Pre-Image as Karcher Mean using Diffusion Maps: Application to shape and image denoising

Abstract : In the context of shape and image modeling by manifold learning, we focus on the problem of denoising. A set of shapes or images being known through given samples, we capture its structure thanks to the Diffusion Maps method. Denoising a new element classically boils down to the key-problem of pre-image determination, i.e.recovering a point, given its embedding. We propose to model the underlying manifold as the set of Karcher means of close sample points. This non-linear interpolation is particularly well-adapted to the case of shapes and images. We define the pre-image as such an interpolation having the targeted embedding. Results on synthetic 2D shapes and on real 2D images and 3D shapes are presented and demonstrate the superiority of our pre-image method compared to several state-of-the-art techniques in shape and image denoising based on statistical learning techniques.