Sparse inverse problems for Fourier imaging

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.