Analysis versus synthesis in signal priors

The concept of prior probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors assign probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. In this paper we describe these two prior classes, focusing on the distinction between them. We show that although when reducing to the complete and under-complete formulations the two become equivalent, in their more interesting overcomplete formulation the two types depart. Focusing on the ℓ¹ denoising case, we present several ways of comparing the two types of priors, establishing the existence of an unbridgeable gap between them.