System and method for radar imaging of antennas with position ambiguity

Abstract

A radar system for generating a radar image of a scene. Radar measurements of reflections for each point in the scene measured by a receiver are received. The radar image is generated using stored data to solve the radar image recovery (RIR) problem. The radar measurements are related to a shift of a reflected field with a receiver shift. The receiver shift defines the error between a stored receiver position and an actual receiver position, the reflected field being generated by reflecting a transmitted field from the scene according to reflections from each point in the scene. The reflected field is related to a shift of an incident field with a transmitter shift. The transmitter shift defines the error between a stored transmitter position and an actual transmitter position. The joint estimation of the reflections, the receiver shift, and the transmitter shift for each point in the scene is solved as a multilinear problem.

Description

Technical Field

[0001] The present disclosure relates generally to radar systems, and more particularly to radar imaging by fusing measurements from various antennas whose positions are not exactly known. Background Art

[0002] High-resolution radar imaging is a requirement for various remote sensing applications, including synthetic aperture radar (SAR) and through-the-wall radar imaging (TWI). The longitudinal range resolution is mainly controlled by the bandwidth of the transmitted pulse, while the lateral range (azimuth) resolution depends on the aperture of the radar array. In practice, a large physical aperture can be achieved by deploying multiple distributed antennas or arrays, each with a relatively small aperture. A distributed setup allows flexibility in platform placement, reduces operating and maintenance costs, and adds robustness to sensor failures. Exploiting a priori knowledge of the scene, such as precise knowledge of antenna positions and perfect synchronization of received signals, has been shown to significantly improve radar imaging resolution. However, the geographical distribution of the array introduces data coherence issues due to ambiguity in antenna positions and / or difficulties in precisely synchronizing antenna clocks. Summary of the invention

[0003] Technical issues

[0004] For example, state-of-the-art models model antenna position errors as phase and amplitude errors in the received data. Among them, the data distortion introduced due to the position error may be equivalent to the phase and amplitude distortion in the data. To this end, traditional methods attempt to estimate and correct the phase error in the data so that coherent imaging techniques can be applied to the corrected data. However, in many problems, at least one common problem of these solutions is that the estimation of phase and amplitude distortion is not simple due to the nonlinearity of the resulting formulas, and there is a lack of models that directly calculate the impact of position ambiguity on the phase and amplitude of the system. For example, typical phase models in the literature (such as subspace restrictions) perform poorly because they fail to capture the true nature of the error. In addition, there is a complication that the phase is a parcel quantity, which makes its estimation more difficult. As a result, these solutions are impractical.

[0005] Therefore, there is a need for radar imaging systems and methods for fusing measurements from various antennas having position errors.

[0006] Solution to the problem

[0007] Embodiments of the present disclosure relate to radar systems and methods for performing radar imaging by fusing measurements from various antennas with position errors. Some embodiments also relate to radar systems and methods for performing radar imaging by fusing measurements from various antennas with synchronized or asynchronous clocks.

[0008] The embodiments described below are described as two-dimensional examples. However, they are not specific to two dimensions. Other embodiments rely on the same implementation and algorithm, replacing two-dimensional quantities with one-dimensional or three-dimensional quantities, such as images or unknown shifts and shift kernels, as required by the specific application.

[0009] Some systems and methods of the present disclosure provide an optimization-based solution to the problem of distributed radar imaging using antennas with position ambiguity. In a distributed setting, it is challenging to know the positions of different antennas to achieve the accuracy required for radar imaging. At least one aspect of the present disclosure assumes that these positions are not precisely known, and attempts to determine the position of each antenna in addition to recovering an image of the scene that the radar is illuminating. By way of non-limiting example, the solution also allows for modeling and recovery of timing ambiguities in situations where the antennas are not perfectly synchronized.

[0010] Some systems and methods of the present disclosure explicitly and separately model the position uncertainty of the transmitter antenna as an unknown position shift of the transmitted field, and the position uncertainty of the receiver antenna as an unknown position shift of the reflected field. Thus, by way of non-limiting example, the method accurately models each shift as a convolution with a spatial shift operator. This formulation leads to an optimization problem that simultaneously recovers all position errors and the radar image acquired by the radar. Furthermore, the solution provides the resolution benefits of coherent imaging, even if the measurements are incoherent.

[0011] In addition, other beneficial aspects of the solution include taking advantage of the recognition that the impact of position errors is different for the transmitting antenna and the receiving antenna. The solution further relies on separating the typical radar algorithm into two independent operators, one describing the incident field and the other describing the reflected field, so that the transmitter and receiver errors can be described as simple transformations of the two incident and reflected fields, respectively. In addition, this method takes advantage of the recognition that a transformation in space is equivalent to convolution with a shifted kernel operating in space on the fields, and that the shifted kernel is sparse. Therefore, sparse optimization methods can recover the kernel very efficiently.

[0012] In particular, position errors in the transmitting antennas cause the fields introduced into the scene to be shifted by the same amount. Similarly, position errors in the receiving antennas cause the data received to appear as if the reflected fields were shifted by the same amount in the opposite direction. Therefore, a multilinear optimization problem can be formulated that simultaneously recovers all antenna position errors, as well as the sparse scene being imaged.

[0013] To achieve some of the above implementations, some embodiments provide radar imaging based on measurements provided by antennas with position ambiguity, where the phrase position ambiguity indicates that the true position of the antenna (which may or may not be accurately known) may be different from the assumed position of the antenna. To resolve when some antennas have position ambiguity, some embodiments work under the additional assumption that the antennas have clocks that are not perfectly synchronized, for example.

[0014] Some embodiments are based on the recognition that the radar imaging problem of distributed antennas with position ambiguity can be an ill-posed problem with a large number of unknowns. Specifically, when the radar imaging problem is formulated as determining a radar image from measurements associated with the radar image by a radar operator with uncertainty, the antenna position ambiguity is encoded, and each measurement of an unknown scene includes an error caused by the position error of the antenna. In addition, since the relationship between the measurement and the position error of the antenna is nonlinear, each sample of measurements from the same antenna may have a different error, thereby increasing the number of unknowns in the model of the radar imaging problem formed by multiple measurements from multiple antennas. For this reason, the formulation of the radar imaging problem that aims to recover a correct radar image from incorrect measurements caused by an incorrect radar operator is an ill-posed problem that is difficult to model accurately.

[0015] Some embodiments are based on another recognition that the effect of position error is different for a transmit antenna than for a receive antenna. In particular, the effect of a transmit antenna position error is a shift of the incident field introduced into the scene by the same amount as the position error by that transmit antenna. The scene interacts with the incident field, creating a reflected field. Each receive antenna measures the reflected field at the location of the receive antenna. The effect of a position error in a receive antenna is equivalent to measuring the field at a different point, which in turn is equivalent to shifting the field by the same amount in the opposite direction as measured by an antenna without position error (i.e., at the assumed location).

[0016] Some embodiments are based on another recognition that the radar image recovery problem under position ambiguity is to determine the unknown scene, the displacement of the reflected field generated by the unknown scene and measured by the receiving antenna, and the displacement of the incident field introduced by the transmitting antenna and interacting with the scene to generate the reflected field, so that the scene and the displacement explain the radar measurement results of the receiving antenna. Such a problem transformation allows the effect of the unknown displacement of the transmitting antenna to be decoupled from the effect of the unknown displacement of the receiving antenna and from the unknown scene that generates the unknown ideal signal.

[0017] In addition, some embodiments are based on the recognition that an unknown shift of the incident field (which is an unknown translation of the incident field in space) can be represented as a convolution with an unknown two-dimensional convolution shift kernel, and further based on the recognition that the kernel is a unit pulse in two dimensions. This representation is counterintuitive because representing such an unknown shift as an unknown two-dimensional kernel expands the dimensionality of the problem. In particular, the two-dimensional translation can be represented using a two-dimensional vector (i.e., two parameters), and the size of the two-dimensional kernel depends on the maximum ambiguity of the position error in each dimension as a function of the size of the representation grid used to represent the kernel. However, the unknown translation affects the data in a nonlinear way, while the convolution of the unknown pulse is a linear operation. In this way, the decoupling of the unknowns combined with the linearization obtained from the convolution representation leads to a multilinear radar image recovery problem. Although multilinear problems are not convex and are not as easy to solve as convex problems, they are still easier to solve than more general non-convex problems using methods such as lifting or alternating optimization.

[0018] In addition, some embodiments are based on another recognition that the unknown shift of the reflection field (which is the unknown translation of the reflection field in space) can also be represented as a convolution with an unknown convolution shift kernel, and based on the further recognition that the kernel is a two-dimensional unit pulse. In a similar manner, such a representation is counterintuitive because representing such an unknown shift as an unknown two-dimensional kernel further expands the dimensionality of the problem. However, the unknown translation affects the data in a nonlinear way, while the convolution with the unknown pulse is a linear operation. In this way, the decoupling of the unknowns combined with the linearization obtained from the convolution representation leads to a multilinear radar image recovery problem.

[0019] However, even after decoupling, this multilinear radar image restoration problem remains difficult to solve because the resulting problem has a large number of solutions, only one of which is desirable. For this reason, some implementations are based on the alternative realization that the unknown convolution shift kernel can be represented as a shifted pulse, which is an unknown two-dimensional signal that is singly sparse in space. In fact, these implementations allow the original nonlinear image restoration problem to be converted into a multilinear sparse restoration problem, which in turn allows the use of sparse reconstruction techniques to reduce the size of the solution space of the radar image restoration problem.

[0020] To this end, some embodiments solve the multilinear radar image restoration problem to produce a radar image of a scene. Radar image restoration individually restores the shift of the incident field, the reflection of the scene observed by the radar, and the equivalent shift of the reflected field, which, when combined together, describe the measurement data. The shifted incident field and reflected field are represented as the convolution of the shifted field with a spatial shift kernel in space, which is singly sparse in space. In some embodiments, the scene is also sparse in space, or when transformed by an appropriate transform (such as a wavelet transform, a gradient transform, or a Fourier transform).

[0021] During the experiments, some experimental methods involved explicitly modeling the position ambiguity on the distributed antennas as a single unknown shift on the image for each transmitter-receiver antenna pair. However, by modeling these ambiguities as unknown shifts on the image, rather than separate shifts for the incident and reflected fields, these methods only allowed for a single shift per transmit-receive antenna pair.

[0022] In fact, these experimental methods implicitly assume that the position errors of the transmitter and receiver are the same. In practice, this is a very limited model that is accurate only in a few distributed sensing systems. In order for this model to be accurate, the transmitter and receiver should be co-located (i.e., located on the same platform), move together, and experience the same position error. This model is not accurate even if they are on the same platform but at different locations on the platform, and the platform rotates or changes orientation.

[0023] In reality, the experimental model is only accurate in a real system that is configured so that the antenna acts as both a transmitter and a receiver, and no other antenna acts as a receiver for the same transmission. Using such a system configuration reduces the benefits and effectiveness of a distributed radar system because it cannot take advantage of the availability of multiple receiver antennas. To acquire more data and improve system performance, additional receiving antennas located at different locations from the transmitting antenna should receive reflections from the scene.

[0024] However, these experimental methods fail to recognize the separation of the incident and reflected fields in the radar operator. This separation is necessary in order to model the transmitter position error as different from the receiver position error. Otherwise, a single unknown translation error describes the position error of each transmitter-receiver antenna pair, which means that they all exhibit the same common error. Therefore, these experimental methods attempt to recover this common position error for each transmitter / receiver pair, modeling the common shift of the positions of the transmitter and receiver in the pair compared to their assumed positions as a corresponding shift of the same amount in opposite directions of the scene imaged by the radar. As mentioned above, under this formulation, the model is accurate only if the transmitter and receiver are located jointly for each transmission and reception, which is an unrealistic assumption in most distributed systems.

[0025] Furthermore, modeling the position error for each antenna pair results in an exponential growth in the number of unknown position errors as the number of transmitter or receiver antennas increases. In contrast, by recognizing the separation of incident and reflected fields in the radar operator and modeling the position errors of the transmit and receive antennas separately, the number of unknown position errors grows only linearly with the number of antennas. For example, in a system with 3 transmit antennas and 5 receive antennas, where all receive antennas receive reflections from all transmit antennas, there are 3×5=15 transmitter-receiver antenna pairs and, therefore, 15 unknown position errors, one for each pair. Increasing the number of transmit antennas to 4 will create 5 more pairs, one for each receive antenna, and thus 5 more unknown position errors, for a total of 4×5=20 unknown position errors. In contrast, if the position errors of the transmit antennas are treated separately from the position errors of the receive antennas, there are only 3+5=8 and 4+5=9 unknown position errors, respectively. Reducing the number of unknowns significantly reduces the computational complexity of solving the problem, as well as the amount of data required to find the correct solution. Both issues are important in most practical applications. Therefore, separating the incident and reflected fields is also preferred from the perspective of computation and data acquisition complexity.

