 52a and 52b. In the adaptive grid, the mass density analysis system 14 may provide relatively more locations 55 (e.g., smaller pixels) in areas where there is determined change in mass density. As described with reference to FIG. 18, the change in mass density may indicate that an object of interest in present. Accordingly, by adding more locations 55 in an area where there is a change in mass density, the mass density analysis system 14 may obtain more granular information to determine whether an object is present. By not providing more locations 55 in an area where the target mass 52 is likely not present, the mass density analysis system 14 may not waste computational resources and, instead, utilize those computational resources elsewhere.

[0175] In some embodiments, the measurement collection system 12 may be utilized to obtain gravity measurements 64 of a relatively large volume. It is presently recognized that it may be advantageous to solve the equations or matrices (e.g., solve Cx — b) through multiple iterations, each time using a pixel allocation 56 for a discretized domain 54 that includes a portion with a high density of locations 55, while the remaining portion of the discretized domain 54 has a low density of locations 55. In each interaction, the portion that includes the high density of locations 55 may be moved to a different portion of the volume. As such, by solving the equations or matrices with each new discretized domain 54, this provides effectively a scan of the masses within a volume. Solving the equations or matrices in this manner may be more efficient than solving for a discretized domain that includes a high density of locations 55 throughout the discretized domain 54. To illustrate this, FIG. 33 shows a volume 620 that includes an object 622 to be detected using the gravity measurement system 10. The volume 620 may be represented using a discretized domain 54 that includes a high-resolution portion 624 and multiple low-resolution portions 626. The mass density analysis system 14 may record gravity measurements multiple times, and each time, thegravity measurements may be processed with an adjusted discretized domain 54. For example, the mass density analysis system 14 may adjust the discretized domain 54 such that the high-resolution portion 624 corresponds to a different location within the volume 620, thereby producing new low-resolution portions 626. The mass density analysis system 14 may iteratively solve the matrix equation Cx — b, thereby obtaining a solution for the high-resolution portion 624 in each iteration. If the solution for the high-resolution portion 624 does not indicate that the object 622 is present, the mass density analysis system 14 may allocate fewer locations 55 to that portion in the following iteration. Accordingly, the mass density analysis system may allocate locations only in portions of the volume 620 where the object 622 likely exists. This may improve the efficiency of solving the matrix equation Cx — b by preventing the matrix x from being too large.

[0176] The foregoing description, for purpose of explanation, has been described with reference to specific embodiments. However, the illustrative discussions above are not intended to be exhaustive or to limit the disclosure to the precise forms disclosed. Many modifications and variations are possible in view of the above teachings. Moreover, the order in which the elements of the methods described herein are illustrated and described may be re-arranged, and / or two or more elements may occur simultaneously. The embodiments were chosen and described in order to best explain the principals of the disclosure and its practical applications, to thereby enable others skilled in the art to best utilize the disclosure and various embodiments with various modifications as are suited to the particular use contemplated.

[0177] Finally, the techniques presented and claimed herein are referenced and applied to material objects and concrete examples of a practical nature that demonstrably improve the present technical field and, as such, are not abstract, intangible or purely theoretical. Further, if any claims appended to the end of this specification contain one or more elements designated as “means for [perform]ing [a function]...” or “step for [perform]ing [a function]...”, it is intended that such elements are to be interpreted under 35 U.S.C. 112(f). However, for any claims containing elements designated in any other manner, it is intended that such elements are not to be interpreted under 35 U.S.C. 112(f).

Claims

CLAIMS1. An apparatus for determining a density distribution of a mass based on gravity data, the apparatus comprising:a plurality of gravity sensors, each of the plurality of sensors generating a signal based on the mass; anda processing system operatively coupled to the plurality of gravity sensors, the processing system configured to:receive each respective signal from each respective gravity sensor; anddetermine the density distribution of the mass based on the plurality of signals.

