An efficient method for finding intersections of many manifolds with application to patch based image processing

Solving inverse problems in signal processing often involves making prior assumptions about the signal being reconstructed. Here the appropriateness of the chosen model greatly determines the quality of the final result. Recently it has been proposed to model images by representing them as sets of smaller patches arising from an underlying manifold. This model has been shown to be surprisingly effective in tasks such as denoising, inpainting, and superresolution. However, such a representation is fraught with the difficulty of finding the intersection of many presumably non-linear manifolds in high-dimensional space, which precludes much of its potential use. This paper proposes an efficient method of solving this problem using a kernel methods variant of the projection onto convex sets algorithm to quickly find the intersection of many manifolds while learning their non-linear structure. Indeed the final solution can even be expressed in closed form. We foresee our method allowing a patch-based regularization to be applied across a wide variety of inverse problems, including compressive sensing, inpainting, deconvolution, etc. Indeed, as a proof of concept of our approach, we show how it can be employed in the regularization of an image denoising problem. Here, it even outperforms a state-of-the-art denoising technique, non-local means.