# Distortion matrix approach for ultrasound imaging of random scattering media

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 1, 2020 (received for review December 7, 2019)

## Significance

Ultrasound is a flexible and powerful medical imaging tool. However, variations in organ tissue structure cause propagating waves to undergo unexpected phase shifts, resulting in aberration (blurring) of an image. Compensation for these shifts is possible if the tissue microarchitecture is known; however, this becomes extremely difficult without any such prior knowledge. While adaptive focusing methods have been able to overcome some aberration, they are only effective over small aberration-invariant regions. Here, we present a noninvasive reflection method to access a transmission matrix, which connects any point inside the medium with a sensor array outside. This matrix is the holy grail for imaging: here, we show that it enables in vivo imaging with close-to-ideal resolution and contrast at every pixel.

## Abstract

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.

Light traveling through soft tissues, ultrasonic waves propagating through the human skull, or seismic waves in the Earth’s crust are all examples of wave propagation through inhomogeneous media. Short-scale inhomogeneities of the refractive index, referred to as scatterers, cause incoming waves to be reflected. These backscattered echoes are those which enable reflection imaging; this is the principle of, for example, ultrasound imaging in acoustics and optical coherence tomography for light or reflection seismology in geophysics. However, wave propagation between the sensors and a focal point inside the medium is often degraded by 1) wave front distortions (aberrations) induced by long-scale heterogeneities of the wave velocity or 2) multiple scattering if scatterers are too bright and/or concentrated. Because both phenomena can strongly degrade the resolution and contrast of the image, they constitute the most fundamental limits for imaging in all domains of wave physics.

Astronomers were the first to deal with aberration issues in wave imaging. Their approach to improve image quality was to measure and compensate for the wave front distortions induced by the spatial variations of the optical index in the atmosphere; this is the concept of adaptive optics, proposed as early as the 1950s (1). Subsequently, ultrasound imaging (2) and optical microscopy (3) have also drawn on the principles of adaptive optics to compensate for the aberrations induced by uneven interfaces or tissues’ inhomogeneities. In ultrasound imaging, for instance, arrays of transducers are employed to emit and record the amplitude and phase of broadband wave fields. Wave front distortions can be compensated for by adjusting the time delays added to each emitted and/or detected signal in order to focus at a certain position inside the medium (Fig. 1*A*).

Conventional adaptive focusing methods generally require the presence of a dominant scatterer (guide star) from which the signal to be optimized is reflected. While it is possible in some cases to generate an artificial guide star, the subsequent optimization of focus will nevertheless be imperfect for a heterogeneous medium. This is because a wave front returning from deep within a complex biological sample is composed of a superposition of echoes coming from many unresolved scatterers (resulting in a speckle image), and its interpretation is thus not at all straightforward. A first alternative to adaptive focusing, derived from stellar speckle interferometry (4), is to extract the aberrating phase law from spatial/angular correlations of the reflected wave field (5⇓⇓⇓⇓–10). A second alternative is to correct the aberrations not by measuring the wave front but by simply optimizing the image quality [i.e., by manipulating the incident and/or reflected wave fronts in a controlled manner in order to converge toward an optimal image (11⇓⇓⇓⇓⇓–17)]. However, both methods generally imply a time-consuming iterative focusing process. More importantly, these alternatives rely on the hypothesis that aberrations do not change over the entire field of view (FOV). This assumption of spatial invariance is simply incorrect at large imaging depths for biological media (18, 19). High-order aberrations induced by small-scale variations in the speed of sound of the medium are only invariant over small regions of the image, often referred to as isoplanatic patches in the literature (Fig. 1 *B*–*D*). Conventional adaptive focusing methods thus suffer from a very limited FOV at large depths, which severely limits their performance for in-depth imaging. Recently, however, several acoustic imaging groups have demonstrated convincing approaches for heterogeneous media, whether by mapping the speed of sound distribution in the medium and using it to reconstruct an image (20, 21) or via estimation of (and compensation for) time delays for a local correction of aberrations (22⇓–24). Each of these methods leverages the multielement capabilities of ultrasonic transducers to extract spatial coherence or travel time difference between signals recorded by each array element. In this paper, we propose a more general solution to optimize the information offered by transducer arrays—a universal matrix approach for wave imaging. We develop a rigorous mathematical formalism for our approach and apply the theoretical results to aberration correction for in vivo imaging of the human body.

