A Sub-Riemannian Modular Approach for Diffeomorphic Deformations

Abstract

Abstract. We develop a generic framework to build large deformations from a combination of base modules. These modules constitute a dynamical dictionary to describe transformations. The method, built on a coherent sub-Riemannian framework, defines a metric on modular deformations and characterises optimal deformations as geodesics for this metric. We will present a generic way to build local affine transformations as deformation modules, and display examples. IntroductionA central aspect of Computational Anatomy is the comparison of different shapes, which are encoded as meshes or images. A common approach is the study of deformations matching one shape onto another, so that the differences between the two shapes are encoded by the deformation parameters [9,11,17]. In order to study differences between subjects on a particular structure, it should be useful to constrain locally the deformation, to favour realistic anatomic deformations, or to introduce some anatomical priors. For instance, for cortical surfaces with different sulci topography, one can prefer to favour lateral displacement over the creation of new sulci. Large deformations are commonly obtained through the integration of a vector field [4,10,13,18] and a natural route is to introduce the constraints in the vector fields instead of the final diffeomorphism [19]. The vector field could be restricted, via a finite dimensional control variable, to a state dependent finite dimensional subspace generated by a finite basis and conceptualized in structures called hereafter deformation modules. Deformation modules should create interpretable deformations, and several modules should be allowed to combine to form more complex compound deformation modules in the spirit of Grenander's Pattern Theory [7].Preliminary instantiations of the concept of deformation modules can be found in several early works. In the poly-affine framework [3,14,20], deformations are created by the integration of a poly-affine stationary vector field. This vector field is a sum of few local affine transformations, which are then easily interpretable and share some features of deformation modules even if regions of each affine component are not updated during the deformation. In the LDDMM framework, a Riemannian structure is defined on the group of diffeomorphisms and optimal matchings are geodesics for this metric [2,10]. Several discretization schemes based on landmarks induce examples of finite dimensional state dependent representations of the velocity fields updated along the deformation and could be rephrased inside our definition of deformation modules. Note that the discretization scheme are thought of as approximations of the unconstrained nonparametrized infinite dimensional case. In a recently developed sparse-LDDMM framework [6,12,15], the vector field is constrained to be a sum of a fixed number of local translations, carried by control points, with a more clearer focus on finite-dimensionality and local interpretability. Extension to locally more complex transformation a...

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