Convex recovery from interferometric measurements

This note formulates a deterministic recovery result for vectors $x$ from quadratic measurements of the form $(Ax)_i \overline{(Ax)_j}$ for some left-invertible $A$. Recovery is exact, or stable in the noisy case, when the couples $(i,j)$ are chosen as edges of a well-connected graph. One possible way of obtaining the solution is as a feasible point of a simple semidefinite program. Furthermore, we show how the proportionality constant in the error estimate depends on the spectral gap of a data-weighted graph Laplacian. Such quadratic measurements have found applications in phase retrieval, angular synchronization, and more recently interferometric waveform inversion

Interferometric inversion: a robust approach to linear inverse problems

In this abstract, we present a new approach to linear wavebased inverse problems (e.g. inverse source, inverse Born scattering). Instead of looking directly at the data, we propose to match cross-correlations and other quadratic data combinations that generalize cross-correlations. This approach is expected to be robust to a wide variety of modeling uncertainties. In deriving a method to perform inversion using data pairs, a non-convex optimization problem is first advanced. It is then convexified by lifting the problem to a higher dimension space. The lifted problem is studied and sufficient conditions for invertibility are obtained. The lifted formulation is however too computationally intensive to be used for imaging, so a less-expansive non-convex approximation is considered. We illustrate the remarkable robustness of interferometric inversion numerically.Massachusetts Institute of Technology. Earth Resources LaboratoryNational Science Foundation (U.S.)United States. Air Force. Office of Scientific ResearchUnited States. Office of Naval ResearchAlfred P. Sloan Foundatio

Inverse Transport Theory of Photoacoustics

We consider the reconstruction of optical parameters in a domain of interest from photoacoustic data. Photoacoustic tomography (PAT) radiates high frequency electromagnetic waves into the domain and measures acoustic signals emitted by the resulting thermal expansion. Acoustic signals are then used to construct the deposited thermal energy map. The latter depends on the constitutive optical parameters in a nontrivial manner. In this paper, we develop and use an inverse transport theory with internal measurements to extract information on the optical coefficients from knowledge of the deposited thermal energy map. We consider the multi-measurement setting in which many electromagnetic radiation patterns are used to probe the domain of interest. By developing an expansion of the measurement operator into singular components, we show that the spatial variations of the intrinsic attenuation and the scattering coefficients may be reconstructed. We also reconstruct coefficients describing anisotropic scattering of photons, such as the anisotropy coefficient $g(x)$ in a Henyey-Greenstein phase function model. Finally, we derive stability estimates for the reconstructions

Modelisation et Simulation en Photo-acoustique

This thesis deals with the problem of photoacoustic imaging. In this imaging setting, one heats a medium up with an electromagnetic wave. The medium dilates and emits an ultrasonic wave. The aim is then to reconstruct inner properties of the medium from boundary measurements. It is an inverse problem on the initial condition for the wave equation. In an idealized frame, the reconstruction procedure has been thoroughly studied. The main goal of this thesis is to stray from the standard model by considering less restrictive assumptions. For each of them (boundary condition, partial view, attenuation, inhomogeneous sound speed) the thesis proposes a correction based on adapted mathematical tools (asymptotic analysis, dual approach , correlation...). Reconstruction of the initial condition of the wave equation is however not enough. It depends on the electromagnetique illumination. A second inverse problem has to be solved on the electromagnetic wave propagation to acces the physical coefficients of interest. The thesis presents algorithmic results in the frame of the diffusion equation and theoretic estimates in the frame of the transport equation. The thesis also presents an improving result for a topological derivative based imaging approach.Cette these traite du probleme de l'imagerie photo-acoustique. Dans ce systeme d'imagerie, on chauffe un milieu avec une onde electromagnetique. Le milieu se dilate et emet une onde ultrasonique qu'on mesure. Le but est de reconstruire les caracteristiques internes du milieu a partir des mesures de l'onde acoustique sur son bord. C'est un probleme inverse sur la condition initiale pour l'equation des ondes. Dans un cadre idealise, la procedure de reconstruction est connue et a ete etudiee en profondeur. Le but premier de cette these est de s'eloigner du cadre standard en considerant des hypotheses moins restrictives. Pour chaque hypothese (conditions de bord, vue partielle, attenuation, vitesse non-homogene ) la these propose une correction basee sur des outils mathematiques adaptes (analyse asymptotique, approche duale, correlation...). La reconstruction de la condition initiale de l'equation des ondes n'est cependant pas suffisante. Elle depend de l'illumination electromagnetique. Un second probleme inverse doit etre resolu sur la propagation de l'onde electromagnetique pour avoir acces aux coefficients physiques d'interet. La these presente des resultats algorithmiques dans le cadre de l'equation de de diffusion et des estimations theoriques dans le cadre de l'equation de transfert radiatif. La these presente aussi un resultat d'amelioration d'une approche d'imagerie par derivee topologique

Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging

International audienc

Coherent Interferometry Algorithms for Photoacoustic Imaging

The aim of this paper is to develop new coherent interferometry (CINT) algorithms to correct the effect of an unknown cluttered sound speed (random fluctuations around a known constant) on photoacoustic images. By back-propagating the correlations between the preprocessed pressure measurements, we show that we are able to provide statistically stable photoacoustic images. The preprocessing is exactly in the same way as when we use the circular or the line Radon inversion to obtain photoacoustic images. Moreover, we provide a detailed stability and resolution analysis of the new CINT--Radon algorithms. We also present numerical results to illustrate their performance and to compare them with Kirchhoff--Radon migration functions.European Research Council (MULTIMOD–267184

How to Register a Live onto a Liver ? Partial Matching in the Space of Varifolds

30 pages, 11 figures, Special Issue: Information Processing in Medical Imaging (IPMI) 2021, Accepted for publication at the Journal of Machine Learning for Biomedical Imaging (MELBA) https://www.melba-journal.orgPartial shapes correspondences is a problem that often occurs in computer vision (occlusion, evolution in time...). In medical imaging, data may come from different modalities and be acquired under different conditions which leads to variations in shapes and topologies. In this paper we use an asymmetric data dissimilarity term applicable to various geometric shapes like sets of curves or surfaces, assessing the embedding of a shape into another one without relying on correspondences. It is designed as a data attachment for the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, allowing to compute a meaningful deformation of one shape onto a subset of the other. We refine it in order to control the resulting non-rigid deformations and provide consistent deformations of the shapes along with their ambient space. We show that partial matching can be used for robust multi-modal liver registration between a Computed Tomography (CT) volume and a Cone Beam Computed Tomography (CBCT) volume. The 3D imaging of the patient CBCT at point of care that we call live is truncated while the CT pre-intervention provides a full visualization of the liver. The proposed method allows the truncated surfaces from CBCT to be aligned non-rigidly, yet realistically, with surfaces from CT with an average distance of 2.6mm(+/- 2.2). The generated deformations extend consistently to the liver volume, and are evaluated on points of interest for the physicians, with an average distance of 5.8mm (+/- 2.7) for vessels bifurcations and 5.13mm (+/- 2.5) for tumors landmarks. Such multi-modality volumes registrations would help the physicians in the perspective of navigating their tools in the patient's anatomy to locate structures that are hardly visible in the CBCT used during their procedures. Our code is available at https://github.com/plantonsanti/PartialMatchingVarifolds