Historically, the matrix approach for imaging was inspired by the advent of multielement technology in acoustics and by the insight that a matrix formalism is a natural way to describe ultrasonic wave propagation between arrays of transducers (25⇓–27). In the 2010s, the emergence of spatial light modulators allowed the extension of this transmission matrix approach to optics (28). Experimental access to the transmission matrix then enabled researchers to take advantage of multiple scattering for optimal light focusing (28, 29) and communication across a diffusive layer (30) or multimode fiber (31). However, a transmission configuration is not adapted to noninvasive and/or in vivo imaging of biological media, motivating the development of approaches in an epi-illumination configuration.

The reflection matrix has already been shown to be a powerful tool for focusing in multitarget media (25, 32, 33), target detection (34⇓–36), and energy delivery (37, 38) in scattering media. A few studies have also looked at reflection imaging under a matrix formalism (9, 39⇓–41); however, as with most conventional adaptive focusing methods, their effectiveness is limited to a single isoplanatic patch (Fig. 1). Spatially distributed aberrations have not been addressed under a matrix approach until very recently (10, 24, 42). Inspired by the pioneering work of Robert and Fink (40), the concept of the distortion matrix, D, has been introduced in optical imaging (42). Whereas the reflection matrix R holds the wave fronts that are reflected from the medium, D contains the deviations from an ideal reflected wave front that would be obtained in the absence of inhomogeneities. In addition, while R typically contains responses between inputs and outputs in the same basis [e.g., responses between individual ultrasonic transducer elements (43, 44) or between focal points inside the medium (45)], D is concerned with the “dual basis” responses between a set of incident plane waves (46) and a set of focal points inside the medium (40). In optical imaging, Badon et al. (42) recently showed that, for a large specular reflector, the matrix D exhibits long-range correlations in the focal plane. Such spatial correlations can be taken advantage of to decompose the FOV into a set of isoplanatic modes and their corresponding wave front distortions in the far field. The Shannon entropy H of D is also shown to yield an effective rank of the imaging problem (i.e., the number of isoplanatic patches in the FOV). This decomposition was then used to correct for output aberrations when imaging planar specular objects through a scattering medium.

In this paper, we develop the distortion matrix approach for acoustic imaging. In view of medical ultrasound applications, this requires a method that can go beyond imaging specular reflectors in order to tackle the more challenging case of random scattering media. Ultrasonic wave propagation in soft tissues gives rise to a speckle regime in which scattering is often due to a random distribution of unresolved scatterers. Apart from specular reflections at interfaces of tissues and organs, the reflectivity of the medium can be considered to be continuous and random. In this paper, we demonstrate 1) how projecting the reflection matrix into the far field allows the suppression of specular reflections and multiple reverberations (clutter noise), enabling access to a purely random speckle regime; 2) how, in this regime, the far-field correlations of D enable discrimination between and correction for input and output aberrations over each isoplanatic patch; 3) how a position-dependent distortion matrix enables noninvasive access to the transmission matrix T between the plane wave basis and the entire set of image voxels; and 4) how a minimization of the entropy H enables a quantitative measurement of the wave velocity (or refractive index) in the FOV.

Throughout the paper, our theoretical developments are supported by an ultrasonic experiment using a tissue-mimicking phantom and further applied to in vivo ultrasound imaging of the human body. Due to its experimental flexibility, ultrasound imaging is an ideal modality for our proof of concept. Nevertheless, the distortion matrix approach is by no means limited to one particular type of wave but can be extended to any situation in which the amplitude and phase of the medium response can be recorded between multiple inputs and outputs. This study thus opens important perspectives in various domains of wave physics such as acoustics, optics, radar, and seismology.

