Second Workshop on Mathematical Analysis of Images in Bordeaux

Registration Venue Accomodation

Andrés Almansa: Covariance Trees for efficiently learning and quering local Gaussian models in non-local patch-based image and surface reconstruction problems.
co-authors: Thierry Guillemot, Tamy Boubekeur


Gaussian Mixture Models have become one of the major tools in modern statistical image processing, and allowed performance breakthroughs in patch-based image denoising and restoration problems. Nevertheless, their adoption level was kept relatively low because: (i) the number of models is a critical parameter that is difficult to set, and (ii) the computational cost associated to learning such models on large image databases. This work provides a flexible and generic tool for dealing with such models without the computational penalty or parameter tuning difficulties associated to a naive implementation of GMM-based image restoration tasks. It does so by organising the data manifold in a hirerachical multiscale structure (the Covariance Tree) that can be queried at various scale levels around any point in feature-space. We start by explaining how to construct a Covariance Tree from a subset of the input data, how to enrich its statistics from a larger set in a streaming process, and how to query it efficiently, at any scale. We then demonstrate its usefulness on several applications, including non-local image filtering, data-driven denoising, reconstruction from random samples and surface modeling from unorganized 3D points sets. We end with a discussion on possible applications in image and surface inpainting.

chapter 4 & 5 in preprint