2. The apparatus of claim 1, wherein the plurality of gravity sensors comprises accelerometers, absolute gravimeters, relative gravimeters, spring gravimeters, magnetic field gravimeters, or any combination thereof.

3. The apparatus of any of claims 1 and 2, wherein each respective signal is processed together to determine the density distribution of the mass.

4. The apparatus of any of claims 1 to 3, wherein the plurality of gravity sensors is arranged in a linear array.

5. The apparatus of any of claims 1 to 3, wherein the plurality of gravity sensors is arranged in a two-dimensional array.

6. The apparatus of any of claims 1 to 3, wherein the plurality of gravity sensors is arranged in a three-dimensional array.

7. The apparatus of any claims 1 to 6, comprising a system to hold each of the plurality of gravity sensors in a fixed position relative to one another.

8. The apparatus of any of claims 1 to 7, comprising a system to determine a respective location of each respective gravity sensor relative to at least one other of the plurality of gravity sensors.

9. The apparatus of any of claims 1 to 8, wherein the processor is further configured to solve one or more equations based on the signals from the plurality of gravity sensors to determine the density distribution of the mass.

10. The apparatus of claim 9, wherein the equations comprise one or more matrices that are solved to determine the density distribution of the mass.

11. The apparatus of any of claims 1 to 10, comprising generating an image of the mass based on the density distribution.

12. The apparatus of any of claims 1 to 11, comprising generating a location of the mass based on the density distribution.

13. The apparatus of any of claims 1 to 12, comprising generating an indication of a change in density distribution of the mass based on a previous determination of density distribution of the mass.

14. The apparatus of any of claims 1 to 13, wherein each of the plurality of gravity sensors generates a respective signal related to gravitational force associated with the mass.

15. A method for determining a density distribution of a mass based on gravity data, the method comprising:obtaining a plurality of data signals generated by a respective plurality of gravity sensors, wherein each of the plurality of data signals is related to a respective gravitational force associated with the mass; andprocessing each of the respective data signals together to determine the density distribution of the mass.

16. The method of claim 15, comprising generating a three-dimensional volume of the mass based on the density distribution.

17. The method of any of claims 15 and 16, wherein the processing comprises solving one or more equations based on the data signals to determine the density distribution of the mass.

18. The method of claim 17, wherein solving the one or more equations comprises solving one or more matrices to determine the density distribution of the mass.

19. The method of any of claims 16 to 18, comprising generating an image of the mass based on the three-dimensional volume.

20. The method of claim 19, wherein generating the image comprises generating a two- dimensional image of the three-dimensional volume.

21. The method of any of claims 19 and 20, wherein generating the image comprises:generating a blurred image of the mass; andapplying one or more deblurring techniques to the blurred image to generate a clearer image of the mass.

22. The method of claim 21, generating the clearer image of the mass comprises:providing the blurred image to a trained neural network architecture; and receiving, as an output of the trained neural network model, the clearer image.

23. The method of any of claims 19 to 22, wherein generating the image comprises generating a three-dimensional image of the three-dimensional volume.

24. The method of claim 23, wherein generating the image comprises:generating a blurred image of the three-dimensional volume; and applying one or more deblurring techniques to the blurred image to generate a clearer image of the three-dimensional volume.

25. The method of claim 24, wherein generating the clearer image of the mass comprises:providing the blurred image to a trained neural network architecture; and receiving, as an output of the trained neural network model, the clearer image.

26. The method of any of claims 15 to 25, comprising generating a location of three- dimensional volume of the mass.

27. The method of any of claims 15 to 26, comprising generating an indication of a change in density distribution of the mass based on a previous determination of density distribution of the mass.

28. The method of any of claims 15 to 27, comprising generating a four-dimensional density distribution of the mass based on volume of the mass and time.

