Möbius Registration

• Published in: Computer graphics forum Vol. 37; no. 5; pp. 211 - 220
• Main Authors: Baden, Alex, Crane, Keenan, Kazhdan, Misha
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.08.2018
• Subjects: 1.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems; Alignment; Categories and Subject Descriptors (according to ACM CCS); Data transfer (computers); Parameterization; Symmetry detection
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.13503

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.