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E IGEN D EFORMATION OF 3 D M ODELS. Tamal K. Dey , Pawas Ranjan , Yusu Wang [The Ohio State University] (CGI 2012). Problem. Perform deformations without asking the user for extra structures (like cages, skeletons etc ). Previous Work. Skeleton based [YBS03], [DQ04], [BP07],...

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e igen d eformation of 3 d m odels

EIGEN DEFORMATIONOF 3D MODELS

Tamal K. Dey, PawasRanjan, Yusu Wang

[The Ohio State University]

(CGI 2012)

problem
Problem
  • Perform deformations without asking the user for extra structures (like cages, skeletons etc)
previous work
Previous Work
  • Skeleton based [YBS03], [DQ04], [BP07],...
  • Cage based [FKR05], [JMGDS07], [LLC08],...
  • Constrained vertices and energy minimization [SA07], [YZXSB04], [ZHSLBG05] ,etc.
cage less deformation
Cage-less deformation
  • Skeleton and cage based methods
    • very fast, but need extra structures
  • Energy based methods
    • do not require extra structures, but are usually slow

Need to perform fast deformations without asking the user for extra structures like skeletons or cages

the laplace beltrami operator
The Laplace-Beltrami operator
  • A popular operator defined for surfaces
    • Isometry invariant
    • Robust against noise and sampling
    • Changes smoothly with changes in shape
  • Its eigenvectors form an orthonormal basis for functions defined on the surface
eigen skeleton
Eigen-skeleton
  • Treat x, y and z coordinates as functions
  • Reconstruct them using the eigenvectors, ignoring high frequencies
eigen skeleton for deformation
Eigen-skeleton for deformation
  • User specifies a shape along with:
    • A region on the shape
    • Deformation desired on that region
  • We:
    • Create the eigen-skeleton
    • Apply the deformation to the entire region
    • Smooth out the skeleton
    • Add details to get the deformed shape
choice of number of eigenvectors
Choice of number of eigenvectors
  • Need to be able to capture the feature to be deformed
  • Use the size of region of interest to choose the number of eigenvectors to use
    • Smaller features need more eigenvectors
skeleton energy
Skeleton energy
  • Let <ϕ1, ϕ2, ... ϕm> be the top m eigenvectors
  • We wish to find new weights for the deformed shape
skeleton energy1
Skeleton energy
  • Taking partial derivatives and re-arranging the terms, we get the following linear system
skeleton energy2
Skeleton energy
  • Solving for the unknown weights Ai, we get a smooth representation of the deformed skeleton
recovering shape details
Recovering Shape Details
  • Using few eigenvectors causes loss of details
  • Once smooth deformed skeleton is obtained, these details need to be added back
  • Use the one-to-one correspondence between the shape and skeleton to recover the details
conclusion
Conclusion
  • Fast deformations using implicit skeleton
  • No need for user to provide extra structures
  • Software coming very soon!
  • Result not necessarily free of self-intersections
  • Computing the eigenvectors of the Laplace operator can be time-consuming