Computational anatomy: an emerging discipline
Quarterly of Applied Mathematics - Special issue on current and future challenges in the applications of mathematics
A Mathematical Introduction to Robotic Manipulation
A Mathematical Introduction to Robotic Manipulation
Linear combination of transformations
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Deformation transfer for triangle meshes
ACM SIGGRAPH 2004 Papers
The Scaling and Squaring Method for the Matrix Exponential Revisited
SIAM Journal on Matrix Analysis and Applications
SCAPE: shape completion and animation of people
ACM SIGGRAPH 2005 Papers
Geometric modeling in shape space
ACM SIGGRAPH 2007 papers
Pattern Theory: From Representation to Inference
Pattern Theory: From Representation to Inference
A simple geometric model for elastic deformations
ACM SIGGRAPH 2010 papers
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part VI
Linear Local Models for Monocular Reconstruction of Deformable Surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Statistics of shape via principal geodesic analysis on lie groups
CVPR'03 Proceedings of the 2003 IEEE computer society conference on Computer vision and pattern recognition
Detailed human shape and pose from images
Detailed human shape and pose from images
Home 3D body scans from noisy image and range data
ICCV '11 Proceedings of the 2011 International Conference on Computer Vision
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Three-dimensional object shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Existing models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, summing the shape deformations for two people does not necessarily yield a deformation corresponding to a valid human shape, nor does the Euclidean difference of these two deformations provide a meaningful measure of shape dissimilarity. Consequently, we define a novel manifold for shape representation, with emphasis on body shapes, using a new Lie group of deformations. This has several advantages. First we define triangle deformations exactly, removing non-physical deformations and redundant degrees of freedom common to previous methods. Second, the Riemannian structure of Lie Bodies enables a more meaningful definition of body shape similarity by measuring distance between bodies on the manifold of body shape deformations. Third, the group structure allows the valid composition of deformations. This is important for models that factor body shape deformations into multiple causes or represent shape as a linear combination of basis shapes. Finally, body shape variation is modeled using statistics on manifolds. Instead of modeling Euclidean shape variation with Principal Component Analysis we capture shape variation on the manifold using Principal Geodesic Analysis. Our experiments show consistent visual and quantitative advantages of Lie Bodies over traditional Euclidean models of shape deformation and our representation can be easily incorporated into existing methods.