Open AccessStable recovery of compressed sensing signals via optimal dual frame based ℓ q ‐minimisation for 0 < q ≤ 1
Author(s) -
Cao Chunhong,
Gao Xieping
Publication year - 2019
Publication title -
iet signal processing
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.384
H-Index - 42
eISSN - 1751-9683
pISSN - 1751-9675
DOI - 10.1049/iet-spr.2018.5008
Subject(s) - minimisation (clinical trials) , compressed sensing , signal recovery , computer science , algorithm , mathematics , statistics
Compressed sensing with sparse frame representation has received much greater attention than the orthonormal bases for its practical application in signal processing. One can expect exact recovery from undersampled data via an ℓ q ‐minimisation under some proper conditions imposed on the sensing/measurement matrix and sparse representation matrix. In this study, the authors first introduce a recovery condition named by B ‐ GRIP , which is actually a generalisation of the well‐known restricted isometry property that most of the compressed sensing problems depend on. Under this condition, they expand the performance analysis of compressed sensing problem by an ℓ q ‐minimisation problem for 0 < q ≤ 1 via the optimal dual frame. Finally, they present an iterative algorithm concerning the restoration of compressed sensing signals based on ℓ q ‐minimisation for 0 < q ≤ 1 as well as the convergence of the algorithm, and prove that the iterative sequence can not only approximate the original signal, but be the exact one.