Three dimensional shape modeling: Segmentation, reconstruction and registration

Accounting for uncertainty in three-dimensional (3D) shapes is important in a large number of scientific and engineering areas, such as biometrics, biomedical imaging, and data mining. It is well known that 3D polar shaped objects can be represented by Fourier descriptors such as spherical harmonics and double Fourier series. However, the statistics of these spectral shape models have not been widely explored. This thesis studies several areas involved in 3D shape modeling, including random field models for statistical shape modeling, optimal shape filtering, parametric active contours for object segmentation and surface reconstruction. It also investigates multi-modal image registration with respect to tumor activity quantification.; Spherical harmonic expansions over the unit sphere not only provide a low dimensional polarimetric parameterization of stochastic shape, but also correspond to the Karhunen-Loeve (K-L) expansion of any isotropic random field on the unit sphere. Spherical harmonic expansions permit estimation and detection tasks, such as optimal shape filtering, object registration, and shape classification, to be performed directly in the spectral domain with low complexities. An issue which we address is the effect of center estimation accuracy on the accuracy of polar shape models. A lower bound is derived for the variance of ellipsoid fitting center estimator. Simulation shows that the performance of a maximum likelihood center estimator can approach the bound in low noise situations.; Due to the large number of voxels in 3D images, 3D parametric active contour techniques have very high computational complexity. A novel parametric active contour method with lower computational complexity is proposed in this thesis. A spectral method using double Fourier series as an orthogonal basis is applied to solving elliptic partial differential equations over the unit sphere, which control surface evolution. The complexity of the spectral method is O(N2 log N) for a grid size of N x N as compared to O(N3) for finite element methods and finite difference methods. A volumetric penalization term is introduced in the energy function of the active contour to prevent the contour from leaking through blurred boundaries.; Multi-modal medical image registration is widely used to quantify tumor activity in radiation therapy patients. Rigid global registration sometimes cannot perfectly overlay the tumor volume of interest (VOI), e.g. segmented from a CT anatomical image, with the apparent position of a tumor in a SPELT functional image. We investigate a new local registration method which aligns the CT and SPELT tumor volumes by maximizing the SPELT intensity within the CT-segmented tumor VOI.