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],... Cage based [FKR05], [JMGDS07], [LLC08],... Constrained vertices and energy minimization [SA07], [YZXSB04], [ZHSLBG05],etc.

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 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 Treat x, y and z coordinates as functions Reconstruct them using the eigenvectors, ignoring high frequencies

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

Eigen-skeleton for deformation

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 Let be the top m eigenvectors We wish to find new weights for the deformed shape

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

Skeleton energy Solving for the unknown weights A i, we get a smooth representation of the deformed skeleton

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

Algorithm

Results

Arbitrary genus

Comparison

Timing (in seconds)

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