Most stereo techniques compute disparity assuming that it varies slowly along surfaces. We quantify and justify this assumption, using weak assumptions about surface orientation distributions in the world to derive the density of disparity surface orientations. The small disparity change assumption is justified by the orientation density's heavy bias toward disparity surfaces that are nearly parallel to the image plane. In addition, the bias strengthens with smaller baselines, larger focal lengths, and as surfaces move farther from the cameras. To analyze current stereo techniques, we derive three densities from the first density, those of the disparity gradient magnitude, the directional derivative of disparity, and the difference in disparity between neighboring surface points. The latter may be used in Bayesian algorithms computing dense disparity fields. The directional derivative density and the disparity difference density both show that feature-based algorithms should strongly favor small disparity changes, contrary to several well-known algorithms. Finally, we use our original surface orientation density and the gradient magnitude density to derive two new “surfaces-from-stereo” techniques, techniques combining feature-based matching and surface reconstruction. The first uses the densities to severely restrict the search range for the optimum fit. The second incorporates the surface orientation density into the optimization criteria, producing a Bayesian formulation. Both algorithms are shown to be efficient and effective.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Arnold, R.D. and Binford, T.O. 1980. Geometric constraints in stereo vision. In Proc. SPIE, 238:281–292.
Ayache N. and Faverjon B. 1987. Efficient registration of stereo images by matching graph descriptions of edge segments. International Journal of Computer Vision, 1:107–131.
Baker, H.H. and Binford, T.O. 1981. Depth from edge and intensity based stereo. In Proceedings Seventh International Joint Conference on Artificial Intelligence, pp. 631–636.
Ballard, D.H. and Brown, C.M. 1982. Computer Vision. Prentice-Hall.
Barnard S.T. 1989. Stochastic stereo matching over scale. International Journal of Computer Vision, 3:17–32.
Barnard S.T. and ThompsonW.B. 1980. Disparity analysis of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2:333–340.
Barnard S.T. and Fischler M.A. 1982. Computational stereo. Computing Surveys, 14:553–572.
Beaton A.E. and Tukey J.W. 1974. The fitting of power series, meaning polynomials, illustrated on bandspectroscopic data. Technometrics, 16:147–185.
Belhumeur, P.N. 1993. A bionocular stereo algorithm for reconstructing sloping, creased, and broken surfaces in the presence of half-occlusion. In Proceedings of the IEEE International Conference on Computer Vision, pp. 431–438.
Belhumeur, P.N. and Mumford, D. 1992. A Bayesian treatment of the stereo correspondence problem using half-occluded regions. In Proceedings IEEE Conference on Computer Vision and Pattern Recognition, pp. 506–512.
Besl, P.J., Birch, J.B., and Watson, L.T. 1988. Robust window operators. In Proceedings of the IEEE International Conference on Computer Vision, pp. 591–600.
Bolles, R.C., Baker, H.H., and Hannah, M.J. 1993. The JISCT stereo evaluation. In Proceedings of the DARPA Image Understanding Workshop, pp. 263–274.
Boult, T.E. and Chen, L.-H. 1988. Synergistie smooth surface stereo. In Proceedings of the IEEE International Conference on Computer Vision, pp. 118–122.
Canny J. 1986. A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8:679–698.
Dhond U.R. and Aggarwal J.K. 1989. Structure from stereo—A review. IEEE Transactions on Systems, Man and Cybernetics, 19:1489–1510.
Drumheller, M. and Poggio, T. 1986. On parallel stereo. In Proceedings IEEE International Conference on Robotics and Automation, pp. 1439–1448.
Eastman R.D. and Waxman A.M. 1987. Using disparity functionals for stereo correspondence and surface reconstruction. Computer Vision, Graphics, and Image Processing, 39:73–101.
Geman S. and Geman D. 1984. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721–741.
Grimson, W.E.L. 1981. From Images to Surfaces: A Computational Study of the Human Early Visual System, MIT Press.
Grimson W.E.L. 1985. Computational experiments with a feature based stereo algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 7:17–34.
Grimson W.E.L. and Mundy J.L. 1994. Computer vision applications. Communications of the ACM, 37(3):45–51.
Hampe lF.R., Rousseeuw P.J., and Ronchetti E., 1981. The change-of-variance curve and optimal redescending M-estimators. Journal of the American Statistical Association, 76:643–648.
