Green's function estimation in speckle using the decomposition of the time reversal operator: application to aberration correction in medical imaging

Three-dimensional ultrasound matrix imaging

Matrix imaging paves the way towards a next revolution in wave physics. Based on the response matrix recorded between a set of sensors, it enables an optimized compensation of aberration phenomena and multiple scattering events that usually drastically hinder the focusing process in heterogeneous media. Although it gave rise to spectacular results in optical microscopy or seismic imaging, the success of matrix imaging has been so far relatively limited with ultrasonic waves because wave control is generally only performed with a linear array of transducers. In this paper, we extend ultrasound matrix imaging to a 3D geometry. Switching from a 1D to a 2D probe enables a much sharper estimation of the transmission matrix that links each transducer and each medium voxel. Here, we first present an experimental proof of concept on a tissue-mimicking phantom through ex-vivo tissues and then, show the potential of 3D matrix imaging for transcranial applications.

Deep learning for hetero-homo conversion in channel-domain for phase aberration correction in ultrasound imaging

Echo imaging in ultrasound computed tomography (USCT) using the synthetic aperture technique is performed with the assumption that the speed of sound is constant in the system. However, tissue heterogeneity causes a mismatch between the predicted arrival time and the actual arrival time of the echo signal, which will result in phase aberration, leading to the quality degradation of the reconstructed B-mode image. The conventional correction methods that use the correlation of each different channel require the presence of strong point scatterers and involve the problem of local solutions due to excessive correction. In this study, we propose a novel approach to correcting the signal distortion due to sound speed heterogeneity using a deep neural network (DNN). The DNN was trained to convert the distorted radio frequency (RF) inputs for the heterogeneous medium to the distortion-free RF outputs for the homogeneous medium. The network with U-net architecture using ResNet-34 as a backbone was trained using the hetero-homo corresponding channel-domain RF data generated via numerical simulations. The trained network performed phase aberration correction in the channel-domain RF, with the B-mode images reconstructed with the corrected RF demonstrating a higher contrast and an improved resolution compared with uncorrected cases. It was also demonstrated that the DNN model is robust to both varied reflection intensities and varied sound speed heterogeneities. The successful results demonstrated that the proposed DNN-based method is effective for phase aberration correction in US imaging.

Ultrasound Matrix Imaging-Part II: The Distortion Matrix for Aberration Correction Over Multiple Isoplanatic Patches

This is the second article in a series of two which report on a matrix approach for ultrasound imaging in heterogeneous media. This article describes the quantification and correction of aberration, i.e. the distortion of an image caused by spatial variations in the medium speed-of-sound. Adaptive focusing can compensate for aberration, but is only effective over a restricted area called the isoplanatic patch. Here, we use an experimentally-recorded matrix of reflected acoustic signals to synthesize a set of virtual transducers. We then examine wave propagation between these virtual transducers and an arbitrary correction plane. Such wave-fronts consist of two components: (i) An ideal geometric wave-front linked to diffraction and the input focusing point, and; (ii) Phase distortions induced by the speed-of-sound variations. These distortions are stored in a so-called distortion matrix, the singular value decomposition of which gives access to an optimized focusing law at any point. We show that, by decoupling the aberrations undergone by the outgoing and incoming waves and applying an iterative strategy, compensation for even high-order and spatially-distributed aberrations can be achieved. After a numerical validation of the process, ultrasound matrix imaging (UMI) is applied to the in-vivo imaging of a gallbladder. A map of isoplanatic modes is retrieved and is shown to be strongly correlated with the arrangement of tissues constituting the medium. The corresponding focusing laws yield an ultrasound image with drastically improved contrast and transverse resolution. UMI thus provides a flexible and powerful route towards computational ultrasound.

Ultrasound Matrix Imaging-Part I: The Focused Reflection Matrix, the F-Factor and the Role of Multiple Scattering

This is the first article in a series of two dealing with a matrix approach for aberration quantification and correction in ultrasound imaging. Advanced synthetic beamforming relies on a double focusing operation at transmission and reception on each point of the medium. Ultrasound matrix imaging (UMI) consists in decoupling the location of these transmitted and received focal spots. The response between those virtual transducers form the so-called focused reflection matrix that actually contains much more information than a confocal ultrasound image. In this paper, a time-frequency analysis of this matrix is performed, which highlights the single and multiple scattering contributions as well as the impact of aberrations in the monochromatic and broadband regimes. Interestingly, this analysis enables the measurement of the incoherent input-output point spread function at any pixel of this image. A fitting process enables the quantification of the single scattering, multiple scattering and noise components in the image. From the single scattering contribution, a focusing criterion is defined, and its evolution used to quantify the amount of aberration throughout the ultrasound image. In contrast to the state-of-the-art coherence factor, this new indicator is robust to multiple scattering and electronic noise, thereby providing a contrasted map of the focusing quality at a much better transverse resolution. After a validation of the proof-of-concept based on time-domain simulations, UMI is applied to the in-vivo study of a human calf. Beyond this specific example, UMI opens a new route for speed-of-sound and scattering quantification in ultrasound imaging.