## Results

### Confocal Imaging with the Reflection Matrix.

The sample under study is a tissue-mimicking phantom with a speed of sound *A*). The phantom also contains eight subwavelength nylon monofilaments of diameter 0.1 mm placed perpendicularly to the probe (white point-like targets). The bright circular target located at depth *B*) is a section of a hyperechoic cylinder composed of a higher density of unresolved scatterers. A 15-mm-thick layer of plexiglass [*A*).

Our matrix approach begins with the experimental acquisition of the reflection matrix R using an ultrasonic transducer array placed in direct contact with the plexiglass layer (Fig. 2*A*). The reflection matrix is built by plane wave beamforming in emission and reception by each individual element (46). Acquired in this way, the reflection matrix is denoted *Materials and Methods*. A conventional ultrasound image consists of a map of the local reflectivity of the medium. This information can be obtained from *A*. They are 1) the recording basis, which here corresponds to the transducer array elements located at u; 2) the illumination basis, which is composed of the incident plane waves with angle θ; 3) the spatial Fourier basis, mapped by the transverse wave number

We first apply a temporal Fourier transform to the experimentally acquired reflection matrix to obtain *A*). **2** simulates focused beamforming in postprocessing in both emission and reception. For broadband signals, ballistic time gating can be performed to select only the echoes arriving at the ballistic time (*A*); we denote this subspace of the focused reflection matrix as **2**). This significantly improves the accuracy and spatial resolution of the subsequent analysis.

Note that this matrix could have been directly formed in the time domain from the recorded matrix **3** consists, in the time domain, of only keeping the echoes arriving at the expected ballistic time.

In a recent work, the broadband focused reflection matrix

Fig. 3*B* shows *B* displays the resulting image **1a** and **1b**). This value of c was not chosen based on any a priori knowledge of the medium but rather, as the value that gives the least aberrated image by eye—a trial and error method typically used by medical practitioners and technicians. However, even with this optimal value for c, the image in Fig. 2*B* remains strongly degraded by the plexiglass layer for two reasons: 1) multiple reverberations between the plexiglass walls and the probe have induced strong horizontal specular echoes and 2) the input and output focal spots are strongly distorted (Fig. 1 *C* and *D*) because of the mismatch between the homogeneous propagation model and the heterogeneous reality. In the following, we show that a matrix approach to wave imaging is particularly appropriate to correct for these two issues. A flow chart summarizing all of the mathematical operations involved in this process is provided in *SI Appendix*, Fig. S1.

### Removing Multiple Reverberations with the Far-Field Reflection Matrix.

Reverberation signals are a common problem in medical ultrasound imaging, often originating from multiple reflections at tissue interfaces or between bones in the human body. Here, we observe strong horizontal artifacts at shallow depths of the image (Fig. 2*B*), which are due to waves that have undergone multiple reflections—often called reverberations in the literature—between the parallel walls of the plexiglass layer. In the following, we show that these signals can be isolated and suppressed using the reflection matrix.

To project the reflection matrix into the far field, we define a free space transmission matrix, *D*. Surprisingly, this matrix is dominated by a strongly enhanced reflected energy along its main antidiagonal (

In *SI Appendix*, section S1, a theoretical expression of *C*). Hence, signatures of such reflections should arise along the main antidiagonal (

We can take advantage of this sparse feature in *Materials and Methods*). Then, the inverse operation of Eq. **6** can be applied to the filtered matrix *E* shows an example of *B* shows that the low-spatial frequency components of the reflected wave field have been removed from the diagonal of *C* shows the full images calculated from **4**). The removal of multiple reflections has enabled the discovery of previously hidden bright targets at shallow depths. However, the confocal image still suffers from aberrations, especially at small and large depths (Fig. 2*C*).

### Distortion Matrix in the Speckle Regime.

In ref. 42, the distortion matrix concept was introduced for optical imaging of extended specular reflectors in a strong aberration regime. Here, we show how this distortion matrix approach can be extended to the speckle regime.

