Method, system, and product for overcoming cycle skipping for full-waveform inversion

Abstract

A method for generating an image comprises providing an imaging device comprising a multi-channel radiofrequency data acquisition system comprising a plurality of receive elements and a transmission system, transmitting a signal pulse including content at first and second high frequencies from the transmission system, recording a time-domain channel waveform from each receive element in the multi-channel radiofrequency data acquisition system, calculating a set of frequency-difference signals for each receive element based on the first and second high frequencies, and applying a frequency-difference waveform inversion (FDWI) to the set of frequency difference signals to generate an image. An imaging system is also disclosed.

Description

Attorney Docket No.204606-0170-00WO 6-24009 METHOD, SYSTEM, AND PRODUCT FOR OVERCOMING CYCLE SKIPPING FOR FULL-WAVEFORM INVERSION CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. provisional application No.63 / 517,476 filed on Aug.03, 2023, the contents of which are incorporated herein by reference in its entirety. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under EB028652 and EB034589 awarded by the National Institutes of Health. The government has certain rights in the invention. BACKGROUND OF THE INVENTION

[0003] Most ultrasound transducer designs are driven towards high frequencies by the need to maximize the resolution achievable by B-mode reflectivity imaging. Because full-waveform inversion (FWI) requires low frequencies to overcome cycle skipping, it is often difficult to apply FWI to the same high-frequency transducer hardware.

[0004] The lack of low-frequency signal on most ultrasound transducers currently designed for B-mode imaging prevents full-waveform inversion (FWI) from becoming more broadly applicable. In ultrasound tomography (UST), FWI often requires low-frequency signals (<1 MHz) to overcome cycle skipping caused by errors in a starting model. This is because FWI directly minimizes the error between simulated and measured signals. If the simulated signal is delayed or time-advanced by more than a half cycle relative to the measured signal, the underlying model used in the simulation will update in the wrong direction due to cycle skipping. Low frequencies lengthen this half-cycle window, allowing FWI to overcome errors in a starting model (see Ali, R., et al., In Medical Imaging 2023.).Attorney Docket No.204606-0170-00WO 6-24009

[0005] Recent work on time-domain adaptive waveform inversion (AWI) addresses this based on a deconvolution approach to overcome cycle skipping. However, most clinical implementations of ultrasound tomography rely on frequency-domain FWI to quickly reconstruct images on clinically relevant time scales.

[0006] Adaptive waveform inversion (AWI) overcomes this dependence on low frequencies by modeling the mismatch between simulated and measured signals as a bank of impulse responses (see Warner, M., et al., Geophysics, 2016). Rather than directly minimizing the error between simulated and measured signals, AWI drives these impulse responses toward zero-time lag, circumventing the cycle skipping problem. However, the main difficulty limiting AWI is its time-domain methodology as most clinical UST scanners rely on frequency-domain FWI for fast image reconstruction (see Sandhu, G. Y., et al., Physics in Medicine & Biology, 2015; and Wiskin, J., et al., The Journal of the Acoustical Society of America, 2012). Fortunately, time lag is equivalent to a linear phase response in the frequency domain, so a frequency-domain implementation of AWI can be achieved by flattening this linear phase response.

[0007] One way this can be achieved is by using a mathematical trick from sonar known as frequency-differencing (H. Abadi, et al., The Journal of the Acoustical Society of America, 2012.). By taking the ratio of signals at two different frequencies^^and^, a new signal is produced at frequency difference Δ^ = |^^−^|. By minimizing the error between simulated and measured signals after frequency differencing, cycle skipping can be overcome by using a sufficiently small frequency difference Δ^.

[0008] Although the broadband nature of AWI makes it difficult to translate to the frequency domain, there exists a need to approximate the properties of AWI by using a frequency- differencing approach. SUMMARY OF THE INVENTION

[0009] In one aspect, a method for generating an image comprises providing an imaging device comprising a multi-channel radiofrequency data acquisition system comprising a plurality of receive elements and a transmission system, transmitting a signal pulse including content at first and second high frequencies from the transmission system, recording a time-domain channelAttorney Docket No.204606-0170-00WO 6-24009 waveform from each receive element in the multi-channel radiofrequency data acquisition system, calculating a set of frequency-difference signals for each receive element based on the first and second high frequencies, and applying a frequency-difference waveform inversion (FDWI) to the set of frequency difference signals to generate an image.

[0010] In one embodiment, the method further comprises calculating a set of phasors from each time domain channel waveform at the first and second high frequencies, wherein the phasors are calculated using a discrete-time Fourier transform. In one embodiment, the imaging device comprises a ring array. In one embodiment, the imaging device comprises at least one linear array. In one embodiment, the imaging device comprises at least one ultrasound transducer. In one embodiment, the step of applying a FDWI to the set of frequency difference signals further comprises calculating an objective function associated with a difference between the first and second high frequencies. In one embodiment, the objective function includes calculation of a simulation model of tissue mapping. In one embodiment, the objective function comprises calculating a Helmholtz equation at the difference (Δ^) between the first and second high frequencies. In one embodiment, the method further comprises the step of augmenting the set of frequency difference signals with a convolutional neural network to bring the frequency difference signals closer to true low-frequency data.

