Elastic shape matching of parameterized surfaces using square root normal fields.

In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple \ensuremathL2\ensuremathL2 metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods

Imposing Hard Constraints on Soft Snakes

An approach is presented for imposing generic hard constraints on deformable models at a low computational cost, while preserving the good convergence properties of snake-like models. We believe this capability to be essential not only for the accurate modeling of individual objects that obey known geometric and semantic constraints but also for the consistent modeling of sets of objects. Many of the approaches to this problem that have appeared in the vision literature rely on adding penalty terms to the objective functions. They rapidly become untractable when the number of constraints increases. Applied mathematicians have developed powerful constrained optimization algorithms that, in theory, can address this problem. However, these algorithms typically do not take advantage of the specific properties of snakes. We have therefore designed a new algorithm that is closely related to Lagrangian methods but is tailored to accommodate the particular brand of deformable models..

3D Point Correspondence by Minimum Description Length in Feature Space

Abstract. Finding point correspondences plays an important role in automatically building statistical shape models from a training set of 3D surfaces. For the point correspondence problem, Davies et al. [1] proposed a minimum-descriptionlength-based objective function to balance the training errors and generalization ability. A recent evaluation study [2] that compares several well-known 3D point correspondence methods for modeling purposes shows that the MDL-based approach [1] is the best method. We adapt the MDL-based objective function for a feature space that can exploit nonlinear properties in point correspondences, and propose an efficient optimization method to minimize the objective function directly in the feature space, given that the inner product of any vector pair can be computed in the feature space. We further employ a Mercer kernel [3] to define the feature space implicitly. A key aspect of our proposed framework is the generalization of the MDL-based objective function to kernel principal component analysis (KPCA) [4] spaces and the design of a gradient-descent approach to minimize such an objective function. We compare the generalized MDL objective function on KPCA spaces with the original one and evaluate their abilities in terms of reconstruction errors and specificity. From our experimental results on different sets of 3D shapes of human body organs, the proposed method performs significantly better than the original method.

Morphometric analysis of brain structures for improved discrimination

Abstract. We perform discriminative analysis of brain structures using morphometric information. Spherical harmonics technique and point distribution model are used for shape description. Classification is performed using linear discriminants and support vector machines with several feature selection approaches. We consider both inclusion and exclusion of volume information in the discrimination. We perform extensive experimental studies by applying different combinations of techniques to hippocampal data in schizophrenia and achieve best jackknife classification accuracies of 95 % (whole set) and 90 % (right-handed males), respectively. Our results find that the left hippocampus is a better predictor than the right in the complete dataset, but that the right hippocampus is a stronger predictor than the left in the right-handed male subset. We also propose a new method for visualization of discriminative patterns.