E IGEN D EFORMATION OF 3 D M ODELS

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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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EIGEN DEFORMATIONOF 3D MODELS

Tamal K. Dey, PawasRanjan, 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
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 <ϕ1, ϕ2, ... ϕm> 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 Ai, 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
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