Performance analysis of &#x2113;<inf>1</inf>-synthesis with coherent frames

Abstract

Signals with sparse frame representations comprise a much more realistic model of nature than that with orthonomal bases. Studies about the signal recovery associated with such sparsity models have been one of major focuses in compressed sensing. In such settings, one important and widely used signal recovery approach is known as ℓ 1synthesis (or Basis Pursuit). We present in this article a more effective performance analysis (than what are available) of this approach in which the dictionary D may be highly, and even perfectly correlated. Under suitable conditions on the sensing matrix Φ, an error bound of the recovered signalf (by the ℓ 1 -synthesis method) is established. Such an error bound is governed by the decaying property ofD * o f , where f is the true signal andD o denotes the optimal dual frame of D in the sense that D * of 1 produces the smallest D * f 1 in value among all dual framesD of D and all feasible signalsf . This new performance analysis departs from the usual description of the combo ΦD, and places the description on Φ. Examples are demonstrated to show that when the usual analysis fails to explain the working performance of the synthesis approach, the newly established results do.

A null space analysis of the ℓ1-synthesis method in dictionary-based compressed sensing

“…Third, it appears that the ℓ 1 -analysis method is a sub-problem of ℓ 1 -synthesis by the work of Li et al [21], which we will elaborate below.…”

“…In all of the above discussions, it is assumed that the signal z 0 is sparse with respect to an orthonormal basis. A recent direction of interest in compressed sensing concerns problems where signals are sparse in an overcomplete dictionary D instead of a basis, see [24,4,21,2,11]. Here D is a d × n matrix with full column rank.…”

“…But it requires the sparsity of D * z 0 , which does not seem to fit into the original setting very well. The work in [21] proposes an optimal dual based ℓ 1 -analysis approach along with an efficient algorithm, but the stability result does not hold universally for all frame-sparse signals.…”

The &#x2113;<inf>1</inf> analysis approach by sparse dual frames for sparse signal recovery represented by frames

“…In a related article also presented at ISIT 2012 [14], a notion of optimal-dual-based analysis approach is also outlined,…”

“…It is important that we outline a fundamental difference between the sparse-dual approach (2.2) and the optimal-dual approach (2.3). We have shown in Theorem 2 of [14], that the optimal-dual-based analysis problem (2.3) is equivalent to the 1 synthesis approach (1.2). Because it minimizes the 1 norm of D * f simultaneously over the dual frame D and the signal f .…”