Programming and Computer Software

, Volume 44, Issue 2, pp 112–119 | Cite as

Application of Computer Algebra to Photometric Stereo with Two Light Sources

  • R. Kozera
  • A. N. Prokopenya


This paper addresses the problem of reconstructing the shape of an unknown Lambertian surface from its two photometric images obtained by successive illumination of the surface with two different remote light sources. Using computer algebra methods, we investigate the conditions of existence and uniqueness of a solution to a system of algebraic equations that determine the gradient of a function of two variables given by the equation u(x, y) − z = 0. We also analyze necessary and sufficient conditions for unique determination of a second-order algebraic surface from its two images in the general case. Correctness of the theoretical results obtained is confirmed by simulating photometric images of various surfaces.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Horn, B.K.P., Robot Vision, MIT Press, 2001.Google Scholar
  2. 2.
    Horn, B.K.P. and Brooks, M.J., Shape from Shading, MIT Press, 1989.zbMATHGoogle Scholar
  3. 3.
    Kozera, R., Existence and uniqueness in photometric stereo, Appl. Math. Comput., 1991, vol. 1, no. 44, pp. 1–104.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Deift, P. and Sylvester, J., Some remarks on shapefrom-shading in computer vision, J. Math. Anal. Appl., 1991, vol. 84, no. 1, pp. 235–248.CrossRefzbMATHGoogle Scholar
  5. 5.
    Oliensis, J., Uniqueness in shape from shading, Int. J. Comput. Vision, 1991, vol. 2, no. 6, pp. 75–104.CrossRefzbMATHGoogle Scholar
  6. 6.
    Brooks, M.J., Chojnacki, W., and Kozera, R., Shading without shape. Q. Appl. Math., 1992, vol. 1, no. 50, pp. 27–38.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kozera, R., Uniqueness in shape from shading revisited, Int. J. Math. Imaging Vision, 1997, vol. 7, pp. 123–138.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kozera, R., On complete integrals and uniqueness in shape from shading, Appl. Math. Comput., 1995, vol. 1, no. 73, pp. 1–37.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bruss, A.R., The eikonal equation: Some results applicable to computer vision, J. Math. Phys., 1982, vol. 23, no. 5, pp. 890–896.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brooks, M.J., Chojnacki, W., and Kozera, R., Circularly symmetrical eikonal equations and nonuniqueness in computer vision, J. Math. Anal. Appl., 1992, vol. 1, no. 165, pp. 192–215.CrossRefzbMATHGoogle Scholar
  11. 11.
    Kimmel, R. and Bruckstein, A.M., Tracking level sets by level sets: A method of solving the shape from shading problem, Comput. Vision Graphics Image Understanding, 1995, no. 62, pp. 47–58.CrossRefGoogle Scholar
  12. 12.
    Woodham, R.J., Photometric stereo: A reflectance map technique for determining surface orientation from multiple images, Opt. Eng., 1980, vol. 1, no. 19, pp. 139–144.Google Scholar
  13. 13.
    Kozera, R., On shape recovery from two shading patterns, Int. J. Pattern Recognit. Artif. Intell., 1992, vol. 4, no. 6, pp. 673–698.CrossRefGoogle Scholar
  14. 14.
    Onn, R. and Bruckstein, A.M., Integrability disambiguates surface recovery in two-image photometric stereo, Int. J. Comput. Vision, 1990, vol. 1, no. 5, pp. 105–113.CrossRefGoogle Scholar
  15. 15.
    Kozera, R. and Prokopenya, A., Orthogonal illuminations in two light-source photometric stereo, Lect. Notes Comput. Sci., 2016, vol. 9842, pp. 402–415.CrossRefGoogle Scholar
  16. 16.
    Kozera R. and Prokopenya, A.N., Application of computer algebra for the reconstruction of surfaces from their photometric stereo images, Program. Comput. Software, 2017, vol. 43, no. 2, pp. 98–104.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wolfram, S., The Mathematica Book, Cambridge University Press, 2003, 5th ed.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Warsaw University of Life Sciences – SGGWWarsawPoland
  2. 2.School of Computer Science and Software EngineeringUniversity of Western AustraliaCrawley, PerthAustralia

Personalised recommendations