Theory And Method For Compressed Sensing Under Tight Frames

A large number of applications in signal and image processing point to problems where signals are not sparse, but sparsity is expressed in terms of tight frames. That is, an unknown signal x∈Rn can be expressed by a tight frame x=Dx, where D∈Rn×d (n<<d) is a tight frame, x∈Rd is a k-sparse vector.There are two part of our main result. Firstly, the paper aims to give new conditions for reconstructing signals from undersampled data in the situation that signals are sparse in terms of a tight frame. Let Φ∈Rm×n be a measurement matrix, D∈Rn×d be a Parseval frame, D*∈Rd×n is the Hermitian conjugate of D (satisfies DD*=I), and δk (k<d) be the D-restricted isometry constant (abbreviated D-RIC) with order k of the measurement matrix Φ. We present a new D-RIC bound for recovering signals x∈Rn with D*x∈Rd being sparse, that is, if the measurement matrix∈satisfies the condition δk<1/3or δ2k≤1/2, then signals x∈Rn with D*x∈Rd being k-sparse can be recovered exactly via the l1-analysis approach; secondly, when we give a specific Parseval frame that is Harmonic frame, we can exactly recovery signals via Supp-projection algorithm. The numerical experiments demonstrate that our algorithm behaves very well.