#### Manifestation of aberrations.

In Fig. 3*E*, a significant spreading of energy over off-diagonal coefficients of *A*, which can be understood by rewriting **7** and **8**:*C* and *D*). Eq. **9** tells us that the off-diagonal energy spreading in *E*) occurs when the focusing matrix *C*).

#### The memory effect.

To isolate and correct for these aberration effects, we build upon a physical phenomenon often referred to as the memory effect (50⇓–52) or isoplanatism (1, 53) in wave physics. Usually, this phenomenon is considered in a plane wave basis. When an incident plane wave is rotated by an angle θ, the far-field speckle image is shifted by the same angle θ (50, 51) [or

#### Revealing hidden correlations.

To isolate the effects of aberration in the reflection matrix, *C*, input focusing points at different locations result in wave fronts with different angles in the far field (*SI Appendix*, section S1 and Fig. S2 have further details). This geometric effect hides the correlations that could allow discrimination between isoplanatic patches.

To reveal correlations in *SI Appendix*, Fig. S2): 1) a geometric component that would be obtained for a perfectly homogeneous medium (represented by the black dashed line in Fig. 4*B*) and that can be directly extracted from the reference matrix *C*, *Left*). The key idea of this paper is to isolate the latter contribution by subtracting, from the experimentally measured wave front, its ideal counterpart. Mathematically, this operation can be expressed as a Hadamard product between the normalized reflection matrix **11**. All input focusing points *A*) do not play the role of guide stars.

Compared with *B*), D exhibits long-range correlations (*SI Appendix*, Fig. S2). While the original reflected wave fronts display a different tilt for each focal point *C*). To support our identification of spatial correlations in D with isoplanatic patches, D is now expressed mathematically. We begin with the simplest case of an isoplanatic aberration that implies, by definition, a spatially invariant input focal spot: **9** and **10** into Eq. **11** gives the following expression for D (*SI Appendix*, section S3):**13** and **14** is the following: removing the geometrical component of the reflected wave field in the far field as done in Eq. **11** is equivalent to shifting each virtual source to the central point *C*, *Right*). This superposition of the input focal spots will enable the unscrambling of the propagation and scattering components in the reflected wave field.

#### Time-reversal analysis.

The next step is to extract and exploit the correlations of D for imaging. In the specular scattering regime, D is dominated by spatial correlations in the input focal plane (42). This is due to the long-range coherence of the sample reflectivity for specular reflectors. Conversely, in the speckle scattering regime, the sample reflectivity **13** is averaged over enough independent realizations of disorder (i.e., if the perturbation term *A*). In the following, we will thus assume a convergence of C toward its covariance matrix C due to disorder self-averaging.

Let us now express the covariance matrix C theoretically. This allows C to be written as (*SI Appendix*, section S4)*D*). Expressed in the form of Eq. **17**, *D*. It is important to emphasize, however, that the induced focal spot is enlarged compared with the diffraction limit (58, 59). For the goal of diffraction-limited imaging, the size of this focal spot should be reduced. In the following, we express this situation mathematically and show how to resolve it.

By the van Cittert–Zernike theorem (6), the correlation coefficients *SI Appendix*, section S4 has details). To reduce the size of the virtual reflector, one can equalize the Fourier spectrum of its scattering distribution. Interestingly, this can be done by normalizing the correlation matrix coefficients as follows:*E*. The normalized correlation matrix **17**, *E*). A reflection matrix associated with such a point-like reflector is of rank 1 (25, 43); this property should also hold for the normalized correlation matrix

Beyond the isoplanatic case, the singular value decomposition (SVD) of

### Isoplanatic Patches and Shannon Entropy.

#### FOV decomposition into isoplanatic patches.

We now apply our theoretical predictions to the experimental ultrasound imaging data. Fig. 5*A* displays the normalized singular values *A*, a few singular values seem to dominate, but it is not clear how many are significantly above the noise background. To solve this problem, we consider the Shannon entropy H of the singular values **16** (42). The singular values of Fig. 5*A* (calculated using the model wave velocity *C*). Hence, only the three first eigenstates should be required to construct an unaberrated image of the medium. Fig. 5*B* shows the phase of the three first eigenvectors *E* and *F* illustrates the benefit of our matrix approach at depth *E*), the corrected reflection matrix **23**) is almost diagonal (Fig. 3*F*). This feature demonstrates that the input and output focal spots are now close to being diffraction limited. In other words, aberrations have been almost fully corrected by the transmission matrix

The resulting ultrasound images, *D*, *F*, and *H*: each estimator *C* and 5*D*). This isoplanatic patch does not require aberration correction because the model wave velocity *B*). While the convex shape of *F*) and shallow depths (*H*).

