Object Recognition Through Topo-Geometric Shape Models Using Error-Tolerant Subgraph Isomorphisms

We propose a method for 3-D shape recognition based on inexact subgraph isomorphisms, by extracting topological and geometric properties of a shape in the form of a shape model, referred to as topo-geometric shape model (TGSM). In a nutshell, TGSM captures topological information through a rigid transformation invariant skeletal graph that is constructed in a Morse theoretic framework with distance function as the Morse function. Geometric information is then retained by analyzing the geometric profile as viewed through the distance function. Modeling the geometric profile through elastic yields a weighted skeletal representation, which leads to a complete shape signature. Shape recognition is carried out through inexact subgraph isomorphisms by determining a sequence of graph edit operations on model graphs to establish subgraph isomorphisms with a test graph. Test graph is recognized as a shape that yields the largest subgraph isomorphism with minimal cost of edit operations. In this paper, we propose various cost assignments for graph edit operations for error correction that takes into account any shape variations arising from noise and measurement errors.

[1]  D. Corneil,et al.  An Efficient Algorithm for Graph Isomorphism , 1970, JACM.

[2]  John E. Hopcroft,et al.  Linear time algorithm for isomorphism of planar graphs (Preliminary Report) , 1974, STOC '74.

[3]  Julian R. Ullmann,et al.  An Algorithm for Subgraph Isomorphism , 1976, J. ACM.

[4]  Douglas C. Schmidt,et al.  A Fast Backtracking Algorithm to Test Directed Graphs for Isomorphism Using Distance Matrices , 1976, J. ACM.

[5]  Birgit Ueberreiter,et al.  Pattern Recognition by Graph Matching - Combinatorial versus continuous Optimization , 1988, Int. J. Pattern Recognit. Artif. Intell..

[6]  Alfred M. Bruckstein,et al.  On Minimal Energy Trajectories , 1990, Comput. Vis. Graph. Image Process..

[7]  Tosiyasu L. Kunii,et al.  Constructing a Reeb graph automatically from cross sections , 1991, IEEE Computer Graphics and Applications.

[8]  Tony DeRose,et al.  Surface reconstruction from unorganized points , 1992, SIGGRAPH.

[9]  D. Mumford.  Algebraic Geometry and Its Applications , 1994 .

[10]  C. Bajaj.  Algebraic Geometry and its Applications , 1994 .

[11]  B. Messmer.  Efficient graph matching algorithms , 1995 .

[12]  V. Jurdjevic.  NON-EUCLIDEAN ELASTICA , 1995 .

[13]  Heinrich Müller,et al.  Interpolation and Approximation of Surfaces from Three-Dimensional Scattered Data Points , 1997, Scientific Visualization Conference (dagstuhl '97).

[14]  K. Mardia,et al.  Statistical Shape Analysis , 1998 .

[15]  Anne Verroust-Blondet,et al.  Level set diagrams of polyhedral objects , 1999, SMA '99.

[16]  Jindong Chen,et al.  Automatic Reconstruction of 3D CAD Models from Digital Scans , 1999, Int. J. Comput. Geom. Appl..

[17]  Paul A. Yushkevich,et al.  Segmentation, registration, and measurement of shape variation via image object shape , 1999, IEEE Transactions on Medical Imaging.

[18]  Taku Komura,et al.  Topology matching for fully automatic similarity estimation of 3D shapes , 2001, SIGGRAPH.

[19]  Sunghee Choi,et al.  The power crust , 2001, SMA '01.

[20]  Shengrui Wang,et al.  A new algorithm for inexact graph matching , 2002, Object recognition supported by user interaction for service robots.

[21]  Bernard Chazelle,et al.  A Reflective Symmetry Descriptor , 2002, ECCV.

[22]  松本 幸夫.  An introduction to Morse theory , 2002 .

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

[24]  Thomas A. Funkhouser,et al.  Harmonic 3D shape matching , 2002, SIGGRAPH '02.

[25]  A. Ben Hamza,et al.  Geodesic Object Representation and Recognition , 2003, DGCI.

[26]  Marcel Körtgen,et al.  3D Shape Matching with 3D Shape Contexts , 2003 .

[27]  Marcin Novotni,et al.  3D zernike descriptors for content based shape retrieval , 2003, SM '03.

[28]  Tony Tung,et al.  Augmented Reeb graphs for content-based retrieval of 3D mesh models , 2004, Proceedings Shape Modeling Applications, 2004..

[29]  Anuj Srivastava,et al.  Interpolations with elasticae in Euclidean spaces , 2004 .

[30]  David W. Jacobs.  Matching 3-D Models to 2-D Images , 2004, International Journal of Computer Vision.

[31]  Mario Vento,et al.  A (sub)graph isomorphism algorithm for matching large graphs , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[32]  Hamid Krim,et al.  3D object representation with topo-geometric shape models , 2005, 2005 13th European Signal Processing Conference.

[33]  Hamid Krim,et al.  2D Shape Modeling using Skeletal Graphs in a Morse Theoretic Framework , 2006, Statistics and Analysis of Shapes.

[34]  Alexander M. Bronstein,et al.  Efficient Computation of Isometry-Invariant Distances Between Surfaces , 2006, SIAM J. Sci. Comput..

[35]  Djamila Aouada,et al.  Geometric modeling of rigid and non-rigid 3D shapes using the global geodesic function , 2008, 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops.

[36]  Jitendra Malik,et al.  Shape matching and object recognition using shape contexts , 2010, 2010 3rd International Conference on Computer Science and Information Technology.