^{1}and Mathias Fink

^{2}

### Abstract

The FDORT method (French acronym for decomposition of the time reversal operator using focused beams) is a time reversal based method that can detect point scatterers in a heterogeneous medium and extract their Green’s function. It is particularly useful when focusing in a heterogeneous medium. This paper generalizes the theory of the FDORT method to random media (speckle), and shows that it is possible to extract Green’s functions from the speckle signal using this method. Therefore it is possible to achieve a good focusing even if no point scatterers are present. Moreover, a link is made between FDORT and the Van Cittert–Zernike theorem. It is deduced from this interpretation that the normalized first eigenvalue of the focused time reversal operator is a well-known focusing criterion. The concept of an equivalent virtual object is introduced that allows the random problem to be replaced by an equivalent deterministic problem and leads to an intuitive understanding of FDORT in speckle. Applications to aberration correction are presented. The reduction of the variance of the Green’s function estimate is discussed. Finally, it is shown that the method works well in the presence of strong interfering scatterers.

I. INTRODUCTION

II. INTERPRETATIONS OF

A. Deterministic objects: time reversal operator

B. Spatial correlation matrix, or Van Cittert Zernike matrix

1. Interpretation of the transmits as realizations of the random backscattered signal

2. Link between and the spatial correlation matrix

3. Link to the VanCittert–Zernike theorem

C. Time reversal operator for an equivalent virtual object

D. Variance and standard deviation of the estimation

E. Interpretation of the first eigenvalue in speckle

1. Interest of the focusing criterion

2. Link between first eigenvalue of and coherent intensity

3. Link between the incoherent intensity and the sum of the eigenvalues

4. Link between C and the normalized first eigenvalue

III. APPLICATION TO FOCUSING IN HETEROGENEOUS MEDIA

A. Equivalent virtual object and iteration of the method

B. Focusing through a far-field phase screen

C. Medical phantom results

1. Results

2. Discussion

IV. GREEN FUNCTION ESTIMATION AND FOCUSING IN THE PRESENCE OF STRONG INTERFERING SIGNALS

A. Simulation

V. CONCLUSION

ACKNOWLEDGMENTS

### Key Topics

- Eigenvalues
- 77.0
- Speckle
- 65.0
- Green's function methods
- 61.0
- Optical aberrations
- 22.0
- Medical imaging
- 8.0

## Figures

(Color online) Acquisition of the transfer matrix in the FDORT method. The focused transmit insonify the medium. The signal received by array element is time gated to select the signal from depth to , and its Fourier coefficient at frequence gives the matrix coefficient . In speckle is taken to be about the pulse length.

(Color online) Acquisition of the transfer matrix in the FDORT method. The focused transmit insonify the medium. The signal received by array element is time gated to select the signal from depth to , and its Fourier coefficient at frequence gives the matrix coefficient . In speckle is taken to be about the pulse length.

(a) A transmit focusing at a point and the corresponding received wave front. In speckle, ideally, to estimate the Green function at point , we would like to fire several times the same transmit focusing at , but with a different speckle distribution each time, to provide a good averaging of the randomness. (b) Instead, we use a neighbor beam focusing at close to (the distance between and is exaggerated here). (c) By properly shifting the wave front of the received signal, it looks like the whole medium is virtually translated and ’, the new position of , corresponds to . Thus a new realization of the signal coming from is obtained, with a different scatterer distribution. By virtually translating the phantom by a different distance, several realizations are obtained. This is valid only if the aberration seen by is the same as the aberration seen by .

(a) A transmit focusing at a point and the corresponding received wave front. In speckle, ideally, to estimate the Green function at point , we would like to fire several times the same transmit focusing at , but with a different speckle distribution each time, to provide a good averaging of the randomness. (b) Instead, we use a neighbor beam focusing at close to (the distance between and is exaggerated here). (c) By properly shifting the wave front of the received signal, it looks like the whole medium is virtually translated and ’, the new position of , corresponds to . Thus a new realization of the signal coming from is obtained, with a different scatterer distribution. By virtually translating the phantom by a different distance, several realizations are obtained. This is valid only if the aberration seen by is the same as the aberration seen by .

(Color online) Theoretical amplitude of the spatial correlation matrix (a) and projection on the antidiagonal (b) in homogeneous media as predicted by the VanCittert–Zernike theorem. The triangular shape that arises from the Fourier transform of is easily identifiable. sixtyfour of the 128 elements were used in transmission, which is why the triangle base is 64 elements wide. (c), (d) Experimental amplitude of the FDORT matrix, for a focus. The differences observed for the edge elements are mainly due to the transducers directivity, which was not taken into account in the VanCittert–Zernike prediction.