The gain in image quality can quantified by the Strehl ratio, S (62). Initially introduced in the context of optics, S is defined as the ratio of the peak intensity of the imaging system point spread function with aberration to that without. Equivalently, it can also be defined in the far field as the squared magnitude of the mean aberration phase factor. S is directly proportional to the focusing parameter introduced by Mallart and Fink in the context of ultrasound imaging (6). Here, we can calculate a spatially resolved Strehl ratio using the distortion matrices *E*, *G*, and *I*. These maps enable direct visualization of the isoplanatic area in which each different aberration correction is effective, allowing quantitative confirmation of our previous qualitative analysis of confocal images. Moreover, *E*, *G–I*).

The results displayed in Fig. 5 show that the decomposition of the imaging problem into isoplanatic patches, originally demonstrated with D for large specular reflectors in optics (42), also holds in a random speckle regime if we consider, this time, the normalized correlation matrix

#### Shannon entropy minimization.

The first path toward full-field imaging is based on a minimization of the distortion entropy **21**). The logic is as follows. 1) In the speckle regime, there is a direct relation between the Shannon entropy *SI Appendix*, Fig. S3). 3) Thus, *SI Appendix*, section S5, the SD of the coefficients of

Fig. 5*C* provides a first proof of concept of this idea. It shows the entropy *SI Appendix*, Fig. S3).

Note that while the entropy **16**: experimental noise and an insufficient number of input focal points can hinder perfect smoothing of the fluctuations caused by the random sample reflectivity. Another potential reason is that imperfections in the probe or plexiglass layer could induce lateral variations of the aberrations upstream of the FOV.

#### Discussion.

To obtain a spatial map of the speed of sound and a full-field image of a heterogeneous medium, one would need to repeat the same entropy minimization process described above but over a finite and moving FOI. The value of c that minimizes the entropy would be the speed of sound averaged over this FOI. However, a compromise must be made between the spatial resolution (FOI size) and the precision of the speed of sound measurement (*SI Appendix*, section S5). Moreover, note that for highly resolved mapping, this approach may prove prohibitively computationally expensive.

### Transmission Matrix Imaging.

#### Phantom imaging: Depth-dependent aberrations.

The second route toward full-field imaging is more general and goes far beyond the case of spatially invariant aberrations. It consists of locally estimating each coefficient of the transmission matrix T that links the far-field and focused bases. The idea is to consider a subdistortion matrix *SI Appendix*, section S5). Here, the dimensions of this window have been empirically set to *E*. The corresponding Strehl ratio map *F*. The clarity of the ultrasound image compared with the initial (Fig. 2*B*) and intermediate (Fig. 2*C*) ones and the marked improvement in *D* demonstrate the effectiveness of this transmission matrix approach. A satisfying Strehl ratio

This proof of concept experiment opens a number of additional questions. First, despite our best efforts, the measured Strehl ratio

#### In vivo ultrasound imaging: Spatially distributed aberrations.

We now apply the aberration correction technique to a dataset acquired in vivo from a human calf (*Materials and Methods*). The uncorrected image is shown in Fig. 6*B*. Larger structures can be clearly identified, such as the vein (white arrow) near *B*). This observation is confirmed by the accompanying Strehl ratio map in Fig. 6*A*, which shows values inferior to 0.1 over most areas of the image. In the previous section, Strehl ratio maps of the phantom/plexiglass systems showed values smaller than 0.1 for areas that were the most strongly affected by aberration. These results suggest that the image in Fig. 6*B* is significantly aberrated over the entire spatial area.

