  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.
