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E IGEN D EFORMATION OF 3 D M ODELS

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

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  1. EIGEN DEFORMATIONOF 3D MODELS Tamal K. Dey, PawasRanjan, Yusu Wang [The Ohio State University] (CGI 2012)

  2. Problem • Perform deformations without asking the user for extra structures (like cages, skeletons etc)

  3. Previous Work • Skeleton based [YBS03], [DQ04], [BP07],... • Cage based [FKR05], [JMGDS07], [LLC08],... • Constrained vertices and energy minimization [SA07], [YZXSB04], [ZHSLBG05] ,etc.

  4. 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

  5. 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

  6. Eigen-skeleton • Treat x, y and z coordinates as functions • Reconstruct them using the eigenvectors, ignoring high frequencies

  7. 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

  8. Eigen-skeleton for deformation

  9. 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

  10. Skeleton energy • Let <ϕ1, ϕ2, ... ϕm> be the top m eigenvectors • We wish to find new weights for the deformed shape

  11. Skeleton energy • Taking partial derivatives and re-arranging the terms, we get the following linear system

  12. Skeleton energy • Solving for the unknown weights Ai, we get a smooth representation of the deformed skeleton

  13. 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

  14. Algorithm

  15. Results

  16. Results

  17. Results

  18. Arbitrary genus

  19. Comparison

  20. Comparison

  21. Timing (in seconds)

  22. 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

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