[0026] Furthermore, separating the transmit antenna position error and the receive antenna position error forces consistency of the solution, which is not the case when the unknown position error is on each transmitter-receiver antenna pair. In fact, at the time of transmission, the transmit antenna is at a single location and has a single unknown position error. Similarly, the receive antennas are each at a single location and exhibit a single unknown position error. Modeling the transmitter-receiver pair position error as a single unknown position error means that the unknown error is a combination of the two individual position errors in the pair, such as an average. Given a single position error, one can combine them and determine the position error for the pair in the model. However, given an estimate of the combined position error for each pair, there is no guarantee that the pairs are consistent, i.e., for each transmitter and receiver in each pair, the individual position errors can be chosen so that when appropriately combined, a given pairwise error can be obtained. Therefore, directly estimating the error for each pair may result in inconsistent estimates. Reconciling these inconsistencies is not simple. In contrast, separating the transmitter and receiver position errors naturally leads to consistent estimates.

[0027] At least one further realization including the separation operator in some embodiments of the present disclosure as described above is the recognition that splitting a typical radar operator into two separate operators, one describing the incident field and the other describing the reflected field, enables a simple conversion of the transmitter error and the receiver error to describe the two incident and reflected fields, respectively. In particular, a key realization is that a shift of the transmitter causes the incident field to be shifted by the same amount in the same direction, and the shifted receiver will receive the same data as a receiver that observes the reflected field shifted by the same amount in the exact opposite direction at the original position.

[0028] Furthermore, this approach exploits the realization that a translation in space is equivalent to a convolution with a shifted kernel operating in the space of the field, and that the shifted kernel is sparse. While this is a significant expansion of the dimension of the unknown parameters, it makes the problem linear in the unknown shift and separates the effects of the shift from the rest of the operator describing the physical properties of the system. Furthermore, since the shifted kernel is sparse, sparse optimization methods can recover the kernel very efficiently. Specifically, if an unknown quantity is sparse, estimating that quantity by enforcing sparsity using sparse optimization can improve the quality of the estimate and reduce the amount of data that should be collected in order to successfully produce that estimate.

[0029] Furthermore, the separation operator has a number of benefits: the ability to accurately model a wider variety of systems and significantly improve reconstruction performance. This is because this separation in the model captures the physical behavior of most real systems even if the transmitter and receiver are not collocated. At the same time, it can also model the errors if the transmitter and receiver are known to be collocated simply by modeling the unknown shift using the same unknown kernel. Furthermore, the model still applies if the transmitter and receiver are on the same platform, but not the same antenna, and the platform exhibits both translation and rotation.

[0030] Other experimental approaches may attempt to correct for synchronization errors between antennas. While accurate synchronization is difficult to achieve in practice, these experiments show that it is possible to describe a convex optimization algorithm that accurately recovers the timing (i.e., synchronization) errors between antennas. However, it is not possible to include position errors in this formulation. Therefore, these experimental approaches are only applicable if the position of each antenna is known exactly and without any errors.

[0031] In contrast, some embodiments of the present disclosure recover synchronization errors in addition to position errors by formulating a multilinear problem. In particular, they exploit the realization that synchronization errors result in unknown nominal time delays and advances in recorded data, which can be modeled as time shifts in the data. Similar to the shifts in position described above, these time shifts in the data can be modeled as convolutions with a shift kernel in the time domain. Thus, they interact linearly with the sensing model. Since the sensing model includes the position errors of the antennas and the unknowns of the unknown scene, the resulting problem is multilinear.

[0032] Some embodiments of the present disclosure rely on the recognition that, although multilinear problems are not convex, they are relatively well studied, well understood, and easy to solve compared to more general convex problems. In particular, while general non-convex problems may be difficult to solve, multilinear problems exhibit a particular structure that makes them amenable to different algorithmic solutions, such as alternating optimization or lifting to higher dimensions. Therefore, they admit that algorithms may be very effective in practical use. Therefore, giving up convexity to flexibly correct both position and synchronization errors is a favorable compromise in many applications.

[0033] In addition, some embodiments of the present disclosure may utilize multilinear formulas to better enforce timing consistency when recovering synchronization errors. In particular, convex formulas in the art recover the timing error between a transmitter and receiver pair. Embodiments of the present disclosure may alternatively recover the timing error of each transmitter and each receiver clock relative to an absolute clock.

[0034] For example, the clock of one transmitter is considered to be the absolute clock, and all clocks are considered to run faster or slower relative to this clock. Consider that the clock of the second transmitter is 2 time units slow, the clocks of the two receivers are 1 time unit and 3 time units fast respectively, and the clock of the third receiver is 1 time unit slow. Embodiments of the present invention only need to recover four absolute clock errors: one for the second transmitter and three for the three receivers. The convex formula tried during the experiments requires recovery of the relative delay of each pair, that is, 2×3=6 unknown relative delays. In the same example, these delays should be consistent. For example, the relative delay between the clock of the first transmitter and the clocks of the three receivers should be equal to the relative delay of the clock of the second transmitter plus the relative delay between the clock of the second transmitter and the clocks of the three receivers.

[0035] Because the convex formula in the above experiment recovers six independent unknown relative delays, the resulting estimates of the relative clock errors may be inconsistent, especially in the presence of noise. The correct solution is to recover the three receivers' clocks to be 1 and 3 time units ahead and 1 unit behind, respectively, relative to the first transmitter's clock, and 3, 5, and 1 time units ahead, respectively, relative to the second transmitter's clock. The recovered times ahead of 3, 5, and 1 time unit as the second set of advances provide inconsistent solutions, which are still possible in the above experiment. Such inconsistencies are not possible using embodiments of the present disclosure that recover the clock errors of each transmitter and receiver individually.

[0036] At least some benefits of the systems and methods of the present disclosure include overcoming problems with conventional distributed radar imaging approaches that use geographical distribution of arrays, which introduce data coherence issues due to ambiguity in antenna positions and / or difficulties in accurately synchronizing antenna clocks. Some embodiments of the present disclosure overcome these problems by separately modeling the position uncertainty of the transmitter antenna as an unknown position shift of the transmitted field, and the position uncertainty of the receiver antenna as an unknown position shift of the reflected field, and further modeling the shift as a convolution with a spatial shift operator. This more accurate formulation also leads to an optimization problem of simultaneously recovering all position errors and the radar image acquired by the radar.

[0037] Another benefit of the present disclosure is that it overcomes traditional methods that attempt to estimate and correct phase errors in the data in order to apply coherent imaging techniques to the corrected data. Some embodiments of the present disclosure overcome these problems by splitting the typical radar operator into two separate operators (one describing the incident field and the other describing the reflected field), thereby being able to describe the transmitter and receiver errors as simple transformations of the two incident and reflected fields, respectively. In addition, this approach exploits the recognition that a translation in space is equivalent to convolution with a shifted kernel operating in space on the field, and that the shifted kernel is sparse. Therefore, sparse optimization methods can recover the kernel very efficiently.

[0038] According to an embodiment of the present disclosure, a radar system for generating a radar image of a scene. Wherein, during operation, the radar system is connected to a group of transmitters configured to transmit radar pulses to the scene and a group of receivers configured to receive reflections of the radar pulses from the scene. The radar system includes a memory configured to store predetermined configuration data indicating radar propagation of the radar pulses to and from the scene. The configuration data includes the position of the transmitter and the position of the receiver, so that the position of at least one receiver is different from the transmitter position of all transmitters. Wherein, the configuration data defines the incident field generated by the radar pulses transmitted by the transmitter from the transmitter position. Wherein, the configuration data defines the structure of the reflection field generated by the reflection of the incident field from the scene and measured by the receiver at the receiver position. The input interface is configured to receive the radar measurement results of the reflection of each point in the scene measured by the group of receivers. The hardware processor is configured to use the configuration data to solve the radar image recovery problem to generate a radar image of the reflection of each point in the scene. Based on associating the received radar measurement results to the displacement of the reflection field with the unknown position displacement of the receiver. wherein the receiver unknown position shift defines the error between the receiver position stored in the memory and the actual position of the receiver. The reflection field is generated from the scene reflection transmission field according to the reflection of each point in the scene. The reflection field is associated to the displacement of the incident field with the transmitter unknown position shift. wherein the transmitter unknown position shift defines the error between the transmitter position stored in the memory and the actual position of the transmitter. The radar image recovery problem is solved as a multilinear problem of joint estimation of reflection, receiver shift and transmitter shift for each point in the scene. The output interface is configured to render one or a combination of the radar image of the reflection, receiver shift or transmitter shift for each point in the scene.

[0039] According to another embodiment of the present disclosure, a method for a radar system to generate a radar image of a region of interest (ROI). The method includes transmitting radar pulses to the ROI using transmitting antennas at different positions for measurement. The receiving antenna is configured to receive reflections of the radar pulses corresponding to the transmitted radar pulses from the ROI. The receiver is caused to measure the reflections of the radar pulses to obtain radar measurements of the reflections of each point in the scene. A hardware processor in communication with a transmitter, a receiver, and a memory is used, and the hardware processor is configured to access the memory, which stores data including predetermined configuration data indicating the propagation of the radar pulses to and from the ROI. The configuration data includes the position of the transmitter and the position of the receiver, so that the position of at least one receiver is different from the transmitter position of all transmitters. Wherein, the configuration data defines the incident field generated by the radar pulses transmitted by the transmitter from the transmitter position. Wherein, the configuration data defines the structure of the reflection field generated by the incident field from the reflection of the scene and measured by the receiver at the receiver position. The radar image recovery problem is solved using the configuration data to generate a radar image of the reflection of each point in the scene. Based on associating the received radar measurement results to the displacement of the reflection field with the unknown position displacement of the receiver. Wherein, the receiver unknown position shift defines the error between the receiver position stored in the memory and the actual position of the receiver. The reflection field is generated from the scene reflection transmission field according to the reflection of each point in the scene. The reflection field is associated with the displacement of the incident field with the transmitter unknown position shift. Wherein, the transmitter unknown position shift defines the error between the transmitter position stored in the memory and the actual position of the transmitter. The radar image recovery problem is solved as a multilinear problem of joint estimation of the reflection of each point in the scene, the receiver shift and the transmitter shift. One or a combination of the reflected radar image, the receiver shift or the transmitter shift of each point in the scene is output via an output interface.

[0040] According to another embodiment of the present disclosure, a non-transitory computer-readable storage medium has implemented thereon a program executable by a processor for performing a method for a radar system to generate a radar image of a region of interest (ROI). The method includes transmitting radar pulses to the ROI for measurement using transmitting antennas at different positions. The receiving antenna is configured to receive reflections of the radar pulses corresponding to the transmitted radar pulses from the ROI. The receiver is caused to measure the reflections of the radar pulses to obtain radar measurements of the reflections of each point in the scene. A hardware processor in communication with a transmitter, a receiver, and a memory is used, and the hardware processor is configured to access the memory, which stores data including predetermined configuration data indicating the propagation of radar pulses to and from the ROI. The configuration data includes the position of the transmitter and the position of the receiver. The position of at least one receiver is different from the transmitter position of all transmitters. Wherein, the configuration data defines an incident field generated by a radar pulse transmitted by the transmitter from the transmitter position. Wherein, the configuration data defines the structure of the reflection field generated by the incident field from the reflection of the scene and measured by the receiver at the receiver position. The radar image recovery problem is solved using the configuration data to generate a radar image of the reflection of each point in the scene. Based on associating received radar measurements to a shift of a reflected field with a receiver unknown position shift. Wherein the receiver unknown position shift defines the error between the receiver position stored in memory and the actual position of the receiver. The reflected field is generated from the scene reflection transmission field according to the reflection of each point in the scene. The reflected field is associated to a shift of an incident field with a transmitter unknown position shift. Wherein the transmitter unknown position shift defines the error between the transmitter position stored in memory and the actual position of the transmitter. The radar image restoration problem is solved as a multilinear problem of joint estimation of reflections, receiver shifts and transmitter shifts for each point in the scene. One or a combination of the reflected radar image, receiver shift or transmitter shift for each point in the scene is output through an output interface.

