  This paper studies a type of image priors that are constructed implicitly
through the alternating direction method of multiplier (ADMM) algorithm, called
the algorithm-induced prior. Different from classical image priors which are
defined before running the reconstruction algorithm, algorithm-induced priors
are defined by the denoising procedure used to replace one of the two modules
in the ADMM algorithm. Since such prior is not explicitly defined, analyzing
the performance has been difficult in the past.
  Focusing on the class of symmetric smoothing filters, this paper presents an
explicit expression of the prior induced by the ADMM algorithm. The new prior
is reminiscent to the conventional graph Laplacian but with stronger
reconstruction performance. It can also be shown that the overall
reconstruction has an efficient closed-form implementation if the associated
symmetric smoothing filter is low rank. The results are validated with
experiments on image inpainting.
