Discretization of Parametrizable Signal Manifolds

Transformation-invariant analysis of signals often requires the computation of the distance from a test pattern to a transformation manifold. In particular, the estimation of the distances between a transformed query signal and several transformation manifolds representing different classes provides essential information for the classification of the signal. In many applications, the computation of the exact distance to the manifold is costly, whereas an efficient practical solution is the approximation of the manifold distance with the aid of a manifold grid. In this paper, we consider a setting with transformation manifolds of known parameterization. We first present an algorithm for the selection of samples from a single manifold that permits to minimize the average error in the manifold distance estimation. Then we propose a method for the joint discretization of multiple manifolds that represent different signal classes, where we optimize the transformation-invariant classification accuracy yielded by the discrete manifold representation. Experimental results show that sampling each manifold individually by minimizing the manifold distance estimation error outperforms baseline sampling solutions with respect to registration and classification accuracy. Performing an additional joint optimization on all samples improves the classification performance further. Moreover, given a fixed total number of samples to be selected from all manifolds, an asymmetric distribution of samples to different manifolds depending on their geometric structures may also increase the classification accuracy in comparison with the equal distribution of samples.

[1]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[2]  Thomas A. Funkhouser,et al.  The Princeton Shape Benchmark , 2004, Proceedings Shape Modeling Applications, 2004..

[3]  David L. Donoho,et al.  Image Manifolds which are Isometric to Euclidean Space , 2005, Journal of Mathematical Imaging and Vision.

[4]  Andrew W. Fitzgibbon,et al.  Joint manifold distance: a new approach to appearance based clustering , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[5]  Alessandro Neri,et al.  Maximum likelihood localization of 2-D patterns in the Gauss-Laguerre transform domain: theoretic framework and preliminary results , 2004, IEEE Transactions on Image Processing.

[6]  M. Farrashkhalvat,et al.  Basic Structured Grid Generation: With an introduction to unstructured grid generation , 2003 .

[7]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[8]  Yann LeCun,et al.  Transformation Invariance in Pattern Recognition-Tangent Distance and Tangent Propagation , 1996, Neural Networks: Tricks of the Trade.

[9]  Philip Ogunbona,et al.  On the computational complexity of the LBG and PNN algorithms , 1997, IEEE Trans. Image Process..

[10]  Pascal Frossard,et al.  Minimum Distance between Pattern Transformation Manifolds: Algorithm and Applications , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[11]  Allen Gersho,et al.  Vector quantization and signal compression , 1991, The Kluwer international series in engineering and computer science.

[12]  Laurent D. Cohen,et al.  Geodesic Remeshing Using Front Propagation , 2003, International Journal of Computer Vision.

[13]  C. D. Kuglin,et al.  The phase correlation image alignment method , 1975 .

[14]  Gabriel Peyré,et al.  Manifold models for signals and images , 2009, Comput. Vis. Image Underst..

[15]  Yuan F. Zheng,et al.  Image Registration Using Adaptive Polar Transform , 2009, IEEE Transactions on Image Processing.

[16]  D. Donoho,et al.  Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Nuno Vasconcelos,et al.  A multiresolution manifold distance for invariant image similarity , 2005, IEEE Transactions on Multimedia.

[18]  Pascal Frossard,et al.  Distance-based discretization of parametric signal manifolds , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[19]  Robert M. Gray,et al.  An Algorithm for Vector Quantizer Design , 1980, IEEE Trans. Commun..

[20]  Richard G. Baraniuk,et al.  The multiscale structure of non-differentiable image manifolds , 2005, SPIE Optics + Photonics.

[21]  Arye Nehorai,et al.  Analysis of a polarized seismic wave model , 1996, IEEE Trans. Signal Process..