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Moebius Registration

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JeGX

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This distribution contains code for constructing and registering conformal spherical parametrizations of water-tight, genus-zero surfaces. Specifically, it provides implementations for:

- Computing a conformal parametrization over the sphere
- Centering the parametrization with respect to Möbius inversions
- Tessellating the conformal parametrization to a regular equirectangular grid
- Performing fast spherical correlation to find the rotation/reflections that best aligning two centered parametrizations
- Using the registered parametrizations to compute dense correspondences from a source mesh to a target

Link: https://github.com/mkazhdan/MoebiusRegistration



Quote
Conformal parameterizations over the sphere provide high-quality maps between genus zero surfaces, and are essential for applications such as data transfer and comparative shape analysis. However, such maps are not unique: to define correspondence between two surfaces, one must find the Möbius transformation that best aligns two parameterizations—akin to picking a translation and rotation in rigid registration problems. We describe a simple procedure that canonically centers and rotationally aligns two spherical maps. Centering is implemented via elementary operations on triangle meshes in R3, and minimizes area distortion. Alignment is achieved using the FFT over the group of rotations. We examine this procedure in the context of spherical conformal parameterization, orbifold maps, non-rigid symmetry detection, and dense point-to-point surface correspondence.

Link: http://www.cs.jhu.edu/~misha/MyPapers/SGP18.pdf