Sparse representations of signals for information recovery from incomplete data

Abstract

Mathematical modeling of inverse problems in imaging, such as inpainting, deblurring and denoising, results in ill-posed, i.e. underdetermined linearsystems. Sparseness constraintis used often to regularize these problems.That is because many classes of discrete signals (e.g. naturalimages), when expressed as vectors in a high-dimensional space, are sparse in some predefined basis or a frame(fixed or learned). An efficient approach to basis / frame learning is formulated using the independent component analysis (ICA)and biologically inspired linear model of sparse coding. In the learned basis, the inverse problem of data recovery and removal of impulsive noise is reduced to solving sparseness constrained underdetermined linear system of equations. The same situation occurs in bioinformatics data analysis when novel type of linear mixture model with a reference sample is employed for feature extraction. Extracted features can be used for disease prediction and biomarker identification