Identifying the shape difference between two groups of anatomical objects is important for medical image analysis and computer-aided diagnosis. A method called "discriminative direction" in the literature has been proposed to solve this problem. In that method, the shape difference between groups is identified by deforming a shape along the discriminative direction. This paper conducts a thorough study about inferring this discriminative direction in an efficient and accurate way. First, finding the discriminative direction is reformulated as a preimage problem in kernel-based learning. This provides a complementary but conceptually simpler solution than the previous method. More importantly, we find that a shape deforming along the original discriminative direction cannot faithfully maintain its anatomical correctness. This unnecessarily introduces spurious shape differences and leads to inaccurate analysis. To overcome this problem, this paper further proposes a regularized discriminative direction by requiring a shape to conform to its underlying distribution when it deforms. Two different approaches are developed to impose the regularization, one from the perspective of probability distributions and the other from a geometric point of view, and their relationship is discussed. After verifying their superior performance through controlled experiments, we apply the proposed methods to detecting and localizing the hippocampal shape difference between sexes. We get results consistent with other independent research, providing a more compact representation of the shape difference compared with the established discriminative direction method.

Matching and registration of shapes is a key issue in Computer Vision, Pattern Recognition, and Medical Image Analysis. This paper presents a shape representation framework based on Gaussian curvature and Markov random fields (MRFs) for the purpose of shape matching. The method is based on a surface mesh model in R3, which is projected into a two-dimensional space and there modeled as an extended boundary closed Markov random field. The surface is homeomorphic to S2. The MRF encodes in the nodes entropy features of the corresponding similarities based on Gaussian curvature, and in the edges the spatial consistency of the meshes. Correspondence between two surface meshes is then established by performing probabilistic inference on the MRF via Gibbs sampling. The technique combines both geometric, topological, and probabilistic information, which can be used to represent shapes in three dimensional space, and can be generalized to higher dimensional spaces. As a result, the representation can be used for shape matching, registration, and statistical shape analysis.