# Synthesis versus analysis in patch-based image priors

## TL;DR

This paper explores the analysis and synthesis approaches in patch-based image priors, establishing their differences, proposing a synthesis model, and demonstrating its practical application in image denoising.

## Contribution

It introduces a synthesis formulation for patch-based image priors, contrasting it with existing analysis models, and provides an ADMM-based method for denoising.

## Key findings

- All existing patch-based models are analysis-based.
- A new synthesis formulation is proposed and formalized.
- The synthesis approach is computationally feasible for denoising.

## Abstract

In global models/priors (for example, using wavelet frames), there is a well known analysis vs synthesis dichotomy in the way signal/image priors are formulated. In patch-based image models/priors, this dichotomy is also present in the choice of how each patch is modeled. This paper shows that there is another analysis vs synthesis dichotomy, in terms of how the whole image is related to the patches, and that all existing patch-based formulations that provide a global image prior belong to the analysis category. We then propose a synthesis formulation, where the image is explicitly modeled as being synthesized by additively combining a collection of independent patches. We formally establish that these analysis and synthesis formulations are not equivalent in general and that both formulations are compatible with analysis and synthesis formulations at the patch level. Finally, we present an instance of the alternating direction method of multipliers (ADMM) that can be used to perform image denoising under the proposed synthesis formulation, showing its computational feasibility. Rather than showing the superiority of the synthesis or analysis formulations, the contributions of this paper is to establish the existence of both alternatives, thus closing the corresponding gap in the field of patch-based image processing.