Anatomy in Curved Spaces - Non-linear Modeling of Deformation and Shape for Medical Imaging

PhD-defence by Stefan Sommer

Abstract

To obtain compact description of deformation while keeping the capacity of the deformation model, we present two results on registration and deformation modeling. We introduce the kernel bundle framework which extends the LDDMM framework to represent deformation at multiple scales while preserving much of the mathematical structure underlying the original framework. The kernel bundle in particular allows application of sparse priors across scales, and we use this property to obtain compact representations while keeping the capacity of the deformation model and its ability to generalize to test data.

In addition, sparse deformation representation with LDDMM is restricted by representing only translational motion. We introduce higher order kernels in the framework to allow modeling of locally affine deformation. We show how the increased description capacity allows registration with very few parameters, and we apply the kernels to register MR scans of patients suffering from Alzheimer's disease.

We present algorithms for computing the differential of the Exponential map and second order derivatives on Riemannian manifolds leading to an algorithm for computing exact Principal Geodesic Analysis, a generalization of PCA to manifolds which is exact as it does not use the common tangent space linearization. We evaluate the results obtained with the exact algorithm against the standard PGA method and provide new insight into when modelling non-linearity is beneficial.

To reduce annotation variation in point based models, we introduce the bicycle chain shape model for 2D-shape representation. We develop tools for performing statistics on the embedded Riemannian manifold comprising the model, and we apply the method to represent and perform statistics on a dataset of human vertebrae X-rays.

Assessment Committee:

  • Chairman: Associate Professor Jon Sporring, Department of Computer Science, Copenhagen University
  • Member 1: Associate Professor Sarang Joshi, Department of Bioengeneering, University of Utah
  • Member 2: Professor Alain Trouvé, Centre de Mathematiques et Leurs Applications, Ecole Normale Superieure, Cahan, France
  • Academic supervisor: Assistant Professor Francois Lauze

For an electronic copy of the thesis, please contact dinariis@diku.dk