High quality deformations of planar and volumetric domains are central to many computer graphics related problems like modeling, character animation, and non-rigid registration. Besides common “as-rigid-as-possible” approaches the class of nearly-isometric deformations is highly relevant to solve this kind of problems. Recent continuous deformation approaches try to find planar first order nearly-isometric defor- mations by integrating along approximate Killing vector fields (AKVFs). In this work we derive a generalized metric energy for deformation vector fields that has close-to-isometric AKVFs as a special case and addi- tionally supports close-to-length-preserving, close-to-conformal as well as close-to-equiareal deformations. Like AKVF-based deformations we minimize nonlinear energies to first order using efficient linear optimizations. Our energy formulation supports nonhomogeneous as well as anisotropic behavior and we show that it is applicable to both planar and volumetric domains. We apply energy specific regularization to achieve smoothness and provide a GPU implementation for interactivity. We compare our approach to AKVF-based deformations for the planar case and demonstrate the effectiveness of our method for the 2d and 3d case.



  title = {Generalized Metric Energies for Continuous Shape Deformation},
  author = { {Martinez~Esturo}, J. and R{\"o}ssl, C. and Theisel, H.},
  journal = {Springer LNCS (Proc. Curves and Surfaces 2012)},
  number = {1},
  pages = {135--157},
  volume = {8177},
  year = {2013}