Hampel, F.R., Rousseeuw, P.J., Ronchetti, E., and Stahel, W.A. 1986. Robust Statistics: The Approach Based on Influence Functions. John Wiley & Sons.
Hoff W. and Ahuja N. 1989. Surfaces from stereo: Integrating feature matching, disparity estimation and contour detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11:121–136.
Holland P.W. and Weisch R.E. 1977. Robust regression using iteratively reweighted least-squares. Commun. Statist.-Theor. Meth., A6:813–827.
Horn, B.K.P. 1986. Robot Vision. MIT Press.
Huber, P.J. 1981. Robust Statistics. John Wiley & Sons.
Illingworth J. and Kittler J. 1988. A survey of the Hough transform. Computer Vision, Graphics, and Image Processing, 44:87–116.
Jenkin M.R.M., Jepson A.D., and Tsotsos J.K. 1991. Techniques for disparity measurement. CVGIP: Image Understanding, 53:14–30.
Jolliffe I.T. 1986. Principal Component Analysis. Springer-Verlag: New York.
Kanade T. and Okutomi M. 1994. A stereo matching algorithm with an adaptive window: Theory and experiment. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16:920–932.
Marr D. and Poggio T. 1976. Cooperative computation of stereo disparity. Science, 194:283–287.
Marr D. and Poggio T. 1979. A computational theory of human stereo vision. Proceedings of the Royal Society of London, B., 204:301–328.
Matthies L. 1992. Stereo vision for planetary rovers: Stochastic modeling to near real-time implementation. International Journal of Computer Vision, 8:71–91.
Mclauchlan, P.F. 1990. Describing Textured Surfaces using Stereo Vision. Ph.D. thesis, University of Sheffield.
Mclauchlan P.F., Mayhew J.E.W., and Frisby J.P. 1991. Stereoscopic recovery and description of smooth textured surfaces. Image and Vision Computing, 9:20–26.
Medioni G. and Nevatia R. 1985. Segment-based stereo matching. Computer Vision, Graphics, and Image Processing, 31:2–18.
Mirza M.J. and Boyer K.L. 1993. Performance evaluation of a class of M-estimators for surface paramter estimation in noisy range data. IEEE Transactions on Robotics and Automation, 9:75–85.
Nielsen, M. 1993. Isotropic regularization. In Proceedings of the 4th British Machine Vision Conference, pp. 135–144.
Nielsen, M. 1995. Surface reconstruction: GNCs and MFA. In Proceedings of the IEEE International Conference on Computer Vision.
Ohta Y. and Kanade T. 1985. Stereo by intra-and interscanline search. IEEE Transactions on Pattern Analysis and Machine Intelligence, 7:139–154, 1985.
Olsen S.I. 1990. Stereo correspondence by surface reconstruction. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12:309–315.
Poggio T., Torre V., and Koch C. 1985. Computational vision and regularization theory. Nature, 317:314–319.
Pollard S.B., Mayhew J.E.W., and Frisby J.P. 1985. PMF: A stereo correspondence algorithm using a disparity gradient limit. Perception, 14:449–470.
Prazdny K. 1985. Detection of binocular disparities. Biological Cybernetics, 52:93–99.
Shah, J. 1993. A nonlinear diffusion model for discontinuous disparity and half-occlusions in stereo. In Proceedings IEEE Conference on Computer Vision and Pattern Recognition, pp. 34–40.
Stewart, C.V. 1992. On the derivation of geometric constraints in stereo. In Proceedings IEEE Conference on Computer Vision and Pattern Recognition, pp. 769–772.
Weisberg, S. 1985. Applied Linear Regression. John Wiley and Sons.
Yuille, A., Geiger, D., and Bulthoff, H. 1990. Stereo integration, mean field theory and psychophysics. In Proceedings First European Conference on Computer Vision, pp. 73–82.
The authors would like to thank James Miller for helpful discussions and comments on earlier versions of this paper and Wes Turner for making the camera system work. They would also like to acknowledge the financial support of the National Science Foundation under grant IRI-9217195.
About this article
Cite this article
Stewart, C.V., Flatland, R.Y. & Bubna, K. Geometric constraints and stereo disparity computation. Int J Comput Vision 20, 143–168 (1996). https://doi.org/10.1007/BF00208717
- Surface Orientation
- Gradient Magnitude
- Bayesian Formulation
- Orientation Density
- Small Baseline