We propose a method for 3-D shape recognition based on inexact subgraph isomorphisms, by extracting topological and geometric properties of a shape in the form of a shape model, referred to as topo-geometric shape model (TGSM). In a nutshell, TGSM captures topological information through a rigid transformation invariant skeletal graph that is constructed in a Morse theoretic framework with distance function as the Morse function. Geometric information is then retained by analyzing the geometric profile as viewed through the distance function. Modeling the geometric profile through elastic yields a weighted skeletal representation, which leads to a complete shape signature. Shape recognition is carried out through inexact subgraph isomorphisms by determining a sequence of graph edit operations on model graphs to establish subgraph isomorphisms with a test graph. Test graph is recognized as a shape that yields the largest subgraph isomorphism with minimal cost of edit operations. In this paper, we propose various cost assignments for graph edit operations for error correction that takes into account any shape variations arising from noise and measurement errors.

Diffusion tensor magnetic resonance imaging (DT-MRI) provides a comprehensive characterization of white matter (WM) in the brain and therefore, plays a crucial role in the investigation of diseases in which WM is suspected to be compromised such as multiple sclerosis and neuropsychiatric disorders like schizophrenia. However changes induced by pathology may be subtle and affected regions of the brain can only be revealed by a group-based analysis of patients in comparison with healthy controls. This in turn requires voxel-based statistical analysis of spatially normalized brain DT images, as in the case of conventional MR images. However this process is rendered extremely challenging in DT-MRI due to the high dimensionality of the data and its inherent non-linearity that causes linear component analysis methods to be inapplicable. We therefore propose a novel framework for the statistical analysis of DT-MRI data using manifold-based techniques such as isomap and kernel PCA that determine the underlying manifold structure of the data, embed it to a manifold and help perform high dimensional statistics on the manifold to determine regions of difference between the groups of patients and controls. The framework has been successfully applied to DT-MRI data from patients with schizophrenia, as well as to study developmental changes in small animals, both of which identify regional changes, indicating the need for manifold-based methods for the statistical analysis of DTI.

Skewness of shape data often arises in applications (e.g., medical image analysis) and is usually overlooked in statistical shape models. In such cases, a Gaussian assumption is unrealistic and a formulation of a general shape model which accounts for skewness is in order. In this paper, we present a novel statistical method for shape modeling, which we refer to as the flexible skew-symmetric shape model (FSSM). The model is sufficiently flexible to accommodate a departure from Gaussianity of the data and is fairly general to learn a "mean shape" (template), with a potential for classification and random generation of new realizations of a given shape. Robustness to skewness results from deriving the FSSM from an extended class of flexible skew-symmetric distributions. In addition, we demonstrate that the model allows us to extract principal curves in a point cloud. The idea is to view a shape as a realization of a spatial random process and to subsequently learn a shape distribution which captures the inherent variability of realizations, provided they remain, with high probability, within a certain neighborhood range around a mean. Specifically, given shape realizations, FSSM is formulated as a joint bimodal distribution of angle and distance from the centroid of an aggregate of random points. Mean shape is recovered from the modes of the distribution, while the maximum likelihood criterion is employed for classification.

In this paper, we propose a numerical algorithm for extracting the topology of a three-dimensional object (2 dimensional surface) embedded in a three-dimensional space /spl Ropf//sup 3/. The method is based on capturing the topology of a modified Reeb graph by tracking the critical points of a distance function. As such, the approach employs Morse theory in the study of translation, rotation, and scale invariant skeletal graphs. The latter are useful in the representation and classification of objects in /spl Ropf//sup 3/.

Three dimensional objects viewed as surfaces or volumes embedded in /spl Ropf//sup 3/, are usually sampled along the z-dimension by planes for rendering or modeling purposes. The resulting intersections are curves or planar shapes which may in turn be modeled for parsimony of representation. Each curve or planar shape may be viewed as a point in a high dimensional manifold, thereby providing the notion of interpolation between two curves or two points on this manifold to reconstruct the subsurface that lies between the two slices. We exploit some recent results in formulating this interpolation problem as an optimization problem in /spl Ropf//sup 3/ to yield a simple interpolating spline, known as elasticae, which when evaluated at intermediate points yields curves which can in turn be instrumental in 3D reconstruction. The approach is particularly suited for interpolation between MRI slices and for modeling and reconstruction of 3D shapes.

Shape modeling and template learning form an important area of research in image analysis. The paper addresses the problem from a novel viewpoint /sup s/ing a new class of semiparametric skew distributions. Given several realizations of a shape, we represent its template as a joint distribution of angle and distance from the centroid for all points on the boundary. The shape boundary may be arbitrary and irregular but simple. Its corresponding distribution is learned from the scattered data points of the available boundary realizations. We first obtained a bimodal distribution of the radii distances for given angles and subsequently synthesize the overall joint distributions according to some prior on the angles. We substantiate our proposed methodology with a number of examples.