[0011] In one aspect, an imaging system comprises a multi-channel radiofrequency data acquisition system comprising a plurality of receive elements, a transmission system configured to generate pulses at first and second high frequencies, a computing device communicative connected to the transmission system and the multi-channel radiofrequency data acquisition system, wherein the computing device comprises a processor and a non-transitory computer- readable medium with instructions stored thereon, which when executed by the processor, performs steps comprising transmitting a signal pulse including content at the first and second high frequencies from the transmission system, recording a time-domain channel waveform from each receive element in the multi-channel radiofrequency data acquisition system, calculating a set of frequency-difference signals for each receive element based on the first and second high frequencies, and applying a frequency-difference waveform inversion (FDWI) to the set of frequency difference signals to generate an image.Attorney Docket No.204606-0170-00WO 6-24009

[0012] In one embodiment, the multi-channel radiofrequency data acquisition system comprises at least one ultrasound transducer. In one embodiment, the at least one ultrasound transducer comprises at least one B-mode reflectivity imaging transducer. In one embodiment, the multi-channel radiofrequency data acquisition system comprises a ring array of receive elements. In one embodiment, the ring array comprises 256 elements. In one embodiment, the imaging device comprises at least one linear array of receive elements. In one embodiment, the first and second high frequencies are between 1.2 MHz and 1.7 MHz. In one embodiment, a difference between the first and second high frequencies is between 300 MHz and 400 MHz. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The foregoing purposes and features, as well as other purposes and features, will become apparent with reference to the description and accompanying figures below, which are included to provide an understanding of the invention and constitute a part of the specification, in which like numerals represent like elements, and in which:

[0014] Fig.1 depicts a computing device.

[0015] Fig.2 depicts a true sound speed ultrasound image.

[0016] Fig.3 depicts an ultrasound image generated using FWI.

[0017] Fig.4 depicts an ultrasound image generated using data-domain frequency differencing.

[0018] Fig.5 depicts an ultrasound image generated using FWI after data-domain frequency differencing.

[0019] Fig.6 depicts an ultrasound image generated using model-domain frequency differencing.

[0020] Fig.7 depicts an ultrasound image generated using FWI after model-domain frequency differencing.Attorney Docket No.204606-0170-00WO 6-24009 DETAILED DESCRIPTION OF THE INVENTION

[0021] It is to be understood that the figures and descriptions of the present invention have been simplified to illustrate elements that are relevant for a clearer comprehension of the present invention, while eliminating, for the purpose of clarity, many other elements found in systems and methods. Those of ordinary skill in the art may recognize that other elements and / or steps are desirable and / or required in implementing the present invention. However, because such elements and steps are well known in the art, and because they do not facilitate a better understanding of the present invention, a discussion of such elements and steps is not provided herein. The disclosure herein is directed to all such variations and modifications to such elements and methods known to those skilled in the art.

[0022] Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. Although any methods and materials similar or equivalent to those described herein can be used in the practice or testing of the present invention, exemplary methods and materials are described.

[0023] As used herein, each of the following terms has the meaning associated with it in this section.

[0024] The articles “a” and “an” are used herein to refer to one or to more than one (i.e., to at least one) of the grammatical object of the article. By way of example, “an element” means one element or more than one element.

[0025] The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limiting. For example, as used herein, the singular forms “a”, “an” and “the” may be intended to include the plural forms as well, unless the context clearly indicates otherwise. The terms “comprises,” “comprising,” “including,” and “having,” are inclusive and therefore specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof. The method steps, processes, and operations described herein are not to be construed as necessarily requiring their performance in the particular order discussed or illustrated, unless specifically identified asAttorney Docket No.204606-0170-00WO 6-24009 an order of performance. It is also to be understood that additional or alternative steps may be employed.

[0026] When an element or layer is referred to as being “on”, “engaged to”, “connected to” or “coupled to” another element or layer, it may be directly on, engaged, connected or coupled to the other element or layer, or intervening elements or layers may be present. In contrast, when an element is referred to as being “directly on,” “directly engaged to,” “directly connected to” or “directly coupled to” another element or layer, there may be no intervening elements or layers present. Other words used to describe the relationship between elements should be interpreted in a like fashion (e.g., “between” versus “directly between,” “adjacent” versus “directly adjacent,” etc.). As used herein, the term “and / or” includes any and all combinations of one or more of the associated listed items.

[0027] In some aspects of the present invention, software executing the instructions provided herein may be stored on a non-transitory computer-readable medium, wherein the software performs some or all of the steps of the present invention when executed on a processor.

[0028] Aspects of the invention relate to algorithms executed in computer software. Though certain embodiments may be described as written in particular programming languages, or executed on particular operating systems or computing platforms, it is understood that the system and method of the present invention is not limited to any particular computing language, platform, or combination thereof. Software executing the algorithms described herein may be written in any programming language known in the art, compiled or interpreted, including but not limited to C, C++, C#, Objective-C, Java, JavaScript, MATLAB, Python, PHP, Perl, Ruby, or Visual Basic. It is further understood that elements of the present invention may be executed on any acceptable computing platform, including but not limited to a server, a cloud instance, a workstation, a thin client, a mobile device, an embedded microcontroller, a television, or any other suitable computing device known in the art.

[0029] Parts of this invention are described as software running on a computing device. Though software described herein may be disclosed as operating on one particular computing device (e.g. a dedicated server or a workstation), it is understood in the art that software is intrinsically portable and that most software running on a dedicated server may also be run, for the purposesAttorney Docket No.204606-0170-00WO 6-24009 of the present invention, on any of a wide range of devices including desktop or mobile devices, laptops, tablets, smartphones, watches, wearable electronics or other wireless digital / cellular phones, televisions, cloud instances, embedded microcontrollers, thin client devices, or any other suitable computing device known in the art.

