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Linear scale-space

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Abstract

The formulation of afront-end or“early vision” system is addressed, and its connection with scale-space is shown. A front-end vision system is designed to establish a convenient format of some sampled scalar field, which is suited for postprocessing by various dedicated routines. The emphasis is on the motivations and implications of symmetries of the environment; they pose natural, a priori constraints on the design of a front-end.

The focus is on static images, defined on a multidimensional spatial domain, for which it is assumed that there are no a priori preferred points, directions, or scales. In addition, the front-end is required to be linear. These requirements are independent of any particular image geometry and express the front-end's pure syntactical, “bottom up” nature.

It is shown that these symmetries suffice to establish the functionality properties of a front-end. For each location in the visual field and each inner scale it comprises a hierarchical family of tensorial apertures, known as the Gaussian family, the lowest order of which is the normalised Gaussian. The family can be truncated at any given order in a consistent way. The resulting set constitutes a basis for alocal jet bundle.

Note that scale-space theory shows up here without any call upon the “prohibition of spurious detail”, which, in some way or another, usually forms the basic starting point for “diffusion-like” scale-space theories.

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References

  1. J. Babaud, A.P. Witkin, M. Baudin, and R.O. Duba, “Uniqueness of the gaussian kernel for scale-space filtering,”IEEE Trans. Pattern Analysis and Machine Intelligence, 8(1):26–33, 1986.

    Google Scholar 

  2. Piet Bijl,Aspects of Visual Contrast Detection. PhD thesis, University of Utrecht, The Netherlands, May 1991.

    Google Scholar 

  3. J. Blom, B.M. ter Haar Romeny, A. Bel, and J.J. Koenderink, “Spatial derivatives and the propogation of noise in gaussian scale-space,”J. of Vis. Comm. and Im. Repr., 4(1):1–13, March 1993.

    Google Scholar 

  4. P.J. Burt, T.H. Hong, and A. Rosenfeld, “Segmentation and estimation of image region properties through cooperative hierarchical computation,”IEEE Trans. Systems, Man, and Cybernetics, 11(12):802–825, 1981.

    Google Scholar 

  5. J.L. Crowley, “Towards continuously operating integrated vision systems for robotics applications,” In Peter Johnsen and Søren Olsen, editors,Proc. 7th Scand. Conf. on Image Analysis, pages 494–505, Aalborg, DK, August 13–16 1991.

  6. J.L. Crowley, A. Chehikian, and F. Veillon, “Definition of image primitives,” VAP Tech. Rep. IR.A.3.1, LIFIA (IMAG)-I.N.P., Grenoble, France, 1990.

    Google Scholar 

  7. James Damon, “Local Morse theory for solutions to the heat equation and Gaussian blurring,” Internal report Department of Mathematics. University of North Carolina, Chapel Hill, North Carolina, USA, 1990.

    Google Scholar 

  8. Burkart Fischer, “Overlap of receptive field centers and representation of the visual field in the cat's optic tract,”Vision Research, 13:2113–2120, 1973.

    Google Scholar 

  9. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, and M.A. Viergever, “Scale and the differential structure of iamges,”Image and Vision Computing, 10(6):376–388, July/August 1992.

    Google Scholar 

  10. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, and M.A. Viergever, “Cartesian differential invariants in scale-space,”Journal of Mathematical Imaging and Vision, 3(4):327–348, November 1993.

    Google Scholar 

  11. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, and M.A. Viergever, “General intensity transformations and differential invariants,”Journal of Mathematical Imaging and Vision, 1993. Accepted.

  12. Joseph Fourier.The Analytical Theory of Heat. Dover Publications, Inc. New York, 1955. Replication of the English translation that first appeared in 1878 with previous corrigenda incorporated into the text, by Alexander Freeman, M.A. Original work: “Théorie Analytique de la Chaleur,” Paris, 1822.

    Google Scholar 

  13. David H. Hubel.Eye, Brain and Vision, volume 22 ofScientific American Library, Scientific American Press, New York, 1988.

    Google Scholar 

  14. R.A. Hummel and B.C. Gidas, “Zero crossings and the heat equation,”Technical Report 111, New York Univ. Courant Institute of Math. Sciences, Computer Science Division, 1984.

  15. R.A. Hummel, B.B. Kimia, and S.W. Zucker, “Deblurring Gaussian blur,”Computer Vision, Graphics, and Image Processing, 38:66–80, 1987.

