Diffusion maps as a framework for shape modeling

Statistical analysis has proven very successful in the image processing community. Linear methods such as principal component analysis (PCA) measure the degree of correlation in datasets to extract meaningful information from high-dimensional data. PCA was successfully applied in several applications such as image segmentation with shape priors and image denoising. The major assumption in these applications is that the dataspace is a linear space. However, this assumption is mainly wrong and as a consequence several non-linear methods were developed, among which diffusion maps were recently proposed. In this paper we develop a variational framework to compute the pre-image using diffusion maps. The key-problem of pre-image determination consists of, given its embedding, recovering a point. Therefore 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 then define the pre-image as an interpolation with the targeted embedding. The new methodology can then be used for regularization in image segmentation as well as for shape and image denoising. We demonstrate our method by testing our new non-linear shape prior for shape segmentation of partially occluded objects. Further, we report results on denoising 2D images and 3D shapes 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.

Research Highlights
► Approximation of an infinite dimensional space (shape space) by a lower dimensional space using diffusion maps.
► We propose two variational models to project data onto the low-dimensional representation.
► We apply our projection algorithm to solve the problem of denoising shapes and images and image segmentation.