To correct for aberration, we apply the technique described in previous sections. Due to the heterogeneity of the tissues examined, it can be expected that there are multiple isoplanatic patches, which should not be assumed to be laterally invariant or of the same spatial extent. For full-field imaging, we thus extend the FOI scanning method to an iterative approach that consists of gradually decreasing the spatial extent of the FOI. Specifically, this entails correcting as in Eq. **27**, recalculating a new D, and performing a new correction with a smaller window size. The process is iterated until optimal focusing is achieved—maximization of the Strehl ratio for each focal point. Four window sizes were used: *SI Appendix*, Fig. S4, the spatial distribution of aberrations is exhibited in the evolution of the phase of

After correction (Fig. 6*C*), the ultrasound image is indeed sharper, with better contrast, and smaller structures can be more easily discerned (highlighted areas in Fig. 6 *B* and *C*). The Strehl ratio map shows that the image resolution has been improved over most regions of the image (Fig. 6*D*), with the most significant improvements being at muscle fibers [e.g., at

#### Discussion.

Unlike conventional adaptive focusing whose efficiency range is limited to a single isoplanatic patch (Fig. 1), a full-field image of the medium under investigation is obtained with diffraction-limited resolution. Note that other recent approaches for acoustic imaging have also proposed analogous spatial sectioning to correct for spatially distributed aberration (24, 63). In this respect, a key parameter is the choice of the FOI at each iteration. As shown in *SI Appendix*, section S5, the SVD of

## Discussion and Conclusion

The distortion matrix approach provides a powerful tool for imaging inside a heterogeneous medium with a priori unknown characteristics. Aberrations can be corrected without any guide stars or prior knowledge of the speed of sound distribution in the medium. While our method is inspired by previous works in ultrasound imaging (5, 6, 8, 39, 40), and is built on the recent introduction of the distortion matrix in optics (42), it features several distinct and important advances.

The first is its primary building block: the broadband focused reflection matrix that precisely selects the echoes originating from a single scattering event at each depth. This operation is decisive in terms of signal-to-noise ratio since it drastically reduces the detrimental contribution of out-of-focus and multiply scattered echoes. Equally importantly, this matrix captures all of the input–output spatial correlations of these singly scattered echoes.

The approach presented here also introduces the projection of the reflection matrix in the far field. This enables the elimination of artifacts from multiple reflections between parallel surfaces, revealing previously hidden parts of the image. Here, we have only examined reflections from surfaces that are parallel to the ultrasound array, which is more relevant for imaging layered materials than it is for imaging human tissue. While signatures of other flat surfaces should be identifiable as correlations in off-antidiagonal lines of

For aberration correction, projection of the reflection matrix into a dual basis allows the isolation of the distorted component. Then, all of the input focal spots can be superimposed onto the same (virtual) location. The normalized correlation of these distorted wave fields, and an average over disorder, then enables the synthesis of a virtual reflector. Unlike related works in acoustics (8, 24, 39, 40), this virtual scatterer is point like (i.e., not limited by the size of the aberrated focal spot). Moreover, this approach constitutes a significant advance over recent works, which were limited to aberration correction at either input (24) or output (42). Here, we demonstrate how the randomness of a scattering medium can be leveraged to identify and correct for aberrations at both input and output. By retrieving the transmission matrix between the elements of the probe and each focal point in the medium, spatially distributed aberrations can be overcome. A full-field and diffraction-limited image is recovered. Our approach is thus straightforward, not requiring a tedious iterative focusing process to be repeated over each isoplanatic patch.

It is important to note that, although the first experimental proof of concept involved a relatively simple multilayered wave velocity distribution, our approach is not at all limited to laterally invariant aberrations. As shown by the in vivo imaging experiment, the distortion matrix approach also corrects for complex position-dependent aberrations caused by an unknown speed of sound distribution in the medium.

