Registering pairs or groups of images is a widely-studied problem. Most of its solutions are variational, using objective functions. These objective functions should satisfy several basic and desired properties. Besides the existed properties, such as symmetry and inverse consistency, we propose two additional properties: (1) invariance of objective function under identical warping of input images and (2) the objective function induces a proper metric on the set of equivalence classes of images, and motivate their importance to registrationa and post-registration analysis.
We also introduce a registration framework that satisfies these properties, using the L2-norm between a novel representation of images. Additionally, for multiple images, the induced metric enables us to compute a mean image, or a template, and perform multiple image registration.

The analysis of the shapes of 3D objects is an important area of research with a wide variety of applications. The need for shape analysis arises in many branches of science, for example, medical image analysis, protein structure analysis, computer graphics, and 3D printing and prototyping. Many of these are especially concerned with capturing variability within and across shape classes, and so the main focus of research has been on statistical shape analysis and on comparing shapes.
These tasks require a fundamental tool called parallel transport of tangent vectors along arbitrary paths. This tool is essential for: (1) computation of geodesic paths using either shooting or path-straightening method, (2) transferring deformations across objects, and (3) modeling of statistical variability in shapes. Using the square-root normal field (SRNF) representation of parameterized surfaces, we propose a method for transporting deformations along paths in the shape space.

In recent years, the increased strengths of MRI scanners have led to thepossibility of measuring diffusion orientations beyond the three canonical directions, and have led to HARDI (High Angular Resolution Diffusion Imaging) technology. It measures fluid flow at each voxel in numerous directions and has been used for studying brain structures, their connectivities, and functionalities.
We propose a Riemannian framework for analyzing orientation distribution functions (ODFs) in HARDI for use in comparing, interpolating, averaging, and denoising. We develop a framework to compute pseudo-geodesics which have better biological interpretation (in terms of interpolating points between given PDFs by preserving shape diffusivity and anisotropy) and provide tools for pairwise comparison and averaging of a collection of ODFs.

Here one uses 2D and/or 3D medical images taken across time, species, or specimens to compare to extract salient differences in anatomical structures, and to analyze and model their variations both within and across biological classes. These differences may result from standard biological growth, abnormalities, inter-specimen variability, or other reasons.
The growth by random iterated diffeomorphisms (GRID) model seeks to decompose large deformations into smaller and biologically-meaningful components. These components are spatially local and parametric, and are characterized by radial deformation patterns around randomly-placed seeds. The actual decomposition requires estimation of GRID parameters from observations of large growth from images. We address the problem of parameter estimation under the original GRID model that advocates sequential composition of arbitrarily interacting components.

When the region of interest is thought to be under the effect of several unobserved putative pollution sources simultaneously, response, such as disease risk, is modeled as a function of spatial locations in relation to point sources. Related works focus on identifying potential focus of risk, e.g. hot spots, and analyzing the radial distance effects from one a priori specified foci. We aim to separate and quantify the influence pattern, such as contagion, from different foci within a given area by creating a smooth random surface. Kernel density functions are adjusted to help model the surface trend (first order component) and at the same time the heterogeneous local precision. Full Bayesian approach is set up for computational accessibility. The New York Leukemia data is analyzed as an example. Simulation study is further carried out to evaluate the sensitivity due to choices of kernels and priors. The convergence behavior is studied as well.