Applications in computer-aided design (CAD), computer graphics, medical imaging, scientific computing, computer vision, and other areas demand algorithms for fine-grained 3D shape analysis. Our team adapts elegant theory from modern geometry to the algorithmic domain, tackling mathematically-intense challenges needed to understand and process real-world geometry.
Among many research directions, some of our recent work has focused on the following problems:
- Hexahedral meshing: The hexahedral (hex) meshing problem involves dividing a volume in 3D into cube-shaped cells, valuable for applications in physical simulation and isogeometric analysis. Our work advances the mathematical understanding of topological and differential geometry aspects of this problem, and we have developed state-of-the-art numerical algorithms for field-guided hex meshing in practice.
- Shape analysis of surfaces and volumes: We work to advance the theory and practice of 3D shape analysis, with applications in computer graphics and medical imaging. Our recent work shows how to apply ideas from spectral shape analysis (popular for two-dimensional surfaces) to higher-dimensional domains using the boundary element method, and we have demonstrated how to link algorithms for machine learning to classical models for 3D shape analysis built from first-principles in mathematics.
- Shape correspondence: A key challenge problem in geometry processing is correspondence, finding corresponding pairs of points on different shapes that differ by noise and deformation. We have published state-of-the-art algorithms and representations for this problem, leading to correspondence algorithms that are resilient to noise, deformation, and other artifacts.