    Google Scholar 

  16. B.B. Kimia,Deblurring gaussian blur, continuous and discrete approaches, Master's thesis, McGill University, Electrical Eng. Dept., Montreal, Canada, 1986.

    Google Scholar 

  17. J.J. Koenderink, “The concept of local sign,” in A.J. vanDoorn, W.A. van de Grind, and J.J. Koenderink, editors,Limits in Perception, pages 495–547. VNU Science Press, Utrecht, 1984.

    Google Scholar 

  18. J.J. Koenderink, “The structure of images,”Biol. Cybern., 50:363–370, 1984.

    PubMed  Google Scholar 

  19. J.J. Koenderink, “The structure of the visual field,” in W. Güttinger, and G. Dangelmayr, editors,The Physics of Structure Formation, Theory and Simulation. Springer-Verlag, 1986. Proceedings of an International Symposium, tübingen, Fed. Rep. of Germany, October 27–November 2.

  20. J.J. Koenderink, “Scale-time,”Biol. Cybern., 58:159–162, 1988.

    Google Scholar 

  21. J.J. Koenderink,Solid Shape, MIT Press, Cambridge, Mass., 1990.

    Google Scholar 

  22. J.J. Koenderink, “Local image structure,” inProc. Scand Conf. on Image Analysis, pages 1–7, Aalborg, DK, August 1991.

  23. J.J.Koenderink and A.J. van Doorn, “Representation of local geometry in the visual system,”Biol. Cybern., 55:367–375, 1987.

    Google Scholar 

  24. J.J. Koenderink and A.J. van Doorn, “Receptive field families,”Biol. Cybern., 63:291–298, 1990.

    Google Scholar 

  25. L.M. Lifshitz and S.M. Pizer, “A multiresolution hierarchical approach to image segmentation based on intensity extrema,”IEEE Trans. Pattern Analysis and Machine Inteligence, 12(6):529–541, 1990.

    Google Scholar 

  26. Alan P. Lightman, William H. Press, Richard H. Price, and Saul A. Teukolsky,Problem Book in Relativity and Gravitation, Princeton University Press, Princeton, New Jersey, 1975.

    Google Scholar 

  27. T. Lindeberg, “Scale-space for discrete signals,”IEEE Trans. Pattern Analysis and Machine Intelligence, 12(3):234–245, 1990.

    Google Scholar 

  28. T. Lindeberg,Discrete Scale-Space Theory and the Scale-Space Primal Sketch, PhD thesis, Royal Institute of Technology, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden, May 1991.

    Google Scholar 

  29. T. Lindeberg and J.O. Eklundh, “Guiding early visual processing with a scale-space primal sketch,”Technical Report CVAP-72, Royal Institute of Technology, University of Stockholm, Dept. of Numerical Analysis and Computing Science, Stockholm, Sweden, 1990.

    Google Scholar 

  30. T. Lindeberg and J.O. Eklundh, “On the computation of a scale-space primal sketch,” Technical Report CVAP-68, Royal Institute of Technology, University of Stockholm, Dept. of Numerical Analysis and Computing Science, Stockholm, Sweden, 1990.

    Google Scholar 

  31. M.M. Lipschutz,Differential Geometry, Schaum's Outline Series. McGraw Hill, New York, 1969.

    Google Scholar 

  32. Lord Rayleigh, “The principle of similitude,”Nature, XCV:66–68, 644, March 1915.

    Google Scholar 

  33. S.G. Mallat, “A theory of multiresolution signal decomposition: The wavelet representation,”IEEE Trans. Pattern Analysis and Machine Intelligence, 11(7):674–694, 1989.

    Google Scholar 

  34. J.B. Martens, “Deblurring digital images by means of polynomial transforms,”Computer Vision, Graphics, and Image and Stochastic Processing, 50:157–176, 1990.

    Google Scholar 

  35. H.E. Moses and A.F. Quesada, “The expansion of physical quantities in terms of the irreducible representations of the scale-euclidean group and applications to the construction of the scale-invariant correlation functions,”Arch. Rat. Mech. Anal., 44:217–248, 1971, Part I. Concepts. One-Dimensional Problems, Communicated by J. Meixner.