[0041] The presently disclosed embodiments will be further explained with reference to the accompanying drawings. The drawings shown are not necessarily drawn to scale, with emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] [ Figure 1A ]

[0043] Figure 1A is a block diagram showing steps of a method according to an embodiment of the present disclosure;

[0044] [ Figure 1B ]

[0045] Figure 1B The present invention is implemented according to the embodiment of the present invention. Figure 1AA block diagram of some components of a system of the method;

[0046] [ Figure 1C ]

[0047] Figure 1C The present invention is implemented according to the embodiment of the present invention. Figure 1A A block diagram of some components of a system of the method;

[0048] [ Figure 1D ]

[0049] Figure 1D is a schematic diagram showing a radar system having a set of antennas at different locations to measure a set of reflections corresponding to a transmitted radar pulse from a region of interest (ROI) according to an embodiment of the present disclosure;

[0050] [ Figure 1E ]

[0051] Figure 1E is a schematic diagram showing a radar system having a set of antennas at different locations to measure a set of reflections corresponding to a transmitted radar pulse from a region of interest (ROI) according to an embodiment of the present disclosure;

[0052] [ Figure 2A ]

[0053] Figure 2A is a schematic diagram of an embodiment of the present disclosure, which, when viewed together, illustrates distortion of a time domain signal affecting the measurement of each antenna due to errors in antenna position when measuring reflections of a single target;

[0054] [ Figure 2B ]

[0055] Figure 2B is a schematic diagram of an embodiment of the present disclosure, which, when viewed together, illustrates distortion of a time domain signal affecting the measurement of each antenna due to errors in antenna position when measuring reflections of a single target;

[0056] [ Figure 2C ]

[0057] Figure 2C is a schematic diagram of an embodiment of the present disclosure, which, when viewed together, illustrates distortion of the measured time domain signal affecting each antenna due to uncertainty in the antenna clock when measuring reflections from a single target;

[0058] [ Figure 2D ]

[0059] Figure 2Dis a schematic diagram of an embodiment of the present disclosure, which, when viewed together, illustrates distortion of the measured time domain signal affecting each antenna due to uncertainty in the antenna clock when measuring reflections from a single target;

[0060] [ Figure 3A ]

[0061] Figure 3A is a schematic diagram of the influence of the error of the transmitting antenna position on the incident field according to an embodiment of the present disclosure;

[0062] [ Figure 3B ]

[0063] Figure 3B is a schematic diagram showing the influence of the error of the receiving antenna position on the measurement of the reflection field according to an embodiment of the present disclosure;

[0064] [ Figure 3C ]

[0065] Figure 3C is a schematic diagram showing how to use convolution to describe the shift of an image or field according to an embodiment of the present disclosure;

[0066] [ Figure 4A ]

[0067] Figure 4A is a schematic diagram illustrating an example of the influence of a clock error on a time stamp of a recorded signal according to another embodiment;

[0068] [ Figure 4B ]

[0069] Figure 4B is a schematic diagram illustrating another example of the influence of a clock error on a time stamp of a transmission signal according to another embodiment;

[0070] [ Figure 4C ]

[0071] Figure 4C is a schematic diagram illustrating an example of the effects of clock errors at both transmitter and receiver local clocks according to some embodiments;

[0072] [ Figure 5A ]

[0073] Figure 5A is a schematic diagram showing how a spatial shift may be represented in a computer as a set of coefficients or equivalently as a shift kernel according to another embodiment;

[0074] [ Figure 5B ]

[0075] Figure 5Bis a schematic diagram illustrating how a delay according to another embodiment may be represented in a computer as a single coefficient or equivalently as a shifted kernel;

[0076] [ Figure 6 ]

[0077] Figure 6 is a diagram of the equivalence of the convolution of two signals and the multiplication of their Fourier transforms according to another embodiment;

[0078] [ Fig. 7A ]

[0079] Fig. 7A is a block diagram of a method for estimating a radar image and a set of transmitter and receiver shifts and delays according to some embodiments of the present disclosure;

[0080] [ Figure 7B ]

[0081] Figure 7B is a block diagram of a method for estimating a radar image and a set of transmitter and receiver shifts and delays according to some embodiments of the present disclosure;

[0082] [ Figure 7C ]

[0083] Figure 7C is a block diagram of a method for estimating a radar image and a set of transmitter and receiver shifts and delays according to some embodiments of the present disclosure;

[0084] [ Fig.7D ]

[0085] Fig.7D It is to illustrate some embodiments according to the present disclosure Fig. 7A Diagram of the transmitted pulse in ;

[0086] [ Fig. 7E ]

[0087] Fig. 7E It is to illustrate some embodiments according to the present disclosure Fig. 7A A graph of radar reflections in;

[0088] [ Fig. 8A ]

[0089] Fig. 8A is a block diagram of a method for estimating an initial radar image from measured radar reflections, transmit pulses, and erroneous antenna positions and delays according to some embodiments of the present disclosure;

[0090] [ Figure 8B ]

[0091] Figure 8BSome embodiments of the present disclosure are shown in Fig. 8A A diagram of the initial radar image in ;

[0092] [ Figure 8C ]

[0093] Figure 8C is a block diagram of a method for updating a radar image using measured radar reflections and estimated antenna positions and delays using one-norm regularization and total variation regularization according to some embodiments of the present disclosure;

[0094] [ Fig.8D ]

[0095] Fig.8D The present invention is shown in the embodiment according to the present disclosure. Fig. 7A , Figure 7B and Figure 7C A diagram of an intermediate update of the radar image before the process shown in ;

[0096] [ Fig.9A ]

[0097] Fig.9A is a block diagram of a method for estimating transmitter shift using measured radar reflections, estimated receiver shifts, estimated time delays, and estimates of radar images according to some embodiments of the present disclosure;

[0098] [ Fig. 9B ]

[0099] Fig. 9B is a block diagram of a method for estimating receiver shift using measured radar reflections, estimated transmitter shift, estimated time delay, and an estimate of a radar image according to some embodiments of the present disclosure;

[0100] [ Fig. 9C ]

[0101] Fig. 9C is a schematic diagram illustrating steps of aligning estimated transmitter or receiver shifts according to average assumed positions of antennas according to some embodiments of the present disclosure;

[0102] [ Fig. 10A ]

[0103] Fig. 10A is a block diagram of a method for estimating relative transmitter / receiver clock errors using measured radar reflections, estimated transmitter and receiver position shifts, and an estimate of a radar image, according to some embodiments of the present disclosure;

[0104] [ Fig. 10B ]

[0105] Fig. 10Bis a block diagram of a method for estimating a transmitter clock shift relative to a global clock using measured radar reflections, estimated transmitter and receiver position shifts, an estimate of a receiver clock shift, and an estimate of a radar image, according to some embodiments of the present disclosure;

[0106] [ Fig. 10C ]

[0107] Fig. 10C is a block diagram of a method for estimating a receiver clock shift relative to a global clock using measured radar reflections, estimated transmitter and receiver position shifts, an estimate of transmitter clock shift, and an estimate of a radar image, according to some embodiments of the present disclosure; and

[0108] [ Fig.11 ]

[0109] Fig.11 is a block diagram of a computer system of a radar system contemplated by the present disclosure, according to some embodiments of the present disclosure. DETAILED DESCRIPTION

[0110] Although the above-mentioned drawings illustrate the embodiments of the present disclosure, other embodiments may be envisioned as noted in the discussion. The present disclosure presents illustrative embodiments by way of illustration and not limitation. Those skilled in the art may devise numerous other modifications and embodiments that fall within the scope and spirit of the principles of the embodiments of the present disclosure.

[0111] Embodiments of the present disclosure relate to radar systems and methods for radar imaging by fusing measurements from various antennas with synchronous or asynchronous clocks.

[0112] Figure 1A is a block diagram illustrating some steps of a method according to an embodiment of the present disclosure.At least one system and method 100A includes a system and method for a radar system to generate a radar image of a region of interest (ROI).

[0113] Step 160A of FIG. 1 includes measuring using a transmit antenna at different locations that transmits radar pulses to the ROI and a receive antenna configured to receive reflections of the radar pulses corresponding to the transmitted radar pulses from the ROI. The receiver measures the reflections of the radar pulses to obtain radar measurements of the reflections for each point in the scene. The measured reflections can be stored in a memory of a hardware processor of each receiver and transmitted to a central controller for processing.

[0114] Figure 1AStep 165A of may include a hardware processor in communication with a transmitter, a receiver, and a memory, the hardware processor being configured to access a memory having data stored therein. The stored data may include predetermined configuration data indicating the propagation of a radar pulse to and from the ROI, and data indicating pulse parameters (such as pulse shape in terms of time or frequency). In addition, the configuration data may include the location of the transmitter and the location of the receiver, which may include errors such that the stored locations are different from the true locations. The configuration data may be used to define an incident field generated by a radar pulse transmitted by the transmitter from the transmitter location. The configuration data may also be used to define a reflected field generated by reflections of the incident field from a scene and measured by a receiver at the receiver location, wherein the reflected field is consistent with the reflections of each point in the scene and the incident field.

[0115] The radar transmitter and receiver can be stationary or move along a pre-designed trajectory. The set of valid positions of each radar transmitter and receiver forms a virtual array. In some embodiments, the virtual array can be positioned on the same side of the region of interest where the target is located, so that the target reflects the pulse emitted by the transmitter back to the receiver.

[0116] For each valid position, each transmitter or receiver is located at a specific real position in space. In addition, each transmitter or receiver has some limited knowledge of its current position, that is, knowledge of the assumed position. The assumed position can be inferred in a variety of ways. For example, the position may be the position that the array is intended to occupy as part of the system design, or a predefined trajectory. Alternatively, the assumed position can be inferred using geolocation measurements such as GPS. In other cases, for example if the array is stationary, the assumed position can be obtained through a calibration process. In other cases, however, an inertial measurement unit (IMU) operating along the antenna path can be used to calculate the assumed position. In many cases, including those mentioned, the assumed position may not be accurate and may be a rough estimate of the true position.

[0117] The assumed position may deviate from the true position of the transmitter or receiver. For all transmitters and receivers in the set of transmitting and receiving antennas that form a virtual array, the displacement is determined by subtracting their assumed positions from the true position of each antenna in the transmitter or receiver set. For example, position deviations are caused by calibration errors of static positions, errors in tracking motion via IMU, or inaccurate GPS. If the deviation (which may be as large as several radar center frequency wavelengths) is not well compensated, the generated radar image will not be focused. If the deviation is well compensated, the subtraction of the true position of the antenna from the virtual array position of the antenna should be zero, and the corresponding fused radar image is well focused. Through appropriate distance compensation, radar reflections can be aligned in time so that they can be spatially superimposed at the target position, thereby forming a focused image of the target in the radar imaging process.

[0118] However, it can be difficult or expensive to know the position of each antenna with sufficient accuracy to produce a well-focused, coherent image. A generally accepted rule of thumb in the art is that the position of each antenna should be known to within a small fraction of the wavelength of the transmitted wave. For example, for a pulse centered at 1 GHz, the corresponding wavelength in free space is 30 cm. In this case, the acceptable position error is less than 1 cm, and preferably lower. In comparison, the accuracy of current GPS systems in optimal use scenarios is about 30 cm.

[0119] Figure 1A Step 170A includes using the configuration data to solve a radar image restoration problem to produce a radar image of the reflection of each point in the scene by:

[0120] Figure 1A Step 175A includes correlating the received radar measurements to a shift of a reflected field having a receiver unknown position shift, wherein the receiver unknown position shift defines an error between a receiver position stored in a memory and an actual position of the receiver, and wherein the reflected field is generated by reflecting the transmitted field from the scene based on reflections at each point in the scene.

[0121] Figure 1A Step 180A includes correlating the reflected field to a shift of the incident field with a transmitter unknown position shift, wherein the transmitter unknown position shift defines an error between the transmitter position stored in the memory and the actual position of the transmitter.

[0122] Figure 1A Step 185A may include solving the radar image recovery problem as a multilinear problem of joint estimation of reflections, receiver shifts, and transmitter shifts for each point in the scene.

[0123] Figure 1AAnother step may include outputting one or a combination of radar images of reflection, receiver displacement, or transmitter displacement for each point in the scene via an output interface.

[0124] Figure 1B A block diagram of a radar system 100B according to some embodiments is shown. The radar system 100B may have multiple interfaces that connect the system 100B to other systems and devices. The network interface controller 150B is suitable for connecting the system 100B to a network 190B that connects the radar system 100B to a sensing device via a bus 106B. For example, the radar system 100B includes a transmitter interface 160B that is configured to command a transmitter 165B to transmit a radar pulse. Using a receiver interface 180B connected to a receiver 185B, the system 100B can receive reflections of a scene corresponding to the transmitted pulse. In some embodiments, the radar system 100B receives radar measurements 195B, which are measurements of reflections of radar pulses transmitted to the scene via the network 190B.

[0125] Radar system 100B includes an output interface 170B configured to render reconstructed radar image 128B. For example, output interface 170B can display reconstructed image 128B on a display device, store the image in a storage medium, and / or transmit the image over a network. For example, system 100B can be linked to a display interface via bus 106B, which is suitable for connecting system 100B to a display device, such as a computer monitor, camera, television, projector, or mobile device. System 100B can also be connected to an application interface, which is suitable for connecting the system to a device for performing various tasks.

[0126] In some implementations, the radar system 100B includes an input interface to receive radar measurements of a scene collected from a set of antennas with clock ambiguity. Examples of input interfaces include a network interface controller (NIC) 150B, a receiver interface 180B, and a human-machine interface 110B. The human-machine interface 110B within the system 100B connects the system to a keyboard 111B and a pointing device 112B, where the pointing device 112B may include a mouse, a trackball, a touchpad, a joystick, a pointing stick, a stylus, or a touch screen, etc.

[0127] The system 100B includes a processor 120B configured to execute stored instructions 130B, and a memory 140B storing instructions executable by the processor. The processor 120B can be a single-core processor, a multi-core processor, a computing cluster, or any number of other configurations. The memory 140B can include a random access memory (RAM), a read-only memory (ROM), a flash memory, or any other suitable memory system. The processor 120B can be connected to one or more input and output devices via a bus 106B.

[0128] Instructions 130B may implement a method for a multilinear sparse recovery problem to produce a radar image 128B of a scene. To this end, the instructions include a sparse recovery problem solver 145B, which is configured to solve a multilinear sparse recovery problem to produce a radar image of a scene. As defined herein, the multilinear convex sparse recovery problem solver associates received radar measurements to a shift of a reflection field with a receiver unknown position shift, wherein the receiver unknown position shift defines an error between a receiver position stored in a memory and an actual position of the receiver, and generates a reflection field from a scene reflection transmission field by reflecting from each point in the scene; associates the reflection field to a shift of an incident field with a transmitter unknown position shift, wherein the transmitter unknown position shift defines an error between a transmitter position stored in a memory and an actual position of the transmitter. The sparse recovery problem solver solves the radar image recovery problem as a multilinear problem of joint estimation of reflections for each point in the scene, receiver shifts, and transmitter shifts.