Full-Matrix Phase Shift Migration Method for Transcranial Ultrasonic Imaging

A spectrum-domain method, called full-matrix phase shift migration (FM-PSM), is presented for transcranial ultrasound phase correction and imaging with ideal synthetic aperture focusing technology. The simulated data obtained using the pseudospectral time-domain method are used to evaluate the feasibility of the method. The experimental data measured from a 3-D printed skull phantom are used to evaluate the algorithm performance in terms of resolution, contrast-to-noise ratio (CNR), and eccentricity comparing with the classical ray-tracing delay and sum (DAS) method. In wire imaging experiment, FM-PSM has a lateral resolution of 0.22 mm and ray-tracing DAS has a lateral resolution of 0.24 mm measured at -6-dB drop using a transducer with a center frequency of 6.25 MHz. In cylinder imaging experiment, FM-PSM has a CNR of 2.14 and ray-tracing DAS has a CNR of 1.82, which illustrates about 17% improvement. For a J -element array and an output image with pixels M ×N (lateral × axial), the computational cost of the DAS is of O(J ×M²×N²) ; on the contrary, it reduces to O(J ×M ×N²) with the proposed FM-PSM. The results suggest that FM-PSM is an efficiency method for transcranial ultrasonic imaging.

The SVD Beamformer: Physical Principles and Application to Ultrafast Adaptive Ultrasound

A shift of paradigm is currently underway in biomedical ultrasound thanks to plane or diverging waves coherent compounding for faster imaging. One remaining challenge consists in handling phase and amplitude aberrations induced during the ultrasonic propagation through complex layers. Unlike conventional line-per-line imaging, ultrafast ultrasound provides backscattering information from the whole imaged area for each transmission. Here, we take benefit from this feature and propose an efficient approach to perform fast aberration correction. Our method is based on the Singular Value Decomposition (SVD) of an ultrafast compound matrix containing backscattered data for several plane wave transmissions. First, we explain the physical signification of SVD and associated singular vectors within the ultrafast matrix formalism. We theoretically demonstrate that the separation of spatial and angular variables, rendered by SVD on ultrafast data, provides an elegant and straightforward way to optimize the angular coherence of backscattered data. In heterogeneous media, we demonstrate that the first spatial and angular singular vectors retrieve respectively the non-aberrated image of a region of interest, and the phase and amplitude of its aberration law. Numerical, in vitro and in vivo results prove the efficiency of the image correction, but also the accuracy of the aberrator determination. Based on spatial and angular coherence, we introduce a complete methodology for adaptive beamforming of ultrafast data, performed on successive isoplanatism patches undergoing SVD beamforming. The simplicity of this method paves the way to real-time adaptive ultrafast ultrasound imaging and provides a theoretical framework for future quantitative ultrasound applications.

Distortion matrix concept for deep optical imaging in scattering media

In optical imaging, light propagation is affected by the inhomogeneities of the medium. Sample-induced aberrations and multiple scattering can strongly degrade the image resolution and contrast. On the basis of a dynamic correction of the incident and/or reflected wavefronts, adaptive optics has been used to compensate for those aberrations. However, it only applies to spatially invariant aberrations or to thin aberrating layers. Here, we propose a global and noninvasive approach based on the distortion matrix concept. This matrix basically connects any focusing point of the image with the distorted part of its wavefront in reflection. A singular value decomposition of the distortion matrix allows to correct for high-order aberrations and forward multiple scattering over multiple isoplanatic modes. Proof-of-concept experiments are performed through biological tissues including a turbid cornea. We demonstrate a Strehl ratio enhancement up to 2500 and recover a diffraction-limited resolution until a depth of 10 scattering mean free paths.

