Scale Invariant Metrics of Volumetric Datasets

Nature reveals itself in similar structures of different scales. A child and an adult share similar organs yet dramatically differ in size. Comparing the two is a challenging task to a computerized approach as scale and shape are coupled. Recently, it was shown that a local measure based on the Gaussian curvature can be used to normalize the local metric of a surface and then to extract global features and distances. In this paper we consider higher dimensions; specifically, we construct a scale invariant metric for volumetric domains which can be used in analysis of medical datasets such as computed tomography (CT) and magnetic resonance imaging (MRI).

[1]  Jean-Michel Morel,et al.  Integral and local affine invariant parameter and application to shape recognition , 1994, Proceedings of 12th International Conference on Pattern Recognition.

[2]  P. Bérard,et al.  Embedding Riemannian manifolds by their heat kernel , 1994 .

[3]  A. Ben Hamza,et al.  Geodesic matching of triangulated surfaces , 2006, IEEE Transactions on Image Processing.

[4]  Ron Kimmel,et al.  Affine Invariant Geometry for Non-rigid Shapes , 2014, International Journal of Computer Vision.

[5]  Leonidas J. Guibas,et al.  One Point Isometric Matching with the Heat Kernel , 2010, Comput. Graph. Forum.

[6]  Ron Goldman,et al.  Curvature formulas for implicit curves and surfaces , 2005, Comput. Aided Geom. Des..

[7]  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.

[8]  Alfred M. Bruckstein,et al.  Similarity-invariant signatures for partially occluded planar shapes , 1992, International Journal of Computer Vision.

[9]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[10]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[11]  T. Funkhouser,et al.  Möbius voting for surface correspondence , 2009, SIGGRAPH 2009.

[12]  Ramesh Raskar,et al.  Evaluating Local Contractions from Large Deformations Using Affine Invariant Spectral Geometry , 2014, STACOM.

[13]  Ron Kimmel,et al.  From High Energy Physics to Low Level Vision , 1997, Scale-Space.

[14]  Ron Kimmel,et al.  Scale Invariant Geometry for Nonrigid Shapes , 2013, SIAM J. Imaging Sci..

[15]  Stéphane Lafon,et al.  Diffusion maps , 2006 .

[16]  Ann B. Lee,et al.  Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Jean-Michel Morel,et al.  ASIFT: A New Framework for Fully Affine Invariant Image Comparison , 2009, SIAM J. Imaging Sci..

[18]  Luc Van Gool,et al.  Recognition of planar shapes under affine distortion , 2005, International Journal of Computer Vision.

[19]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[20]  Guillermo Sapiro,et al.  A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching , 2010, International Journal of Computer Vision.

[21]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

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

[23]  Luc Van Gool,et al.  Foundations of semi-differential invariants , 2005, International Journal of Computer Vision.

[24]  Alexander M. Bronstein,et al.  Expression-invariant face recognition via spherical embedding , 2005, IEEE International Conference on Image Processing 2005.

[25]  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.

[26]  Ehud Rivlin,et al.  Scale space semi-local invariants , 1997, Image Vis. Comput..

[27]  Alexander M. Bronstein,et al.  Affine-Invariant Photometric Heat Kernel Signatures , 2012, 3DOR@Eurographics.

[28]  Guillermo Sapiro,et al.  O(N) implementation of the fast marching algorithm , 2006, Journal of Computational Physics.

[29]  Ron Kimmel,et al.  Texture Mapping via Spherical Multi-dimensional Scaling , 2005, Scale-Space.

[30]  Matthijs C. Dorst.  Distinctive Image Features from Scale-Invariant Keypoints , 2011 .

[31]  A. Bronstein,et al.  Shape Google : a computer vision approach to invariant shape retrieval , 2009 .

[32]  Luc Van Gool,et al.  SURF: Speeded Up Robust Features , 2006, ECCV.

[33]  Leonidas J. Guibas,et al.  A concise and provably informative multi-scale signature based on heat diffusion , 2009 .

[34]  Alfred M. Bruckstein,et al.  Invariant signatures for planar shape recognition under partial occlusion , 1992, [1992] Proceedings. 11th IAPR International Conference on Pattern Recognition.

[35]  Alexander M. Bronstein,et al.  Equi-affine Invariant Geometry for Shape Analysis , 2013, Journal of Mathematical Imaging and Vision.

[36]  Edwin R. Hancock,et al.  Commute Times, Discrete Green's Functions and Graph Matching , 2005, ICIAP.

[37]  Ron Kimmel,et al.  Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[38]  Mark W. Woolrich,et al.  Advances in functional and structural MR image analysis and implementation as FSL , 2004, NeuroImage.