Advertisement

Analytical Image Models and Their Applications

  • Anuj Srivastava
  • Xiuwen Liu
  • Ulf Grenander
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2350)

Abstract

In this paper, we study a family of analytical probability models for images within the spectral representation framework. First the input image is decomposed using a bank of filters, and probability models are imposed on the filter outputs (or spectral components). A two-parameter analytical form, called a Bessel K form, derived based on a generator model, is used to model the marginal probabilities of these spectral components. The Bessel K parameters can be estimated efficiently from the filtered images and extensive simulations using video, infrared, and range images have demonstrated Bessel K form’s fit to the observed histograms. The effectiveness of Bessel K forms is also demonstrated through texture modeling and synthesis. In contrast to numeric-based dimension reduction representations, which are derived purely based on numerical methods, the Bessel K representations are derived based on object representations and this enables us to establish relationships between the Bessel parameters and certain characteristics of the imaged objects. We have derived a pseudometric on the image space to quantify image similarities/differences using an analytical expression for L 2-metric on the set of Bessel K forms. We have applied the Bessel K representation to texture modeling and synthesis, clutter classification, pruning of hypotheses for object recognition, and object classification. Results show that Bessel K representation captures important image features, suggesting its role in building efficient image understanding paradigms and systems.

Keywords

Image features spectral analysis Bessel K forms clutter classification object recognition 

References

  1. 1.
    O. Barndorff-Nielsen, J. Kent, and M. Sorensen, “Normal variance-mean mixtures and z distributions,” International Statistical Review, vol. 50, pp. 145–159, 1982.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    P. N. Belhumeur, J. P. Hepanha, and D. J. Kriegman, “Eigenfaces vs. fisherfaces: Recognition using class specific linear projection,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19(7), pp. 711–720, 1997.CrossRefGoogle Scholar
  3. 3.
    A. J. Bell and T. J. Sejnowski, “The “independent components” of natural scenes are edge filters,” Vision Research, vol. 37(23), pp. 3327–3338, 1997.CrossRefGoogle Scholar
  4. 4.
    P. Comon, “Independent component analysis, a new concept?” Signal Processing, Special issue on higher-order statistics, vo. 36(4), pp. 287–314, 1994.zbMATHGoogle Scholar
  5. 5.
    I. S. Gradshteyn and I. M. Ryzhik, Table of Integral Series and Products, Academic Press, 2000.Google Scholar
  6. 6.
    U. Grenander, General Pattern Theory, Oxford University Press, 1993.Google Scholar
  7. 7.
    U. Grenander, M. I. Miller, and A. Srivastava, “Hilbert-schmidt lower bounds for estimators on matrix lie groups for atr,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20(8), pp. 790–802, 1998.CrossRefGoogle Scholar
  8. 8.
    U. Grenander and A. Srivastava, “Probability models for clutter in natural images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23(4), pp. 424–429, 2001.CrossRefGoogle Scholar
  9. 9.
    U. Grenander, A. Srivastava, and M. I. Miller, “Asymptotic performance analysis of bayesian object recognition,” IEEE Transactions of Information Theory, vol. 46(4), pp. 1658–1666, 2000.zbMATHCrossRefGoogle Scholar
  10. 10.
    D. J. Heeger and J. R. Bergen, “Pyramid-based texture analysis/synthesis,” In Proceedings of SIGGRAPHS, pp. 229–238, 1995.Google Scholar
  11. 11.
    B. Julesz, “A theory of preattentive texture discrimination based on first-order statistics of textons,” Biological Cybernetics, vol. 41, pp. 131–138, 1962.CrossRefMathSciNetGoogle Scholar
  12. 12.
    M. Kirby and L. Sirovich, “Application of the karhunen-loeve procedure for the characterization of human faces,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12(1), pp. 103–108, 1990.CrossRefGoogle Scholar
  13. 13.
    Ann B. Lee and David Mumford, “Occlusion models for natural images: A statistical study of scale-invariant dead leaves model,” International Journal of Computer Vision, vol. 41, pp. 35–59, 2001.zbMATHCrossRefGoogle Scholar
  14. 14.
    M. I. Miller, A. Srivastava, and U. Grenander, “Conditional-expectation estimation via jump-diffusion processes in multiple target tracking/recognition,” IEEE Transactions on Signal Processing, vol. 43(11), pp. 2678–2690, 1995.CrossRefGoogle Scholar
  15. 15.
    D. Mumford, “Empirical investigations into the statistics of clutter and the mathematical models it leads to,” A lecture for the review of ARO Metric Pattern Theory Collaborative, 2000.Google Scholar
  16. 16.
    B. A. Olshausen and D. J. Field, “Sparse coding with an overcomplete basis set: A strategy employed by V1?” Vision Research, vol. 37(23), pp. 3311–3325, 1997.CrossRefGoogle Scholar
  17. 17.
    M. Pontil and A. Verri, “Support vector machines for 3d object recognition,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20(6), pp. 637–646, 1998.CrossRefGoogle Scholar
  18. 18.
    J. Portilla and E. P. Simoncelli, “A parametric texture model based on joint statistics of complex wavelets,” International Journal of Computer Vision, vol. 40(1), pp. 49–71, 2000.zbMATHCrossRefGoogle Scholar
  19. 19.
    S. T. Roweis and L. K. Saul, “Nonlinear dimensionality reduction by locally linear embedding,” Science, vol. 290, pp. 2323–2326, 2000.CrossRefGoogle Scholar
  20. 20.
    A. Srivastava, X. Liu, and U. Grenander, “Universal analytical forms for modeling image probabilities,” IEEE Transactions on Pattern Analysis and Machine Intelligence, in press, 2002.Google Scholar
  21. 21.
    A. Srivastava, M. I. Miller, and U. Grenander, “Bayesian automated target recognition,” Handbook of Image and Video Processing, Academic Press, pp. 869–881, 2000.Google Scholar
  22. 22.
    M. J. Wainwright, E. P. Simoncelli, and A. S. Willsky, “Random cascades on wavelet trees and their use in analyzing and modeling natural images,” Applied and Computational Harmonic Analysis, vol. 11, pp. 89–123, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    M. H. Yang, D. Roth, and N. Ahuja, “Learning to recognize 3d objects with SNoW,” In Proceedings of the Sixth European Conference on Computer Vision, vol. 1, pp. 439–454, 2000.Google Scholar
  24. 24.
    S. C. Zhu, X. Liu, and Y. N. Wu, “Statistics matching and model pursuit by efficient MCMC,” IEEE Transactions on Pattern Recognition and Machine Intelligence, vol. 22, pp. 554–569, 2000.CrossRefGoogle Scholar
  25. 25.
    S. C. Zhu, Y. N. Wu, and D. Mumford, “Minimax entropy principles and its application to texture modeling,” Neural Computation, vol. 9(8), pp. 1627–1660, 1997.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Anuj Srivastava
    • 1
  • Xiuwen Liu
    • 2
  • Ulf Grenander
    • 3
  1. 1.Department of StatisticsFlorida State UniversityTallahassee
  2. 2.Department of Computer ScienceFlorida State UniversityTallahassee
  3. 3.Division of Applied MathematicsBrown UniversityProvidence

Personalised recommendations