Statistical Theory for Shape Analysis of Curves and Surfaces with Applications in Image Analysis, Biometrics, Bioinformatics and Medical Diagnostics

摘要

Over the five year period, this project has mainly been concerned about developing a theory for statistical analysis of shapes of objects, both two and three-dimensional. Focusing on the boundaries of these objects, our framework is for shape analysis of curves and surfaces. The main achievements were development of tools for: (1) Quantifying Shape Differences: Given any two objects, we can quantify differences between their shapes. (2) Achieve Desired Invariance: Our notion of shape is invariant to certain transformations of curves - rigid motion, scaling and re-parameterization. (3) Compute Summary Statistics: Given a collection of shapes and shape classes we can generate summary statistics - mean, covariance, etc, to characterize a shape class. (4) Stochastic Modeling: We have developed probability models that capture observed variability in shape classes. These models form priors for Bayesian inferences. (5) Statistical Inferences: We have studied statistical evaluations, such as hypothesis testing, likelihood ratios, performance bounds, etc, for shape analysis. The salient components of this differential geometric framework are following. First, we define a space of curves or surfaces by choosing a mathematical representation for these objects and establish a Hilbert submanifold(s) for such representations. Then, we choose a Riemannian metric, usually an elastic metric for measuring distances on such manifolds. We arrive at a shape manifold by imposing remaining invariances in the representation. For these shape spaces, we have developed two numerical techniques for computing geodesic paths. Finally, we define and compute empirical statistics, and define probability models on tangent bundles. The resulting statistical models are then used to characterize objects in images according to shapes, for using in object detection, tracking and recognition. We have demonstrated these tools in different application understanding.