Sampling in the Analysis Transform Domain

TL;DR

This paper introduces a novel sampling scheme for analysis model signals that enables the use of existing synthesis-based algorithms for recovery, bridging the gap between analysis and synthesis sparse modeling.

Contribution

It proposes a new sampling strategy for analysis signals that allows existing synthesis algorithms to be used for recovery, expanding their applicability.

Findings

Sampling scheme enables analysis signals to be recovered with synthesis algorithms.

The approach applies to signals with analysis dictionaries like frames or finite difference operators.

It broadens the utility of synthesis methods in analysis sparse modeling.

Abstract

Many signal and image processing applications have benefited remarkably from the fact that the underlying signals reside in a low dimensional subspace. One of the main models for such a low dimensionality is the sparsity one. Within this framework there are two main options for the sparse modeling: the synthesis and the analysis ones, where the first is considered the standard paradigm for which much more research has been dedicated. In it the signals are assumed to have a sparse representation under a given dictionary. On the other hand, in the analysis approach the sparsity is measured in the coefficients of the signal after applying a certain transformation, the analysis dictionary, on it. Though several algorithms with some theory have been developed for this framework, they are outnumbered by the ones proposed for the synthesis methodology. Given that the analysis dictionary is either a frame or the two dimensional finite difference operator, we propose a new sampling scheme for signals from the analysis model that allows to recover them from their samples using any existing algorithm from the synthesis model. The advantage of this new sampling strategy is that it makes the existing synthesis methods with their theory also available for signals from the analysis framework.