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

Anna Auria Rasclosa, Rafael Eduardo Carrillo Rangel, Yves Wiaux
2013
Conference paper

Abstract

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.

Official source

https://infoscience.epfl.ch/record/183885?ln=en

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts

Related publications

Sparsity Averaging Reweighted Analysis (SARA): a novel algorithm for radio-interferometric imaging

Rafael Eduardo Carrillo Rangel, Jason Douglas McEwen, Yves Wiaux

We propose a novel algorithm for image reconstruction in radio interferometry. The ill-posed inverse problem associated with the incomplete Fourier sampling identified by the visibility measurements is regularized by the assumption of average signal sparsity over representations in multiple wavelet bases. The algorithm, defined in the versatile framework of convex optimization, is dubbed Sparsity Averaging Reweighted Analysis (SARA). We show through simulations that the proposed approach outperforms state-of-the-art imaging methods in the field, which are based on the assumption of signal sparsity in a single basis only.

Wiley-Blackwell2012

PURIFY: a new approach to radio-interferometric imaging

Rafael Eduardo Carrillo Rangel, Jason Douglas McEwen, Yves Wiaux

In a recent article series, the authors have promoted convex optimization algorithms for radio-interferometric imaging in the framework of compressed sensing, which leverages sparsity regularization priors for the associated inverse problem and defines a minimization problem for image reconstruction. This approach was shown, in theory and through simulations in a simple discrete visibility setting, to have the potential to outperform significantly CLEAN and its evolutions. In this work, we leverage the versatility of convex optimization in solving minimization problems to both handle realistic continuous visibilities and offer a highly parallelizable structure paving the way to significant acceleration of the reconstruction and high-dimensional data scalability. The new algorithmic structure promoted relies on the simultaneous-direction method of multipliers (SDMM), and contrasts with the current major-minor cycle structure of CLEAN and its evolutions, which in particular cannot handle the state-of-the-art minimization problems under consideration where neither the regularization term nor the data term are differentiable functions. We release a beta version of an SDMM-based imaging software written in C and dubbed PURIFY (http://basp-group.github.io/purify/) that handles various sparsity priors, including our recent average sparsity approach SARA. We evaluate the performance of different priors through simulations in the continuous visibility setting, confirming the superiority of SARA.

Wiley-Blackwell2014

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.

EPFL2017