[0030] Similarly, parts of this invention are described as communicating over a variety of wireless or wired computer networks. For the purposes of this invention, the words “network”, “networked”, and “networking” are understood to encompass wired Ethernet, fiber optic connections, wireless connections including any of the various 802.11 standards, cellular WAN infrastructures such as 3G, 4G / LTE, or 5G networks, Bluetooth®, Bluetooth® Low Energy (BLE) or Zigbee® communication links, or any other method by which one electronic device is capable of communicating with another. In some embodiments, elements of the networked portion of the invention may be implemented over a Virtual Private Network (VPN).

[0031] Fig.1 and the following discussion are intended to provide a brief, general description of a suitable computing environment in which the invention may be implemented. While the invention is described above in the general context of program modules that execute in conjunction with an application program that runs on an operating system on a computer, those skilled in the art will recognize that the invention may also be implemented in combination with other program modules.

[0032] Generally, program modules include routines, programs, components, data structures, and other types of structures that perform particular tasks or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the invention may be practiced with other computer system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, minicomputers, mainframe computers, and the like. The invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.

[0033] Fig.1 depicts an illustrative computer architecture for a computer 100 for practicing the various embodiments of the invention. The computer architecture shown in Fig.1 illustrates aAttorney Docket No.204606-0170-00WO 6-24009 conventional personal computer, including a central processing unit 150 (“CPU”), a system memory105, including a random access memory 110 (“RAM”) and a read-only memory (“ROM”) 115, and a system bus 135 that couples the system memory 105 to the CPU 150. A basic input / output system containing the basic routines that help to transfer information between elements within the computer, such as during startup, is stored in the ROM 115. The computer 100 further includes a storage device 120 for storing an operating system 125, application / program 130, and data.

[0034] The storage device 120 is connected to the CPU 150 through a storage controller (not shown) connected to the bus 135. The storage device 120 and its associated computer-readable media provide non-volatile storage for the computer 100. Although the description of computer- readable media contained herein refers to a storage device, such as a hard disk or CD-ROM drive, it should be appreciated by those skilled in the art that computer-readable media can be any available media that can be accessed by the computer 100.

[0035] By way of example, and not to be limiting, computer-readable media may comprise computer storage media. Computer storage media includes volatile and non-volatile, removable and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EPROM, EEPROM, flash memory or other solid state memory technology, CD-ROM, DVD, or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by the computer.

[0036] According to various embodiments of the invention, the computer 100 may operate in a networked environment using logical connections to remote computers through a network 140, such as TCP / IP network such as the Internet or an intranet. The computer 100 may connect to the network 140 through a network interface unit 145 connected to the bus 135. It should be appreciated that the network interface unit 145 may also be utilized to connect to other types of networks and remote computer systems.Attorney Docket No.204606-0170-00WO 6-24009

[0037] The computer 100 may also include an input / output controller 155 for receiving and processing input from a number of input / output devices 160, including a keyboard, a mouse, a touchscreen, a camera, a microphone, a controller, a joystick, or other type of input device. Similarly, the input / output controller 155 may provide output to a display screen, a printer, a speaker, or other type of output device. The computer 100 can connect to the input / output device 160 via a wired connection including, but not limited to, fiber optic, Ethernet, or copper wire or wireless means including, but not limited to, Wi-Fi, Bluetooth, Near-Field Communication (NFC), infrared, or other suitable wired or wireless connections.

[0038] As mentioned briefly above, a number of program modules and data files may be stored in the storage device 120 and / or RAM 110 of the computer 100, including an operating system 125 suitable for controlling the operation of a networked computer. The storage device 120 and RAM 110 may also store one or more applications / programs 130. In particular, the storage device 120 and RAM 110 may store an application / program 130 for providing a variety of functionalities to a user. For instance, the application / program 130 may comprise many types of programs such as a word processing application, a spreadsheet application, a desktop publishing application, a database application, a gaming application, internet browsing application, electronic mail application, messaging application, and the like. According to an embodiment of the present invention, the application / program 130 comprises a multiple functionality software application for providing word processing functionality, slide presentation functionality, spreadsheet functionality, database functionality and the like.

[0039] The computer 100 in some embodiments can include a variety of sensors 165 for monitoring the environment surrounding and the environment internal to the computer 100. These sensors 165 can include a Global Positioning System (GPS) sensor, a photosensitive sensor, a gyroscope, a magnetometer, thermometer, a proximity sensor, an accelerometer, a microphone, biometric sensor, barometer, humidity sensor, radiation sensor, or any other suitable sensor.

[0040] Although the terms first, second, third, etc., may be used herein to describe various elements, components, regions, layers and / or sections, these elements, components, regions, layers and / or sections should not be limited by these terms. These terms may be only used toAttorney Docket No.204606-0170-00WO 6-24009 distinguish one element, component, region, layer or section from another element, component, region, layer or section. That is, terms such as “first,” “second,” and other numerical terms, when used herein, do not imply a sequence or order unless clearly indicated by the context. Thus, a first element, component, region, layer or section discussed below could be termed a second element, component, region, layer or section without departing from the teachings of the exemplary embodiments.

