A Sub-Riemannian Modular Framework for Diffeomorphism-Based Analysis of Shape Ensembles

Abstract

Abstract. Deformations, and dieormophisms in particular, have played a tremendous role in the eld of statistical shape analysis, as a proxy to measure and interpret dierences between similar objects but with dierent shapes. Dieomorphisms usually result from the integration of a ow of regular velocity elds, whose parameters have not enabled so far a full control of the local behaviour of the deformation.In this work, we propose a new mathematical and computational framework, in which these velocity elds are constrained to be built through a combination of local deformation modules with few degrees of freedom. Deformation modules contribute to the global velocity eld, and interact with it during integration so that the local modules are transported by the global dieomorphic deformation under construction. Such modular dieomorphisms are used to deform shapes and to provide the shape space with a sub-Riemannian metric.We then derive a method to estimate a Fréchet mean from a series of observations, and to decompose the variations in shape observed in the training samples into a set of elementary deformation modules encoding distinctive and interpretable aspects of the shape variability. We show how this approach brings new solutions to long lasting problems in the elds of computer vision and medical image analysis. For instance, the easy implementation of priors in the type of deformations oers a direct control to favor one solution over another in situations where multiple solutions may t the observations equally well. It allows also the joint optimisation of a linear and a non-linear deformation between shapes, the linear transform simply being a particular type of modules.The proposed approach generalizes previous methods for constructing dieomorphisms and opens up new perspectives in the eld of statistical shape analysis.

Incorporation of a Deformation Prior in Image Reconstruction

“…The object of this article is to show how the framework of deformation modules introduced in [14] can be used to incorporate motion prior in image reconstruction.…”

“…We now give a slightly simplified formal definition of a deformation module from the one defined in [14]: Definition 2. Let k, ∈ N * .…”

“…Remark 1. In [14], the deformation module was defined by a five-fold (O, H, ζ, ξ, c) where ξ is the infinitesimal action associated to the shape space O. Here in order to simplify the notations (and as in the examples we present there is no ambiguity about them ), we will denote all the infinitesimal actions v · o and they will be implicitly defined via the shape spaces of geometrical descriptors.…”

“…. As shown in [14], the uniform embedding condition is stable result under combination and then an easy way to build complex deformation modules satisfying the uniform embedding condition is to combine several simple deformation modules satisfying this condition.…”

“…Several frameworks allow to incorporate particular prior in deformation models [5,4,12,22,23,27,28,29,31] so that they are adapted to specific situations. The goal of this article is to show how a generic prior on the set of deformations can be incorporated via the notion of deformation modules [14] so that only the desired solutions are used to reconstruct an image from the observed data and the initial template. For instance in the case of biological images, this framework would ensure that only the deformations that are possible from a biological point of view are considered.…”