Distortion matrix approach for ultrasound imaging of random scattering media

Focusing waves inside inhomogeneous media is a fundamental problem for imaging. Spatial variations of wave velocity can strongly distort propagating wave fronts and degrade image quality. Adaptive focusing can compensate for such aberration but is only effective over a restricted field of view. Here, we introduce a full-field approach to wave imaging based on the concept of the distortion matrix. This operator essentially connects any focal point inside the medium with the distortion that a wave front, emitted from that point, experiences due to heterogeneities. A time-reversal analysis of the distortion matrix enables the estimation of the transmission matrix that links each sensor and image voxel. Phase aberrations can then be unscrambled for any point, providing a full-field image of the medium with diffraction-limited resolution. Importantly, this process is particularly efficient in random scattering media, where traditional approaches such as adaptive focusing fail. Here, we first present an experimental proof of concept on a tissue-mimicking phantom and then, apply the method to in vivo imaging of human soft tissues. While introduced here in the context of acoustics, this approach can also be extended to optical microscopy, radar, or seismic imaging.

Reconstruction of Photoacoustic Tomography Inside a Scattering Layer Using a Matrix Filtering Method

Photoacoustic (PA) tomography (PAT) has potential for use in brain imaging due to its rich optical contrast, high acoustic resolution in deep tissue, and good biosafety. However, the skull often poses challenges for transcranial brain imaging. The skull can cause severe distortion and attenuation of the phase and amplitude of PA waves, which leads to poor resolution, low contrast, and strong noise in the images. In this study, we propose an image reconstruction method to recover the PA image insider a skull-like scattering layer. This method reduces the scattering artifacts by combining a correlation matrix filter and a time reversal operator. Both numerical simulations and PA imaging experiments demonstrate that the proposed approach effectively improves the image quality with less speckle noise and better signal-to-noise ratio. The proposed method may improve the quality of PAT in a complex acoustic scattering environment, such as transcranial brain imaging.

Imaging acoustic sources through scattering media by using a correlation full-matrix filter

In the inhomogeneous medium, acoustic scattering is always a fundamental challenge for photoacoustic imaging. We implement a correlation full-matrix filter (CFMF) combing with a time reversal operator to improve the imaging quality of acoustic sources in complex media. The correlation full-matrix filtering process extracts the direct wave component from the detected signal and preserve all the useful information at the same time. A location factor is considered in the time reversal operator to compensate for the image distortion and false contrast caused by the limited-view detection. The numerical simulations demonstrate that the proposed approach can perform good imaging quality with the higher image signal-noise ratio and better resolution in an acoustic scattering environment. This scheme might be applied to improve the photoacoustic imaging for inhomogeneous biological tissues.

Photoacoustic imaging in scattering media by combining a correlation matrix filter with a time reversal operator

Acoustic scattering medium is a fundamental challenge for photoacoustic imaging. In this study, we reveal the different coherent properties of the scattering photoacoustic waves and the direct photoacoustic waves in a matrix form. Direct waves show a particular coherence on the antidiagonals of the matrix, whereas scattering waves do not. Based on this property, a correlation matrix filter combining with a time reversal operator is proposed to preserve the direct waves and recover the image behind a scattering layer. Both numerical simulations and photoacoustic imaging experiments demonstrate that the proposed approach effectively increases the image contrast and decreases the background speckles in a scattering medium. This study might improve the quality of photoacoustic imaging in an acoustic scattering environment and extend its applications.

En face coherence microscopy [Invited]

En face coherence microscopy or flying spot or full field optical coherence tomography or microscopy (FF-OCT/FF-OCM) belongs to the OCT family because the sectioning ability is mostly linked to the source coherence length. In this article we will focus our attention on the advantages and the drawbacks of the following approaches: en face versus B scan tomography in terms of resolution, coherent versus incoherent illumination and influence of aberrations, and scanning versus full field imaging. We then show some examples to illustrate the diverse applications of en face coherent microscopy and show that endogenous or exogenous contrasts can add valuable information to the standard morphological image. To conclude we discuss a few domains that appear promising for future development of en face coherence microscopy.

Smart optical coherence tomography for ultra-deep imaging through highly scattering media

Multiple scattering of waves in disordered media is a nightmare whether it is for detection or imaging purposes. So far, the best approach to get rid of multiple scattering is optical coherence tomography. This basically combines confocal microscopy and coherence time gating to discriminate ballistic photons from a predominant multiple scattering background. Nevertheless, the imaging-depth range remains limited to 1 mm at best in human soft tissues because of aberrations and multiple scattering. We propose a matrix approach of optical imaging to push back this fundamental limit. By combining a matrix discrimination of ballistic waves and iterative time reversal, we show, both theoretically and experimentally, an extension of the imaging-depth limit by at least a factor of 2 compared to optical coherence tomography. In particular, the reported experiment demonstrates imaging through a strongly scattering layer from which only 1 reflected photon out of 1000 billion is ballistic. This approach opens a new route toward ultra-deep tissue imaging.