[0041] Relative terms such as “horizontal,” “vertical,” “up,” “down,” “top,” and “bottom” as well as derivatives thereof (e.g., “horizontally,” “downwardly,” “upwardly,” etc.) should be construed to refer to the orientation as then described or as shown in the drawing figure under discussion. These relative terms are for convenience of description and normally are not intended to require a particular orientation. Terms including “inwardly” versus “outwardly,” “longitudinal” versus “lateral” and the like are to be interpreted relative to one another or relative to an axis of elongation, or an axis or center of rotation, as appropriate. Terms concerning attachments, coupling and the like, such as “connected” and “interconnected,” refer to a relationship wherein structures are secured or attached to one another either directly or indirectly through intervening structures, as well as both movable or rigid attachments or relationships, unless expressly described otherwise. The term “operatively or operably connected” is such an attachment, coupling or connection that allows the pertinent structures to operate as intended by virtue of that relationship.

[0042] “About” as used herein when referring to a measurable value such as an amount, a temporal duration, and the like, is meant to encompass variations of ±20%, ±10%, ±5%, ±1%, and ±0.1% from the specified value, as such variations are appropriate.

[0043] Ranges: throughout this disclosure, various aspects of the invention can be presented in a range format. It should be understood that the description in range format is merely for convenience and brevity and should not be construed as an inflexible limitation on the scope of the invention. Where appropriate, the description of a range should be considered to have specifically disclosed all the possible subranges as well as individual numerical values within that range. For example, description of a range such as from 1 to 6 should be considered to have specifically disclosed subranges such as from 1 to 3, from 1 to 4, from 1 to 5, from 2 to 4, from 2Attorney Docket No.204606-0170-00WO 6-24009 to 6, from 3 to 6 etc., as well as individual numbers within that range, for example, 1, 2, 2.7, 3, 4, 5, 5.3, and 6. This applies regardless of the breadth of the range.

[0044] The figures and descriptions provided herein may have been simplified to illustrate aspects that are relevant for a clear understanding of the herein described apparatuses, systems, and methods, while eliminating, for the purpose of clarity, other aspects that may be found in typical similar devices, systems, and methods. Those of ordinary skill may thus recognize that other elements and / or operations may be desirable and / or necessary to implement the devices, systems, and methods described herein. But because such elements and operations are known in the art, and because they do not facilitate a better understanding of the present disclosure, for the sake of brevity a discussion of such elements and operations may not be provided herein. However, the present disclosure is deemed to nevertheless include all such elements, variations, and modifications to the described aspects that would be known to those of ordinary skill in the art.

[0045] Embodiments disclosed herein relate to methods and systems for imaging, for example medical imaging. In some embodiments, imaging devices and methods disclosed herein comprise one or more transmission systems for transmitting energy to be used in imaging. Examples of transmitted energy include, but are not limited to, radiofrequency energy, light, laser light, acoustic energy, ultrasound, X-ray energy, or combinations of these, or any other energy suitable for imaging. Energy may be transmitted continuously or as one or more pulses.

[0046] Imaging devices may further comprise a receiver or data acquisition system configured to receive all or a portion of the energy transmitted by the transmission system, either reflected off or transmitted through the imaging target. Examples of receivers include, but are not limited to, charge coupled devices, transducers, antennas, or the like. In some embodiments, the receiver comprises a radiofrequency data acquisition system. In some embodiments, a receiver may comprise multiple receive elements. In some embodiments, one or more receive elements may be configured or configurable to receive a different type of signal or different frequency of signal than one or more other receive elements. In some embodiments, the receiver may be configured to measure and record received energy in the time domain. In some embodiments, multiple receive elements may be arranged in a pattern, for example as a linear array, a two- dimensional array, a three-dimensional array, a ring, an oval, an opposing pair of linear or two-Attorney Docket No.204606-0170-00WO 6-24009 dimensional planar arrays in a pitch catch setup, a multi-row ring array, a hemispherical array, or any other suitable arrangement.

[0047] The present invention relates to a method of reconstruction of an object from projections using quantitative reconstruction discussed above of an object from projection views of the object. Software systems, arrangements, and processes for evaluating an image associated with at least one portion of an anatomical structure are provided.

[0048] The present invention may perform tissue mapping with imaging systems. Tissue mapping may be performed using simulation of quantitative methods (i.e., in silico) as discussed herein or may be performed on actual tissue. By way of example and without limitation, image analysis may be used in disease (e.g., cancer) detection or diagnosis techniques that assess volumes. Alternatively, imaging may be used on a simulated or actual human breast as shown in Figs.2 to 7. Using the quantitative methods discussed above, resulting images may be used to identify dense regions of the tissue such as connective breast tissue or may be used to identify less dense regions such as fat of the breast tissue. The present invention is not limited to its use in imaging breast tissue but encompasses imaging of other animal tissue and / or human tissue.

[0049] A scanning device can be configured to scan a volume of tissue and transmit signals to a server to store an image. The image may be electronically received and recorded by the server for display and may be used at a later point in time by users. Images may be standalone images or video. Images may be displayed on a user device by transmitting over wired or wireless network from the server to the user device. Images may be transmitted in whole or in part as a continuous stream. Images displayed on the user device may be evaluated by a healthcare professional and may receive a scoring criterion.

