Abstract
Intermediate-level vision is central to form perception, and we outline an approach to intermediate-level segmentation based on complexity analysis. We focus on the problem of edge detection, and how edge elements might be grouped together. This is typical because, once the local structure is established, the transition to global structure must be effected and context is critical. To illustrate, consider an edge element inferred from an unknown image. Is this local edge part of a long curve, or part of a texture? If the former, which is the next element along the curve? If the latter, is the texture like a hair pattern, in which nearby elements are oriented similarly, or like a spaghetti pattern, in which they are not? Are there other natural possibilities? Such questions raise issues of dimensionality, since curves are 1-D and textures are 2-D, and also of complexity. Working toward a measure of representational complexity for vision, in this first of a pair of papers we develop a foundation based on geometric measure theory. The main result concerns the distribution of tangents in space and in orientation, which serves as a formal basis for the concrete measure of representational complexity developed in the companion paper.
Similar content being viewed by others
References
Besicovitch, A.S. 1928. On the fundamental geometrical properties of linearly measurable plane sets of points. Mathematische Annalen, 98:422–464.
Binford, T. 1981. Inferring surfaces from images. Artificial Intelligence, 17:205–244.
Boyer, K. and Sarkar, S. 2000. Perceptual Organization for Artificial Vision Systems. Kluwer Academic Publishers.
Burkill, J.C. and Burkill, H. 1970. A Second Course in Mathematical Analysis. Cambridge University Press.
Canny, J.F. 1986. A computational approach to edge detection. IEEE Trans. Pattern Anal. Machine Intell, 8:679–698.
Cesari, L. 1965. Surface Area. Princeton University Press: Princeton, NJ.
Cox, I.J., Rehg, J.M., and Hingorani, S. 1993. A Bayesian multiplehypothesis approach to edge grouping and contour segmentation. International Journal of Computer Vision, 11(1):5–24.
David, C. and Zucker, S.W. 1990. Potentials, valleys, and dynamic global coverings. International Journal of Computer Vision, 5(3):219–238.
Dubuc, B. and Zucker, S.W. 2001. Complexity, confusion, and perceptual grouping. Part II: Mapping complexity. Internation Journal of Computer Vision, 12(1):57–84.
Edgar, G.A. 1990. Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. Springler-Verlag: New York.
Falconer, K.J. 1987. The Geometry of Fractal Sets, Vol. 85 of Cambridge Tracts in Mathematics. Cambridge University Press: Cambridge, England.
Falconer, K.J. 1990. Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons: Chichester, England.
Federer, H. 1985. Geometric Measure Theory. Springler-Verlag: Berlin, Germany.
Freeman, W.T. and Adelson, E.H. 1991. The design and use of steerable filters. IEEE Trans. Pattern Anal. Machine Intell. 13(9):891–906.
Galli, A. and Zama, A. 1931. Untersuchungen über diewahrnehmung ebener geometrischen figuren die ganz oder teilweise von anderen geometrischen figuren verdeckt sind. Zeitschrift f¨ur Psychologie, 123:308–348.
Guy, G. and Medioni, G. 1993. Inferring global perceptual contours from local features. International Journal of Computer Vision, 20(1):113–133.
Guzman, A. 1968. Decomposition of a visual scene into threedimensional bodies. AFIPS Conf. Proc., 33:291–304.
Heitger, F. and von der Heydt, R. 1993. A computational model of neural contour processing: figure-ground segregation and illusory contours. In Proc. of the Fourth ICCV, IEEE Comp. Society (Ed.). pp. 32–40.
Hilbert, D. and Cohn-Vossen, S. 1990. Geometry and the Imagination. Chelsea Publishing Company: New York.
Hubel, D.H. and Wiesel, T.N. 1962. Receptive fields and functional architecture of monkey striate cortex. J. Physiology (London), 195:215–243.
Huggins, P. and Zucker, S. 2000. How folds cut a scene. Center For Computational Vision and Control, Yale University, New Haven, Technical Report.
Hurewicz, W. and Wallman, H. 1948. Dimension Theory. Princeton University Press: Princeton, NJ.
Iverson, L.A. 1993. Toward discrete geometric models for early vision. Ph.D. Thesis, Dept. of Electrical Engineering, McGill University, Montréal.
Iverson, L.A. and Zucker, S.W. 1995. Logical/linear operators for image curves.IEEE Trans. Pattern Anal. Machine Intell., 17(10):982–996.
Kanizsa, G. 1979. Organisation in Vision. Praeger: New York.
Kass, M. and Witkin, A. 1987. Analysing oriented patterns. Computer Vision, Graphics and Image Processing, 37:362–385.
Koenderink, J. 1990. Solid Shape. MIT Press: Cambridge, MA.
