Ricci Flow for 3D Shape Analysis

Ricci flow is a powerful curvature flow method in geometric analysis. This work is the first application of surface Ricci flow in computer vision. We show that previous methods based on conformal geometries, such as harmonic maps and least-square conformal maps, which can only handle 3D shapes with simple topology are subsumed by our Ricci flow based method which can handle surfaces with arbitrary topology. Because the Ricci flow method is intrinsic and depends on the surface metric only, it is invariant to rigid motion, scaling, and isometric and conformal deformations. The solution to Ricci flow is unique and its computation is robust to noise. Our Ricci flow based method can convert all 3D problems into 2D domains and offers a general framework for 3D surface analysis. Large non-rigid deformations can be registered with feature constraints, hence we introduce a method that constrains Ricci flow computation using feature points and feature curves. Finally, we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems, such as 3D shape matching and registration.

[1]  Philip L. Bowers,et al.  Coordinate systems for conformal cerebellar flat maps , 2000, NeuroImage.

[2]  Shing-Tung Yau,et al.  Global Conformal Parameterization , 2003, Symposium on Geometry Processing.

[3]  E. Sharon,et al.  2D-Shape Analysis Using Conformal Mapping , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[4]  Xianfeng Gu,et al.  Discrete Surface Ricci Flow: Theory and Applications , 2007, IMA Conference on the Mathematics of Surfaces.

[5]  Demetri Terzopoulos,et al.  Constraints on Deformable Models: Recovering 3D Shape and Nonrigid Motion , 1988, Artif. Intell..

[6]  Mark Meyer,et al.  Intrinsic Parameterizations of Surface Meshes , 2002, Comput. Graph. Forum.

[7]  Alexander M. Bronstein,et al.  Three-Dimensional Face Recognition , 2005, International Journal of Computer Vision.

[8]  F. P. Gardiner,et al.  Quasiconformal Teichmuller Theory , 1999 .

[9]  Wei Zeng,et al.  3D face matching and registration based on hyperbolic Ricci flow , 2008, 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops.

[10]  Ron Kimmel,et al.  Bending invariant representations for surfaces , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[11]  Jake K. Aggarwal,et al.  Curvature-based representation of objects from range data , 1986, Image Vis. Comput..

[12]  A. Yezzi,et al.  Metrics in the space of curves , 2004, math/0412454.

[13]  Bennett Chow,et al.  The Ricci flow on the 2-sphere , 1991 .

[14]  B. Chow,et al.  COMBINATORIAL RICCI FLOWS ON SURFACES , 2002, math/0211256.

[15]  Baba C. Vemuri,et al.  A fast level set based algorithm for topology-independent shape modeling , 1996, Journal of Mathematical Imaging and Vision.

[16]  Xianfeng Gu,et al.  Computing surface hyperbolic structure and real projective structure , 2006, SPM '06.

[17]  Sen Wang,et al.  High resolution tracking of non-rigid 3D motion of densely sampled data using harmonic maps , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[18]  Paul M. Thompson,et al.  Mutual information-based 3D surface matching with applications to face recognition and brain mapping , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[19]  Patrick J. Flynn,et al.  A Survey Of Free-Form Object Representation and Recognition Techniques , 2001, Comput. Vis. Image Underst..

[20]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[21]  Dongmei Zhang,et al.  Harmonic maps and their applications in surface matching , 1999, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149).

[22]  David G. Lowe,et al.  Distinctive Image Features from Scale-Invariant Keypoints , 2004, International Journal of Computer Vision.

[23]  Marc Levoy,et al.  Efficient variants of the ICP algorithm , 2001, Proceedings Third International Conference on 3-D Digital Imaging and Modeling.

[24]  Alan L. Yuille,et al.  Feature extraction from faces using deformable templates , 2004, International Journal of Computer Vision.

[25]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[26]  Bruno Lévy,et al.  Least squares conformal maps for automatic texture atlas generation , 2002, ACM Trans. Graph..

[27]  Bernard Chazelle,et al.  Shape distributions , 2002, TOGS.

[28]  Linda G. Shapiro,et al.  A new paradigm for recognizing 3-D objects from range data , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[29]  Reinhard Koch,et al.  Invariant-based registration of surface patches , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[30]  Guillermo Sapiro,et al.  Conformal Surface Parameterization for Texture Mapping , 1999 .

[31]  David P. Dobkin,et al.  A search engine for 3D models , 2003, TOGS.

[32]  S. Yau,et al.  Lectures on Harmonic Maps , 1997 .

[33]  Dimitris N. Metaxas,et al.  Motion Analysis of the Right Ventricle From MRI Images , 1998, MICCAI.

[34]  G. Perelman.  Finite extinction time for the solutions to the Ricci flow on certain three-manifolds , 2003, math/0307245.

[35]  D. Mumford,et al.  Riemannian Geometries on Spaces of Plane Curves , 2003, math/0312384.

[36]  Anuj Srivastava,et al.  On Shape of Plane Elastic Curves , 2007, International Journal of Computer Vision.

[37]  Wei Zeng,et al.  3D Non-rigid Surface Matching and Registration Based on Holomorphic Differentials , 2008, ECCV.

[38]  Jitendra Malik,et al.  Efficient shape matching using shape contexts , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[39]  Timothy F. Cootes,et al.  Extraction of Visual Features for Lipreading , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[40]  Guillermo Sapiro,et al.  A Theoretical and Computational Framework for Isometry Invariant Recognition of Point Cloud Data , 2005, Found. Comput. Math..

[41]  Jitendra Malik,et al.  Recognizing Objects in Range Data Using Regional Point Descriptors , 2004, ECCV.

[42]  Jeff Erickson,et al.  Greedy optimal homotopy and homology generators , 2005, SODA '05.

[43]  George Kollios,et al.  BoostMap: A method for efficient approximate similarity rankings , 2004, CVPR 2004.

[44]  Sen Wang,et al.  Conformal Geometry and Its Applications on 3D Shape Matching, Recognition, and Stitching , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[45]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[46]  Ron Kikinis,et al.  On the Laplace-Beltrami operator and brain surface flattening , 1999, IEEE Transactions on Medical Imaging.

[47]  Martial Hebert,et al.  Parts-based 3D object classification , 2004, CVPR 2004.

[48]  Sen Wang,et al.  High Resolution Tracking of Non-Rigid Motion of Densely Sampled 3D Data Using Harmonic Maps , 2008, International Journal of Computer Vision.

[49]  T. Chan,et al.  Genus zero surface conformal mapping and its application to brain surface mapping. , 2004, IEEE transactions on medical imaging.

[50]  Jean-Philippe Pons,et al.  Generalized Gradients: Priors on Minimization Flows , 2007, International Journal of Computer Vision.

[51]  Nikos Paragios,et al.  Establishing Local Correspondences towards Compact Representations of Anatomical Structures , 2003, MICCAI.

[52]  Mongi A. Abidi,et al.  Surface matching by 3D point's fingerprint , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[53]  B. Chow,et al.  The Ricci flow on surfaces , 2004 .

[54]  Song Zhang,et al.  High-Resolution, Real-time 3D Shape Acquisition , 2004, 2004 Conference on Computer Vision and Pattern Recognition Workshop.