We present a method for volumetric shape interpolation with unique shape preserving features. The input to our algorithm are two or more 3-manifolds, immersed into R3 and discretized as tetrahedral meshes with shared connectivity. The output is a continuum of shapes that naturally blends the input shapes, while striving to preserve the geometric character of the input. The basis of our approach relies on the fact that the space of metrics with bounded isometric and angular distortion is convex [Chien et al. 2016b]. We show that for high dimensional manifolds, the bounded distortion metrics form a positive semidefinite cone product space. Our method can be seen as a generalization of the bounded distortion interpolation technique of [Chen et al. 2013] from planar shapes immersed in R2 to solids in R3. The convexity of the space implies that a linear blend of the (squared) edge lengths of the input tetrahedral meshes is a simple yet powerful-and-natural choice. Linearly blending flat metrics results in a new metric which is, in general, not flat, and cannot be immersed into three-dimensional space. Nonetheless, the amount of curvature that is introduced in the process tends to be very low in practical settings. We further design an extremely robust nonconvex optimization procedure that efficiently flattens the metric. The flattening procedure strives to preserve the low distortion exhibited in the blended metric while guaranteeing the validity of the metric, resulting in a locally injective map with bounded distortion. Our method leads to volumetric interpolation with superb quality, demonstrating significant improvement over the state-of-the-art and qualitative properties which were obtained so far only in interpolating manifolds of lower dimensions.
|Journal||ACM Transactions on Graphics|
|State||Published - Nov 2019|
Bibliographical noteFunding Information:
This research was supported by grant number 2012264 from the United States-Israel Binational Science Foundation (BSF) and by the Max Planck Center for Visual Computing and Communication. Authors’ addresses: Ido Aharon, Bar-Ilan University, Israel, firstname.lastname@example.org; Renjie Chen, Max Planck Institute for Informatics, Germany, University of Science and Technology of China, email@example.com; Denis Zorin, New York University, USA, firstname.lastname@example.org; Ofir Weber, Bar-Ilan University, Israel, email@example.com.
© 2019 Association for Computing Machinery.
- Bounded distorted maps
- Convex optimization
- Metric tensor
- Shape interpolation
- Tetrahedral meshes