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Generalized Gaussian Scale-Space Axiomatics Comprising Linear Scale-Space, Affine Scale-Space and Spatio-Temporal Scale-Space

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Abstract

This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms).

The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision.

It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces.

The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives.

The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and/or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area.

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References

  1. Al-Sharif, S., Khalil, R.: On the generator of two parameter semigroups. Appl. Math. Comput. 156(2), 403–414 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Almansa, A., Lindeberg, T.: Fingerprint enhancement by shape adaptation of scale-space operators with automatic scale-selection. IEEE Trans. Image Process. 9(12), 2027–2042 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alvarez, L., Guichard, F., Lions, P.-L., Morel, J.-M.: Axioms and fundamental equations of image processing. Arch. Ration. Mech. Anal. 123(3), 199–257 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babaud, J., Witkin, A.P., Baudin, M., Duda, R.O.: Uniqueness of the Gaussian kernel for scale-space filtering. IEEE Trans. Pattern Anal. Mach. Intell. 8(1), 26–33 (1986)

    Article  MATH  Google Scholar 

  5. Ballester, C., Gonzalez, M.: Affine invariant texture segmentation and shape from texture by variational methods. J. Math. Imaging Vis. 9, 141–171 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baumberg, A.: Reliable feature matching across widely separated views. In: Proc. IEEE Comp. Soc. Conf. on Computer Vision and Pattern Recognition, pp. I:1774–1781. Hilton Head, SC (2000)

    Google Scholar 

  7. Broadbridge, P.: Entropy diagnostics for fourth order partial differential equations in conservation form. Entropy 10, 365–379 (2008). doi:10.3390/e10030365

    Article  MathSciNet  MATH  Google Scholar 

  8. Burt, P.J.: Fast filter transforms for image processing. Comput. Vis. Graph. Image Process. 16, 20–51 (1981)

    Article  Google Scholar 

  9. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Clarendon Press, Oxford (1959)

    Google Scholar 

  10. Crowley, J.L.: A representation for visual information. PhD thesis, Carnegie-Mellon University, Robotics Institute, Pittsburgh, Pennsylvania (1981)

  11. DeAngelis, G.C., Ohzawa, I., Freeman, R.D.: Receptive field dynamics in the central visual pathways. Trends Neurosci. 18(10), 451–457 (1995)

    Article  Google Scholar 

  12. Duits, R., Florack, L., de Graaf, J., ter Haar Romeny, B.: On the axioms of scale space theory. J. Math. Imaging Vis. 22, 267–298 (2004)

    Article  Google Scholar 

  13. Duits, R., Felsberg, M., Florack, L., Platel, B.: α-scale-spaces on a bounded domain. In: Griffin, L., Lillholm, M. (eds.) Proc. Scale-Space Methods in Computer Vision: Scale-Space’03, Isle of Skye, Scotland, Jun. 2003. Lecture Notes in Computer Science, vol. 2695, pp. 494–510. Springer, Berlin (2003)

    Chapter  Google Scholar 

  14. Dunninger, D.R.: Maximum principles for solutions of some fourth-order elliptic equation. J. Math. Anal. Appl. 37, 665–668 (1972)

    Article  MathSciNet  Google Scholar 

  15. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  16. Fagerström, D.: Temporal scale-spaces. Int. J. Comput. Vis. 2–3, 97–106 (2005)

    Article  Google Scholar 

  17. Fagerström, D.: Spatio-temporal scale-spaces. In: Gallari, F., Murli, A., Paragios, N. (eds.) Proc. 1st International Conference on Scale-Space Theories and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 4485, pp. 326–337. Springer, Berlin (2007)

    Chapter  Google Scholar 

  18. Felsberg, M., Sommer, G.: The monogenic scale-space: A unifying approach to phase-based image processing in scale-space. J. Math. Imaging Vis. 21, 5–26 (2004)

    Article  MathSciNet  Google Scholar 

  19. Fleet, D.J., Langley, K.: Recursive filters for optical flow. IEEE Trans. Pattern Anal. Mach. Intell. 17(1), 61–67 (1995)

    Article  Google Scholar 

  20. Florack, L.M.J.: Image Structure. Series in Mathematical Imaging and Vision. Springer, Dordrecht (1997)

    Google Scholar 

  21. Florack, L.M.J., ter Haar Romeny, B.M., Koenderink, J.J., Viergever, M.A.: Scale and the differential structure of images. Image Vis. Comput. 10(6), 376–388 (1992)

    Article  Google Scholar 

  22. Florack, L.M.J., Niessen, W., Nielsen, M.: The intrinsic structure of optic flow incorporating measurement duality. Int. J. Comput. Vis. 27(3), 263–286 (1998)

    Article  Google Scholar 

  23. Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)

    MATH  Google Scholar 

  24. Freeman, W.T., Adelson, E.H.: The design and use of steerable filters. IEEE Trans. Pattern Anal. Mach. Intell. 13(9), 891–906 (1991)

    Article  Google Scholar 

  25. Gårding, J., Lindeberg, T.: Direct computation of shape cues using scale-adapted spatial derivative operators. Int. J. Comput. Vis. 17(2), 163–191 (1996)