[0129] In order to solve the multilinear sparse recovery problem, the radar system 100B stores additional information and / or modules to formulate the problem in an advantageous and / or solvable manner. For example, instruction 130B indicates the physical properties of the hardware implementation of the receiver and transmitter used to determine the radar measurement results. Data 155B captures the propagation law of the transmitted radar pulse in the environment, including the law of how the propagated pulse interacts with the scene. In some embodiments, data 155B includes storing the laws of how the induction field B and the induction field interact with any scene to generate the reflection field, reflecting this propagation. Different configurations of the radar system may have different values ​​of the induction field B and the reflection law. In some embodiments, data 155B includes storing distance fields, which store the distance of each point in the scene from each of the transmitter and receiver. The data can also be used to calculate the induction field and determine the reflection field at the position of the receiver as a function of the scene. In some embodiments, data 155B includes storing the relative positions of the transmitter and receiver. Using this data, the distance field can be calculated for each transmitter and receiver.

[0130] Additionally or alternatively, instructions 130B may store a multilinear sparse recovery problem formulator 135B configured to formulate a sparse recovery problem using radar data 155B and radar measurements 195B. For example, formulator 135B may transform received radar measurements in the Fourier domain, transform a convex sparse recovery problem into a matrix form to simplify the solution, and select and apply various constraints and regularizers to a solution to the multilinear sparse recovery problem.

[0131] Figure 1C 1 is a block diagram of some radar systems and methods 100C, which shows a group of antennas 101C that can communicate with each other and with a computer / processor 141C. The group of antennas 101C can be synchronized and the collected data can be stored in a memory 143C processed by an autofocus imaging processor 145C of the computer 141C. The autofocus imaging processor 145C can implement systems and methods for generating autofocus high-resolution two-dimensional (2D) radar images. The imaging results can be displayed in a user interface 147C of the computer 141C. Note that the computer 141C can be a processor or multiple computers or multiple processors that communicate with other computers or processors.

[0132] Figure 1D An example implementation of the present disclosure is shown. A virtual array 101D includes transmitting radars and receiving radars mounted on platforms 102D, 103D, 104D, 105D, observing a scene 140D using some reflectors 130D. The transmitter can emit pulses to the scene 140D to create an incident field, which is reflected back to the array by the reflector. The receiver measures the reflected field 120D at its location and records the measurement results. The antenna set 101D can be synchronous or asynchronous, and the collected data can be stored in a memory 140B processed by a synchronization and imaging processor 120B of the radar system 100B. The processor 120B can execute systems and methods for generating high-resolution radar images. The imaging results 128B can be displayed through an output interface 170B.

[0133] Figure 1D The system also includes systems and methods for determining and storing estimates of the positions of the transmitting and receiving antennas. For example, GPS receivers may be mounted on the platform, or the platform may include an IMU that tracks their positions as the platform moves. Alternatively, the antennas may be stationary at known fixed positions or mounted on a track with a servo or other mechanism that is able to learn the position of the antenna within the track. Thus, the data collected by the system includes an estimate of the position of each antenna.

[0134] Figure 1Eis a schematic diagram showing a distributed mobile radar imaging system 100E imaging an ROI 140E with a target 130E according to some embodiments. Transmitters and receivers are mounted on mobile platforms 102E, 103E, 104E, 105E, each of which moves on a different path 112E, 113E, 114E, 115E and forms a virtual array 120E. The transmitter transmits a radar pulse, which generates an incident field 170E to the scene. The incident field interacts with a reflector 130E in the scene, generating a reflected field 180E from the scene, which is measured by a receiver on the mobile platform at its specific location. The distributed array of mobile radar platforms 101E includes at least one radar platform having an antenna unit connected to a radar transmitter 102E that generates a radar pulse toward the scene. As described above, the radar transmitter 102E can be combined with the receiver 102E. Radar receivers 102E, 103E, 104E, 105E acquire reflections reflected by targets 130E in a region of interest (ROI) of a scene 140E.

[0135] Incident field B t (where t represents the time relative to the global clock) is a function of time and space. In some embodiments, it can be represented as a function with elements (B t ) n，m , where n and m are spatial coordinates. In addition, in some embodiments, the region of interest and the corresponding incident field can be one-dimensional or three-dimensional, with corresponding coordinate dimensions, and are represented as vectors or three tensors. In addition, in some embodiments, in order to be able to store and process the reflection and field of the ROI as matrices or tensors, the coordinates and time can be discretized, that is, discrete values ​​can be taken on a grid (such as an integer grid). For example, they may take values ​​in m=1,...,M, n=1,...,N, and t=1,...,T, where M×N is the size of the discretized grid in the ROI and T is the total time of observation. According to the specific grid size in each dimension, the integer points on the spatial grid correspond to physical points on the ROI. Similarly, according to a specific time interval, the integer time point corresponds to the actual time. In some embodiments of the present disclosure, for computational convenience, the field can also be converted using Fourier transform for the time evolution of the signal and represented in frequency as (B f ) n，m , where f represents the frequency index, taking the value in f=1,...,F, and the corresponding physical frequency is recorded as ω f , represents the frequency of the components comprising the time domain signal.

[0136] In some embodiments, the incident field as a function of the pulse can be calculated by calculating the distance of each point in the ROI from the emitter and using that distance to calculate the delay of the pulse arriving at that point. In particular, if emitter k transmits in the frequency domain represented as The incident field due to the emitter k is

[0137]

[0138] Among them, d k，m，n is the distance from the grid point (m, n) to the transmitter k, and c is the propagation speed of the wave in the medium. In some embodiments, there is a further attenuation term a that can be taken into account when calculating the field k，m，n , captures the attenuation of the pulse as it propagates to the ROI. In some embodiments, especially when the ROI is small compared to its distance from the antenna, the attenuation is effectively constant and can be ignored.

[0139] The scene 140E interacts multiplicatively with the incident field to generate a reflected field 180E. The reflected field is measured at the location of the receiver to produce received data in the frequency domain. Specifically, at the location of receiver 1, the received field and the corresponding recorded signal is equal to

[0140]

[0141] Here, x is a matrix, also indexed by m and n, representing the reflection of the scene at points n and m. The element product is represented by ⊙, and d I，m，n is the distance of the grid point (m, n) from the receiver l.

[0142] In some embodiments, radar receivers 102E, 103E, 104E, 105E each have a local clock 152E, 153E, 154E, 155E, which may be fast or slow relative to the global clock. The local clock is used to time stamp the received reflections. For example, in some embodiments, the timestamp may use absolute time. In some other embodiments, the timestamp may be relative to one or more commonly agreed start times, commonly referred to as zero. In some other embodiments, the timestamp may be implicit, such as by a sequence of signal samples recorded at regular intervals relative to the start time of the recording.

[0143] A fundamental challenge that arises in distributed array imaging comes from the uncertainty of clock and antenna positions. Advanced positioning and navigation systems (such as Global Navigation Satellite Systems (GPS / GNSS) and Inertial Navigation Systems (INS)) provide somewhat accurate position and timing information, and tracking, timing, and synchronization algorithms can further improve accuracy at a reasonable cost. However, given the operating frequencies of the transmitting and receiving antennas, the uncertainty retained in the true error can be large. As a result, when inaccurate position and clock timing are used as references, the received signal contains gain and phase uncertainties. Therefore, applying standard reconstruction techniques without considering position and timing perturbations produces out-of-focus radar images.

[0144] Figure 2A and Figure 2B is a schematic diagram showing the distortion of the measured time domain signal of each antenna due to the error of the antenna position when measuring the reflection of a single target according to an embodiment of the present disclosure when viewed together. In addition, Figure 2A and Figure 2B yes Figure 1A A schematic diagram of step 160A, Figure 1A The effect of position perturbations on the temporal alignment of radar reflections is shown. Figure 2A is a schematic diagram of the ideal situation where the positions of all radars are completely known.

[0145] Array 201A observes a single reflector 230A by transmitting a pulse 210A using a transmitter mounted on, for example, platform 202A and receiving its reflection using receivers mounted on all platforms. Each platform records its position as 202A, 203A, 204A, 205A, with some error, such that the actual position of the platform is 202x, 203x, 204x, 205x, respectively. The corresponding signals 212A, 213A, 214A, 215A describe the reflections that the platforms would receive if they were located at 202x, 203x, 204x, 205x, respectively (i.e., where they think they are). However, if Figure 2B As shown by the dashed line, instead, the platform receives signals 212B, 213B, 214B, 215B. Generally speaking, if the transmitter or receiver is closer to the reflector, the reflection will arrive earlier. If they are farther away, the reflection will arrive later.

[0146] Typical methods in the art model both transmitter and receiver position errors as a common phase error in the frequency domain and correct for it before reconstructing the radar image. Some experimental methods model both transmitter and receiver position errors as a common shift in the entire acquisition scene and correct for this shift.

[0147] In contrast, various embodiments of the present disclosure model the transmitter position error as a corresponding unknown shift of the incident field. This shift, in turn, is modeled as the convolution of the incident field with a shift kernel representing compensation (i.e., a single sparse signal) and has the same dimensions as the field and the ROI. Similarly, various embodiments of the present disclosure model the receiver position error as a corresponding unknown opposite shift of the reflected field. This shift, in turn, is modeled as the convolution of the reflected field with a shift kernel representing compensation (i.e., a single sparse signal) and has the same dimensions as the field and the ROI. A sparse signal is one in which most of its coefficients are zero and very few of its coefficients are non-zero. In particular, only one of the coefficients of a single sparse signal is non-zero, and the rest are equal to zero. A shifted kernel is a single sparse signal whose non-zero coefficients are located at the positions of the shift implemented by the kernel.

[0148] Figure 2C and Figure 2D is a schematic diagram showing, when viewed together, distortion of the measured time domain signal affecting each antenna due to errors in each radar clock when measuring reflections from a single target according to some embodiments. In addition, Figure 2C is a schematic diagram of an ideal situation where all radar clocks are perfectly synchronized. A distributed array 201C is shown as including a transmitter / receiver platform 202C and receiver-only platforms 203C, 204C, 205C. The transmitter antenna transmits a pulse 210C into a scene containing a single reflecting target 230C. The pulse is reflected by the reflecting target and its reflection 220C is picked up by all receiving antennas in all receiving platforms 202C, 203C, 204C, 205C. Antennas 202C-205C form a distributed array of antennas 201C. If the clocks of all transmitters and receivers are perfectly synchronized, then according to the global time t 0 , the signal 212C, 213C, 214C, 215C received by each antenna 202C, 203C, 204C, 205C includes pulses that are delayed and scaled according to the travel distance from the transmitter antenna to a single reflector and back to each receiver.

[0149] Figure 2D FIG. 1 is a diagram showing a group of antennas presenting clock errors and measuring reflections from a single target according to an embodiment of the present disclosure. Figure 2C Schematic diagram of the mapping between a set of antennas with synchronized clocks and measuring shifted versions of the same target. Figure 2D is a schematic diagram of a signal model showing that the measured reflections of a single target with an erroneous clock are equivalent to the time-shifted measurements of the same target with a clock synchronized with the global time.

[0150] Figure 2DThe effect of clock errors in the acquired data is illustrated by showing how the signals are recorded and time stamped 212D, 213D, 214D, 215D with the local time of each receiver. The local clocks of each system are 252, 253, 254, 255, respectively, where the solid lines represent the local time and the dashed lines represent the global time. In the example in the figure, some clocks may run slower 252, 254, i.e., display time 0 when the global time passes zero, or some clocks may run faster 253, 255, i.e., display time 0 when the global time has not yet reached zero. Depending on the time shift, the corresponding signal may be time-shifted and appear earlier or later (solid line) compared to how the signal would be recorded when all clocks were synchronized (dashed line).

[0151] In this example, the transmitter / receiver clock 252 runs slow. Therefore, the transmit pulse 210C is delayed due to the clock error, and its reflection delays the arrival at the receiver. The receiver antenna of the transmitter / receiver platform 202C exhibits the same clock error, which advances the signal at local time t, thus offsetting the transmission delay of the recorder signal 212D. On the other hand, the clock 253 of the receiver 203C runs fast in this example. Therefore, compared with the signal 213C that should be recorded when all platforms are synchronized with the global clock, the recorder signal 213D is delayed by the accumulated error of the two clocks. Similarly, the clock 254 of the receiver 204C may run slower by an amount smaller than the error of the transmitter clock 252. Therefore, compared with the signal 214C recorded when all platforms are synchronized to the global clock, the recorder signal 214D is delayed by the difference in the errors of the two clocks. Similarly, the clock 255 of the receiver 205C may run faster by an amount greater than the error of the transmitter clock 252. Therefore, compared to the signal 215C that would be recorded if all platforms were synchronized to the global clock, the recorder signal 215D is ahead by the difference in error of two clocks.

