Local Kernels that Approximate Bayesian Regularization and Proximal Operators

In this work, we broadly connect kernel-based filtering (e.g. approaches such as the bilateral filter and nonlocal means, but also many more) with general variational formulations of Bayesian regularized least squares, and the related concept of proximal operators. Variational/Bayesian/proximal formulations often result in optimization problems that do not have closed-form solutions, and therefore typically require global iterative solutions. Our main contribution here is to establish how one can approximate the solution of the resulting global optimization problems using locally adaptive filters with specific kernels. Our results are valid for small regularization strength (i.e. weak noise) but the approach is powerful enough to be useful for a wide range of applications because we expose how to derive a "kernelized" solution to these problems that approximates the global solution in one shot, using only local operations. As another side benefit in the reverse direction, given a local data-adaptive filter constructed with a particular choice of kernel, we enable the interpretation of such filters in the variational/Bayesian/proximal framework.