  Inverse problems play a central role for many classical computer vision and
image processing tasks. Many inverse problems are ill-posed, and hence require
a prior to regularize the solution space. However, many of the existing priors,
like total variation, are based on ad-hoc assumptions that have difficulties to
represent the actual distribution of natural images. Thus, a key challenge in
research on image processing is to find better suited priors to represent
natural images.
  In this work, we propose the Adaptive Quantile Sparse Image (AQuaSI) prior.
It is based on a quantile filter, can be used as a joint filter on guidance
data, and be readily plugged into a wide range of numerical optimization
algorithms. We demonstrate the efficacy of the proposed prior in joint
RGB/depth upsampling, on RGB/NIR image restoration, and in a comparison with
related regularization by denoising approaches.