Time reversal transfer: Exploring the robustness of time reversed acoustics in media with geometry perturbations

In this letter, fundamentals of transferring a time reversal experiment between similar objects are discussed. The time reversal experiment consists of two steps: forward propagation, when a source excites the medium and a complex wave field is created, and back propagation, resulting in time reversal focusing. Here the procedure of performing the first step on one specimen and the second step on another is investigated. The theory of time reversal transfer is explained on an example of object shape variations. However, conclusions of the theoretical analysis are applicable universally. The feasibility of the proposed procedure is validated in experiments modeling conditions in practice.

Photoacoustic tomography extracted from speckle noise in acoustically inhomogeneous tissue

Photoacoustic tomography is usually limited to acoustically homogeneous tissue. A hybrid scheme is developed to break this limitation by utilizing ultrasound to determine the unknown Green's function of inhomogeneous tissue. The method can effectively decrease the distortion and false contrast in images by extracting information from speckle noise. The method does not depend on the prior knowledge of tissue and the medium complexity. Moreover, the estimation of Green's function and the photoacoustic detection are performed by the same transducer. Therefore, the scheme could be easily integrated into a classical photoacoustic tomography system and extend its application in speckle environment.

Timing-error-difference calibration of a two-dimensional array imaging system using the overlapping-subaperture algorithm

Timing errors in the transmitting and receiving electronic channels of an imaging system can generate different transmission and reception phase-aberration profiles. To decide if these two profiles need to be measured separately, an overlapping-subaperture algorithm has been proposed in a previous paper to measure the difference between timing errors in transmitting and receiving channels connected to each element in a two-dimensional array. This algorithm has been used to calibrate a custom built imaging system with a curved linear two-dimensional array, and the results are presented in this paper. The experimental results have demonstrated that the overlapping-subaperture algorithm is capable of calibrating the timing-error-difference profile of this imaging system with a standard deviation of only a few nanoseconds. Experimental results have also shown that the time-error-difference profile of this imaging system is smaller than one tenth of a wavelength and there is no need to measure the transmission and reception phase-aberration profiles separately. The derived average phase-aberration profile using the near-field signal-redundancy algorithm can be used to correct phase aberrations for both transmission and reception.

Correction of tissue-motion effects on common-midpoint signals using reciprocal signals

The near field signal redundancy algorithm for phase-aberration correction is sensitive to tissue motion because several separated transmissions are usually needed to acquire a set of common-midpoint signals. If tissues are moving significantly due to, for example, heart beats, the effects of tissue motion on common-midpoint signals need to be corrected before the phase-aberration profile can be successfully measured. Theoretical analyses in this paper show that the arrival-time difference between a pair of common-midpoint signals due to tissue motion is usually very similar to that between the pair of reciprocal signals acquired using the same two transmissions. Based on this conclusion, an algorithm for correcting tissue-motion effects on the peak position of cross-correlation functions between common-midpoint signals is proposed and initial experimental results are also presented.

Aberration correction by time reversal of moving speckle noise

Focusing a wave through heterogeneous media is an important problem in medical ultrasound imaging. In such aberrating media, in the presence of a small number of point reflectors, iterative time reversal is a well-known method able to focus on the strongest reflector. However, in presence of speckle noise generated by many non-resolved scatterers, iterative time reversal alone does not work. In this paper, we propose the use of the echoes coming from moving particles in a flow, such as red blood cells, to generate a virtual point reflector by iterative time reversal. The construction of the virtual point reflector is performed by a coherent addition of independent realizations of speckle coming from moving particles. After focusing on a virtual point reflector, ultrasound images can be locally corrected inside an isoplanatic patch. An application for the correction of power Doppler images is presented. A theoretical analysis shows that this iterative method allows focusing on the point of maximal insonification of the uncorrected beam.

Time reversal of speckle noise

Focusing a wave in an unknown inhomogeneous medium is an open problem in wave physics. This work presents an iterative method able to focus in pulse-echo mode in an inhomogeneous medium containing a random distribution of scatterers. By performing a coherent summation of the random echoes backscattered from a set of points surrounding the desired focus, a virtual bright pointlike reflector is generated. A time-reversal method enables an iterative convergence towards the optimal wave field focusing at the location of this virtual scatterer. Thanks to this iterative time-reversal process, it is possible to focus at any arbitrary point in the heterogeneous medium even in the absence of pointlike source. An experimental demonstration is given for the correction of strongly distorted images in the field of medical ultrasound imaging. This concept enables envisioning many other applications in wave physics.

