In the framework of large deformation diffeomorphic metric mapping (LDDMM), we present a practical methodology to integrate prior knowledge about the registered shapes in the regularizing metric. Our goal is to perform rich anatomical shape comparisons from volumetric images with the mathematical properties offered by the LDDMM framework. We first present the notion of characteristic scale at which image features are deformed. We then propose a methodology to compare anatomical shape variations in a multi-scale fashion, i.e., at several characteristic scales simultaneously. In this context, we propose a strategy to quantitatively measure the feature differences observed at each characteristic scale separately. After describing our methodology, we illustrate the performance of the method on phantom data. We then compare the ability of our method to segregate a group of subjects having Alzheimer's disease and a group of controls with a classical coarse to fine approach, on standard 3D MR longitudinal brain images. We finally apply the approach to quantify the anatomical development of the human brain from 3D MR longitudinal images of pre-term babies. Results show that our method registers accurately volumetric images containing feature differences at several scales simultaneously with smooth deformations.

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This paper develops a method for higher order parametric regression on diffeomorphisms for image regression. We present a principled way to define curves with nonzero acceleration and nonzero jerk. This work extends methods based on geodesics which have been developed during the last decade for computational anatomy in the large deformation diffeomorphic image analysis framework. In contrast to previously proposed methods to capture image changes over time, such as geodesic regression, the proposed method can capture more complex spatio-temporal deformations. We take a variational approach that is governed by an underlying energy formulation, which respects the nonflat geometry of diffeomorphisms. Such an approach of minimal energy curve estimation also provides a physical analogy to particle motion under a varying force field. This gives rise to the notion of the quadratic, the cubic and the piecewise cubic splines on the manifold of diffeomorphisms. The variational formulation of splines also allows for the use of temporal control points to control spline behavior. This necessitates the development of a shooting formulation for splines. The initial conditions of our proposed shooting polynomial paths in diffeomorphisms are analogous to the Euclidean polynomial coefficients. We experimentally demonstrate the effectiveness of using the parametric curves both for synthesizing polynomial paths and for regression of imaging data. The performance of the method is compared to geodesic regression.