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Shapes of boundaries can play an important role in characterizing objects in images. Shape analysis involves choosing mathematical representations of shapes, deriving tools for quantifying shape differences, and characterizing imaged objects according to the shapes of their boundaries. We describe an approach for statistical analysis of shapes of closed curves using ideas from differential geometry. In this thesis, we initially focus on characterizing shapes of continuous curves, both open and closed, in R^2 and then propose extensions to more general elastic curves in R^n. Under appropriate constraints that remove shape-preserving transformations, these curves form infinite-dimensional, non-linear spaces, called shape spaces. We impose a Riemannian structure on the shape space and construct geodesic paths under different metrics. Geodesic paths are used to accomplish a variety of tasks, including the definition of a metric to compare shapes, the computation of intrinsic statistics for a set of shapes, and the definition of intrinsic probability models on shape spaces. Riemannian metrics allow for the development of a set of tools for computing intrinsic statistics for a set of shapes and clustering them hierarchically for efficient retrieval. Pursuing this idea, we also present algorithms to compute simple shape statistics --- means and covariances, --- and derive probability models on shape spaces using local principal component analysis (PCA), called tangent PCA (TPCA). These concepts are demonstrated using a number of applications: (i) unsupervised clustering of imaged objects according to their shapes, (ii) developing statistical shape models of human silhouettes in infrared surveillance images, (iii) interpolation of endo- and epicardial boundaries in echocardiographic image sequences, and (iv) using shape statistics to test phylogenetic hypotheses. Finally, we present a framework for incorporating prior information about high-probability shapes in the process of contour extraction and object recognition in images. Here one studies shapes as elements of an infinite-dimensional, non-linear quotient space, and statistics of shapes are defined and computed intrinsically using differential geometry of this shape space. Prior models on shapes are constructed using probability distributions on tangent bundles of shape spaces. Similar to the past work on active contours, where curves are driven by vector fields based on image gradients and roughness penalties, we incorporate prior shape knowledge also in form of gradient fields on curves. Through experimental results, we demonstrate the use of prior shape models in estimation of object boundaries, and their success in handling partial obscuration and missing data. Furthermore, we describe the use of this framework in shape-based object recognition or classification. This Bayesian shape extraction approach is found to yield a significant improvement in detection of objects in presence of occlusions or obscurations.
A Dissertation submitted to the Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Includes bibliographical references.
Anuj Srivastava, Professor Co-Directing Dissertation; Anke Meyer-Baese, Professor Co-Directing Dissertation; Eric Klassen, Outside Committee Member; Rodney Roberts, Committee Member; Simon Y. Foo, Committee Member; John W. Fisher, III, Committee Member.
Florida State University
Joshi, S. H. (2007). Inferences in Shape Spaces with Applications to Image Analysis and Computer Vision. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd-3697