Real-time locally injective volumetric deformation

Wentao Liao, Renjie Chen, Yuchen Hua, Ligang Liu, Ofir Weber

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


We present a highly efficient method for interactive volumetric meshless shape deformation. Our method operates within a low dimensional sub-space of shape-aware C8 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 with unique quality guarantees at real-time rates which were possible only in 2D so far.

Original languageEnglish
Article number74
JournalACM Transactions on Graphics
Issue number4
StatePublished - 1 Jul 2021

Bibliographical note

Publisher Copyright:
© 2021 ACM.


  • GPU
  • NEWTON method
  • harmonic maps
  • injective maps
  • shape deformation


Dive into the research topics of 'Real-time locally injective volumetric deformation'. Together they form a unique fingerprint.

Cite this