Kernel PCA for similarity invariant shape recognition

We present in this paper a novel approach for shape description based on kernel principal component analysis (KPCA). The strength of this method resides in the similarity (rotation, translation and particularly scale) invariance of KPCA when using a family of triangular conditionally positive definite kernels. Beside this invariance, the method provides an effective way to capture non-linearities in shape geometry. A given two-dimensional curve is described using the eigenvalues of the underlying manifold modeled in a high-dimensional Hilbert space. Using Fourier analysis, we will show that this eigenvalue description captures low to high variations of the shape frequencies. Experiments conducted on standard databases including the SQUID, the Swedish and the Smithsonian leaf databases, show that the method is effective in capturing invariance and generalizes well for shape matching and retrieval.

[1]  David B. Cooper,et al.  Practical Reliable Bayesian Recognition of 2D and 3D Objects Using Implicit Polynomials and Algebraic Invariants , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[2]  Bernhard Schölkopf,et al.  Nonlinear Component Analysis as a Kernel Eigenvalue Problem , 1998, Neural Computation.

[3]  Song-Chun Zhu,et al.  A multi-scale generative model for animate shapes and parts , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[4]  C. Micchelli.  Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

[5]  Marc G. Genton,et al.  Classes of Kernels for Machine Learning: A Statistics Perspective , 2002, J. Mach. Learn. Res..

[6]  Zhengyou Zhang,et al.  New Measurements and Corner-Guidance for Curve Matching with Probabilistic Relaxation , 2002, International Journal of Computer Vision.

[7]  Josef Kittler,et al.  Robust and Efficient Shape Indexing through Curvature Scale Space , 1996, BMVC.

[8]  Christos Faloutsos,et al.  FastMap: a fast algorithm for indexing, data-mining and visualization of traditional and multimedia datasets , 1995, SIGMOD '95.

[9]  David J. Kriegman,et al.  Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection , 1996, ECCV.

[10]  David C. Hogg,et al.  An Adaptive Eigenshape Model , 1995, BMVC.

[11]  Marcel Worring,et al.  Content-Based Image Retrieval at the End of the Early Years , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[12]  F. Fleuret,et al.  Scale-Invariance of Support Vector Machines based on the Triangular Kernel , 2001 .

[13]  Benjamin B. Kimia,et al.  Curves vs skeletons in object recognition , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[14]  Oskar Söderkvist,et al.  Computer Vision Classification of Leaves from Swedish Trees , 2001 .

[15]  P. Toft.  The Radon Transform - Theory and Implementation , 1996 .

[16]  K. W. Cattermole.  The Fourier Transform and its Applications , 1965 .

[17]  Daniel Cremers,et al.  Nonlinear Shape Statistics via Kernel Spaces , 2001, DAGM-Symposium.

[18]  James M. Coggins.  Statistical approach to multiscale medial vision , 1992, Optics & Photonics.

[19]  Bernhard Schölkopf,et al.  The Kernel Trick for Distances , 2000, NIPS.

[20]  Rama Chellappa,et al.  Classification of Partial 2-D Shapes Using Fourier Descriptors , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[21]  Haim J. Wolfson.  On curve matching , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  Bernhard Schölkopf,et al.  Improving the accuracy and speed of support vector learning machines , 1997, NIPS 1997.

[23]  Benjamin B. Kimia,et al.  On the Local Form and Transitions of Symmetry Sets, Medial Axes, and Shocks , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[24]  C. Berg,et al.  Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions , 1984 .

[25]  David B. Cooper,et al.  The Complex Representation of Algebraic Curves and Its Simple Exploitation for Pose Estimation and Invariant Recognition , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[26]  Nello Cristianini,et al.  An Introduction to Support Vector Machines and Other Kernel-based Learning Methods , 2000 .

[27]  Zhengyou Zhang,et al.  Iterative point matching for registration of free-form curves and surfaces , 1994, International Journal of Computer Vision.

[28]  Alex Pentland,et al.  View-based and modular eigenspaces for face recognition , 1994, 1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[29]  Bernhard Schölkopf,et al.  Comparing support vector machines with Gaussian kernels to radial basis function classifiers , 1997, IEEE Trans. Signal Process..

[30]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[31]  Nello Cristianini,et al.  An introduction to Support Vector Machines , 2000 .

[32]  Timothy F. Cootes,et al.  Active Shape Models-Their Training and Application , 1995, Comput. Vis. Image Underst..

[33]  William T. Freeman,et al.  Orientation Histograms for Hand Gesture Recognition , 1995 .

[34]  Laurent Younes,et al.  Computable Elastic Distances Between Shapes , 1998, SIAM J. Appl. Math..

[35]  Song-Chun Zhu,et al.  Stochastic Jump-Diffusion Process for Computing Medial Axes in Markov Random Fields , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[36]  Azriel Rosenfeld,et al.  Axial representations of shape , 1986, Computer Vision Graphics and Image Processing.

[37]  Alex Pentland,et al.  Modal Matching for Correspondence and Recognition , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[38]  R. Brubaker.  Models for the perception of speech and visual form: Weiant Wathen-Dunn, ed.: Cambridge, Mass., The M.I.T. Press, I–X, 470 pages , 1968 .

[39]  Josef Kittler,et al.  A survey of the hough transform , 1988, Comput. Vis. Graph. Image Process..

[40]  Ralph Roskies,et al.  Fourier Descriptors for Plane Closed Curves , 1972, IEEE Transactions on Computers.

[41]  David B. Cooper,et al.  Improving the stability of algebraic curves for applications , 2000, IEEE Trans. Image Process..

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

[43]  Euripides G. M. Petrakis,et al.  Shape retrieval based on dynamic programming , 2000, IEEE Trans. Image Process..

[44]  Remco C. Veltkamp,et al.  A Straight Skeleton Approximating the Medial Axis , 2004, ESA.

[45]  PaperNo.  Recognition of shapes by editing shock graphs , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[46]  Bernhard Schölkopf,et al.  Improving the Accuracy and Speed of Support Vector Machines , 1996, NIPS.

[47]  Gérard G. Medioni,et al.  Generic Shape Learning and Recognition , 1996, Object Representation in Computer Vision.

[48]  Benjamin B. Kimia,et al.  On the Local Form and Transitions of Symmetry Sets, Medial Axes, and Shocks , 2004, International Journal of Computer Vision.