29. A method of generating a mass density distribution of an object, the method comprising:obtaining a plurality of data signals from a respective plurality of gravity sensors, wherein the plurality of data signals is related to the object;pre-processing the plurality of data signals to obtain a plurality of pre-processed data signals;forming one or more matrices based on the plurality of pre-processed data signals; andsolving the one or more matrices to obtain the mass density distribution of the object.

30. The method of claim 29, wherein pre-processing comprises:determining a time period corresponding to the plurality of data signals; obtaining one or more pre-process inputs corresponding to sensor noise correction factors; andgenerating the pre-processed data signals based on the one or more pre-process inputs and the time period.

31. The method of any of claims 29 and 30, wherein forming one or more matrices comprises:identifying metadata associated with the pre-processed data signals; and determining the one or more matrices based at least in part on the metadata and the pre-processed data signals.

32. The method of any of claims 29 to 31, wherein pre-processing the plurality of data signals and forming one or more matrices comprises:obtaining a sensor mapping of the plurality of gravity sensors;determining a domain relative to the object and to the plurality of gravity sensors based on the sensor mapping;determining a pixel allocation based on the domain; andgenerating the one or more matrices based at least in part on the pixel allocation.

33. The method of claim 32, wherein determining the pixel allocation comprises: identifying a boundary of the domain bound by the plurality of gravity sensors based on the sensor mapping; anddetermining a pixel allocation based on the boundary of the domain.

34. The method of any of claims 32 and 33, wherein determining the pixel allocation comprises:determining a first pixel allocation comprising pixels having pixel values corresponding to mass densities;identifying a mass density variation between the pixel values of the first pixel allocation;determining a subset of the pixels having a mass density variation that exceed a threshold; anddetermining a second pixel allocation based on the subset of the pixels, wherein the second pixel allocation is the pixel allocation.

35. The method of any of claims 32 to 34, wherein determining the pixel allocation comprises:determining a first pixel allocation comprising pixels having pixel values corresponding to mass densities;integrating a scalar gravitational potential field over the pixel values of the first pixel allocation;identifying a subset of the pixels having pixel values within one or more threshold ranges;determining a second pixel allocation based on the subset of the pixel values, wherein the second pixel allocation is the pixel allocation.

36. The method of any of claims 32 to 35, wherein determining the pixel allocation comprises:allocating more pixels in the domain nearest the plurality of gravity sensors and allocating fewer pixels in the domain furthest from the plurality of gravity sensors.

37. The method of any of claims 32 to 36, wherein determining the pixel allocation comprises:allocating pixels more pixels in portions of the domain indicating the mass density distribution of the object and fewer pixels in portions of the domain not indicating the mass density distribution of the object.

38. The method of any of claims 32 to 37, wherein the domain is bound on at least one side by the plurality of gravity sensors.

39. The method of claim 38, wherein the domain is bound on more than one side by the plurality of gravity sensors.

40. The method of any of claims 29 to 39, wherein solving the one or more matrices to determine the mass density distribution of the object comprises:generating a plurality of block submatrices based on the one or more matrices; determining the inverse of the plurality of block submatrices; andsolving the one or more matrices using the inverse of the plurality of block submatrices.

41. The method of any of claims 29 to 40, wherein solving the one or more matrices to determine the mass density distribution of the object comprises:obtaining one or more regularization parameters based on one or more constraints; andsolving the one or more matrices using the one or more regularization parameters.

42. The method of any of claims 29 to 41, comprising generating an image of the object based on the mass density distribution of the object.

43. The method of claim 42, wherein generating the image of the object comprises:generating a blurred image of the object based on the mass density distribution of the object; andapplying one or more deblurring techniques to the blurred image to generate a clearer image of the object.

44. The method of claim 42, wherein generating an image of the object comprises:generating a blurred image of the object based on the mass density distribution of the object;obtaining a trained neural network model architecture;providing the blurred image to the trained neural network model architecture; receiving, as an output of the trained neural network model architecture, a clearer image of the object.

45. The method of claim 44, wherein the trained neural network model architecture comprises a U-net architecture, a GAN architecture, or an autoencoder architecture.

