Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction
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Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling, continuous surface deformation, shape morphing, reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.
Keywordsshape modeling spherical topology parametric surfaces splines differential geometry
This work was funded by the Swiss National Science Foundation under Grant 200020-162343. We are grateful to Zsuzsanna Püspöki for help with the figures and to Irina Radu for help with the video. We also appreciate the interesting discussions on the subject that we had with Masih Nilchian and Emrah Bostan. We thank Mike McCann for proof-reading the manuscript.
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