[0050] Imaging devices contemplated herein may further include or be communicatively connected to computing devices as described herein, the computing devices configured to control and / or receive imaging data from the imaging devices. In some embodiments, the computing device may be configured to execute instructions to perform one or more steps of one or more methods disclosed herein, for example image or signal processing to generate images from signals received from the receiver or data acquisition system.Attorney Docket No.204606-0170-00WO 6-24009

[0051] One challenge in generalizing waveform inversion to a wide variety of ultrasound imaging platforms is the fact that most ultrasound imaging systems do not transmit and receive signals at low enough frequencies to overcome cycle skipping. Frequency differencing is a mathematical trick where two high‐frequency signals can be used to synthesize low frequency content. This trick is currently used in sonar for direction‐of‐arrival estimation with sparse arrays, where it is known as frequency‐difference beamforming. Rather than apply this trick to beamforming, the systems and methods disclosed herein use the same frequency differencing method to overcome cycle skipping in waveform inversion. The implementation of frequency‐ difference waveform inversion requires a careful regularized division between signals at different frequencies to achieve robust results. Higher frequency data acquisition is typically needed for improved resolution in B-mode reflectivity imaging; however, this poses a challenge for waveform inversion techniques that typically rely on low frequency data to overcome cycle skipping. The present invention demonstrates a frequency-differencing method that overcomes the cycle skipping encountered in conventional frequency-domain waveform inversion. The parameters and variations of the basic frequency-differencing approach are disclosed herein.

[0052] Most clinical implementations of ultrasound tomography rely on frequency-domain FWI to quickly reconstruct images on clinically relevant time scales. Although the broadband nature of AWI makes it difficult to translate to the frequency domain, the properties of AWI are approximated by using a frequency-differencing approach.

[0053] Disclosed herein is a frequency-differencing method to overcome cycle skipping when using FWI on data collected at high frequencies, typical of high-frequency transducers for B-mode reflection ultrasound imaging. As used herein, “high-frequency” generally refers to any frequency beyond some upper limit, which results in cycle skipping artifacts. This upper limit is dependent on the exact imaging geometry. For example, if 1 MHz is this upper limit for an 11 cm diameter ring array, then the same upper limit would be 0.5 MHz for 22 cm diameter ring array because sound waves would travel twice the distance. This differencing approach produces a difference-frequency that is low enough to overcome the cycle skipping artifacts encountered in waveform inversion. In some embodiments, high frequency signals used in the disclosed systems and methods may be between 1 MHz and 2 MHz, or between 1.2 MHz and 1.7 MHz, orAttorney Docket No.204606-0170-00WO 6-24009 between 1.2 MHz and 2 MHz, or between 1.5 MHz and 2 MHz, or between 1 MHz and 1.5 MHz, or any other suitable range.

[0054] As contemplated herein, the difference between first and second high frequencies used to create the lower difference frequency is generally less than 500 kHz. In some embodiments, for example in the case of a 22 cm ring array geometry, suitable frequency differences may be in the range of 100 kHz – 1.25 MHz, or 100 kHz – 500 kHz, or 200 kHz – 500 kHz, or 200 kHz – 400 kHz, or 300 kHz -400 kHz. In different embodiments, exact frequency ranges may vary depending on the array configuration and geometry. Numerical experiments discussed below show that frequency differencing overcomes the cycle skipping encountered in FWI when using the same frequency content.

[0055] In some embodiments, during the acquisition of an image or set of images as contemplated herein, the same pair of high frequencies may be used during the entire image acquisition process. In some embodiments, different high frequencies may be used, while still maintaining a constant or roughly constant difference between the pair of high frequencies. In some embodiments, one of the pair of high frequencies may be held at a constant value while the other of the pair of high frequencies is varied, for example according to a linearly or monotonically increasing series of frequency differences.

[0056] As a starting point, and using finite difference methods (see Treeby, B. E., et al., Journal of biomedical optics, 2010), the 2D Helmholtz equation for FWI may be solved to obtain a pressure field ^ for a given source ^, frequency ^ = 2^^, and slowness ^ (i.e., the reciprocalof sound speed):^∇ଶ + ^ଶ^ଶ^^^^, ^^ = ^^^^Equation 1

[0057] After discretization, in some embodiments the ^∇ଶ+ ^ଶ^ଶ^ operator becomes a sparse matrix ^ that encodes the finite difference scheme so that Equation 1 can be described using the following matrix-vector formulation (where ^ and ^ are now vectors over spatial locations on the discretized grid):Attorney Docket No.204606-0170-00WO 6-24009 ^^^, ^^^^^, ^^ = ^^^^

[0058] Differentiating both sides of this equation with respect to ^ gives: ^^ି^ ^^ ି ^^= −^ ^ ି^^^ ^^^ = −^^^^ ^Equation 3

[0059] Rather than observing the full pressure field ^ at all locations in space, the pressure field ^ may be observed at the ring array. The relationship between ^ and ^ may be: ^ = ^^, where ^ is a sampling operator that extracts the pressure field at each element of the ring array. The relationship between changes in slowness and changes in the simulated pressure field at the ring array is ^^ି^ ^^ ି ^^= −^^ ^ = −^^ ^ ^ି^^^ ^^ ^^^

[0060] The cost function for full waveform inversion (FWI) is: ே ^^1^^ − ଶ

[0061] where ^^and ^^^^,^may be the simulated and observed pressure measurements on the ring array for the ^௧^single-element transmission ^^(^ ranges from 1 to ^, the number of elements on the ring array). The gradient of ^^^, ^^ with respect to slowness ^ is given by: ே^^ ு ே ு^ ^^Attorney Docket No.204606-0170-00WO 6-24009

[0062] Referring now to complex value source scaling, the waveform inversion described in the above assumes that the correct phase and amplitude may be used for each single-element source ^^; however, in practice, the phase and amplitude of each source can be estimated from the measured waveform. In one embodiment, the following projection formula may provide a scaling ^^that accounts for both the amplitude and phase from each source ^ (e.g., complex-valued source scaling): ^ =^^^ ^^^^,^^^^^ ^^Equation 7

[0063] The source vector and pressure fields may be scaled to {^^ → ^^^^}, and {^^ → ^^^^},{^^ → ^^^^}, respectively.