[0152] Instead of modeling the timing error as a phase error in the frequency domain and correcting it before reconstructing the radar image, various embodiments model the timing error as a convolution with a shifted kernel (i.e., a single sparse signal) that represents the compensation. A sparse signal is one in which most of its coefficients are zero and very few of its coefficients are non-zero. In particular, only one of the coefficients of a single sparse signal is non-zero, and the rest are equal to zero. A shifted kernel is a single sparse signal whose non-zero coefficients are located at time instances that are time delayed or advanced by the kernel implementation.

[0153] Figure 3Aillustrates how the incident field introduced by the transmitting antenna is affected by the error in the antenna position. In particular, the actual antenna position 311 exhibits a shift 313 compared to the assumed antenna position 312. The actual incident field 321 introduced by the antenna to the scene 301 is the assumed incident field 322 shifted 323 by the same amount as the shift 313 in the antenna position due to the error. The shift can be represented as the convolution 330 of the assumed incident field with a shift kernel 331. Specifically, the incident field at the perturbed position of the transmitter is equal to

[0154]

[0155] Among them, h k is the unknown shift kernel that models the position error.

[0156] Similarly, Figure 3B illustrates how the reflected field measured by the receiving antenna is affected by the antenna position error. Specifically, the measurement result obtained at the real antenna position 351 of the actual reflected field 361 reflected by the real scene 371 is equivalent to the measurement result obtained at the assumed antenna position 352 from the equivalent reflected field 362 reflected from the equivalent scene 372, so that the antenna displacement 353 is the same as the displacement of the reflected field 363 and the scene 373. Therefore, the reflected field obtained at the real position of the receiver 1 due to the pulse transmitted by the transmitter k is equal to

[0157]

[0158] in, is the kernel that models the opposite position shift of the reflected field due to the position error of receiver l. If multiple transmitters transmit simultaneously, the field measurement recorded and stored by receiver 1 is the sum of the reflected fields due to all transmissions, i.e., over all k. Therefore, the data recorded by receiver 1 in the frequency domain is equal to

[0159]

[0160] Figure 3CIt is shown how convolution with a shift kernel can be used to model the spatial shift of a signal. In particular, when convolved 345 with a shift kernel 341, the signal 340 is equal to the shifted signal 342, so that the shift 348 of the signal relative to its original position 347 is equal to the shift indicated by the shift kernel 346. Therefore, determining the correct shift kernel 341 is equivalent to determining the shift itself. The shift kernel for a two-dimensional spatial shift is a two-dimensional signal in space that contains only a single pulse 349 located at the right shift 346. Similarly, the shift kernel for a three-dimensional spatial shift is a three-dimensional signal in space that contains only a single pulse located at the right shift. In all cases, the shift kernels are 1-sparse and sum to 1.

[0161] To recover the radar image and the shifted kernel, the system should determine the radar reflection image x and all kernels h corresponding to the transmitter and receiver shifts, respectively. k and In order to determine the radar reflection image and the kernel interprets the data recorded by the receiver, i.e.,

[0162] Sure

[0163] And make the radar reflection image x sparse in the appropriate domain, and all kernels h k and is 1-sparse and sums to 1.

[0164] Multilinear Optimization

[0165] This is a multilinear problem because it has k and Each of is linear in , but they all interact with each other multiplicatively. The solution to this problem involves three sets of unknowns to be determined, namely, x, h, and 1 for all k and 1. k and In some embodiments, a solution is determined using a sparse multilinear optimization problem so that a penalty function is minimized at the correct solution. If the solution does not explain the data recorded by the receiver, the penalty function may include a component that increases the penalty. In some embodiments, the cost function may include a term that penalizes the solution if one or more components of the solution are not sparse. In some embodiments, the cost function may include a term that penalizes the solution if the sum of any shifted kernels is not equal to 1. Other embodiments may impose hard constraints on solutions that ensure that the sum of each shifted kernel is equal to 1. Some embodiments may further enforce that each shifted kernel has a positive component. Other embodiments may explicitly enforce the constraint that the shifted kernels are exactly 1-sparse.

[0166] For example, one implementation may solve the following optimization problem

[0167]

[0168] Among them, the form The secondary l 2 The term penalizes solutions that do not explain the data, and the form ||·|| 1 l 1 This penalizes non-sparse solutions.

[0169] Some embodiments of the present disclosure may further use some solutions of the total variation (TV) norm as l 1 If the region of interest contains extended objects, the TV norm usually improves performance because it is more efficient than the l norm. 1 The norm is better able to model their range.

[0170] To solve this multilinear optimization problem, some embodiments use alternating minimization. This allows the problem to be separated into multiple linear and therefore convex and easier to solve problems, which are solved in an alternating manner. In particular, the problem is solved in three sets of unknown variables x, h k and Each set of is linear, assuming that in each subproblem, one is considered unknown and the other two are considered known. Therefore, given the other estimates that are known and fixed, the alternating optimization method will improve x, h k and Alternate between the estimates of each in , one at a time. In general the order of optimization does not matter and can be randomized in each iteration. The alternating optimization continues cycling through each set of unknowns until some convergence criterion is met. Fig. 7A A process in an example implementation is shown.

[0171] There are several ways to solve each linear subproblem. For example, the fast iterative shrinkage threshold algorithm (FISTA) or its variants can be used to efficiently enforce the sparsity of the constraints. Other implementations may use greedy algorithms such as iterative hard thresholding (IHT) or matching pursuit (MP).

[0172] Since only the estimate needs to be updated in each update step of the alternating minimization, some embodiments may not solve each linear subproblem to completion, but instead only compute a few steps to improve each estimate. The overall stopping criterion of the alternating minimization is used to ensure that the final estimate converges, even if the estimates computed while solving the linear subproblems did not converge. Other embodiments may solve each subproblem to completion.

[0173] While multilinear optimization problems are easier to solve than general nonconvex problems and are well studied, they are still harder than linear problems. While the latter are convex and therefore guaranteed to have a solution, multilinear problems are not. Furthermore, the more multiplicatively coupled sets there are in a problem, the harder it is. For example, a typical bilinear problem with only two sets of multiplicatively coupled variables is easier to solve than a typical trilinear problem with three sets of multiplicatively coupled variables. Therefore, it is desirable to keep the number of sets of coupled variables as small as possible.

[0174] In some embodiments, the only interesting part of the solution is the unknown radar image x. In other embodiments, the shift kernel h k and It can be used to extract information about the true position of the transmitting and receiving antennas, respectively, and to help the platform on which the antennas are mounted to correct its estimate of its own position.

[0175] Clock and synchronization ambiguity

[0176] In some embodiments of the present disclosure, in addition to the position uncertainty, the antennas may also be imprecisely synchronized. This can introduce additional phase ambiguity in the signal. However, one of the key realizations in the present disclosure is that these uncertainties can also be explicitly and accurately modeled and taken into account when reconstructing the radar image.

[0177] Figure 4A is a schematic diagram illustrating an example of the effect of clock errors on the timestamps of recorded signals used by some embodiments. In particular, the receiver has a local clock 420, which maintains a local time 422. The local clock exhibits an error ∈ 425 relative to the global time 421. In this example, the local clock runs slowly, that is, at any time instance, the local time is lower than the global time. In other words, when the global clock reaches a certain time mark 423, for example at 34 time units, the local clock has not yet reached the time mark, for example, it has reached the time mark 424 at 32.5 time units and ∈ is 1.5 time units ahead. A typical convention in the art is to express delays as positive time shifts and advances as negative time shifts, that is, in this example, ∈ = -1.5 time units.

[0178] Still reference Figure 4A, the signal 410 recorded and time stamped using the time stamp of the local clock 420 appears to have a smaller time stamp than the same signal recorded and time stamped using the global clock 411. For example, the global time instance of the middle negative peak of the pulse in the graph 413 is 35, while the recording using the local clock time stamps the same negative peak at 33.5, 414. If the local clock is slow, then the recorded timestamp is advanced by ∈ compared to the timestamp of the global clock for the same point in the signal. If the local clock is fast, then the recorded timestamp is delayed by ∈ compared to the timestamp of the global clock.

[0179] Figure 4A It is further illustrated by example that time shifting is equivalent to convolution with a shift kernel, and that the shift kernel is a single sparse signal. In particular, the ideal signal time stamped with a global clock 411 is advanced by time ∈ to produce the actual recorded signal and time stamped with a local clock 412. As is well known in the art, time shifting can be represented using a convolution 450 with an impulse δ(t-∈) 415, i.e., an impulse function centered at a time instance ∈. Since the impulse has only one non-zero coefficient, at time ∈, it is a single sparse signal.

[0180] Some embodiments are based on the recognition that using a single sparse signal to represent a delay expands the dimensionality of the representation. For example, using a discrete time representation of a single sparse signal with a sampling interval Δ time units to represent a maximum time shift of T time units requires a shift kernel of size 2T / Δ+1, rather than a single time parameter ∈. If the application requires estimating the time delay, then representing the time delay as a single value ∈ requires estimating a single parameter. In contrast, representing the time delay using a shift kernel requires estimating 2T / Δ+1 parameters, i.e., a significantly larger problem is required. In addition, if the maximum uncertainty of the time shift increases, the size of the representation based on the shift kernel increases proportionally, i.e., the number of parameters to be estimated increases, and more data is required to obtain sufficient information. For comparison, if the delay is represented as a single parameter, the problem size is constant regardless of the maximum uncertainty of the time shift. The constraint that the shift kernel is single sparse helps reduce the data requirements of the estimation problem, despite the higher dimensionality, but it does not eliminate the dependence on the maximum length, nor does it reduce the computational or storage complexity of the estimation.

[0181] Figure 4Bis a schematic diagram of another example of the effect of clock errors on the timestamp of a transmitted signal according to another embodiment. In particular, the transmitter has a local clock 480, which maintains a local time 482. The local clock exhibits an error ∈′ 485 relative to the global time 421. In this example, the local clock runs slowly, that is, at any time instance, the local time is lower than the global time. In other words, when the global clock reaches a certain time mark 483, for example, at 34 time units, the local clock has not yet reached the time mark, for example, has reached the time mark 484 at 32.5 time units and ∩′ is 1.5 time units ahead, that is, similar to the example above, in this example, ∈′ = -1.5 time units.

[0182] Still reference Figure 4B , a signal (such as a pulse) 470 transmitted with timing managed by the local clock 480 will appear to be transmitted later relative to the global clock, that is, to have been delayed. For example, the transmitter may time the transmission of the signal so that the middle negative peak of the pulse occurs at time 34.5 in the local clock 474. In this case, due to the local clock error, the transmitted signal will have the middle negative peak of the pulse occurring at time 36 in the global clock 473. Therefore, the time of the signal transmitted relative to the global clock 471 is advanced by ∈′ compared to the signal transmitted relative to the local clock 472. Because ∩′ is a negative number in this example, the time advance of ∈′ is actually a delay of -∈′=1.5 time units. As already described, this time shift can be represented as a convolution 450 with a shift kernel 416. Since the signal transmitted relative to the global clock 471 is advanced by ∈′ compared to the signal transmitted relative to the local clock 472, the shift kernel corresponds to a delay ∈′.

[0183] Conversely, if the local clock runs faster, the signal transmitted relative to the global clock will be advanced compared to the signal relative to the transmitter's local clock.

[0184] Figure 4C 4 is a diagram illustrating an example of the effect of clock errors at a transmitter local clock and a receiver local clock on a signal measured by a receiver and time stamped according to the receiver's local clock according to some embodiments when the signal is a reflection of a transmitted signal transmitted by the transmitter according to its local clock in a scene. In particular, the transmitter has a local clock 489 that maintains a local time 487. The local transmitter clock exhibits an error ∈′ 486 relative to a global time 421. The receiver has a local clock 429 that maintains a local time 427. The local receiver clock exhibits an error ∈ 426 relative to the global time 421.

[0185] In this example, both the transmitter local clock and the receiver local clock are slow relative to the global clock 495, albeit with different errors. For example, the receiver time 427 may be 1.5 time units slow relative to the global time 421, while the transmitter time 487 may be 1 time unit slow. In other words, the receiver clock 429 is 0.5 units slow relative to the transmitter clock 489, i.e., has a relative clock error 496 equal to ∈″=∈-∈′=-0.5 time units.

[0186] Still reference Figure 4C In the example in , a signal (such as a pulse) 490 transmitted by a transmitter with timing managed by the transmitter's local clock 489 appears to have been transmitted later, i.e., delayed relative to the receiver's local clock 429. For example, the transmitter may time the transmission of the signal so that the middle negative peak of the pulse occurs at time 35 in the transmitter's local clock 435. In this case, due to the relative clock error, the transmitted signal has the middle negative peak of the pulse occurring at time 34.5 in the receiver's local clock 433. Therefore, the signal transmitted relative to the receiver's local clock 492 is delayed in time ∈″ compared to the signal transmitted relative to the transmitter's local clock 491. Since ∈″ is a negative number in this example, the time delay of ∈″ is actually an advance of -∈″=0.5 time units. As already described, this time shift can be represented as a convolution 450 with a shift kernel 417. Since the signal transmitted relative to the transmitter's local clock 491 is advanced by ∈″ compared to the signal transmitted relative to the receiver's local clock 492, the shift kernel corresponds to a delay ∈″.

