Real-time Locally Injective Volumetric Deformation

Описание: Accompanying video for:

"Real-time Locally Injective Volumetric Deformation"

Journal: ACM Transactions on Graphics Vol.40, No. 4 (SIGGRAPH 2021)

Authors:

Wentao Liao - University of Science and Technology of China
Renjie Chen - University of Science and Technology of China
Yuchen Hua - University of Science and Technology of China
Ligang Liu - University of Science and Technology of China
Ofir Weber - Bar-Ilan University

Abstract:

We present a highly efficient method for interactive volumetric meshless shape deformation. Our method operates within a low dimensional subspace of shape-aware 𝐶∞ harmonic maps, and is the first method that is guaranteed to produce a smooth locally injective deformation in 3D. Unlike
mesh-based methods in which local injectivity is enforced on tetrahedral elements, our method enforces injectivity on a sparse set of domain samples.
The main difficulty is then to certify the map as locally injective throughout the entire domain. This is done by utilizing the Lipschitz continuity property of the harmonic basis functions. We show a surprising relation between the Lipschitz constant of the smallest singular value of the map Jacobian and the norm of the Hessian. We further carefully derive a Lipschitz constant
for the Hessian, and develop a sufficient condition for the injectivity certification. This is done by utilizing the special structure of the harmonic basis functions combined with a novel regularization term that pushes the Lipschitz constants further down. As a result, the injectivity analysis can be performed on a relatively sparse set of samples. Combined with a parallel GPU-based implementation, our method can produce superior deformations.