[0064] In one embodiment, which may be referred to as model domain frequency differencing, frequency-differencing may involve making the following ratios of the simulated and measured signals at frequencies ^^= 2^^^and ^ଶ= 2^^^ന (^ , )^^(^ଶ, ^)^^ ^ଶ, ^ =

[0065] In one embodiment, the cost function for frequency difference waveform inversion (FDWI) is 1ேฮଶଶAttorney Docket No.204606-0170-00WO 6-24009

[0066] However, one difficulty with the ratios defined in Equation 8 and Equation 9 is division by a number close to zero, so the following regularized division may be used in some embodiments with regularization ^ଶ: (∗( )^^ ^ଶ, ^) ∘ ^^(^^, ^)^ന^^^,^ଶ, ^ = ^^(^^, ^) ∘ ^∗^(^^, ^) + ^ଶ

[0067] where ∘ refers to element-wise multiplication and the division is also element wise.

[0068] The gradient of ^(^^,^ଶ, ^)with respect to slowness ^ is given by: ே^^ന ு∇^^ = ^^^ ^^^^ ൫^ന^ −^)

[0069] In another embodiment, which may be referred to as data domain frequency differencing, ^^= ^ and ^ଶ= ^ + Δ^, ^ന^^^,^may be re-expressed in terms of ^ and Δ^ as ^^^^,^(^ + Δ^) ∘ ^∗^^^,^ (^)ଶAttorney Docket No.204606-0170-00WO 6-24009

[0070] However, it is preferred that the dependence of ^ന^^^,^on ^ may be eliminated so that it solely depends on the frequency difference Δ^. This is achieved by integrating the numerator and denominator over ^ from ^^^௪to ^^^^^. ^ఠ^^^^ఠ^ ^^^^,^(^ + Δ^) ∘ ^∗^^^,^ (^) ^^^ന ,^( )^^^^^ Δ^ =^ଶ ^ఠ^^^^ ^ ^∗ ^^

[0071] At this point, the standard FWI algorithm may be used with the following objective function: ^ Δ^, ^) =1 ே( ^ฮ^^(Δ^, ^) − ^ന (Δ^)ฮଶଶ

[0072] The objective function in Equation 17 is identical to the original FWI objectivefunction in Equation 5, except that Δ^ replaces ^ and ^ന^^^,^(Δ^)replaces ^^^^,^(^). In otherwords, a single Helmholtz equation at frequency Δ^ is used to simulate ^^(Δ^, ^); however,rather than compare ^^(Δ^, ^)to actual content ^^^^(Δ^)at frequency Δ^ (which may not bepresent with sufficient SNR in the recorded signal), instead ^^(Δ^, ^)is compared to ^ന^^^,^(Δ^)resulting from Equation 16. The process that generates ^ന^^^,^(Δ^)from existing data is like how frequency-difference beamforming (H. Abadi, et al., The Journal of the Acoustical Society of America, 2012. ) overcomes the same lack of low-frequency content in existing signals.

[0073] As opposed to above, where the FWI was modified to precisely model the ratio of data at two different frequencies, here the focus is on generating the data so that it can fit the existing FWI model. Of course, the process of generating ^ന^^^,^(Δ^)via Equation 16 would not match the actual physics (i.e., diffraction) at frequency Δ^, thereby introducing systematic discrepancies with the simulated results of the Helmholtz equation. Despite this limitation, it is believed that data-domain frequency differencing may provide a more accurate starting model than ray tomography for initiating conventional FWI.Attorney Docket No.204606-0170-00WO 6-24009

[0074] Frequency-domain FWI is typically performed from the lowest possible frequency to the highest possible frequency to avoid cycle skipping while also reconstructing images at high resolution, where the lowest possible and highest possible frequencies are dependent on the imaging geometry. FWI may be limited in its ability to avoid cycle skipping due to the absence of low frequency data, ultimately requiring the use of more accurate starting models to overcome cycle skipping. In one aspect of the invention, frequency differencing may be considered as a starting model for FWI so that FWI can be started at a higher frequency. Model-domain and data-domain frequency differencing may therefore be used to initialize FWI. The result of each frequency differencing scheme, before and after FWI, is compared to FWI starting from a homogeneous starting model. To compare cycle skipping in each case, FWI is started at the same frequency discussed below.

[0075] The goal of data-domain frequency-differencing is to synthesize low-frequency information that is not present in the data. The process described by Equation 16 introduces systematic error because it cannot accurately reproduce true content at this frequency. This systematic error limits the imaging resolution of data-domain differencing in the same way that the inaccuracy of time-of-flight picks limits the resolution of ray tomography. However, large- scale features are generally reconstructed accurately.

