Shape models based on elliptic PDES, associated energies, and their applications in 2D and 3D

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2018
Gençtav, Aslı
By using an elliptic PDE or its modifications, we develop implicit shape representations and demonstrate their two- and three-dimensional applications. In the first part of the thesis, we present a novel shape characterization field that provides a local measure of roundness at each shape point. The field is computed by comparing the solution of the elliptic PDE on the shape domain and the solution of the same PDE on the reference disk. We demonstrate its potential via illustrative applications including global shape characterization, context-dependent categorization, and shape partitioning. In the second part, we solve the elliptic PDE multiple times varying either the diffusion parameter or the right hand side function and construct high-dimensional feature space. We then apply low-dimensional reduction to assign a distinctness value to each shape point. We use the obtained distinctness values for non-structural representation of two-dimensional shapes and saliency measurement of surfaces of three-dimensional shapes. In the third and the final part, we use the elliptic PDE modifications for bringing a pair of 3D shapes into comparable topology.

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Citation Formats
A. Gençtav, “Shape models based on elliptic PDES, associated energies, and their applications in 2D and 3D,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.