Geodesic image regression with a sparse parameterization of diffeomorphisms

Abstract

Image regression allows for time-discrete imaging data to be modeled continuously, and is a crucial tool for conducting statistical analysis on longitudinal images. Geodesic models are particularly well suited for statistical analysis, as image evolution is fully characterized by a baseline image and initial momenta. However, existing geodesic image regression models are parameterized by a large number of initial momenta, equal to the number of image voxels. In this paper, we present a sparse geodesic image regression framework which greatly reduces the number of model parameters. We combine a control point formulation of deformations with a L¹ penalty to select the most relevant subset of momenta. This way, the number of model parameters reflects the complexity of anatomical changes in time rather than the sampling of the image. We apply our method to both synthetic and real data and show that we can decrease the number of model parameters (from the number of voxels down to hundreds) with only minimal decrease in model accuracy. The reduction in model parameters has the potential to improve the power of ensuing statistical analysis, which faces the challenging problem of high dimensionality.

Fig. 1

Synthetic image evolution generated by shooting the baseline image (far left) along predefined initial momenta, constraining the resulting evolution to be geodesic.

Fig. 2

Baseline shape and initial momenta estimated for several values of sparsity parameter γ_sp. An increase in the sparsity parameter leads to a more compact representation, with momenta located in areas of dynamic change over time.

Fig. 3

Impact of the sparsity parameter for the synthetic experiment (left) and for the developing brain (right). There is a range of values of the sparsity parameter which result in a considerable decrease in the number of control points for only minimal increase in the relative error of the data matching term.

Fig. 4

Observed data acquired from the same child at 6, 12, and 24 months of age.

Fig. 5

Baseline shape and initial momenta estimated for several values of sparsity parameter γ_sp. The baseline shape was estimated at 24 months of age and evolution was followed backward in time. As the sparsity parameter is increased, the momenta cluster around the perimeter of the brain and around the lateral ventricles. From this, we can infer that the scale of the brain and ventricles are the most salient features describing the development of this child.