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Publication# Sparse inverse problems for Fourier imaging

Abstract

Many natural images have low intrinsic dimension (a.k.a. sparse), meaning that they can be represented with very few coefficients when expressed in an adequate domain. The recent theory of Compressed Sensing exploits this property offering a powerful framework for sparse signal recovery from undetermined linear systems. In this thesis, we deal with two different applications of remote Fourier sensing, for which the available measurements relate to the Fourier coefficients of our concerned sig- nal: optical interferometry and diffusion Magnetic Resonance Imaging (dMRI). In both applications, we face challenging problems due to a restricted number of available measure- ments and the nonlinearity of the direct model for the data. Inspired by the Compressed Sensing framework, our strategy to solve these nonlinear and ill-posed problems resorts to reformulating them as linear inverse problems and propose novel priors to leverage the intrinsic low dimensionality of the solution. The first part of this thesis is devoted to image reconstruction from optical interferom- etry data. State-of-the-art methods are nonconvex due to the intrinsic data nonlinearity and are therefore known to suffer from a strong sensitivity to initialization. We reformu- late the problem as a tensor completion problem, where the aim is to recover a tensor from which we have information through some linear mapping. We propose two different alternatives to solve it, one being a purely convex approach. An original nonconvex alter- nate minimization method has also been explored. We present results on synthetic data and compare pros and cons for both approaches. Our original formulation can be seen as a generalization of the Phase Lift approach and can potentially be applied to other partial phase retrieval problems. In the second part, we tackle the problem of fiber reconstruction in dMRI. dMRI exploits the anisotropy of the water diffusion in the brain to study the organization of its tissue. Particularly, the goal of our work is to recover the local properties of the axon tracts, i.e. their orientation and microstructural features in every voxel of the brain. We resort to a reweighting scheme to leverage the structured sparsity of the solution, where ï¿Œthe structure originates from the spatial coherence of the fiber characteristics between neighbor voxels. Imposing this original prior promotes a powerful regularization that guarantees a strong robustness to undersampling. Due to a time-consuming measuring process, this ability to solve the imaging problem from few dMRI data points is crucial to guarantee the feasibility of this technique in a clinical context. We present results on real and simulated data and compare our approach to other state-of-the-art methods. We also discuss how our novel approach can actually be applied in a more generic framework for multiple correlated sparse signal recovery.

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Compressed sensing

Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Nyquist–Shannon sampling theorem. There are two conditions under which recovery is possible.

Fourier series

A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.

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