2D Affine and Projective Shape Analysis, and Bayesian Elastic Active Contours

• Other Authors: Bryner, Darshan W. (authoraut)
• Format: Others
• Language: English; English
• Published: Florida State University
• Subjects: Statistics
• Online Access: http://purl.flvc.org/fsu/fd/FSU_migr_etd-8534

An object of interest in an image can be characterized to some extent by the shape of its external boundary. Current techniques for shape analysis consider the notion of shape to be invariant to the similarity transformations (rotation, translation and scale), but often times in 2D images of 3D scenes, perspective effects can transform shapes of objects in a more complicated manner than what can be modeled by the similarity transformations alone. Therefore, we develop a general Riemannian framework for shape analysis where metrics and related quantities are invariant to larger groups, the affine and projective groups, that approximate such transformations that arise from perspective skews. Highlighting two possibilities for representing object boundaries -- ordered points (or landmarks) and parametrized curves -- we study different combinations of these representations (points and curves) and transformations (affine and projective). Specifically, we provide solutions to three out of four situations and develop algorithms for computing geodesics and intrinsic sample statistics, leading up to Gaussian-type statistical models, and classifying test shapes using such models learned from training data. In the case of parametrized curves, an added issue is to obtain invariance to the re-parameterization group. The geodesics are constructed by particularizing the path-straightening algorithm to geometries of current manifolds and are used, in turn, to compute shape statistics and Gaussian-type shape models. We demonstrate these ideas using a number of examples from shape and activity recognition. After developing such Gaussian-type shape models, we present a variational framework for naturally incorporating these shape models as prior knowledge in guidance of active contours for boundary extraction in images. This so-called Bayesian active contour framework is especially suitable for images where boundary estimation is difficult due to low contrast, low resolution, and presence of noise and clutter. In traditional active contour models curves are driven towards minimum of an energy composed of image and smoothing terms. We introduce an additional shape term based on shape models of prior known relevant shape classes. The minimization of this total energy, using iterated gradient-based updates of curves, leads to an improved segmentation of object boundaries. We demonstrate this Bayesian approach to segmentation using a number of shape classes in many imaging scenarios including the synthetic imaging modalities of SAS (synthetic aperture sonar) and SAR (synthetic aperture radar), which are notoriously difficult to obtain accurate boundary extractions. In practice, the training shapes used for prior-shape models may be collected from viewing angles different from those for the test images and thus may exhibit a shape variability brought about by perspective effects. Therefore, by allowing for a prior shape model to be invariant to, say, affine transformations of curves, we propose an active contour algorithm where the resulting segmentation is robust to perspective skews. === A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. === Fall Semester, 2013. === November 5, 2013. === Affine Shape, Bayesian Active Contours, Elastic Shape Analysis, Image Segmentation, Projective Shape, Riemannian Geometry === Includes bibliographical references. === Anuj Srivastava, Professor Directing Dissertation; Eric Klassen, Professor Directing Dissertation; Kyle Gallivan, University Representative; Fred Huffer, Committee Member; Wei Wu, Committee Member; Jinfeng Zhang, Committee Member.