[0076] The goal of model-domain frequency differencing is to take two existing frequencies in the data, which would be subject to cycle skipping if FWI were applied to each frequency in isolation, and model their ratio to overcome cycle skipping. Although it may appear that accurately modeling this ratio would lead to a more accurate inversion that overcomes cycle skipping, the results shown in Fig.6 and Fig.7 would indicate that this model may be poorly conditioned for inversion, especially in low signal-to-noise-ratio (SNR) conditions. One potential reason for this ill-conditioning is alluded by prior work on AWI (see Warner, M., et al., Geophysics, 2016) which stipulates that AWI is fundamentally a finite-bandwidth concept that is unlikely to work on sparse-frequency data. It may be that more than a single ratio of two frequencies is necessary to overcome the issue of ill-conditioning. This raises the question of how many frequencies would need to be modeled without substantially increasing computational cost.Attorney Docket No.204606-0170-00WO 6-24009

[0077] Although preliminary results show that the application of FWI to frequency- difference (FD) data generated by Equation 16 can overcome cycle skipping, the spatial resolution and accuracy of this approach may be limited by the mismatch between the simulation model ^^(Δ^, ^) and FD data ^ന^^^,^(Δ^). To overcome this model mismatch, in another embodiment, which may be referred to as an addition to the model domain approach with an integrated frequency difference full-waveform inversion, the simulation model ^^(Δ^, ^)may be substituted with a new model ^ന^(Δ^, ^) that accurately captures the entire FD computation with the physics at each frequency used to generate the FD data: ே ^(Δ^, ^) =1(Δ^, ^) − ^ന^^^,^(Δ^)ଶ

[0078] This approach may be referred to as integrated frequency-difference full-waveform inversion (IFD-FWI). IFD-FWI shows promise because (1) the preliminary data shows that the FD process produces the low-frequency signal needed to overcome cycle skipping, and (2) explicitly modeling the entire FD computation recovers the resolution and accuracy lost by applying the unmodified FWI model to the FD data.

[0079] In the above embodiment, IFD-FWI brings the model closer to FD data. The following embodiment, which may be referred to as an addition to the data-domain approach as low frequency extraposition based on deep learning, considers using deep learning (DL) to augment the FD data so that it accurately captures the higher-order physics (i.e., diffraction), thereby bringing the data closer to the model used in conventional FWI:Attorney Docket No.204606-0170-00WO 6-24009 ^(Δ^, ^)=1 ே2^ฮ^^(Δ^, ^)− ^^^^^^,^(Δ^)ฮଶଶ

[0080] This objective function replaces ^ന^^^,^(Δ^)in Equation 17 with ^^^^^^,^(Δ^), the DL-augmented FD data. In the DL approach, a convolutional neural thetransformation between FD data (and its precursors in the FD low- frequency data needed for accurate FWI. EXPERIMENTAL EXAMPLES

[0081] The invention is now described with reference to the following Examples. These Examples are provided for the purpose of illustration only and the invention should in no way be construed as being limited to these Examples, but rather should be construed to encompass any and all variations which become evident as a result of the teaching provided herein.

[0082] Without further description, it is believed that one of ordinary skill in the art can, using the preceding description and the following illustrative examples, make and utilize the present invention and practice the claimed methods. The following working examples therefore, specifically point out exemplary embodiments of the present invention, and are not to be construed as limiting in any way the remainder of the disclosure.

[0083] In-silico studies show that frequency differencing overcomes cycle skipping when applied at the same frequencies that cause cycle skipping in waveform inversion. Simulations show that that this frequency differencing approach overcomes cycle skipping and reduces the root-mean-square error in sound speed estimation by 13.7 m / s.

[0084] The k-Wave simulation toolbox (see Treeby, B. E., et al., Journal of biomedical optics, 2010) was used to simulate the received channel data from an 11 cm diameter ring array with a 2.5 MHz center frequency transmit pulse from each element (256 elements total). Noise was added to the simulated data. A discrete-time Fourier transform (DTFT) was used to isolate the measured signal at each frequency to apply waveform inversion using the HelmholtzAttorney Docket No.204606-0170-00WO 6-24009 equation. For the simulated data, each reconstruction began with a 1515 m / s homogeneous model. The model-domain differencing uses several frequency pairs whose values range from 1.2 to 1.7 MHz with frequency differences ranging from 0.3 to 0.4 MHz. Data-domain differencing was used to construct frequency differences ranging from 0.3 to 1.25 MHz. For each choice of starting model (model-domain differencing, data-domain differencing, or homogeneous model), conventional FWI starts at 1.2 MHz and ends at 1.7 MHz.

[0085] Referring now to Figs.2 to 7, shown are demonstrations of Frequency Differencing as a Starting Model for Conventional Frequency-Domain FWI in a Breast Imaging Simulation. In all reconstructions, no frequency content below 1.2 MHz was used; however, each frequency differencing method was used to effectively synthesize frequency differences well below 1.2 MHz to overcome cycle skipping. Both data-domain and model-domain frequency differencing provide sufficiently accurate starting models for FWI to overcome the cycle skipping seen in FWI when starting from a homogeneous starting model.