[0187] Still reference Figure 4C , for this particular transmitter-receiver pair in the example, the receiver receives reflections of the signal transmitted by the transmitter from the scene according to the transmitter's local clock. Since these reflections are triggered by the transmitter transmitting the signal, they are time shifted compared to the receiver's local clock by the same time shift that the transmitted signal exhibits compared to the receiver's local clock, namely ∈″. Therefore, for a given transmitter-receiver pair, given a relative time error ∈″ between the transmitter clock and the receiver clock, the receiver will receive a received signal time stamped by the receiver's local clock that is time delayed ∈″, compared to the ideal signal that the receiver would receive and time stamped if the transmitter and receiver were perfectly synchronized.

[0188] Some embodiments of the present disclosure use a single time delay to model the relative time shift between each pair of transmitters and receivers. The model is accurate if a single transmitter transmits at once. The advantage of this model is that each transmitter-receiver pair has a single unknown time delay estimate. In addition, one of the subproblems of the multilinear problem can be made convex, as described in U.S. Patent Application No. 16 / 295,102, thereby reducing the subproblems to the same number as if the transmitters were completely synchronized. A disadvantage of this model is that it introduces an unknown time delay for each transmitter-receiver pair, i.e., given K transmitters and L receivers, the model has KL unknown delays. In addition, the model may not produce a consistent solution, as described in paragraph

[0032] . On the other hand, this lack of consistency may be beneficial when there is clock drift between transmissions.

[0189] Other embodiments of the present disclosure model the time shift of each of the transmitters and receivers relative to a reference global clock separately, typically assuming that one of the transmitter or receiver's clocks is that reference. An advantage of this approach is that the model enforces global consistency of the clocks and their relative delays. Furthermore, given K transmitters and L receivers, the model has only K+L-1 unknown delays: one for each transmitter and one for each receiver, excluding the transmitter or receiver that is treated as a global time reference. A disadvantage of this approach is that the unknown transmitter time shift kernel is multiplicatively coupled with the unknown receiver time shift kernel, increasing the number of sets of multilinear unknowns in the multilinear problem, making it more difficult to solve. Furthermore, if there is clock drift between transmissions, the strong global consistency enforcement may be too strong to capture the clock drift.

[0190] Shift kernel in frequency

[0191] Figure 5A is a schematic diagram showing how a two-dimensional shift may be represented in a computer as a pair of coefficients 502 or equivalently as a shift kernel 501 according to some embodiments. Representing a delay as a single coefficient uses two storage cells. On the other hand, representing a delay as a shift kernel requires a two-dimensional array of coefficients, such as stored in a storage cell bank 503, indexed by two position indices 504 for each coefficient, representing a shift in each dimension, which may be explicit or implicit, relative to a zero shift, where the index may correspond to a time unit of a particular shift amount. Each coefficient in the representation is the value of the shift kernel at the corresponding spatial position. For example, in the figure, the kernel is equal to 0 for all position indices except for the index (-1,2) (where the kernel is equal to 1). This kernel is equivalent to a shift of (-1,2) spatial units. Therefore, for a three-dimensional shift, the shift kernel can be represented as three coefficients or a three-dimensional array.

[0192] The two-dimensional coefficient array used to represent the shift kernel can also be used to represent other signals, i.e., general convolution kernels, such as 505. However, if these convolution kernels are not single sparse, they may not be shifts. The representation of a single sparse convolution kernel includes coefficients that are all zero except for a single coefficient with a non-zero value. A single sparse convolution kernel represents a shift that may be scaled depending on the value of the non-zero coefficient. If this value is equal to 1, the convolution kernel is a simple shift without scaling. A single sparse convolution kernel with coefficient values ​​that sum 506 to 1 must have a single coefficient with a value equal to 1, i.e., it will be a shift kernel.

[0193] Figure 5B is a schematic diagram showing how a time delay may be represented in a computer as a single coefficient 510, or equivalently as a shift kernel 512, according to some embodiments. Representing the delay as a single coefficient uses a single memory cell 511. Representing the delay as a shift kernel, on the other hand, requires a series of coefficients, such as stored in a memory cell bank 513, indexed by a timestamp 514 of each coefficient, which may be explicit or implicit, relative to zero time, where the index may correspond to a time unit of a particular duration. Each coefficient in the representation is the value of the shift kernel at the corresponding time index. For example, in the figure, the kernel is equal to 0 for all time indices except for index -5, where the kernel is equal to 1. This kernel is equivalent to a delay of -5 time units.

[0194] The series of coefficients used to represent the shift kernel can also be used to represent other signals, i.e., general convolution kernels, such as 515. However, if these convolution kernels are not single sparse, they may not be delays. The representation of a single sparse convolution kernel includes coefficients that are all zero except for a single coefficient with a non-zero value. The single sparse convolution kernel representation has a delay that may be scaled depending on the value of the non-zero coefficient. If this value is equal to 1, the convolution kernel is a simple delay without scaling. A single sparse convolution kernel with coefficient values ​​that sum 516 to 1 must have a single coefficient with a value equal to 1, i.e., it will be a shift kernel.

[0195] Figure 6 An example schematic diagram of transforming the components of FIG. 3 and FIG. 4 into a resulting frequency domain model according to some embodiments is shown. For example, the time domain model of the relationship 610 can be obtained by using Fourier transform 610 is represented in the frequency domain. Using the Fourier transform, convolutions with time delays or advances in the time domain model are converted to multiplications 650 with complex exponentials in the frequency domain model, which reduces the complexity of the calculation. Using two-dimensional or three-dimensional Fourier transforms, the same relationship applies to two-dimensional and three-dimensional signals, such as incident fields and reflected fields. Therefore, when calculating the convolution of a field with a corresponding transmitter or receiver spatial shift, some embodiments use a Fourier transform to convert the field and the shift kernel to the spatial frequency domain, where the convolution is transformed into a multiplication. This transformation makes the calculation of the cost simpler.

[0196] Therefore, when the relative delay between the transmitter clock and the receiver clock is taken into account, the resulting frequency domain model is

[0197]

[0198] in, is the data recorded by receiver 1 during the transmission of transmitter k, and ∈ lk is the relative clock error between transmitter k and receiver 1. Using represents the shifted kernel corresponding to the time shift due to this error, and Represents its Fourier transform, the model becomes

[0199]

[0200] Under this model, the multilinear optimization problem becomes

[0201]

[0202] Some implementations may further move the unknown relative time delay to the data side, transforming the problem into

[0203] in, express The complex conjugate

[0204] This formulation delays the shift kernel by the set of unknowns from the other unknown variables, allowing them to be estimated simultaneously with one of the other sets of variables (e.g., the image x), thereby reducing the number of multilinear components to three.

[0205] Alternatively, some embodiments consider both the transmitter clock error and the receiver clock error separately, resulting in the following frequency domain model:

[0206]

[0207] Among them, ∈ l is the clock error introduced by the first receiver, ∈ kis the clock error introduced by the kth transmitter. and denote the shift kernels corresponding to the time shifts caused by these errors, and Representing their corresponding Fourier transform, the model becomes

[0208]

[0209] Under this model, the multilinear optimization problem becomes

[0210]

[0211] Some embodiments may further move the unknown time delay due to receiver clock error to the data side, transforming the problem into

[0212]

[0213] This formulation transforms the unknown number of receiver clock errors into from other unknown variables, allowing them to be correlated with one of the other sets of variables (e.g., receiver clock error ) are estimated simultaneously, thus reducing the number of multilinear components to four.

[0214] Of course, if Figure 6 As described in paragraph

[0094] , all spatial convolutions can also be calculated in the spatial frequency domain for more efficient calculation. In this case, some embodiments of the present disclosure may pre-calculate the incident field The Fourier transform of is obtained and stored in memory to avoid repeated calculation when restoring the image.

[0215] Alternating Optimization

[0216] To solve all of these minimization problems, some embodiments of the present disclosure use alternating optimization, where each set of unknown variables in the multilinear problem is estimated sequentially by solving a set of smaller convex linear problems.

[0217] Fig. 7A is a block diagram of an alternating optimization method for estimating a radar image and a set of transmitter and receiver shifts according to an embodiment of the present disclosure. In particular, step 740 shows an iterative process for obtaining an estimate of the radar image and an estimate of the set of image shifts, and involves Figure 1A Steps 165A, 170A, 175A, 180A and 185A. Fig. 7A Some embodiments are shown in which estimation of transmitter and receiver clock errors is not necessary, for example because the clocks are sufficiently synchronized.

[0218] Fig. 7A Step 110D shows the pulses emitted, where Fig.7D A graph representing the transmitted pulses is provided in .

[0219] Fig. 7A Step 120D shows radar reflections, where Fig. 7E A diagram showing radar reflections is provided in .

[0220] Fig. 7A Step 700 represents the process of combining radar reflection 120D with Fig. 8A The estimation process of the initial image obtained by minimizing the difference between the transmitted pulse 110D and the modeled measurement results produced by the wrong antenna position (see Fig. 8A 801A and Figure 8B ).

[0221] Fig. 7A Step 701 represents the process of combining the radar reflection with the transmission pulse, the initial transmitter and the receiver shift, and the Fig.9A The process shown is a process of estimating a set of emitter shifts by minimizing the differences between synthetic modeled measurements of radar images.

[0222] Fig. 7A Step 702 represents the process of combining the radar reflection with the transmission pulse, the initial transmitter and the receiver shift, and the Fig. 9B The process shown is a process for estimating a set of receiver shifts by minimizing the differences between synthetic modeled measurements of radar images.

[0223] Fig. 7A Step 706 of FIG. 7 shows that the radar reflection is combined with the image shift estimated according to the transmitted pulse, and ... Figure 8C The process shown is a process of updating the estimate of the radar image by minimizing the difference between the estimated radar image and the synthesized modeled measurements.

[0224] Fig. 7A Step 715 indicates iteratively repeating the steps of estimating a set of transmitter shifts, then updating a set of receiver shifts, and then updating the estimate of the radar image until a convergence criterion is reached that the estimate of the radar image does not change. In some embodiments, the order in which steps 701, 702, 706 are performed may be different, as it is not important in determining the result. In some embodiments, the order of execution may even be random.

[0225] Fig. 7A Step 710 represents the process of outputting a converged radar image of the ROI.

[0226] Figure 7Bis a block diagram of an alternating optimization method for estimating a radar image and a set of transmitter and receiver shifts according to other embodiments of the present disclosure. Figure 7B The step 703 comprises combining the radar reflection with the transmitted pulse, the spatial displacement of the transmitter and the receiver, the initial relative delay between the transmitter and the receiver pair, and the Fig. 9C The process shown minimizes the difference between the estimated synthetic modeled measurements of the radar image to estimate the relative delay between the transmitter and receiver pair. The remaining steps are similar to Fig. 7A The steps are the same as in , with the relative delay between the transmitter and receiver pair additionally used in steps 701, 702, 706, 710. Fig. 7A As shown, in some embodiments, the execution order of steps 701, 702, 703, 706 may be different, because it is not important in determining the result. In some embodiments, the execution order may even be random.

[0227] Figure 7C is a block diagram of an alternating optimization method for estimating a radar image and a set of transmitter and receiver shifts according to other embodiments of the present disclosure. Figure 7C Step 704 is included of estimating the transmitter time shift by minimizing the difference between the radar reflections and modeled measurements synthesized from the transmitted pulses, transmitter and receiver spatial shifts, receiver time shift, initial transmitter time shift and an estimate of the radar image. Figure 7C Also included is step 705, which estimates the receiver time shift by minimizing the difference between the radar reflection and the modeled measurement result synthesized from the transmit pulse, the transmitter and receiver spatial shift, the transmitter time shift, the initial receiver time shift and the estimate of the radar image. The remaining steps are similar to Fig. 7A The steps are the same as those of , with the transmitter and receiver time shifts additionally used in steps 701, 702, 706, 710. Fig. 7A As shown, the execution order of steps 701, 702, 704, 705, 706 may be different in some embodiments, because it is not important in determining the result. In some embodiments, the execution order may even be random.

[0228] Fig. 8A is a block diagram of a method for estimating an initial radar image from measured radar reflections, transmit pulses, possibly unsynchronized transmitters and receivers, and erroneous antenna positions in accordance with an embodiment of the present disclosure. Fig. 8A Involved Fig. 7A , Figure 7B , Figure 7C Step 700 of the invention is related to estimating an initial radar image.

[0229] Figure 8BThe present invention is a diagram showing an embodiment of the present invention. Fig. 8A FIG. 8 is a diagram of an initial radar image related to step 801A in FIG. 8 .

[0230] For example, the initial antenna position is then used and an update of the image is calculated by calculating the difference 850A between the measured data and the modeled data synthesized using the current estimate Fig. 8A The image is then filtered by applying, for example, a one-norm regularizer 860 and a total variation regularizer 870, and the process is repeated until convergence 815A. Some embodiments may apply only one of the two regularizers or neither of the two regularizers, or use an alternative regularizer.

[0231] Figure 8C is a block diagram of a method for updating a radar image using measured radar reflections, estimated antenna positions, and estimated timing errors according to an embodiment of the present disclosure, wherein one-norm regularization and total variation regularization may be applied. Then, a method similar to Fig. 8A Method update Figure 8C The radar image 803, where the difference between the radar reflections and the modeled measurements is again minimized 850C to produce a new estimate of the radar image. The new estimate is also filtered by applying a one-norm regularization 860 and a total variation norm regularization 870 until a Figure 8C The output image 803 converges to 815C.

