Eigen Deformation of 3D Models

EIGEN DEFORMATION OF 3D MODELSTamal K. Dey, Pawas Ranjan, Yusu Wang[The Ohio State University](CGI 2012)

ProblemPerform deformations without asking the user for extra structures (like cages, skeletons etc)

Previous WorkSkeleton based [YBS03], [DQ04], [BP07],...

Cage based [FKR05], [JMGDS07], [LLC08],...

Constrained vertices and energy minimization [SA07], [YZXSB04], [ZHSLBG05] ,etc.Cage-less deformationSkeleton and cage based methodsvery fast, but need extra structures

Energy based methodsdo not require extra structures, but are usually slow

Need to perform fast deformations without asking the user for extra structures like skeletons or cagesThe Laplace-Beltrami operatorA popular operator defined for surfacesIsometry invariantRobust against noise and samplingChanges smoothly with changes in shape

Its eigenvectors form an orthonormal basis for functions defined on the surfaceEigen-skeletonTreat x, y and z coordinates as functionsReconstruct them using the eigenvectors, ignoring high frequencies

Eigen-skeleton for deformationUser specifies a shape along with:A region on the shapeDeformation desired on that regionWe:Create the eigen-skeletonApply the deformation to the entire regionSmooth out the skeletonAdd details to get the deformed shapeEigen-skeleton for deformation

Choice of number of eigenvectorsNeed to be able to capture the feature to be deformedUse the size of region of interest to choose the number of eigenvectors to useSmaller features need more eigenvectors

Skeleton energyLet be the top m eigenvectors

We wish to find new weights for the deformed shape

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

Skeleton energySolving for the unknown weights Ai, we get a smooth representation of the deformed skeleton

Recovering Shape DetailsUsing few eigenvectors causes loss of detailsOnce smooth deformed skeleton is obtained, these details need to be added backUse the one-to-one correspondence between the shape and skeleton to recover the details

Algorithm

Results

Results

Results

Arbitrary genus

Comparison

Comparison

Timing (in seconds)

ConclusionFast deformations using implicit skeletonNo need for user to provide extra structuresSoftware coming very soon!

Result not necessarily free of self-intersectionsComputing the eigenvectors of the Laplace operator can be time-consuming