Shape-adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D brightness structure

  • Tony Lindeberg
  • Jonas Gårding
Shape Estimation
Part of the Lecture Notes in Computer Science book series (LNCS, volume 800)


Rotationally symmetric operations in the image domain may give rise to shape distortions. This article describes a way of reducing this effect for a general class of methods for deriving 3-D shape cues from 2-D image data, which are based on the estimation of locally linearized distortion of brightness patterns. By extending the linear scale-space concept into an affine scale-space representation and performing affine shape adaption of the smoothing kernels, the accuracy of surface orientation estimates derived from texture and disparity cues can be improved by typically one order of magnitude. The reason for this is that the image descriptors, on which the methods are based, will be relative invariant under affine transformations, and the error will thus be confined to the higher-order terms in the locally linearized perspective mapping.


Integration Scale Moment Matrix Shape Distortion Slant Angle Scale Selection 
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. L. Alvarez, F. Guichard, P.-L. Lions, and J.-M. Morel: Axioms and fundamental equations of image processing. Arch. Rat. Mech., (to appear).Google Scholar
  2. J. Babaud, A.P. Witkin, M. Baudin, and R.O. Duda: Uniqueness of the Gaussian kernel for scale-space filtering. IEEE-PAMI, 8(1):26–33, 1986.Google Scholar
  3. R. Bajcsy and L. Lieberman: Texture gradients as a depth cue. CVGIP, 5:52–67, 1976.Google Scholar
  4. D. Blostein and N. Ahuja: Shape from texture: integrating texture element extraction and surface estimation. IEEE-PAMI, 11(12):1233–1251, 1989.Google Scholar
  5. L.G. Brown and H. Shvaytser: Surface orientation from projective foreshortening of isotropic texture autocorrelation. IEEE-PAMI, 12(6):584–588, 1990.Google Scholar
  6. R. Cipolla, Y. Okamoto, and Y. Kuno: Robust structure from motion using motion parallax. 4th ICCV, 374–382, 1993.Google Scholar
  7. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, and M.A. Viergever: Scale and the differential structure of images. IVC, 10(6):376–388, 1992.Google Scholar
  8. -: Non-linear scale-space. (submitted), 1993.Google Scholar
  9. M.A. Förstner and E. Gülch: A fast operator for detection and precise location of distinct points, corners and centers of circular features. ISPRS, 1987.Google Scholar
  10. J. Gårding: Shape from texture for smooth curved surfaces in perspective projection. J. Math. Im. Vis., 2:329–352, 1992.Google Scholar
  11. J. Gårding and T. Lindeberg: Direct computation of shape cues by multi-scale retinotopic processing IJCV, (to appear). TRITA-NA-P9304, Royal Inst. Tech., 1993a.Google Scholar
  12. -: Direct estimation of local surface shape in a fixating binocular vision system. 3rd ECCV, (Stockholm, Sweden), (these proceedings), 1993b.Google Scholar
  13. D.G. Jones and J. Malik: Determining three-dimensional shape from orientation and spatial frequency disparities. 2nd ECCV, 661–669, 1992.Google Scholar
  14. J.J. Koenderink: The structure of images. Biol. Cyb., 50:363–370, 1984.Google Scholar
  15. J.J. Koenderink and A.J. van Doorn: Geometry of binocular vision and a model for stereopsis. Biol. Cyb., 21:29–35, 1976.Google Scholar
  16. -: Receptive field families. Biol. Cyb., 63:291–298, 1990.Google Scholar
  17. -: Affine structure from motion. J. Opt. Soc. Am., 377–385, 1991.Google Scholar
  18. T. Lindeberg: Scale-space for discrete signals. IEEE-PAMI, 12(3):234–254, 1990.Google Scholar
  19. -: Scale selection for differential operators. 8th Scand. Conf. Im. An., 857–866, 1993b.Google Scholar
  20. -: Scale-Space Theory in Computer Vision. Kluwer Academic Publishers, 1994.Google Scholar
  21. T. Lindeberg and J. Gårding: Shape from texture from a multi-scale perspective. 4th ICCV, 683–691, 1993a.Google Scholar
  22. -: Shape-adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D brightness structure. TRITA-NA-P9335, Royal Inst. Tech., 1993b.Google Scholar
  23. J. Malik and R. Rosenholtz: A differential method for computing local shape-from-texture for planar and curved surfaces. CVPR, 267–273, 1993.Google Scholar
  24. M. Nitzberg and T. Shiota: Non-linear image filtering with edge and corner enhancement. IEEE-PAMI, 14(8):826–833, 1992.Google Scholar
  25. N. Nordström: Biased anisotropic diffusion: A unified regularization and diffusion approach to edge detection. IVC, 8:318–327, 1990.Google Scholar
  26. P. Perona and J. Malik: Scale-space and edge detection using anisotropic diffusion. IEEE-PAMI, 12(7):629–639, 1990.Google Scholar
  27. G. Sapiro and A. Tannenbaum: Affine invariant scale-space. IJCV, 11(1):25–44, 1993.Google Scholar
  28. J.V. Stone: Shape from texture: textural invariance and the problem of scale in perspective images of surfaces. Brit. Machine Vision Conf, pp. 181–186, 1990.Google Scholar
  29. J. Weber and J. Malik: Robust computation of optical flow in a multi-scale differential framework. 4th ICCV, 12–20, 1993.Google Scholar
  30. R.P. Wildes: Direct recovery of three-dimensional scene geometry from binocular stereo disparity. IEEE-PAMI, 13(8):761–774, 1981.Google Scholar
  31. A. Witkin: Recovering surface shape and orientation from texture. AI, 17:17–45, 1981.Google Scholar
  32. -: Scale-space filtering. 8th IJCAI, 1019–1022, 1983.Google Scholar
  33. A. Yuille and T. Poggio: Scaling theorems for zero-crossings. IEEE-PAMI, 8:15–25, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Tony Lindeberg
    • 1
  • Jonas Gårding
    • 1
  1. 1.Computational Vision and Active Perception Laboratory (CVAP) Department of Numerical Analysis and Computing ScienceRoyal Institute of Technology (KTH)StockholmSweden

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