We study unique recovery of cosparse signals from limited-angle tomographic measurements of two- and three-dimensional domains. Admissible signals belong to the union of subspaces defined by all cosupports of maximal cardinality @math with respect to the discrete gradient operator. We relate @math both to the number of measurements and to a nullspace condition with respect to the measurement matrix, so as to achieve unique recovery by linear programming. These results are supported by comprehensive numerical experiments that show a high correlation of performance in practice and theoretical predictions. Despite poor properties of the measurement matrix from the viewpoint of compressed sensing, the class of uniquely recoverable signals basically seems large enough to cover practical applications, like contactless quality inspection of compound solid bodies composed of few materials.
We overcome this difficulty by adopting the recently introduced from @cite_9 , that provides an alternative viewpoint to the classical and is more suitable to the problem class considered in this paper. Our present work applies and extends the results from @cite_9 to the 3D recovery problem from few tomographic projections of three-dimensional images consisting of few homogeneous regions. We give a theoretical relation between the image and sufficient sampling, validate it empirically and conclude that TV-reconstructions of a class of synthetic phantoms exhibit a well-defined recovery curve similar to the study in @cite_25 @cite_21 .
@cite_9 @cite_21 @cite_25
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Hoeffding's U-statistics model combinatorial-type matrix parameters (appearing in CS theory) in a natural way. This paper proposes using these statistics for analyzing random compressed sensing matrices, in the non-asymptotic regime (relevant to practice). The aim is to address certain pessimisms of worst-case restricted isometry analyses, as observed by both Blanchard and Dossal, We show how U-statistics can obtain average-case analyses, by relating to statistical restricted isometry property (StRIP) type recovery guarantees. However unlike standard StRIP, random signal models are not required; the analysis used here holds in the almost sure (probabilistic) sense. For Gaussian bounded entry matrices, we show that both @math -minimization and LASSO essentially require on the order of @math measurements to respectively recover at least @math fraction, and @math fraction, of the signals. Noisy conditions are considered. Empirical evidence suggests our analysis to compare well to Donoho and Tanner's recent large deviation bounds for @math -equivalence, in the regime of block lengths @math with high undersampling ( @math measurements); similar system sizes are found in recent CS implementation. In this work, it is assumed throughout that matrix columns are independently sampled. Conventional approaches to sampling signals or images follow Shannon's theorem: the sampling rate must be at least twice the maximum frequency present in the signal (Nyquist rate). In the field of data conversion, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation - the signal is uniformly sampled at or above the Nyquist rate. This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use.