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For analyzing shapes of planar, closed curves, we propose differential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinite-dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces. We use a Fourier(More)
We construct examples of nonresolvable generalized n-manifolds, n ≥ 6, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed n-manifold. We further investigate the structure of generalized manifolds and present a program for understanding their topology. By a generalized n-manifold we will mean a finite-dimensional(More)
Using a differential-geometric treatment of planar shapes, we present tools for: 1) hierarchical clustering of imaged objects according to the shapes of their boundaries, 2) learning of probability models for clusters of shapes, and 3) testing of newly observed shapes under competing probability models. Clustering at any level of hierarchy is performed(More)
We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments non-uniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape-preserving transformations. The elastic properties of the curves(More)
1. Introduction Among the many problems attendant to the discovery of exotic generalized man-ifolds 2, 3 is the normal bundle" problem, that is, the classiication of neighborhoods of generalized manifolds tamely embedded in generalized manifolds with the disjoint disks property. In this paper we study the normal structure of tame embeddings of a closed(More)
Quantitative analysis of gene expression domains and investigation of relationships between gene expression and developmental and phenotypic outcomes are central to advancing our understanding of the genotype–phenotype map. Gene expression domains typically have smooth but irregular shapes lacking homologous landmarks, making it difficult to analyze shape(More)
We develop a new framework for the quantitative analysis of shapes of planar curves. Shapes are modeled on elastic strings that can be bent, stretched or compressed at different rates along the curve. Shapes are treated as elements of a space obtained as the quotient of an infinite-dimensional Riemannian manifold of elastic curves by the action of a(More)
We employ 3D arrangements of curves to represent and analyze biological shapes, in particular, the anatomy of the human brain. The arrangements of curves may vary from fairly sparse - such as a collection of sulcal lines that coarsely approximates the global shape of the brain - to very dense decompositions of the cortical surface into space curves. A space(More)
We develop a model of continuous spherical shapes and use it to analyze the anatomy of the hippocampus. To account for the geometry of bends and folds, the model relies on a geodesic metric that is sensitive to first-order deformations. We construct an atlas of the hippocampus as a mean shape and develop statistical models to characterize quantitative and(More)
We develop a computational approach to non-parametric Fisher information geometry and algorithms to calculate geodesic paths in this geometry. Geodesics are used to quantify divergence of probability density functions and to develop tools of data analysis in information manifolds. The methodology developed is applied to several image analysis problems using(More)