46. A system, comprising:a plurality of gravity sensors, wherein each of the gravity sensors is configured to generate respective data relating to a mass of an object;a mass density analysis system comprising one or more processors, wherein the mass density analysis system is configured to:receive the respective data generated by each of the plurality of gravity sensors;pre-process the respective data to generate pre-processed data; determine one or more matrices based on the pre-processed data; andsolve the one or more matrices to determine a mass density distribution of the object.

47. The system of claim 46, wherein the mass density analysis system is configured to pre-process the respective data to generate the pre-processed data by:determining a time period corresponding to respective data;obtaining one or more pre-process inputs corresponding to sensor noise correction factors; andgenerating the pre-processed data based on the one or more pre-process inputs and the time period.

48. The system of any of claims 46 and 47, wherein the mass density analysis system is configured to determine one or more matrices by:identifying metadata associated with the pre-processed data;determining reference information of the plurality of gravity sensors based on the metadata; anddetermining the one or more matrices based at least in part on the reference information and the pre-processed data.

49. The system of any of claims 46 to 48, wherein the mass density analysis system is configured to pre-process the respective data and determine one or more matrices by:obtaining a sensor mapping of the plurality of gravity sensors;determining a domain to be monitored by the plurality of gravity sensors based on the sensor mapping;determining a pixel allocation based on the domain; andgenerating the matrix based at least in part on the pixel allocation.

50. The system of claim 49, wherein the mass density analysis system is configured to determine the pixel allocation by:identifying a boundary of the domain to be monitored by the plurality of gravity sensors based on the sensor mapping; anddetermining a pixel allocation based on the boundary of the domain.

51. The system of any of claims 49 and 50, wherein the mass density analysis system is configured to determine the pixel allocation by:determining a first pixel allocation comprising pixels having pixel values corresponding to mass densities;identifying a mass density variation between the pixel values of the first pixel allocation;determining a subset of the pixels having a mass density variation that exceeds a threshold; anddetermining a second pixel allocation based on the subset of the pixels, wherein the second pixel allocation is the pixel allocation.

52. The system of any of claims 49 to 51, wherein the mass density analysis system is configured to determine the pixel allocation by:determining a first pixel allocation comprising pixels having pixel values corresponding to mass densities;integrating a scalar gravitational potential field over the pixel values of the first pixel allocation;identifying a subset of the pixels having pixel values within one or more threshold ranges; anddetermining a second pixel allocation based on the subset of the pixel values, wherein the second pixel allocation is the pixel allocation.

53. The system of any of claims 46 to 52, wherein the mass density analysis system is configured to solve the one or more matrices to determine the mass density distribution of the object by:generating a plurality of block submatrices based on the one or more matrices; determining the inverse of the plurality of block submatrices; andsolving the one or more matrices using the inverse of the plurality of block submatrices.

54. The system of any of claims 46 to 53, wherein the mass density analysis system is configured to solve the one or more matrices to determine the mass density distribution of the object by:obtaining one or more regularization parameters based on the one or more constraints; andsolving the one or more matrices using the one or more one or more regularization parameters.

55. The system of any of claims 46 to 54, wherein the mass density analysis system is configured to generate an image of the object by:generating a blurred image of the object based on the mass density distribution of the object; andapplying one or more deblurring techniques to the blurred image to generate a clearer image of the object.

56. The system of any of claims 46 to 55, wherein the mass density analysis system is configured to generate an image of the object by:generating a blurred image of the object based on the mass density distribution of the object;obtaining a trained neural network model architecture;providing the blurred image to the trained neural network model architecture; receiving, as an output of the trained neural network model architecture, a clearer image of the object.

57. The system of claim 56, wherein the trained neural network model architecture comprises a U-net architecture or an autoencoder architecture58. The system of any of claims 46 to 57, comprising determining one or more constraints relating to the one or more matrices.

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