Conformal Geometry and Its Applications on 3D Shape Matching, Recognition, and Stitching
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Superresolution Framework for High-Accuracy Multiview Reconstruction
Proceedings of the 31st DAGM Symposium on Pattern Recognition
Optimized Conformal Surface Registration with Shape-based Landmark Matching
SIAM Journal on Imaging Sciences
Semi-supervised segmentation of 3D surfaces using a weighted graph representation
GbRPR'11 Proceedings of the 8th international conference on Graph-based representations in pattern recognition
Brain surface conformal parameterization with algebraic functions
MICCAI'06 Proceedings of the 9th international conference on Medical Image Computing and Computer-Assisted Intervention - Volume Part II
A graph-based technique for semi-supervised segmentation of 3D surfaces
Pattern Recognition Letters
A Super-Resolution Framework for High-Accuracy Multiview Reconstruction
International Journal of Computer Vision
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We propose a general method that parameterizes general surfaces with complex (possible branching) topology using Riemann surface structure. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. We can then automatically partition the surface using a critical graph that connects zero points in the global conformal structure on the surface. The trajectories of iso-parametric curves canonically partition a surface into patches. Each of these patches is either a topological disk or a cylinder and can be conformally mapped to a parallelogram by integrating a holomorphic 1-form defined on the surface. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. For surfaces with similar topology and geometry, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other using constrained harmonic maps. The surface similarity can be measured by direct computation of distance between each pair of corresponding points on two surfaces. To illustrate the technique, we computed conformal structures for anatomical surfaces in MRI scans of the brain and human face surfaces. We found that the resulting parameterizations were consistent across subjects, even for branching structures such as the ventricles, which are otherwise difficult to parameterize. Our method provides a surface-based framework for statistical comparison of surfaces and for generating grids on surfaces for PDE-based signal processing.