[0232] After each iteration, the difference in 850C decreases until convergence occurs. Process 706 is part of a larger interactive process 740, such as Fig. 7A , Figure 7B , Figure 7C As shown, the image is updated, the transmitter and receiver are shifted, and the corresponding time delays are alternated. Therefore, process 706 will be performed several times in iterative process 740. Some embodiments may reduce the number of iterations in 706 and terminate earlier to reduce computational cost, because the approximate estimate 803 in step 740 is usually sufficient. The convergence condition in 715 will ensure the convergence of the main process 740, thereby ensuring the accuracy of the solution.

[0233] Fig.8D The present invention is shown in the embodiment according to the present disclosure. Figure 8C After the process converges Fig. 7A , Figure 7B , Figure 7C Step 710 and Figure 8C 803 in FIG.

[0234] Fig.9Ais a block diagram of a method for estimating transmitter displacement using measured radar reflections and estimates of radar images, receiver displacement, and estimates of time shift (if available) according to an embodiment of the present disclosure. In particular, Fig.9A Involved Fig. 7A , Figure 7B , Figure 7C Step 701 and Figure 1A Step 180A.

[0235] refer to Fig.9A , in the calculation Fig. 7A , Figure 7B , Figure 7C After the initial image 801A, by using the radar reflection and Fig.9A The method shown updates the shift kernel corresponding to the transmitter position shift by minimizing the difference between the synthetic or modeled measurements 950A. The shift kernels are filtered to make them sparser by applying, for example, a one-norm regularizer 960, and the process is repeated until convergence 915A. Alternatively, a hard threshold may replace the one-norm regularizer to further improve the sparsity of the estimate. The calculated transmitter shift 990A can then be calculated based on the average assumed antenna position using Fig. 9C The illustrated method realigns 900A to produce a new estimate of the shift 902A.

[0236] Fig. 9B is a block diagram of a method for estimating receiver displacement using measured radar reflections and estimates of radar images, transmitter displacement, and estimates of time shift (if available) according to an embodiment of the present disclosure. In particular, Fig. 9B Involved Fig. 7A , Figure 7B , Figure 7C Step 702 and Figure 1A Step 175A.

[0237] refer to Fig. 9B , in the calculation Fig. 7A , Figure 7B , Figure 7C After the initial image 801A, by using radar reflection and Fig. 9B The method shown updates the shift kernel corresponding to the receiver position shift by minimizing the difference between the synthetic or modeled measurements 950B. The shift kernels are filtered to make them sparser by applying, for example, a one-norm regularizer 960, and the process is repeated until convergence 915B. Alternatively, a hard threshold may replace the one-norm regularizer to further improve the sparsity of the estimate. The calculated receiver shift 990B can then be calculated based on the average assumed antenna position using Fig. 9C The method shown in aligns 900B to produce an estimate of a new shift 902B.

[0238] Fig. 9C 900C in accordance with an embodiment of the present disclosure. The input displacement 990 is used to calculate the average of the displacement in each dimension and compare it to the target average. The displacement is then repositioned so that the average displacement is equal to the target average so that their relative positions to each other are the same to produce a new displacement 902 estimate with the target average displacement.

[0239] Fig. 10A is a block diagram of a method for estimating relative time delays using measured radar reflections and estimates of radar images and transmitter and receiver spatial shifts in accordance with an embodiment of the present disclosure. In particular, Fig. 10A Involved Figure 7B Step 703.

[0240] Reference Fig. 10A , in the calculation Figure 7B After the initial image 801A, by making the radar reflect with the use Fig. 10A The method shown in 1015A is used to update the time shift kernels corresponding to the relative transmitter / receiver delays by minimizing the difference between the synthetic or modeled measurements 1050A. The time shift kernels are filtered to make them sparser by applying, for example, a one-norm regularizer 1060, and the process is repeated until convergence 1015A to produce a new estimate of the time shift 1002A. Alternatively, a hard threshold may replace the one-norm regularizer to further improve the sparsity of the estimate.

[0241] Fig. 10B is a block diagram of a method for estimating transmitter time shift using measured radar reflections and estimates of radar images, receiver time shift, and transmitter and receiver spatial shifts according to an embodiment of the present disclosure. In particular, Fig. 10B Involved Figure 7C Step 704.

[0242] refer to Fig. 10B , in the calculation Figure 7C After the initial image 801A, by using radar reflection and Fig. 10B The illustrated method updates the time shift kernel corresponding to the transmitter time shift by minimizing the difference between the synthetic or modeled measurements 1050B. The time shift kernels are filtered to make them sparser by applying, for example, a one-norm regularizer 1060, and the process is repeated until convergence 1015B to produce a new estimate 1002B of the transmitter time shift. Alternatively, a hard threshold may replace the one-norm regularizer to further improve the sparsity of the estimate.

[0243] Fig. 10Cis a block diagram of a method for estimating receiver time shift using measured radar reflections and estimates of radar images, transmitter time shift, and transmitter and receiver spatial shifts according to an embodiment of the present disclosure. In particular, Fig. 10C Involved Figure 7C Step 705.

[0244] Reference Fig. 10C , in the calculation Figure 7C After the initial image 801A, by making the radar reflect with the use Fig. 10C The illustrated method updates the time shift kernel corresponding to the receiver time shift by minimizing the difference between the synthetic or modeled measurements 1050C. The time shift kernels are filtered to make them sparser by applying, for example, a one-norm regularizer 1060, and the process is repeated until convergence 1015C to produce a new estimate 1002C of the receiver time shift. Alternatively, a hard threshold may replace the one-norm regularizer to further improve the sparsity of the estimate.

[0245] feature

[0246] Aspects of the present disclosure may include a hardware processor configured to use configuration data to solve a radar image recovery problem to produce a radar image of reflections for each point in a scene, including associating received data to a correlated transmitter / receiver unknown clock delay, wherein the correlated transmitter / receiver unknown clock delay defines an error between a transmitter clock and a receiver clock.

[0247] Another aspect is a hardware processor configured to solve a radar image recovery problem using configuration data to generate a radar image of reflections for each point in a scene, and includes associating received data to a transmitter unknown clock delay, wherein the transmitter unknown clock delay defines an error between a transmitter clock and a global clock. Together with associating received data to a receiver unknown clock delay, wherein the receiver unknown clock delay defines an error between a receiver clock and a global clock. Another aspect is an aspect that may include determining a related transmitter / receiver clock delay by determining an unknown delay kernel, whose time domain convolution with the received data delays the data according to the related transmitter / receiver clock delay. Another aspect is an aspect that may include determining a transmitter clock delay and a receiver clock delay by determining an unknown delay kernel, whose time domain convolution with a transmitted pulse and received data delays the transmitted pulse and received data according to the transmitter clock delay and the receiver clock delay, respectively.

[0248] Another aspect is that the error between the receiver position stored in the memory and the actual position of the receiver is determined jointly and independently of the error between the transmitter position stored in the memory and the actual position of the transmitter. One aspect may be that the effect of a transmit antenna position error is a shift that the transmit antenna introduces into an incident field of a scene by the same amount as the transmit antenna position error, causing the scene to interact with the incident field to create a reflected field. Wherein the receive antenna measures the reflected field at the receive antenna position, resulting in the effect of the receive antenna position error being equivalent to a reflected field measured at a different point, which in turn is equivalent to a reflected field shifted by the same amount in the opposite direction as measured by a receive antenna without position error.

[0249] Another aspect may be that associating the received radar measurements to a shift of the reflected field with an unknown receiver position shift comprises convolution of the reflected field at the assumed position of the receiver with a convolution kernel corresponding to the unknown receiver position shift. Another aspect may be that associating the reflected field to a shift of the incident field with an unknown transmitter position shift comprises convolution of the incident field due to the assumed position of the transmitter with a convolution kernel corresponding to the unknown transmitter position shift.

[0250] Another aspect is to solve the radar image restoration problem as a multilinear problem of joint estimation of the reflection of each point in the scene, including regularization of the reflection of the scene. Regularization improves the sparsity of the scene. Regularization improves the restored scene with low total variation.

[0251] Another aspect is that position errors in the transmitting antennas cause the incident field introduced into the scene to be shifted by the same amount, and position errors in the receiving antennas cause the received data to appear as if the reflected field was shifted by the same amount in the opposite direction, and based on such a configuration, a multilinear problem or a multilinear optimization problem is configured to simultaneously recover all antenna position errors, as well as the sparse scene being imaged.

[0252] Another aspect is a radar image restoration problem of associating received radar measurements with reflections from each point in a scene to shift a reflected field with an unknown position shift of a receiver and to shift an incident field with an unknown position shift of a transmitter, wherein the shift of the reflected field is independent of the unknown position shift of the transmitter. wherein determining the shift of the incident field is performed by determining an unknown shift kernel whose convolution with the incident field shifts the incident field by an unknown shift, and determining the shift of the reflected field is performed by determining an unknown shift kernel whose convolution with the reflected field shifts the reflected field by an unknown shift. wherein the unknown shift kernel is sparse.

[0253] Fig.11 1 is a block diagram of a computer system of a radar system contemplated by the present disclosure according to some embodiments of the present disclosure. Figure 1C The computer system 1100 may communicate with the antenna set 101, 102, 103, 104, 105, and the collected data may be stored in the memory 1130 processed by the processor 1140 of the computer 1100. The computer system 1100 may include a human-machine interface or user interface 1150 that may connect the computer system to a keyboard 1151 and a display device 1152. The computer system 1100 may be linked to a display interface 1157 via a bus 1156, which is suitable for connecting the system 1100 to a display device 1148, wherein the display device 1148 may include a computer monitor, a camera, a television, a projector, or a mobile device, etc.

[0254] The computer system 1100 may include a power supply 1154, which may optionally be located outside the computer system, depending on the application. The auto-focus imaging processor 1140 may be one or more processors that may be configured to execute stored instructions and communicate with a memory 1130 storing instructions executable by the auto-focus imaging processor 1140. The auto-focus imaging processor 1140 may be a single-core processor, a multi-core processor, a computing cluster, or any number of other configurations. The auto-focus imaging processor 1140 is connected to one or more input and output devices via a bus 1156. The memory 1130 may include a random access memory (RAM), a read-only memory (ROM), flash memory, or any other suitable memory system.

[0255] Refer to Fig.11 The computer system 1100 may also include a storage device 1158 suitable for storing supplemental data and / or software modules used by the auto-focus imaging processor 1140. For example, the storage device 1158 may store historical data related to pre-designed radar platform tracks, radar operating frequency bandwidth, transmit waveforms, estimated signal-to-noise ratios, image data related to target identification, imaging results using simulated noise data and different methods to handle position errors, etc. The storage device 1158 may include a hard drive, an optical drive, a thumb drive, a drive array, or any combination thereof.

[0256] Still reference Fig.11, a printer interface 1159 may also be connected to the computer system 1100 via the bus 1156, and is suitable for connecting the computer system 1100 to a printing device 1132, wherein the printing device 1132 may include a liquid inkjet printer, a solid ink printer, a large-scale commercial printer, a thermal printer, a UV printer, or a dye sublimation printer, etc. A network interface controller (NIC) 1134 is suitable for connecting the computer system 1100 to a network 1136 via the bus 1156. Image data or related image data, etc. can be rendered on a display device, an imaging device, and / or a printing device via the network 1136.

[0257] Still reference Fig.11 , image data or related image data, etc. can be transmitted through the communication channel of the network 1136, and / or stored in the storage device 1158 of the computer for storage and / or further processing. In addition, image data or related image data can be received from the receiver 1141 in a wireless or wired manner or transmitted via the transmitter 1143 in a wireless or wired manner, and the receiver 1141 and the transmitter 1143 are both connected to the computer system 1100 via the bus 1156.

[0258] The computer system 1100 may be connected to external sensors 1131, one or more input devices 1133, other computers 1142, and other devices 1144. The external sensors 1131 may include motion sensors, inertial sensors, a type of measurement sensor, etc. The external sensors 1131 may include sensors for speed, direction, air flow, distance to an object or location, weather conditions, etc. The input devices 1141 may include, for example, a keyboard, a scanner, a microphone, a stylus, a touch-sensitive pad, or a display.

[0259] Implementation

[0260] The following description provides only exemplary embodiments and is not intended to limit the scope, applicability or configuration of the present disclosure. On the contrary, the following description of exemplary embodiments will provide those skilled in the art with an effective description for implementing one or more exemplary embodiments. Various changes may be made to the functions and arrangements of the elements without departing from the spirit and scope of the subject matter disclosed in the appended claims.

[0261] Specific details are given in the following description to provide a thorough understanding of the embodiments. However, it will be appreciated by those skilled in the art that the embodiments may be implemented without these specific details. For example, the systems, processes, and other elements in the disclosed subject matter may be displayed as components in block diagram form to avoid confusing the embodiments in unnecessary details. In other cases, well-known processes, structures, and techniques may be shown without unnecessary details to avoid confusing the embodiments. In addition, the same reference numerals and names in the various drawings represent the same elements.

