Through the development of this thesis, starting from the curvature tensor, we have been able to understand the variation of tangent vectors to define a shape analysis operator and also a relationship between the classical shape operator and the curvature tensor on a triangular surface. In continuation, the first part of the thesis analyzed the variation of surface normals and introduced a shape analysis operator, which is further used for mesh and point set denoising. In the second part of the thesis, mathematical modeling and shape quantification algorithms are introduced for retinal shape analysis. In the first half, this thesis followed the concept of the variation of surface normals, which is termed as the normal voting tensor and derived a relation between the shape operator and the normal voting tensor. The concept of the directional and the mean curvatures is extended on the dual representation of a triangulated surface. A normal voting tensor is defined on each triangle of a geometry and termed as the element-based normal voting tensor (ENVT). Later, a deformation tensor is extracted from the ENVT and it consists of the anisotropy of a surface and the mean curvature vector is defined based on the ENVT deformation tensor. The ENVT-based mesh denoising algorithm is introduced, where the ENVT is used as a shape operator. A binary optimization technique is applied on the spectral components of the ENVT that helps the algorithm to retain sharp features in the concerned geometry and improves the convergence rate of the algorithm. Later, a stochastic analysis of the effect of noise on the triangular mesh based on the minimum edge length of the elements in the geometry is explained. It gives an upper bound to the noise standard deviation to have minimum probability for flipped element normals. The ENVT-based mesh denoising concept is extended for a point set denoising, where noisy vertex normals are filtered using the vertex-based NVT and the binary optimization. For vertex update stage in point set denoising, we added different constraints to the quadratic error metric based on features (edges and corners) or non-feature (planar) points. This thesis also investigated a robust statistics framework for face normal bilateral filtering and proposed a robust and high fidelity two-stage mesh denoising method using Tukey’s bi-weight function as a robust estimator, which stops the diffusion at sharp features and produces smooth umbilical regions. This algorithm introduced a novel vertex update scheme, which uses a differential coordinate-based Laplace operator along with an edge-face normal orthogonality constraint to produce a high-quality mesh without face normal flips and it also makes the algorithm more robust against high-intensity noise. The second half of thesis focused on the application of the proposed geometric processing algorithms on the OCT (optical coherence tomography) scan data for quantification of the human retinal shape. The retina is a part of the central nervous system and comprises a similar cellular composition as the brain. Therefore, many neurological disorders affect the retinal shape and these neuroinflammatory conditions are known to cause modifications to two important regions of the retina: the fovea and the optical nerve head (ONH). This thesis consists of an accurate and robust shape modeling of these regions to diagnose several neurological disorders by detecting the shape changes. For the fovea, a parametric modeling algorithm is introduced using Cubic Bezier and this algorithm derives several 3D shape parameters, which quantify the foveal shape with high accuracy. For the ONH, a 3D shape analysis algorithm is introduced to measure the shape variation regarding different neurological disorders. The proposed algorithm uses triangulated manifold surfaces of two different layers of the retina to derive several 3D shape parameters. The experimental results of the fovea and the ONH morphometry confirmed that these algorithms can provide an aid to diagnose several neurological disorders.