(Color online) Theoretical amplitude of the spatial correlation matrix (a) and projection on the antidiagonal (b) in homogeneous media as predicted by the VanCittert–Zernike theorem. The triangular shape that arises from the Fourier transform of is easily identifiable. sixtyfour of the 128 elements were used in transmission, which is why the triangle base is 64 elements wide. (c), (d) Experimental amplitude of the FDORT matrix, for a focus. The differences observed for the edge elements are mainly due to the transducers directivity, which was not taken into account in the VanCittert–Zernike prediction.

In speckle, the standard deviation of the estimation is inversely proportional to the square root of the number of realizations used in the estimation. However, the number of lateral realizations is limited by the size of the isoplanatic patch, which is the size of the region where all the points *see* the same aberration, and thus where the Green functions are similar enough to be averaged. Therefore, to increase the number of realizations, additional windows can be selected using the depth dimension.

In speckle, the standard deviation of the estimation is inversely proportional to the square root of the number of realizations used in the estimation. However, the number of lateral realizations is limited by the size of the isoplanatic patch, which is the size of the region where all the points *see* the same aberration, and thus where the Green functions are similar enough to be averaged. Therefore, to increase the number of realizations, additional windows can be selected using the depth dimension.

(Color online) Standard deviation of the phase in function of the lateral size of the speckle region. In the first numerical experiment (solid blue plot), the transmits are very close to each other while in a second one (red cross), the transmits used in the estimation were separated by the beam width . Taking more transmits than required by the condition to have independent realizations does not improve the estimation.

(Color online) Standard deviation of the phase in function of the lateral size of the speckle region. In the first numerical experiment (solid blue plot), the transmits are very close to each other while in a second one (red cross), the transmits used in the estimation were separated by the beam width . Taking more transmits than required by the condition to have independent realizations does not improve the estimation.

The phase of the estimated Green functions is unwrapped and the geometrical delay removed to show an estimate of the aberrator delay profile. The estimate (dash line) is compared with the true profile (solid line), after the first iteration (a) and after five iterations (b). The estimate is not very good at the first iteration but converges after a few iterations.

The phase of the estimated Green functions is unwrapped and the geometrical delay removed to show an estimate of the aberrator delay profile. The estimate (dash line) is compared with the true profile (solid line), after the first iteration (a) and after five iterations (b). The estimate is not very good at the first iteration but converges after a few iterations.

(Color online) Simulated transmit fields at in presence of the near field phase aberrator for a different number of iterations of the algorithm. (a) Initial transmit: it is based on the Green function in homogeneous medium, and therefore the focusing is very poor. It is the equivalent virtual object for the first FDORT iteration (b) The first eigenvector obtained in the first iteration is back-propagated. It focuses mainly on the brightest spot of the first transmit, but with significant interferences. The focusing criterion is only 0.3. It is used to correct the transmit for the second iteration of FDORT. (c) First eigenvector from the second iteration. The focusing has improved. It is used to correct the transmit for the third iteration. (d) Fifth iteration: the first eigenvector now yields a very good focusing. It is an accurate estimate of the Green function in presence of the phase aberrator. is now equal to 0.7.

(Color online) Simulated transmit fields at in presence of the near field phase aberrator for a different number of iterations of the algorithm. (a) Initial transmit: it is based on the Green function in homogeneous medium, and therefore the focusing is very poor. It is the equivalent virtual object for the first FDORT iteration (b) The first eigenvector obtained in the first iteration is back-propagated. It focuses mainly on the brightest spot of the first transmit, but with significant interferences. The focusing criterion is only 0.3. It is used to correct the transmit for the second iteration of FDORT. (c) First eigenvector from the second iteration. The focusing has improved. It is used to correct the transmit for the third iteration. (d) Fifth iteration: the first eigenvector now yields a very good focusing. It is an accurate estimate of the Green function in presence of the phase aberrator. is now equal to 0.7.

Evolution of the normalized first eigenvalue in speckle with a simulated near-field phase screen. The normalized first eigenvalue is equal to the focusing criterion . At the first iteration, is about 0.3, which is the sign of a bad focusing related to the phase screen. After a few iterations, the algorithm converges and passes 0.7, which means that the focusing is excellent. The corresponding transmit fields are displayed in Fig. 7.