Last but not least, we furthermore exploit concepts from information theory. In particular, we introduce the idea that, by minimizing the Shannon entropy of the correlation matrix

Despite all these exciting perspectives, our matrix approach still suffers from several drawbacks that should be tackled in the near future. One limitation is its restriction to a speckle scattering regime. Theoretically, for specular reflection, the SVD of D should be examined rather than of

To conclude, the distortion matrix concept can be applied to any field of wave physics for which multielement technology is available. A reflection matrix approach to wave imaging has already been initiated in optical microscopy (9, 10, 35, 41, 42), multiple-input multiple-output radar imaging (67), and seismology (36). The ability to apply the distortion matrix to random media (not just specular reflectors) should be valuable for optical deep imaging in biological tissues (68). At the other end of the spatial scale, volcanoes and fault zones are particularly heterogeneous areas (36) in which the distortion matrix concept could be fruitful for a bulk seismic imaging of the Earth’s crust beyond a few kilometers in depth. The reflection/distortion matrix concept is thus universal. The potential range of applications of this approach is wide and highly promising, whether it be for a direct imaging of the medium reflectivity or a quantitative and local characterization of the wave speed (45, 65), absorption (69), and scattering (70, 71) parameters.

## Materials and Methods

### Tissue-Mimicking Phantom Experiment.

The experimental setup consisted of a 1D ultrasound phased-array probe (SuperLinear SL15-4) connected to an ultrafast scanner (Aixplorer; SuperSonic Imagine). The array contains 256 elements with pitch

### In Vivo Ultrasonic Data.

The in vivo ultrasound dataset was collected by the SuperSonic Imagine company on a healthy volunteer from which informed consent had been obtained. Before being put at our disposal, this dataset was previously fully anonymized following standard practice defined by Commission nationale de l’information et des libertées (CNIL). The ultrasonic probe consisted of a 1D 5- to 18-MHz linear transducer array (SL18-5; SuperSonic Imagine) connected to an ultrafast scanner (Aixplorer Mach-30; SuperSonic Imagine). The array contains 192 elements with pitch

### Multiple Reflection Filter.

The multiple reflection filter consists of applying an adaptive Gaussian filter to remove the specular contribution that lies along the main antidiagonal of *F*). When there is no peculiar specular contribution, the parameter α tends to zero, and the Gaussian filter is not applied: the main antidiagonal of

### Data Availability.

Data used in this manuscript have been deposited at Figshare, https://figshare.com/projects/Distortion_matrix_approach_for_full-field_imaging_of_random_scattering_media_2020/78141.

## Acknowledgments

We thank Victor Barolle, Amaury Badon, and Thibaud Blondel whose own research works in optics and seismology inspired this study. We are grateful for the funding provided by Labex WIFI (Laboratory of Excellence within the French Program Investments for the Future, Waves and Imaging: From Fundamentals to Innovation) Grants ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*. W.L. acknowledges financial support from the SuperSonic Imagine company. L.A.C. acknowledges financial support from the European Union’s Horizon 2020 Research and Innovation Program under Marie Skłodowska-Curie Grant 744840. This project has received funding from the European Research Council under the European Union’s Horizon 2020 Research and Innovation Program Grant 819261 (REMINISCENCE: REflection Matrix ImagiNg In wave SCiENCE).

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: alexandre.aubry{at}espci.fr.

Author contributions: M.F. and A.A. designed research; W.L., L.A.C., T.F., and A.A. performed research; W.L., L.A.C., and A.A. analyzed data; and W.L., L.A.C., and A.A. wrote the paper.

Competing interest statement: W.L., L.A.C, M.F., and A.A. are named inventors on patents filed by the CNRS related to the distortion matrix approach. M.F. is cofounder of the SuperSonic Imagine company, which is commercializing the ultrasound platform used in this study. W.L. has his PhD funded by the SuperSonic Imagine company, and T.F. is an employee of this company.

This article is a PNAS Direct Submission.

Data deposition: Data used in this manuscript have been deposited at Figshare, https://figshare.com/projects/Distortion_matrix_approach_for_full-field_imaging_of_random_scattering_media_2020/78141.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1921533117/-/DCSupplemental.

Published under the PNAS license.

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