Harmonic source wavefront aberration correction for ultrasound imaging

A method is proposed which uses a lower-frequency transmit to create a known harmonic acoustical source in tissue suitable for wavefront correction without a priori assumptions of the target or requiring a transponder. The measurement and imaging steps of this method were implemented on the Duke phased array system with a two-dimensional (2-D) array. The method was tested with multiple electronic aberrators [0.39π to 1.16π radians root-mean-square (rms) at 4.17 MHz] and with a physical aberrator 0.17π radians rms at 4.17 MHz) in a variety of imaging situations. Corrections were quantified in terms of peak beam amplitude compared to the unaberrated case, with restoration between 0.6 and 36.6 dB of peak amplitude with a single correction. Standard phantom images before and after correction were obtained and showed both visible improvement and 14 dB contrast improvement after correction. This method, when combined with previous phase correction methods, may be an important step that leads to improved clinical images.

Spatio-temporal invariants of the time reversal operator

The decomposition of the time reversal operator, known by the French acronym DORT, is a technique to extract point scatterers' monochromatic Green's functions from a medium. It is used to detect, locate, and focus on scatterers in various domains such as underwater acoustics, medical ultrasound, and nondestructive evaluation. A limitation of the method arises from its single-frequency nature, when the signals used in acoustics are often broadband. Reconstruction of the broadband Green's functions from the single-frequency Green's functions can be very difficult when numerous scatterers are present in the medium. Moreover, the method does not take advantage of the axial resolution associated with broadband signals. Time domain methods are investigated here as an answer to these problems. It is shown that the time reversal operator in the time domain takes the form of a tensor. The properties of the invariants are discussed. It is shown they do not have all the expected properties. Another method is proposed that requires a priori information on the medium.

Synchronized time-reversal focusing with application to remote imaging from a distant virtual source array

Time-reversing the transfer function between a time-reversal mirror (TRM) and a distant probe source location generates an acoustic spatio-temporal focus at the location. It is shown that a TR focus behaves as a "virtual" source (in the far-field limit) in the outbound direction with respect to the TRM. By extension, a collection of TRM-to-probe source transfer functions constitutes a virtual source array (VSA) that can serve as a remote platform for active imaging methods such as beam-steering and other coherent wavefront techniques. As a demonstration, a set of a-priori sampled TRM-to-VSA transfer functions are steered to coherently focus at a selected location beyond the VSA for which the transfer function is not known a-priori. In this case the VSA acts as a lens that refocuses the TRM field to the target location. Under proper conditions, the resolution is comparable to that of standard TR. While the specific application of active focusing is presented as a validation of the concept, the relationship between coherent focusing and the transfer function implies that the virtual array concept may find use in a range of imaging methods, both active and passive. Possible applications are discussed, and simulation and experimental results are presented.

The prolate spheroidal wave functions as invariants of the time reversal operator for an extended scatterer in the Fraunhofer approximation

The decomposition of the time reversal operator, known by the French acronym DORT, is widely used to detect, locate, and focus on scatterers in various domains such as underwater acoustics, medical ultrasound, and nondestructive evaluation. In the case of point-scatterers, the theory is well understood: The number of nonzero eigenvalues is equal to the number of scatterers, and the eigenvectors correspond to the scatterers Green's function. In the case of extended objects, however, the formalism is not as simple. It is shown here that, in the Fraunhofer approximation, analytical solutions can be found and that the solutions are functions called prolate spheroidal wave-functions. These functions have been studied in information theory as a basis of band-limited and time-limited signals. They also arise in optics. The theoretical solutions are compared to simulation results. Most importantly, the intuition that for an extended objects, the number of nonzero eigenvalues is proportional to the number of resolution cell in the object is justified. The case of three-dimensional objects imaged by a two-dimensional array is also dealt with. Comparison with previous solutions is made, and an application to super-resolution of scatterers is presented.

The time-reversal operator with virtual transducers: application to far-field aberration correction

The decomposition of the time-reversal operator (DORT) is a detection and focusing technique using an array of transmit receive transducers. It can extract Green's functions of scatterers in a medium. A variant consists in transmitting focused beams (FDORT). It is shown here that the FDORT method can be interpreted as the decomposition of a time-reversal operator between an array of virtual transducers located at the transmit beams' foci and the physical array. The receive singular vectors correspond to scatterers' Green's functions expressed in the physical array while the transmit singular vectors correspond to Green's functions expressed in the virtual array. The position of the virtual array can be changed by varying the position of the foci, thus offering different points of view. Parameters and performance of some transmit schemes are discussed. Appropriately positioning the virtual transducers can simplify some problems. One application is measuring and correcting aberration in the case of a far-field phase screen model. Placing the virtual transducers near the phase screen transforms the problem in a simpler near-field phase screen problem.