Tools for statistical analysis on shape spaces of three-dimensional object

With the increasing popularity of information technology, especially electronic imaging techniques, large amount of high dimensional data such as 3D shapes become pervasive in science, engineering and even people's daily life, in the recent years. Though the data quantity is huge, the extraction of relevant knowledge on those data is still limited. How to understand data in a meaningful way is generally an open problem. The specific challenges include finding adequate mathematical representations of data and designing proper algorithms to process them. The existing tools for analyzing high-dimensional data, including 3D shape data, are found to be insufficient as they usually suffer from many factors, such as misalignments, noise, and clutter. This thesis attempts to develop a framework for processing, analyzing and understanding high-dimensional data, especially 3D shapes, by proposing a set of statistical tools including theory, algorithms and optimization applied to practical problems. In particular, the following aspects of shape analysis are considered:;1. A framework adopting the SRNF representation, based on parallel transport of deformations across surfaces in the shape space, leads to statistical analysis on shape data. Three main analyses are conducted under this framework: (1) computing geodesics when either two end surfaces or the starting surface and an initial deformation are given; (2) parallel transporting deformation across surfaces; and (3) sampling random surfaces.;2. Computational efficiency plays an important role in performing statistical shape analysis on large datasets of 3D objects. To speed up the previous method, a framework with numerical solution is introduced by approximating the inverse mapping, and it reduces the computational cost by an order of magnitude.;3. The geometrical and morphological information, or their shapes, of 3D objects can be analyzed explicitly using boundaries extracted from original image scans. An alternative idea is to consider variability in shapes directly from their embedding images. A novel framework is proposed to unify three important tasks, registering, comparing and modeling images.;4. Finally, the spatial deformations learned from registering images are modeled using the GRID based decomposition. This specific model provides a way to decompose a large deformation into local and fundamental ones so that shape differences between images are easily interpretable.;We conclude this thesis with conclusions drawn in this research and discuss potential future directions of statistical shape analysis in the last chapter, both from methodological and application aspects.