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.

Sparse regularization methods in ultrafast ultrasound imaging

Olivier Bernard, Adrien Georges Jean Besson, Rafael Eduardo Carrillo Rangel, Jean-Philippe Thiran, Yves Wiaux

Ultrafast ultrasound (US) imaging based on plane wave (PW) insonification is a widely used modality nowadays. Two main types of approaches have been proposed for image reconstruction either based on classical delay-and-sum (DAS) or on Fourier reconstruction. Using a single PW, these methods lead to a lower image quality than DAS with multi-focused beams. In this paper we review recent beamforming approaches based on sparse regularization methods. The imaging problem, either spatial-based (DAS) or Fourier-based, is formulated as a linear inverse problem and convex optimization algorithms coupled with sparsity priors are used to solve the ill-posed problem. We describe two applications of the framework namely the sparse inversion of the beamforming problem and the compressed beamforming in which the framework is combined with compressed sensing. Based on numerical simulations and experimental studies, we show the advantage of the proposed methods in terms of image quality compared to classical methods.

Ieee2016

Data-driven Measurement Designs for Magnetic Resonance Imaging

Baran Gözcü

Despite being a powerful medical imaging technique which does not emit any ionizing radiation, magnetic resonance imaging (MRI) always had the major problem of long scanning times that can take up to an hour depending on the application. It also requires uncomfortable breath-holds due to the slow acquisition, sedation of children and repeated scans in the cases of degraded image quality due to body motion. Recent years have seen new image reconstruction techniques that need less amount of acquired data (i.e., accelerated scans) for reconstructing MRI images: parallel imaging and compressed sensing (CS). Although much work has been done on the reconstruction side, there has been relatively less work on experimental design, i.e., which parts of the Fourier domain to acquire during the scan, an essential factor that considerably affects the performance of image reconstructions. The state-of-the-art experimental designs use random subsampling either based on parametric models or heuristical adaptive models. The requirement of extensive parameter tuning and the random nature of the performance render these methods impractical and unreliable for clinical use. Can we systematically use the data acquired during past MRI scans for the design of accelerated scans with a reliable image quality? This problem is the focus of this thesis which proposes a data-driven scan design approach and training procedures which efficiently and effectively learn the structure inherent in the data, and accordingly, design the scans that directly acquire only the most relevant information during acquisition given an acceleration rate constraint. As a result, this boosts the performance of the existing state-of-the-art compressive sensing techniques on real-world datasets. Moreover, this approach provides strong theoretical guarantees by using tools from statistical learning theory. The intensive computational training procedures of our approach are made feasible by large-scale implementations on a parallel computing cluster. In return, this approach avoids any dependence on parametric or heuristic models and provides a reliably consistent image reconstruction performance for accelerated scans. Our approach is flexible and capable of giving deterministic scan designs specific to the anatomy, to the acceleration rate in use, to the reconstruction algorithm and the scan settings such as static/dynamic and parallel imaging. Apart from measurement designs for MRI, this thesis also considers the reconstruction problem. In particular, we focus on the inverse problems that involve a mixture of regularizers in the objective function, exploiting multiple structures at the same time. For these problems, we propose a reliable and systematic optimization framework and illustrate its effectiveness. Finally, in the last part of the thesis, we present a data-driven model and an optimization method for the design of nearly isometric, linear and dimensionality reducing embeddings.

EPFL2019

A convex optimization approach for image recovery from nonlinear measurements in optical interferometry

Anna Auria Rasclosa, Rafael Eduardo Carrillo Rangel, Yves Wiaux

Image recovery in optical interferometry is an ill-posed nonlinear inverse problem arising from incomplete power spectrum and bi-spectrum measurements. We formulate a linear version of the problem for the order-3 tensor formed by the tensor product of the signal with itself. This linear problem is regularized by standard convex \ell_1 relaxations of sparsity and low rank constraints and solved using the most advanced algorithms in convex optimization. We show preliminary results on small size synthetic images as a proof of concept.

2013

Data-driven Measurement Designs for Magnetic Resonance Imaging

Baran Gözcü

EPFL2019