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This paper presents a novel approach based on spectral geometry to quantify and visualize non-isometric deformations of 3D surfaces by mapping two manifolds. The proposed method can determine multi-scale, non-isometric deformations through the variation of Laplace-Beltrami spectrum of two shapes. Given two triangle meshes, the spectra can be varied from one to another with a scale function defined on each vertex. The variation is expressed as a linear interpolation of eigenvalues of the two shapes. In each iteration step, a quadratic programming problem is constructed, based on our derived spectrum variation theorem and smoothness energy constraint, to compute the spectrum variation. The derivation of the scale function is the solution of such a problem. Therefore, the final scale function can be solved by integral of the derivation from each step, which, in turn, quantitatively describes non-isometric deformations between two shapes. To evaluate the method, we conduct extensive experiments on synthetic and real data. We employ real epilepsy patient imaging data to quantify the shape variation between the left and right hippocampi in epileptic brains. In addition, we use longitudinal Alzheimer data to compare the shape deformation of diseased and healthy hippocampus. In order to show the accuracy and effectiveness of the proposed method, we also compare it with spatial registration-based methods, e.g., non-rigid Iterative Closest Point (ICP) and voxel-based method. These experiments demonstrate the advantages of our method. [ABSTRACT FROM AUTHOR]

In order to visualize and analyze complex collective data, complicated geometric structure of each data is desired to be mapped onto a canonical domain to enable map-based visual exploration. This paper proposes a novel volume-preserving mapping and registration method which facilitates effective collective data visualization. Given two 3-manifolds with the same topology, there exists a mapping between them to preserve each local volume element. Starting from an initial mapping, a volume restoring diffeomorphic flow is constructed as a compressible flow based on the volume forms at the manifold. Such a flow yields equality of each local volume element between the original manifold and the target at its final state. Furthermore, the salient features can be used to register the manifold to a reference template by an incompressible flow guided by a divergence-free vector field within the manifold. The process can retain the equality of local volume elements while registering the manifold to a template at the same time. An efficient and practical algorithm is also presented to generate a volume-preserving mapping and a salient feature registration on discrete 3D volumes which are represented with tetrahedral meshes embedded in 3D space. This method can be applied to comparative analysis and visualization of volumetric medical imaging data across subjects. We demonstrate an example application in multimodal neuroimaging data analysis and collective data visualization. [ABSTRACT FROM PUBLISHER]

Parameterization of complex surfaces constitutes a major means of visualizing highly convoluted geometric structures as well as other properties associated with the surface. It also enables users with the ability to navigate, orient, and focus on regions of interest within a global view and overcome the occlusions to inner concavities. In this paper, we propose a novel area-preserving surface parameterization method which is rigorous in theory, moderate in computation, yet easily extendable to surfaces of non-disc and closed-boundary topologies. Starting from the distortion induced by an initial parameterization, an area restoring diffeomorphic flow is constructed as a Lie advection of differential 2-forms along the manifold, which yields equality of the area elements between the domain and the original surface at its final state. Existence and uniqueness of result are assured through an analytical derivation. Based upon a triangulated surface representation, we also present an efficient algorithm in line with discrete differential modeling. As an exemplar application, the utilization of this method for the effective visualization of brain cortical imaging modalities is presented. Compared with conformal methods, our method can reveal more subtle surface patterns in a quantitative manner. It, therefore, provides a competitive alternative to the existing parameterization techniques for better surface-based analysis in various scenarios. [ABSTRACT FROM AUTHOR]