Leung, T. and Malik, J. 1998. Contour continuity in region based image segmentation. In Proc. 5th European Conf. on Computer Vision, pp. 544–559.
Malik, J., Belongie, S., Leung, T. and Shi, J. 2000. Contour and texture analysis for image segmentation. In Perceptual Organization for Artificial Vision Systems, K. Boyer and S. Sarkar (Eds.). Kluwer Academic Publishers.
Malik, J. and Perona, P. 1990. Preattentive texture discrimination with early vision mechanisms. J. of the Opt. Soc. of America A, 7:923–932.
Mandelbrot, B.B. 1982. The Fractal Geometry of Nature. Freeman: San Francisco.
Marr, D. and Hildreth, E. 1980. Theory of edge detection. Proc. Royal Society Lond., 207:187–217.
Mendès-France, M. 1991. The Planck constant of a curve. In Fractal Geometry and Analysis, S. Dubuc and J.Bélair (Eds.). Kluwer Academic Publishers, pp. 325–366.
Morel, J.-M. and Solimini, S. 1995. Variational Methods in Image Segmentation, Vol. 14 of Progress in Nonlinear Differential Equations and Their Applications, Birkhaüser: Boston.
Morgan, F. 1987. Geometric Measure Theory. Academic Press, Inc.
Mumford, D. 1992. Elastica and computer vision. In Algebraic Geometry and Applications. Springler-Verlag: Heidelberg.
Munkres, J.R. 1975. Topology: A First Course. Prentice-Hall.
Nitzberg, M., Mumford, D., and Shiota, T. 1993. Filtering, Segmentation and Depth, Vol. 662 of Lecture Notes in Computer Science. Springler-Verlag.
Peitgen, H.-O., Jürgens, H., and Saupe, D. 1992. Chaos and Fractals: New Frontiers of Science. Springer-Verlag: New York. 82 Dubuc and Zucker
Poincaré, H. 1926. Pourquoi l'espace a trois dimensions. In Dernières Pensées. Flammarion.
Rogers, C.A. 1970. Hausdorff Measures. Cambridge University Press: London.
Saund, E. 1993. Identifying salient circular arcs on curves. Computer Vision, Graphics and Image Processing, 58(3):327–337.
Simmons, G.F. 1963. Introduction to Topology and Modern Analysis. McGraw-Hill: New York.
Simon, H.A. 1968. The Sciences of the Artificial. MIT Press.
Smith, K.T. 1971. Primer of Modern Analysis. Bogden and Quigdley, Inc.
Tricot, C. 1991. Rectifiable and fractal sets. In Fractal Geometry and Analysis, S. Dubuc and J.Bélair (Eds.). Kluwer Academic Publishers, pp. 367–403.
Tricot, C. 1995. Curves and Fractal Dimension. Springler-Verlag: New York.
Ullman, S. 1990. Three-dimensional object recognition. In Cold Spring Harbor Symposia on Quantitative Biology, Vol. LV, Cold Spring Harbor Laboratory Press: Cold Spring Harbor, NY, pp. 889–898.
Waltz, D. 1975. Understanding line drawings of scenes with shadows. In The Psychology of Computer Vision, P. Winston (Ed.). McGraw-Hill: New York, pp. 19–91.
Williams, L. and Hanson, A. 1996. Perceptual completion of occluded surfaces. Computer Vision and Image Understanding, 64:1–20.
Williams, L. and Jacobs, D. 1997. Stochastic completion fields: A neural model of illusory contour shape and salience. Neural Computation, 9:837–858.
Yuille, A. and Coughlin, J. 1998. Convergence rates of algorithms for visual search: Detecting visual contours. Advances in Neural Information Processing, pp. 641–647.
Zucker, S.W. 1985. Early orientation selection: Tangent fields and the dimensionality of their support. Computer Vision, Graphics and Image Processing, 32:74–103.
Zucker, S.W. 1993. Endowing AI with Vision: A biological and computational perspective. Current Science, 64(6):407–418.
Zucker, S.W., Dobbins, A., and Iverson, L.A. 1989. Two stages of curve detection suggest two styles of visual computation. Neural Computation, 1:68–81.
Zucker, S.W., Hummel, R.A., and Rosenfeld, A. 1977. An application of relaxation labelling to line and curve enhancement. IEEE Transactions on Computers, 26:394–403.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dubuc, B., Zucker, S.W. Complexity, Confusion, and Perceptual Grouping. Part I: The Curve-like Representation. International Journal of Computer Vision 42, 55–82 (2001). https://doi.org/10.1023/A:1011189501276
Issue Date:
DOI: https://doi.org/10.1023/A:1011189501276