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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.

#### Link to back-propagation

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.

**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.

**Table of contents :**

**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

III.D Link to back-propagation

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**