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Conjugate Gradient in Noisy Photometric Stereo

  • Ryszard Kozera
  • Felicja Okulicka-Dłużewska
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8671)

Abstract

This paper discusses the problem of reconstructing the Lambertian surface from noisy three-light source Photometric Stereo. In the continuous image setting the shape recovery process is divided into two steps: an algebraic one (gradient computation) and analytical one (gradient integration). The digitized case with added noise has it discrete analogue in which also perturbed gradient from three noisy images is first computed. Generically such non-integrable vector field is subsequently rectified to the ”closest” integrable one. Finally, numerical integration scheme yields the unknown surface. The process of vector field rectification is reduced to the corresponding linear optimization task of very high dimension (comparable with the image resolution). Standard methods based on matrix pseudo-inversion suffer from heavy computation due to the necessity of large matrix inversion. A possible alternative is to set up an iterative scheme based on local snapshots’ optimizations (e.g. 2D-Leap-Frog). Another approach which is proposed in this paper is solving the above global optimization scheme by Conjugate Gradient with no inversion of matrices of large dimension. The experimental results from this paper show that the application of Conjugate Gradient forms a computationally feasible alternative in denoising Photometric Stereo.

Keywords

Shape Reconstruction Photometric Stereo noise removal Conjugate Gradient numerical computation 

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References

  1. 1.
    Brooks, M.J., Chojnacki, W., Kozera, R.: Impossible and ambiguous shading patterns. Int. J. Comp. Vision 7(2), 119–126 (1992)CrossRefGoogle Scholar
  2. 2.
    Castelán, M., Hancock, E.R.: Imposing integrability in geometric shape-from-shading. In: Sanfeliu, A., Ruiz-Shulcloper, J. (eds.) CIARP 2003. LNCS, vol. 2905, pp. 196–203. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. (2013)Google Scholar
  4. 4.
    Frankot, R.T., Chellappa, R.: A method of enforcing integrability in shape from shading algorithms. IEEE Trans. Patt. Rec. Machine Intell. 10(4), 439–451 (1988)CrossRefzbMATHGoogle Scholar
  5. 5.
    Horn, B.K.P.: Robot Vision. McGraw-Hill, New York (1986)Google Scholar
  6. 6.
    Horn, B.K.P., Brooks, M.J.: Shape from Shading. MIT Press, CambridgeGoogle Scholar
  7. 7.
    Kozera, R.: Existence and uniqueness in Photometric Stereo. Appl. Math. Comput. 44(1), 1–104 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kozera, R.: On shape recovery from two shading patterns. Int. J. Patt. Rec. Art. Intel. 6(4), 673–698 (1992)CrossRefGoogle Scholar
  9. 9.
    Kozera, R.: On complete integrals and uniqueness in shape from shading. Appl. Math. Comput. 73(1), 1–37 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Kozera, R.: Uniqueness in shape from shading revisited. J. Math. Imag. Vision 7, 123–138 (1997)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Noakes, L., Kozera, R.: The 2-D leap-frog: Integrability, noise, and digitization. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 352–364. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Noakes, L., Kozera, R.: Denoising images: Non-linear leap-frog for shape and light-source recovery. In: Asano, T., Klette, R., Ronse, C. (eds.) Geometry, Morphology, and Computational Imaging. LNCS, vol. 2616, pp. 419–436. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Noakes, L., Kozera, R.: Nonlinearities and noise reduction in 3-source Photometric Stereo. J. Math. Imag. Vision 18(3), 119–127 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Oliensis, J.: Uniqueness in shape from shading. Int. J. Comp. Vision 6(2), 75–104 (1991)CrossRefGoogle Scholar
  15. 15.
    Onn, R., Bruckstein, A.: Uniqueness in shape from shading. Int. J. Comp. Vision 5(1), 105–113 (1990)CrossRefGoogle Scholar
  16. 16.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM (2003)Google Scholar
  17. 17.
    Simchony, T., Chellappa, R., Shao, M.: Direct analytical methods for solving Poisson equations in computer vision problems. IEEE Trans. Patt. Rec. Machine Intell. 12(5), 435–446 (1990)CrossRefGoogle Scholar
  18. 18.
    van der Vorst, H.A.: Iterative Krylov Methods for Large Linear Systems. Cambridge Monographs on Applied and Computational Mathematics (2009)Google Scholar
  19. 19.
    Wei, T., Klette, R.: On depth recovery from gradient vector field. In: Bhattacharaya, B.B., Sur-Kolay, S., Nandy, S.C., Bagch, A. (eds.) Algorithms. in Architectures and Information Systems Security, pp. 765–797 (2009)Google Scholar
  20. 20.
    Wöhler, C.: 3D Computer Vision: Efficient Methods and Applications. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Ryszard Kozera
    • 1
  • Felicja Okulicka-Dłużewska
    • 2
  1. 1.Faculty of Applied Informatics and MathematicsWarsaw University of Life Sciences-SGGWWarsawPoland
  2. 2.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

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