# RECONSTRUCTION OF THE TEMPORAL GREEN’S FUNCTION BY CORRELATION OF MONOCHROMATIC GREEN’S FUNCTIONS

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## Sector scan (or phase scan)

In a sector scan, the full aperture is usually used for a transmit. The consecutive transmits are then obtained by steering (rotating) the beam. Thus, the foci are located on a circle at a constant radius from the array center. This is shown in Figure 20. Therefore, the virtual array is a curved array. The sector scan is especially interesting when the zone to image is a sector.
Resolution: A geometrical derivation similar to the one in Figure 19 shows that with this scheme the resolution of the virtual array is the same as the real array. Indeed, as shown in Figure 20, the angle between the blue scatterer and the virtual array is the same as the angle between the scatterer and the real array. Thus with such a transmit sequences, the performance of FDORT in terms of scatterers separation is identical to DORT, and is equal to D.

### The focused transmits as an orthogonal basis

The DORT method has been implemented in the past with transmits sequences other than single elements transmits. In these cases, orthogonal combinations of array elements were used. The interest of using this kind of transmits is increasing the SNR.
The focused beams can themselves be considered as an orthogonal basis. Each focused beam is the Green’s function of the corresponding focal point. For one frequency, it is given by )()(PreHiPjkrii= . The condition of orthogonality of Green’s functions is given by Eq.2. 5 : the Green’s functions of two points P and Q are orthogonal if it is possible to focus with the array on P without sending energy on Q. For the linear scan, in the focal plane, the field is given by DZXcλπsin.

Let Q be the position of a scatterer. Its Green’s function is H(Q). Let Pm be the foci of a focused beam. Focusing on Pm is equivalent to delay the signals of the physical array by ri(Pm)/c and summing. In a monochromatic formalism, it is equivalent to project the signals on Pm Green’s function. Therefore, the echo from Q heard by Pm is expressed as mmPHQHS(|)(=. According to Eq.2. 5, Sm is also the field at position Pm when the echo from the scatterer Q is back-propagated from the array. Again, this shows that the focused beams act like virtual transducers that probe the value of the field at distance. The difference with the previous virtual transducer interpretation is that here the virtual transducers do not have a directivity, they are isotropic. However, the field is now the back-propagated field, which means that the signal is non-zero only in a cone. To summarize, there are 2 ways to interpret the virtual transducers:
– either as transducers with a directivity probing the free space field from the scatterer.
– or as isotropic transducer probing the back-propagated field of the transducer.
The interest of this interpretation is to give a physical interpretation of the change of basis between the canonical basis (physical array) and the focused basis (virtual array). The change of basis is obtained by a back-propagation of the signals from the array plan to the focal plan.

Steering from the virtual array

With the far-field phase screen, the knowledge of the Green’s function of the 3 scatterers enables to focus only in the immediate neighborhood (isoplanatic patch) of the point. The size of the isoplanatic patch, which is the area where the aberration effect is constant, depends on the strength of the phase screen and the distance to the array. With the phase screen chosen in the simulations, the isoplanatic patch is very narrow, as seen in Figure 32. The Green’s function of a scatterer located at -12 mm was used to focus at this point and at neighbor points. To focus at neighbor points, a steering term was added to the phase of the Green’s function, which corresponds to the difference in geometrical delays between the reference point and the new point. This is equivalent to consider that the term due to the aberration is the same for all points. Both amplitude and phase of the Green’s function are used.

Determination of the phase screen position

In a practical setting, the depth of the phase screen is not necessarily known. We investigate here a few criterion to determine this depth.

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Criterion based on the amplitude

If the phase screen only delays the wave-front, and does not introduce any attenuation, the amplitude of the wave-front immediately after going through the phase screen is approximately constant. During the propagation after the phase screen, interferences lead to amplitude variation. The 1st criterion is then based on the field amplitude. The monochromatic Green’s function of the scatterer at x=-2mm of the previous example is numerically back-propagated in an homogeneous medium, as shown in Figure 35. As the wave-front gets closer from the phase-screen plane (z=30mm), the amplitude variation becomes smoother. After the phase-screen plane, amplitude variation reappears, because the phase screen was not modeled in the back-propagation.