[0086] Also shown is the reconstruction of sound speed in a numerical breast phantom (true sound speed shown) based on data-domain and model-domain frequency differencing. When conventional frequency-domain FWI is started at 1.2 MHz from a homogeneous starting model, cycle skipping corrupts the entire image. When either frequency differencing method is used as the starting model for FWI, cycle skipping can be avoided even if FWI starts as high as 1.2 MHz. Qualitatively, data-domain and model-domain frequency differencing produce different results. Data-domain frequency differencing (see Fig.4) appears to reconstruct a low-resolution but quantitative reconstruction of sound speed, reminiscent of ray tomography. On the other hand, model-domain frequency differencing (see Fig.6) appears to reconstruct high-resolution features of the image while struggling to converge on exact quantitative values at all points in the image. REFERENCES

[0087] Ali, R., Mitcham, T., & Duric, N. (2023, April). Impact of starting model on waveform inversion in ultrasound tomography. In Medical Imaging 2023: Ultrasonic Imaging and Tomography (Vol.12470, pp.89-98). SPIE.Attorney Docket No.204606-0170-00WO 6-24009

[0088] Warner, M., & Guasch, L. (2016). Adaptive waveform inversion: Theory. Geophysics, 81(6), R429-R445.

[0089] Sandhu, G. Y., Li, C., Roy, O., Schmidt, S., & Duric, N. (2015). Frequency domain ultrasound waveform tomography: breast imaging using a ring transducer. Physics in Medicine & Biology, 60(14), 5381.

[0090] Wiskin, J., Borup, D. T., Johnson, S. A., & Berggren, M. (2012). Non-linear inverse scattering: High resolution quantitative breast tissue tomography. The Journal of the Acoustical Society of America, 131(5), 3802-3813.

[0091] H. Abadi, S., Song, H., & R. Dowling, D. (2012). Frequency-difference beamforming with sparse arrays. The Journal of the Acoustical Society of America, 132(3_Supplement), 2082- 2082.

[0092] Treeby, B. E., & Cox, B. T. (2010). K-Wave: MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields. Journal of biomedical optics, 15(2), 021314.

[0093] The disclosures of each and every patent, patent application, and publication cited herein are hereby incorporated herein by reference in their entirety. While this invention has been disclosed with reference to specific embodiments, it is apparent that other embodiments and variations of this invention may be devised by others skilled in the art without departing from the true spirit and scope of the invention.

Claims

Attorney Docket No.204606-0170-00WO 6-24009 CLAIMS What is claimed is:

1. A method for generating an image, comprising: providing an imaging device comprising a multi-channel radiofrequency data acquisition system comprising a plurality of receive elements and a transmission system; transmitting a signal pulse including content at first and second high frequencies from the transmission system; recording a time-domain channel waveform from each receive element in the multi- channel radiofrequency data acquisition system; calculating a set of frequency-difference signals for each receive element based on the first and second high frequencies; and applying a frequency-difference waveform inversion (FDWI) to the set of frequency difference signals to generate an image.

2. The method of claim 1 or 2, further comprising calculating a set of phasors from each time domain channel waveform at the first and second high frequencies, wherein the phasors are calculated using a discrete-time Fourier transform.

3. The method of claim 1 or 2, wherein the imaging device comprises a ring array.

4. The method of claim 1 or 2, wherein the imaging device comprises at least one linear array.

5. The method of claim 1 or 2, wherein the imaging device comprises at least one ultrasound transducer.

6. The method of any one of claims 1-5, wherein the step of applying a FDWI to the set of frequency difference signals further comprises calculating an objective function associated with a difference between the first and second high frequencies.Attorney Docket No.204606-0170-00WO 6-24009 7. The method of claim 6, wherein the objective function includes calculation of a simulation model of tissue mapping.

8. The method of claim 6, wherein the objective function comprises calculating a Helmholtz equation at the difference (Δ^) between the first and second high frequencies.

9. The method of any one of claims 1-8, further comprising the step of augmenting the set of frequency difference signals with a convolutional neural network to bring the frequency difference signals closer to true low-frequency data to capture the physics of diffraction.

10. An imaging system, comprising: a multi-channel radiofrequency data acquisition system comprising a plurality of receive elements; a transmission system configured to generate pulses at first and second high frequencies; a computing device communicatively connected to the transmission system and the multi-channel radiofrequency data acquisition system, said computing device comprising a processor and a non-transitory computer-readable medium with instructions stored thereon, which when executed by the processor, performs steps comprising: transmitting a signal pulse including content at the first and second high frequencies from the transmission system; recording a time-domain channel waveform from each receive element in the multi-channel radiofrequency data acquisition system; calculating a set of frequency-difference signals for each receive element based on the first and second high frequencies; and applying a frequency-difference waveform inversion (FDWI) to the set of frequency difference signals to generate an image.

11. The imaging system of claim 10, wherein the multi-channel radiofrequency data acquisition system comprises at least one ultrasound transducer.Attorney Docket No.204606-0170-00WO 6-24009 12. The imaging system of claim 11, wherein the at least one ultrasound transducer comprises at least one B-mode reflectivity imaging transducer.

13. The imaging system of claim 10, wherein the multi-channel radiofrequency data acquisition system comprises a ring array of receive elements.

14. The imaging system of claim 13, wherein the ring array comprises 256 elements.

15. The imaging system of any one of claims 10-14, wherein the imaging device comprises at least one linear array of receive elements.

16. The imaging system of any one of claims 10-15, wherein the first and second high frequencies are between 1.2 MHz and 1.7 MHz.

17. The imaging system of claim 16, wherein a difference between the first and second high frequencies is between 300 MHz and 400 MHz.

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