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Sparse components of images and optimal atomic decompositions. (English) Zbl 0995.65150

This paper is stimulated by recent interactions between computational neuroscience, visual physiology, and statistical analysis. Sparse components of images have elongated shapes. Their findings assume a wide range of positions, orientations, and scales. Heuristic procedures for sparse components analysis (SCA) are computationally intensive and have been mainly used on small image patches. The author develops a general framework for a mathematical model of SCA and gives an approximate solution for a synthetic image model.
Let \({\mathcal F}\subset L^2([0, 1]^2)\) be a class of objects. The objects are represented by linear combinations of atoms from an overcomplete dictionary (e.g., wavelet packets, collection of multiscale Gabor functions, wedgelet dictionary). The sparsity of representation of an object is measured by the \(l^p\)-norm \((p>0)\) of the coefficients in the linear combination. Let \({\mathcal F}= \text{START}^\alpha\) be the class of black and white images with the black region consisting of a star-shaped set with an \(\alpha\)-smooth boundary \((1< \alpha\leq 2)\). It is shown that there is an optimal sparsity of representation of objects of \(\text{STAR}^\alpha\). There are decompositions with finite \(l^p\)-norm for \(p> 2/(\alpha+1)\) but not for \(p< 2/(\alpha+1)\).
Further, the optimal degree of sparsity is nearly attained using atomic decompositions based on the wedgelet dictionary [see the author, Ann. Stat. 27, 859-897 (1999; Zbl 0957.62029)]. The fine-scaled atoms used in the adaptive atomic decomposition are highly anisotropic and occupy a range of positions, scales, and locations. This agrees qualitatively with the visual appearance of empirically determined sparse components of natural images.

MSC:

65T60 Numerical methods for wavelets
68U10 Computing methodologies for image processing
62C20 Minimax procedures in statistical decision theory
92C55 Biomedical imaging and signal processing
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory

Citations:

Zbl 0957.62029
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