Stable image reconstruction using total variation minimization

This article presents near-optimal guarantees for accurate and robust image recovery from under-sampled noisy measurements using total variation minimization. In particular, we show that from O(slog(N)) nonadaptive linear measurements, an image can be reconstructed to within the best s-term approximation of its gradient up to a logarithmic factor, and this factor can be removed by taking slightly more measurements. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of suitably incoherent matrices.Comment: 25 page

Robust analysis ℓ1\ell_1-recovery from Gaussian measurements and total variation minimization

Analysis $\ell_1$-recovery refers to a technique of recovering a signal that is sparse in some transform domain from incomplete corrupted measurements. This includes total variation minimization as an important special case when the transform domain is generated by a difference operator. In the present paper we provide a bound on the number of Gaussian measurements required for successful recovery for total variation and for the case that the analysis operator is a frame. The bounds are particularly suitable when the sparsity of the analysis representation of the signal is not very small

High-quality Image Restoration from Partial Mixed Adaptive-Random Measurements

A novel framework to construct an efficient sensing (measurement) matrix, called mixed adaptive-random (MAR) matrix, is introduced for directly acquiring a compressed image representation. The mixed sampling (sensing) procedure hybridizes adaptive edge measurements extracted from a low-resolution image with uniform random measurements predefined for the high-resolution image to be recovered. The mixed sensing matrix seamlessly captures important information of an image, and meanwhile approximately satisfies the restricted isometry property. To recover the high-resolution image from MAR measurements, the total variation algorithm based on the compressive sensing theory is employed for solving the Lagrangian regularization problem. Both peak signal-to-noise ratio and structural similarity results demonstrate the MAR sensing framework shows much better recovery performance than the completely random sensing one. The work is particularly helpful for high-performance and lost-cost data acquisition.Comment: 16 pages, 8 figure

TV-min and Greedy Pursuit for Constrained Joint Sparsity and Application to Inverse Scattering

This paper proposes a general framework for compressed sensing of constrained joint sparsity (CJS) which includes total variation minimization (TV-min) as an example. TV- and 2-norm error bounds, independent of the ambient dimension, are derived for the CJS version of Basis Pursuit and Orthogonal Matching Pursuit. As an application the results extend Cand`es, Romberg and Tao's proof of exact recovery of piecewise constant objects with noiseless incomplete Fourier data to the case of noisy data.Comment: Mathematics and Mechanics of Complex Systems (2013

Sampling in the Analysis Transform Domain

Many signal and image processing applications have benefited remarkably from the fact that the underlying signals reside in a low dimensional subspace. One of the main models for such a low dimensionality is the sparsity one. Within this framework there are two main options for the sparse modeling: the synthesis and the analysis ones, where the first is considered the standard paradigm for which much more research has been dedicated. In it the signals are assumed to have a sparse representation under a given dictionary. On the other hand, in the analysis approach the sparsity is measured in the coefficients of the signal after applying a certain transformation, the analysis dictionary, on it. Though several algorithms with some theory have been developed for this framework, they are outnumbered by the ones proposed for the synthesis methodology. Given that the analysis dictionary is either a frame or the two dimensional finite difference operator, we propose a new sampling scheme for signals from the analysis model that allows to recover them from their samples using any existing algorithm from the synthesis model. The advantage of this new sampling strategy is that it makes the existing synthesis methods with their theory also available for signals from the analysis framework.Comment: 13 Pages, 2 figure