Segmentation and Interpolation Algorithms Based on Information Geometry: Theory and Case Studies from Brain Imaging to Musculoskeletal Simulation

Abstract

In this thesis, we consider multivariate normal distributions for a statistical model as the points of a Riemannian manifold using Fisher information metric and investigate algorithms for distance calculation between two normal distributions and interpolation with several covariance matrices based on information geometry. These algorithms are applied to diffusion tensor magnetic resonance images (DT-MRI) segmentation and musculoskeletal dynamics simulation using shape-varying muscle mass models.

In the first topic of this thesis, we explore DT-MRI segmentation problem. Existing clustering-based methods for segmentation of diffusion tensor magnetic resonance images are based on a formulation of a similarity measure between diffusion tensors, or measures that combine translational and diffusion tensor distances in some ad hoc way. In this thesis we propose to use the Fisher information-based geodesic distance on the space of multivariate normal distributions as an intrinsic distance metric. An efficient and numerically robust shooting method is developed for computing the minimum geodesic distance between two normal distributions, together with an efficient graph-clustering algorithm for segmentation. Extensive experimental results involving both synthetic data and real DT-MRI images demonstrate that in many cases our method leads to more accurate and intuitively plausible segmentation results vis-a-vis existing methods.

In the second topic of this thesis, we explore musculoskeletal dynamics simulation reflecting shape-varying muscles. In contrast to existing algorithms that assumes the muscle as a rigid body, we have proposed shape-varying muscle mass model for musculoskeletal simulation which takes into account the changes in inertia that occur during movement. The mass matrix of the muscle is derived from kinetic energy considerations and the muscle Jacobian. We model each muscle deformation with a volume preserving linear transformation and show that our model can be applied to various muscle shapes. Using our model, it is possible to perform more accurate dynamic simulations of musculoskeletal model, but considerably increased computational cost is required. To make the computation more efficient, we propose an approximate dynamics algorithm consisting of two parts: in the offline stage, A parameterized B-Spline surface in P(n) that fits the sampled mass matrix values is first constructed. In order to interpolate the mass matrix, we examine the use of three different distance metrics on P(n) that ensure that the symmetric and positive-definite requirements of the mass matrix are always satisfied. In the online computation stage, an approximated mass matrix is constructed by a weighted average of the control points

this method effectively ensures that the online computational costs remain fixed regardless of the system complexity. The Coriolis forces can also be evaluated straightforwardly in terms of the partial derivatives of the mass matrix. Our model and algorithm lend itself to applications like motion optimization taking into account muscle mass deformations.