Estimation of Facial Angular Information Using a Complex-Number-Based Statistical Model

  • Mario Castelan
  • Edwin R. Hancock
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3773)


In this paper we explore the use of complex numbers as means of representing angular statistics for surface normal data. Our aim is to use the representation to construct a statistical model that can be used to describe the variations in fields of surface normals. We focus on the problem of representing facial shape. The fields of surface normals used to train the model are furnished by range images. We compare the complex representation with one based on angles, and demonstrate the advantages of the new method. Once trained, we illustrate how the model can be fitted to brightness images by searching for the set of parameters that both satisfy Lambert’s law and minimize the integrability error.


Zenith Angle Azimuth Angle Range Image Integrability Error Statistical Shape Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Atick, J., Griffin, P., Redlich, N.: Statistical Approach to Shape from Shading: Reconstruction of Three-Dimensional Face Surfaces from Single Two-Dimensional Images. Neural Computation 8, 1321–1340 (1996)CrossRefGoogle Scholar
  2. 2.
    Dovgard, R., Basri, R.: Statistical symmetric shape from shading for 3D structure recovery of faces. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3022, pp. 99–113. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Erens, R.G.F., Kappers, A.M.L., Koenderink, J.J.: Perception of Local Shape from Shading. Perception and Psychophysics 54(2), 145–156 (1993)CrossRefGoogle Scholar
  4. 4.
    Frankot, R.T., Chellapa, R.: A Method for Enforcing Integrability in Shape from Shading Algorithms. IEEE Trans. Pattern Analysis and Machine Intelligence 10(4), 438–451 (1988)CrossRefGoogle Scholar
  5. 5.
    Mardia, K.V.: Statistics of Directional Data. Academic Press, London (1972)zbMATHGoogle Scholar
  6. 6.
    Horn, B.K.P.: Understanding Image Intensities. Artificial Intelligence 8, 201–231 (1977)zbMATHCrossRefGoogle Scholar
  7. 7.
    Sirovich, L., Everson, R.: Management and Analysis of Large Scientific Datasets. The International Journal of Supercomputer Applications 6(1), 50–68 (1992)Google Scholar
  8. 8.
    Turk, M.A., Pentland, A.P.: Face Recognition Using Eigenfaces. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 586–591 (1991)Google Scholar
  9. 9.
    Blanz, V., Vetter, T.: A Morphable model for the synthesis of 3D Faces. In: Proceedings of SIGGRAPH 1999, pp. 187–194 (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Mario Castelan
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK

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