A Note on Sparsification by Frames

Abstract

The purpose of this note is to establish a new generalized Dictionary-Restricted Isometry Property (D-RIP) sparsity bound constant for compressed sensing. For fulfilling D-RIP, the constant ${\delta }_k$ is used in the definition: $(1-{\delta }_k)\Vert Dv{\Vert }_2^2\le \Vert \Phi Dv{\Vert }_2^2\le (1+{\delta }_k)\Vert Dv{\Vert }^2$. We prove that signals with $k$-sparse $D$-representation can be reconstructed if ${\delta }_{2k}<\frac{2}{3}$. The approach in this note can be extended to obtain other D-RIP bounds (i.e., ${\delta }_{tk}$).

References in corpus (2)

• On Recovery of Sparse Signals via ${\ell }_1$ Minimization
• New Bounds for Restricted Isometry Constants

Cited by in corpus (1)

• The ${\ell }_1$-analysis with redundant dictionary in phase retrieval