    Google Scholar 

  36. H.E. Moses and A.F. Quesada, “The expansion of physical quantities in terms of the irreducible representations of the scale-euclidean group and applications to the construction of scale-invariant correlation functions,”Arch. Rat. Mech. Anal., 50:194–236, 1973. Part II. Three-Dimensional Problems; Generalizations of the Helmholtz Vector Decomposition Theorem, Communicated by J. Meixner.

    Google Scholar 

  37. P.J. Olver,Applications of Lie Groups to Differential Equations, volume 107 ofGraduate Texts in Mathematics, Springer-Verlag, 1986.

  38. S.M. Pizer, J.J. Koenderink, L.M. Lifshitz, L. Helmink, and A.D.J. Kaasjager, “An image description for object defintion, based on extremal regions in the stack,”Information Processing in Medical Imaging, Proceedings of the 8th conference, pages 24–37, 1985.

  39. T. Poston and I. Steward,Catastrophe Theory and its Applications, Pitman, London, 1978.

    Google Scholar 

  40. M. Spivak,A Comprehensive Introduction to Differential Geometry, volume I–V, Publish or Perish, Inc., Houston, Texas, 1970.

    Google Scholar 

  41. B.M. ter Haar Romeny and L.M.J. Florack, “A multiscale geometric model of human vision, “In William R. Hendee and Peter N.T. Wells, editors,Perception of Visual Information, chapter 4, pages 73–114. Springer-Verlag, Berlin, 1993.

    Google Scholar 

  42. B.M. ter Haar Romeny and L.M.J. Florack, J.J. Koenderink, and M.A. Viergever, “Scale-space: Its natural operators and differential invariants,” in A.C.F. Colchester and D.J. Hawkes, editors,Information Processing in Medical Imaging, volume 511 ofLecture Notes in Computer Science, pages 239–255, Berlin, July 1991, Springer-Verlag.

    Google Scholar 

  43. Massimo Tistarelli and Giulio Sandini, Dynamic aspects in active vision,”CVGIP: Image Understanding, 56(1):108–129, July 1992.

    Google Scholar 

  44. A.J. van Door, J.J. Koenderink, and M.A. Bouman, “The influence of the retinal inhomogeneity on the perception of spatial patterns,”Kybernetik, 10:223–230, 1972.

    Google Scholar 

  45. H. Wässle, U. Grünert, J. Röhrenbeck, and B. Boycott, “Retinal ganglion cell density and cortical magnification factor in the primat,”Vision Research, 30:1897–1911, 1990.

    Google Scholar 

  46. H. Weyl,The Classical Groups, their Invariants and Representations, Princeton University Press, Princeton, NJ, 1946.

    Google Scholar 

  47. A.P. Witkin. “Scale space filtering,” inProc. International Joint Conference on Artificial Intelligence, pages 1019–1023, Karlsruhe, W. Germany, 1983.

  48. R.A. Young. The Gaussian derivative theory of spatial vision: Analysis of cortical cell receptive field line-weighting profiles. Publication gmr-4920, General Motors Research Labs, Computer Science Dept., 30500 Mound Road, Box 9055, Warren, Michigan 48090-9055, May 28 1985.

  49. R.A. Young, “The Gaussian derivative model for machine vision: Visual cortex simulation,”Journal of the Optical Society of America, July 1986.

  50. R.A. Young, “Simulation of human retinal function with the Gaussian derivative model,” inProc. IEEE CVPR CH2290-5, pages 564–569, Miami, Fla. 1986.

  51. A.L. Yuille and T.A. Poggio, “Scaling theorems for zerocrossings,”IEEE Trans. Pattern Analysis and Machine Intelligence, 8:15–25, 1986.

    Google Scholar 

  52. A.L. Yuille and T.A. Paggio, “Scaling and fingerprint theorems for zero-crossings,” in C. Brown, editor,Advances in Computer Vision, pages 47–78. Lawrence Erlbaum, 1988.

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Authors and Affiliations

  1. 3D Computer Vision Research Group, Utrecht University Hospital, Room E02.222, Heidelberglaan 100, 3584 CX, Utrecht, The Netherlands

    L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink & M. A. Viergever

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  1. L. M. J. Florack
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  2. B. M. ter Haar Romeny
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  3. J. J. Koenderink
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  4. M. A. Viergever
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Florack, L.M.J., ter Haar Romeny, B.M., Koenderink, J.J. et al. Linear scale-space. J Math Imaging Vis 4, 325–351 (1994). https://doi.org/10.1007/BF01262401

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