Evolution of the normalized first eigenvalue in speckle with a simulated near-field phase screen. The normalized first eigenvalue is equal to the focusing criterion . At the first iteration, is about 0.3, which is the sign of a bad focusing related to the phase screen. After a few iterations, the algorithm converges and passes 0.7, which means that the focusing is excellent. The corresponding transmit fields are displayed in Fig. 7.

(Color online) Simulated transmit fields at in the presence of the far field phase aberrator (drawn on each transmit in white) located at depth, for a different number of iteration of the algorithm. (a) Initial transmit: it is based on the Green function in homogeneous medium, and therefore the focusing is very poor. It is the equivalent virtual object for the first FDORT iteration (b) Field obtained after back-propagation of the first eigenvector obtained in the first iteration. It focuses mainly on the brightest spot of the first transmit, but with interferences. is 0.35. (c) First eigenvector after the fourth iteration. is 0.7. (d) Evolution of the normalized eigenvalue, or factor. Even if the screen is not close to the array, the convergence is reached quickly.

(Color online) Simulated transmit fields at in the presence of the far field phase aberrator (drawn on each transmit in white) located at depth, for a different number of iteration of the algorithm. (a) Initial transmit: it is based on the Green function in homogeneous medium, and therefore the focusing is very poor. It is the equivalent virtual object for the first FDORT iteration (b) Field obtained after back-propagation of the first eigenvector obtained in the first iteration. It focuses mainly on the brightest spot of the first transmit, but with interferences. is 0.35. (c) First eigenvector after the fourth iteration. is 0.7. (d) Evolution of the normalized eigenvalue, or factor. Even if the screen is not close to the array, the convergence is reached quickly.

(Color online) Image of the phantom with a rubber aberrator before (left) and after correction (right) using the FDORT method to improve the focusing in transmit and receive.

(Color online) Image of the phantom with a rubber aberrator before (left) and after correction (right) using the FDORT method to improve the focusing in transmit and receive.

(Color online) Setup for the simulation: the goal is to estimate the Green’s function of point in speckle (dash blue wave front) using focused transmit, in the presence of a strong interferer signal (solid red wave front). The signal from point travels through a heterogeneity, but the interferer does not. In order to estimate the parameters of the heterogeneity it is necessary to isolate the wave front of from the interferers. In the simulation, the interferer is 1000 times stronger than the signal.

(Color online) Setup for the simulation: the goal is to estimate the Green’s function of point in speckle (dash blue wave front) using focused transmit, in the presence of a strong interferer signal (solid red wave front). The signal from point travels through a heterogeneity, but the interferer does not. In order to estimate the parameters of the heterogeneity it is necessary to isolate the wave front of from the interferers. In the simulation, the interferer is 1000 times stronger than the signal.

(Color online) (Top) Phase for the first and second eigenvector of FDORT, corresponding to the setup of Fig. 11. The first eigenvector (left) corresponds clearly to the interference, while the second corresponds to the speckle, and carries information on the heterogeneity. For clarity, a geometrical delay has been removed from the second eigenvector phase. (Bottom) Fields after numerical backpropagation of the first (left) and second eigenvectors. This confirms that the first eigenvector corresponds to the interference, while the second eigenvector corresponds to the desired speckle signal. The eigenvector corresponding to the speckle is a very good estimate of the Green’s function and leads to a good focusing through the heterogeneity despite the presence of a strong interfering signal.

(Color online) (Top) Phase for the first and second eigenvector of FDORT, corresponding to the setup of Fig. 11. The first eigenvector (left) corresponds clearly to the interference, while the second corresponds to the speckle, and carries information on the heterogeneity. For clarity, a geometrical delay has been removed from the second eigenvector phase. (Bottom) Fields after numerical backpropagation of the first (left) and second eigenvectors. This confirms that the first eigenvector corresponds to the interference, while the second eigenvector corresponds to the desired speckle signal. The eigenvector corresponding to the speckle is a very good estimate of the Green’s function and leads to a good focusing through the heterogeneity despite the presence of a strong interfering signal.

(Color online) Phase estimate obtained with the 1-lag cross-correlation method. The estimate is completely dominated by the interference, and it is not possible to extract any information about the heterogeneity.

(Color online) Phase estimate obtained with the 1-lag cross-correlation method. The estimate is completely dominated by the interference, and it is not possible to extract any information about the heterogeneity.

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