Information geometry for shape analysis: Probabilistic models for shape matching and indexing

The study of shape analysis is a core field in computer vision. It is a fundamental building block of higher level cognitive tasks such as recognition. This research introduces novel approaches to basic shape analysis tasks, including shape matching and defining metrics for shape similarity. Our investigations into these methods yielded two supporting statistical tools that have general applicability outside the realm of shape analysis: a new wavelet density estimation procedure and a geometrically motivated model selection criterion to select the wavelet's multiscale decomposition levels. All of the derived techniques are theoretically grounded in the framework of information geometry. Information geometry is an emerging math discipline that applies differential geometry to space of probability distributions. This work will for the first time illustrate a systematic approach to applying information geometry to shape analysis.;Our basic approach to shape analysis is simple: represent shapes as probability densities, then use the intrinsic geometry of the space of densities to establish geodesics between shapes. We can obtain valid intermediate densities (shapes) by walking along the geodesics and the length of the geodesic immediately gives us a similarity measure between shapes. We always assume an unstructured, point-set representation for the underlying shape. Hence, unlike many contemporary methods, there are no topological restrictions (like requiring shapes to be closed curves) on our shape models. We illustrate these concepts by using two types of models to represent the densities: Gaussian mixture models and wavelet densities.;Our development of a wavelet-density, shape model also resulted in a new density estimation procedure. We expand the square root of the density in a multiscale, wavelet basis and then obtain a bona fide density by squaring the expansion. This new method estimates the coefficients of the wavelet expansion using a constrained maximum likelihood objective. It is shown that under this representation wavelet densities are essentially points on a unit hypersphere. The choice of the number of decomposition levels density estimation is determined using a model selection framework. We use the geometry of the space to apply the MDL criterion that selects the best model among set of competing ones by judiciously balancing a model's accuracy and complexity.