[0262] In addition, each embodiment can be described as a process, and the process can be described as a flow chart, a flow diagram, a data flow diagram, a structure diagram or a block diagram. Although a flow chart can describe an operation as a sequential process, many operations can be performed in parallel or simultaneously. In addition, the order of the operations can be rearranged. A process may terminate when its operation is completed, but may have other steps that are not discussed or included in the figure. In addition, not all operations in any specifically described process can occur in all embodiments. Processing can correspond to a method, a function, a process, a subprocess, a subroutine, etc. When processing corresponds to a function, the termination of the function can correspond to the function returning to the calling function or the main function.

[0263] In addition, embodiments of the disclosed subject matter may be implemented at least in part manually or automatically. Manual or automatic implementations may be performed or at least assisted in implementation by machines, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware, or microcode, program code or code segments that perform the necessary tasks may be stored in a machine-readable medium. A processor may perform the necessary tasks.

[0264] Embodiments of the present invention include coherent distributed radar imaging by allowing position ambiguity and autofocusing a single sensor array by utilizing distributed sensing of multiple sensors. In particular, a multi-static radar imaging method in which a transmitting / receiving radar platform and multiple receiving radar platforms are moving towards a region of interest (ROI) with position perturbations. Embodiments of the present disclosure detect targets within the ROI. Due to inaccurate positioning and motion errors, the actual array position is perturbed by up to several times the center radar wavelength. Although the image resolution of each sensor array may be low due to its small aperture size, a high-resolution image can be formed by jointly processing the outputs of all distributed arrays and compensating well for position errors. Embodiments of the present disclosure assume a sparse scene and are iteratively implemented by solving a series of optimization problems for compensating for position-induced phase errors, utilizing target feature maps, and estimating antenna positions.

[0265] Embodiments of the present disclosure also provide auto-focus radar imaging for generating radar images of targets located in a region of interest using a single mobile transmitting radar platform or transmitter / receiver combination along with multiple spatially distributed mobile radar receiver platforms or receivers. The mobile radar receivers are subject to unknown position errors of up to several radar wavelengths.

[0266] In addition, the various methods or processes outlined herein may be encoded as software that can be executed on one or more processors using any of a variety of operating systems or platforms. In addition, such software may be written using any of a variety of suitable programming languages ​​and / or programming or scripting tools, and may be compiled into executable machine language code or intermediate code that is executed on a framework or virtual machine. Typically, the functions of program modules may be combined or distributed as desired in various embodiments.

[0267] In addition, the embodiments of the present disclosure may be embodied as a method, examples of which have been provided. The actions performed as part of the method may be ordered in any suitable manner. Therefore, embodiments in which actions are performed in an order different from that illustrated may be constructed, which may include performing some actions simultaneously, even if they are shown as sequential actions in illustrative embodiments. In addition, the use of ordinal terms such as "first", "second" and the like in the claims to modify claim elements does not itself imply any priority, priority or order of one claim element relative to another claim element, or the temporal order of the actions of the method of execution, but is merely used as a label to distinguish a claim element having a specific name from another element having the same name (but using a serial number) to distinguish the claim elements.

[0268] Although the present disclosure has been described with reference to certain preferred embodiments, it should be understood that various other modifications and changes may be made within the spirit and scope of the present disclosure. Therefore, an aspect of the appended claims is to cover all changes and modifications that fall within the true spirit and scope of the present disclosure.

Claims

1. A radar system for generating a radar image of a scene, wherein: During operation, the radar system is connected to a set of transmitters configured to transmit radar pulses to the scene and a set of receivers configured to receive reflections of the radar pulses from the scene, the radar system comprising: a memory configured to store predetermined configuration data indicative of radar pulse propagation to and from the scene, the configuration data comprising positions of the transmitters and positions of the receivers such that the position of at least one receiver is different from the transmitter positions of all transmitters, wherein the configuration data defines an incident field generated by radar pulses transmitted by the transmitters from the transmitter positions, and wherein the configuration data defines a structure of a reflected field generated by reflections of the incident field from the scene and measured by the receivers at the receiver positions; an input interface configured to receive radar measurements of reflections at each point in the scene measured by the set of receivers; A hardware processor configured to solve a radar image recovery problem using the configuration data to generate a radar image of the reflection for each point in the scene by: associating received radar measurements to a shift of the reflection field with a receiver unknown position shift, wherein the receiver unknown position shift defines an error between a receiver position stored in the memory and an actual position of the receiver, and wherein the reflection field is generated by reflecting a transmitted field from a scene according to reflections at each point in the scene; correlating the reflected field to a shift of the incident field with a transmitter unknown position shift, wherein the transmitter unknown position shift defines an error between a transmitter position stored in the memory and an actual position of the transmitter; solving the radar image restoration problem as a multilinear problem of joint estimation of the reflectance, the receiver shift and the transmitter shift for each point in the scene; and An output interface configured to render one or a combination of the reflected radar image, the receiver displacement, or the transmitter displacement for each point in the scene.

2. The radar system according to claim 1, wherein: The hardware processor is configured to use the configuration data to solve a radar image recovery problem to produce a radar image of reflections for each point in the scene, including correlating received radar measurements to an associated transmitter / receiver unknown clock delay, wherein the associated transmitter / receiver unknown clock delay defines an error between the transmitter clock and the receiver clock.

3. The radar system according to claim 1, wherein: The hardware processor is configured to solve a radar image recovery problem using the configuration data to generate a radar image of the reflection for each point in the scene, including: correlating the received radar measurements to a transmitter unknown clock delay, wherein the transmitter unknown clock delay defines an error between a transmitter clock and a global clock; and Received radar measurements are correlated to a receiver unknown clock delay, wherein the receiver unknown clock delay defines an error between a receiver clock and a global clock.

4. The radar system according to claim 3, wherein: Determining the associated transmitter / receiver clock delay is performed by determining an unknown delay kernel whose time-domain convolution with the received radar measurements delays the radar measurements according to the associated transmitter / receiver clock delay.

5. The radar system according to claim 3, wherein: Determining the transmitter clock delay and the receiver clock delay is performed by determining an unknown delay kernel whose time domain convolution with a transmitted pulse and a received radar measurement result delays the transmitted pulse and the received radar measurement result according to the transmitter clock delay and the receiver clock delay, respectively.

6. The radar system according to claim 1, wherein: The error between the receiver position stored in the memory and the actual position of the receiver is determined in conjunction with or independently of the error between the transmitter position stored in the memory and the actual position of the transmitter.

7. The radar system according to claim 6, wherein: the effect of a transmit antenna position error is that the transmit antenna will introduce a shift of the incident field to the scene by the same amount as the transmit antenna position error, causing the scene to interact with the incident field to create the reflected field, Among them, the receiving antenna measures the reflected field at the position of the receiving antenna, resulting in the influence of the receiving antenna position error, and the influence of the receiving antenna position error is equivalent to the reflected field measured at different points, and the reflected field measured at different points is equivalent to the reflected field shifted by the same amount in the opposite direction measured by a receiving antenna without position error.

8. The radar system according to claim 1, wherein: Associating the received radar measurements to the shift of the reflected field with the unknown receiver position shift includes convolution of the reflected field at the assumed position of the receiver with a convolution kernel corresponding to the unknown receiver position shift.

9. The radar system according to claim 1, wherein: Correlating the reflected field to a shift of the incident field with an unknown position shift of the emitter includes convolving the incident field due to the assumed position of the emitter with a convolution kernel corresponding to the unknown emitter position shift.

10. The radar system according to claim 1, wherein: Solving the radar image restoration problem as a multilinear problem of joint estimation of reflectance at each point in the scene includes regularization of the reflectance of the scene.

11. The radar system according to claim 10, wherein: The regularization improves the sparsity of the scene.

12. The radar system according to claim 10, wherein: The regularization exploits low total variation to improve the restored scene.

13. The radar system of claim 1, wherein: Position errors of the transmitting antennas result in a shift of the incident field introduced into the scene by the same amount, and position errors of the receiving antennas result in the received radar measurements being shifted by the same amount in the opposite direction as the reflected field, and based on such a configuration, the multilinear problem or multilinear optimization problem is configured to simultaneously recover all antenna position errors, as well as the imaged sparse scene.

14. The radar system of claim 1, wherein: The radar image restoration problem relates received radar measurements having reflections for each point in the scene via received radar measurements to a shift of the reflected field with a shift of the receiver unknown position and a shift of the incident field with a shift of the transmitter unknown position, wherein the shift of the reflected field is independent of the transmitter unknown position shift.

15. The radar system of claim 14, wherein: Determining the shift of the incident field is performed by determining an unknown shift kernel, the convolution of the unknown shift kernel with the incident field causes the incident field to be shifted by the unknown shift, and determining the shift of the reflected field is performed by determining an unknown shift kernel, the convolution of the unknown shift kernel with the reflected field causes the reflected field to be shifted by the unknown shift.

16. The radar system of claim 15, wherein: The unknown shift kernel is sparse.

17. A method for a radar system to generate a radar image of a region of interest (ROI), the method comprising the following steps: performing measurements using a transmit antenna that transmits radar pulses at different locations toward the ROI and a receive antenna configured to receive reflections of the radar pulses from the ROI corresponding to the transmitted radar pulses, such that a receiver measures the reflections of the radar pulses to obtain radar measurements of the reflections for each point in the scene; Using a hardware processor in communication with a transmitter, the receiver, and a memory, the hardware processor being configured to: accessing the memory having stored therein data, the data comprising predetermined configuration data indicative of radar pulse propagation to and from the ROI, the configuration data comprising locations of the transmitters and locations of the receivers such that at least one receiver location is different than transmitter locations of all transmitters, wherein the configuration data defines an incident field generated by radar pulses transmitted by the transmitters from the transmitter locations, and wherein the configuration data defines a structure of a reflected field resulting from reflections of the incident field from the scene and measured by the receivers at receiver locations; The configuration data is used to solve the radar image restoration problem to produce a radar image of the reflection for each point in the scene by: associating received radar measurements to a shift of a reflected field having a receiver unknown position shift, wherein the receiver unknown position shift defines an error between a receiver position stored in the memory and an actual position of the receiver, and wherein the reflected field is generated by reflecting a transmitted field from the scene according to reflections at each point in the scene; correlating the reflected field to a shift of the incident field with a transmitter unknown position shift, wherein the transmitter unknown position shift defines an error between a transmitter position stored in the memory and an actual position of the transmitter; solving the radar image restoration problem as a multilinear problem of joint estimation of the reflectance, the receiver shift and the transmitter shift for each point in the scene; and One or a combination of the reflected radar image, the receiver displacement, or the transmitter displacement for each point in the scene is output via an output interface.

18. The method according to claim 17, wherein: Associating the received radar measurements to the shift of the reflected field with the unknown receiver position shift comprises convolving the reflected field at the assumed position of the receiver with a convolution kernel corresponding to the unknown receiver position shift, and Wherein associating the reflected field to a shift of the incident field with an unknown position shift of the emitter comprises convolving the incident field due to the assumed position of the emitter with a convolution kernel corresponding to the unknown emitter position shift.

19. The method according to claim 17, wherein: Solving the radar image restoration problem as a multilinear problem of joint estimation of reflectances at various points in the scene includes regularizing the reflectances of the scene.

20. The method according to claim 17, wherein: Position errors in the transmitting antennas result in a shift of the incident field introduced into the scene by the same amount, and position errors in the receiving antennas result in the received radar measurements being shifted by the same amount in the opposite direction as the reflected field, and based on such a configuration, the multilinear problem or multilinear optimization problem is configured to simultaneously recover all antenna position errors, as well as the imaged sparse scene.

21. A non-transitory computer-readable storage medium having a program executable by a processor implemented thereon for executing a method for generating a radar image of a region of interest (ROI) by a radar system, the method comprising: performing measurements using a transmit antenna that transmits radar pulses at different locations toward the ROI and a receive antenna configured to receive reflections of the radar pulses from the ROI corresponding to the transmitted radar pulses, such that a receiver measures the reflections of the radar pulses to obtain radar measurements of the reflections for each point in the scene; Using a hardware processor in communication with a transmitter, the receiver, and a memory, the hardware processor being configured to: accessing a memory having stored therein data, the data comprising predetermined configuration data indicative of radar pulse propagation to and from the ROI, the configuration data comprising locations of the transmitters and locations of the receivers such that at least one receiver location is different than transmitter locations of all transmitters, wherein the configuration data defines an incident field generated by radar pulses transmitted by the transmitters from the transmitter locations, and wherein the configuration data defines a structure of a reflected field generated by reflections of the incident field from the scene and measured by the receivers at the receiver locations; The configuration data is used to solve the radar image restoration problem to produce a radar image of the reflection for each point in the scene by: associating received radar measurements to a shift of a reflected field having a receiver unknown position shift, wherein the receiver unknown position shift defines an error between a receiver position stored in the memory and an actual position of the receiver, and wherein the reflected field is generated by reflecting a transmitted field from the scene according to reflections at each point in the scene; correlating the reflected field to a shift of the incident field with a transmitter unknown position shift, wherein the transmitter unknown position shift defines an error between a transmitter position stored in the memory and an actual position of the transmitter; solving the radar image restoration problem as a multilinear problem of joint estimation of the reflection, the receiver shift, and the transmitter shift for each point in the scene; and One or a combination of the radar image of the reflection, the receiver displacement, or the transmitter displacement for each point in the scene is output via an output interface.

Images