Continuous Compressed Sensing Hilbert-Schmidt Integral Operator

Abstract

Continuous Compressed-Sensing-Karhunen-Loéve Expansion (CS-KLE) has been proposed. Compressed sensing has been proposed as a highly efficient computational method to represent compressible signals using a few numbers of linear functional. On the other hand, KLE is known to be the optimum orthogonal decomposition. While both methodologies have been addressed comprehensively and independently in the literature, their relationship has not been studied. In this work, we study the relation between random sampling and KLE. In particular, we examine how the doubly orthogonal property is affected by the mutual coherency and RIP of the compressed sensing. We conduct a detailed theoretical study of random sampling and KLE. We prove Compressed Sensing Hilbert-Schmidt integral operator as double integral acting on the signal space and its dual space. The proof of the discussed integral operator follows from the Kolmogorov Conditional Expectation theorem. Then, we propose two formulations to compute CS-KLE relation, one through Mercer’s theorem, ans second through Green’s theorem. We also prove the convergence of CS-KLE with respect to RIP. We show that there is a transition point in the spectral overlap between the estimated and actual signal spaces. The transition point occurs for the optimum subspace of the given compressible signal. Numerical simulation is presented by applying CS-KLE to semi-infinite and infinite-dimensional signals, and also Magnetic Resonance Image (MRI).