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.
In discrete tomography, images to be reconstructed are sampled along lines. Thus, sampling patterns are quite different from random and non-adaptive measurements that are favourable from the viewpoint of compressed sensing. In @cite_24 , we showed that structured sampling patterns as used in commercial scanners do not satisfy the CS conditions, like the nullspace property and the , that guarantee accurate recovery of sparse (or compressible) signals. In fact, these recovery conditions predict a quite poor worst-case performance of tomographic measurements, due to the high nullspace sparsity of a tomographic projection matrix @math . Moreover, the gap between available recovery results of CS @cite_34 and results from tomographic projections in @cite_24 is dramatic.
@cite_24 @cite_34
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We study the discrete tomography problem in Experimental Fluid Dynamics—Tomographic Particle Image Velocimetry (TomoPIV)—from the viewpoint of Compressed Sensing (CS). The problem results in an ill‐posed image reconstruction problem due to undersampling. Ill‐posedness is also intimately connected to the particle density. Higher densities ease subsequent flow estimation but also aggravate ill‐posedness of the reconstruction problem. A theoretical investigation of this trade‐off is studied in the present work. 1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random high-dimensional polytopes; a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications - in statistics, probability, information theory, and signal processing - with potential impacts in practical subjects like medical imaging and digital communications. Before involving the reader in our lengthy analysis of high-dimensional face counting, we describe three implications of our results. 1.1.1. Convex Hulls of Gaussian Point Clouds. Consider a random point cloud of n points xi, i = 1, . . . , n, sampled independently and identically from a Gaussian distribution in R d with nonsingular covariance. This is a standard model of multivariate data; its properties are increasingly important in a wide range of applications. At the same time, it is an attractive and in some sense timeless object for theoretical study. Properties of the convex hull of the random point cloud X = xi have attracted interest for several decades, increasingly so in recent years; there is a nowvoluminous literature on the subject. The results could be significant for understanding outlier detection, or classification problems in machine learning.