Chapter 1. Introduction
I. FOCALISATION DANS DES MILIEUS HOMOGENES ET HETEROGENES
I.A. Focalisation en milieu homogène
I.B Focalisation en milieu heterogene
II. METHODES D’ESTIMATION DE FONCTIONS DE GREEN
II.A Methode reposant sur le modèle écran en champ-proche (estimation de loi de retards seulement)
II.B. Methodes estimant la fonction de Green complète
III. MODELES POUR LES SIGNAUX
III.A Formule de Rayleigh-Sommerfeld
III.A.2 Approximation de Fresnel en coordonnes cartésiennes
III.A.3 Approximation de Fresnel en coordonnées polaires
III.D La propagation comme un filtre passe-bas
IV. PLAN DE LA THESE
Chapter 2. Le Retournement Temporel a partir de Transducteurs Virtuels: FDORT
II. INTRODUCTION AND PRESENTATION OF THE METHOD
III. THE DORT METHOD BETWEEN TWO DIFFERENT ARRAYS
II.A. The transfer matrix and the time reversal operator
II.B. Case of isotropic, pointlike scatterers and single scattering
IV. FOCUSED BEAMS AND VIRTUAL TRANSDUCERS
III.A. Virtual transducer model
III.B. The DORT method between a real array and a virtual array
III.C. The focused transmits as an orthogonal basis
V. APPLICATION TO FOCUSING THROUGH A FAR-FIELD PHASE SCREEN.69
IV.A Changing a far-field phase screen problem into a near-field phase screen problem using a virtual array
IV.B Steering from the virtual array
IV.C Practical implementation
VI. FDORT WITH TIME GATING
V.A. Influence of noise on eigenvectors and eigenvalues
V.B. A solution: FDORT with time gating
VII. LOCAL FDORT
VI.A. FDORT in a limited region of space
VI.B Application to moving scatterers
VI.C. Application to small objects detection
VIII. CONCLUSION
REFERENCES
Chapter 3. Objets Etendus
I. INTRODUCTION
II. THEORY
II.A Expression of the time reversal operator for an extended object
II.B Invariants of the Time Reversal Operator: The Prolate Spheroidal functions
III. RESULTS
III.A. Results with a one-dimensional array
III.B Application to Green’s function estimation and focusing
III.B Results with a 2D array
IV. COMPARISON WITH PREVIOUS WORK
V. APPLICATIONS: SUPER-RESOLUTION, TOMOGRAPHY AND MICROCALCIFICATION DIAGNOSIS
VI. CONCLUSION
APPENDIX A: DERIVATION OF THE KERNEL
APPENDIX B: LARGER OBJECTS
APPENDIX C: INVARIANTS FOR 3D OBJECTS WITH 2D ARRAYS
DORT with a 2D array
Analytical solutions for separable kernels
Chapter 4. FDORT dans le Speckle
I. INTRODUCTION
II. BASIC STATISTICAL PROPERTIES OF SPECKLE SIGNALS
A. Randomness of the speckle
B. First order statistics
C. Second order statistics
D. Basics of estimation theory
III. INTERPRETATIONS OF KKH
A. Spatial correlation matrix, or Van Cittert Zernike matrix
B. Time Reversal Operator for an equivalent virtual object
C. Variance and standard deviation of the estimation
D. Interpretation of the first eigenvalue in speckle
IV. APPLICATION TO FOCUSING IN HETEROGENEOUS MEDIUM
A. Equivalent virtual object and iteration of the method
B. Focusing through a far-field phase screen
C. Medical phantom results
V. LINK WITH OTHER ABERRATION CORRECTION METHOD IN SPECKLE.202
A. 1-lag cross-correlation (O’Donnell)
B. Maximum Speckle Brightness
C. Eigenfunction analysis of backscattering signal
D. Multi-lag cross-correlation (LMS algorithm)
VI. GREEN FUNCTION ESTIMATION AND FOCUSING IN PRESENCE OF STRONG INTERFERING SIGNALS
VII. CONCLUSION
APPENDIX A COHERENT INTENSITY AND FOCUSING CRITERION IN FUNCTION OF THE SPATIAL CORRELATION FUNCTION
Appendix B. VARIANCE OF THE ESTIMATION
A. Variance of the amplitude of the spatial correlation coefficients
B. Variance of the phase
Chapter 5. Signaux bandes larges: Invariants spatio-temporel du Retournement Temporel
I. INTRODUCTION
II. SPATIO-TEMPORAL INVARIANTS OF THE TIME REVERSAL
II.A Heuristics
II.B The Time Reversal Operator in the Time Domain
II.C. Decomposition of the Tensor
II.D. Practical implementation and results
III. DECOMPOSITION OF THE FOCUSED TENSOR
III.A The Focused Tensor
III.B Decomposition of the Focused Tensor
III.C. Results
IV. RECONSTRUCTION OF THE TEMPORAL GREEN’S FUNCTION BY CORRELATION OF MONOCHROMATIC GREEN’S FUNCTIONS
IV.1 Theory
IV.2 Experiments
V.CONCLUSION
REFERENCES

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