    Article  Google Scholar 

  26. Goldstein, J.A.: Semigroups of Linear Operators and Applications. Oxford Mathematical Monographs. Oxford Science, Oxford (1985)

    MATH  Google Scholar 

  27. Griffin, L.: Critical point events in affine scale space. In: Sporring, J., Nielsen, M., Florack, L., Johansen, P. (eds.) Gaussian Scale-Space Theory: Proc. PhD School on Scale-Space Theory, Copenhagen, Denmark, May 1996, pp. 165–180. Springer, Berlin (1996)

    Google Scholar 

  28. Guichard, F.: A morphological, affine, and Galilean invariant scale-space for movies. IEEE Trans. Image Process. 7(3), 444–456 (1998)

    Article  Google Scholar 

  29. Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. American Mathematical Society Colloquium Publications, vol. 31 (1957)

  30. Hummel, R.A., Moniot, R.: Reconstructions from zero-crossings in scale-space. IEEE Trans. Acoust. Speech Signal Process. 37(12), 2111–2130 (1989)

    Article  Google Scholar 

  31. Iijima, T.: Observation theory of two-dimensional visual patterns. Technical report, Papers of Technical Group on Automata and Automatic Control, IECE, Japan (1962)

  32. Koenderink, J.J.: The structure of images. Biol. Cybern. 50, 363–370 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  33. Koenderink, J.J.: Scale-time. Biol. Cybern. 58, 159–162 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  34. Koenderink, J.J., van Doorn, A.J.: Receptive field families. Biol. Cybern. 63, 291–298 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  35. Koenderink, J.J., van Doorn, A.J.: Generic neighborhood operators. IEEE Trans. Pattern Anal. Mach. Intell. 14(6), 597–605 (1992)

    Article  Google Scholar 

  36. Laptev, I., Lindeberg, T.: Velocity-adapted spatio-temporal receptive fields for direct recognition of activities. Image Vis. Comput. 22(2), 105–116 (2004)

    Article  Google Scholar 

  37. Laptev, I., Caputo, B., Schuldt, C., Lindeberg, T.: Local velocity-adapted motion events for spatio-temporal recognition. Comput. Vis. Image Underst. 108, 207–229 (2007)

    Article  Google Scholar 

  38. Lindeberg, T.: Scale-space for discrete signals. IEEE Trans. Pattern Anal. Mach. Intell. 12(3), 234–254 (1990)

    Article  Google Scholar 

  39. Lindeberg, T.: Scale-Space Theory in Computer Vision. The Kluwer International Series in Engineering and Computer Science. Springer, Dordrecht (1994)

    Google Scholar 

  40. Lindeberg, T.: Scale-space theory: A basic tool for analysing structures at different scales. J. Appl. Stat. 21(2), 225–270 (1994). Also available from http://www.csc.kth.se/~tony/abstracts/Lin94-SI-abstract.html

    Article  Google Scholar 

  41. Lindeberg, T.: On the axiomatic foundations of linear scale-space. In: Sporring, J., Nielsen, M., Florack, L., Johansen, P. (eds.) Gaussian Scale-Space Theory: Proc. PhD School on Scale-Space Theory, Copenhagen, Denmark, May, 1996. Springer, Berlin (1996)

    Google Scholar 

  42. Lindeberg, T.: On automatic selection of temporal scales in time-casual scale-space. In: Sommer, G., Koenderink, J.J. (eds.) Proc. AFPAC’97: Algebraic Frames for the Perception-Action Cycle, Kiel, Germany, Sep., 1997. Lecture Notes in Computer Science, vol. 1315, pp. 94–113. Springer, Berlin (1997)

    Chapter  Google Scholar 

  43. Lindeberg, T.: Linear spatio-temporal scale-space. In: ter Haar Romeny, B.M., Florack, L.M.J., Koenderink, J.J., Viergever, M.A. (eds.) Scale-Space Theory in Computer Vision: Proc. First Int. Conf. Scale-Space’97, Utrecht, The Netherlands, July, 1997. Lecture Notes in Computer Science, vol. 1252, pp. 113–127. Springer, Berlin (1997). Extended version available as technical report ISRN KTH NA/P–01/22–SE from KTH

    Google Scholar 

  44. Lindeberg, T.: Feature detection with automatic scale selection. Int. J. Comput. Vis. 30(2), 77–116 (1998)

    Google Scholar 

  45. Lindeberg, T.: Linear spatio-temporal scale-space. Report, ISRN KTH/NA/P–01/22–SE, Dept. of Numerical Analysis and Computing Science, KTH, Nov. (2001)

  46. Lindeberg, T.: Time-recursive velocity-adapted spatio-temporal scale-space filters. In: Johansen, P. (ed.) Proc. 7th European Conference on Computer Vision, Copenhagen, Denmark, May 2002. Lecture Notes in Computer Science, vol. 2350, pp. 52–67. Springer, Berlin (2002)

    Google Scholar 

  47. Lindeberg, T.: Scale-space. In: Wah, B. (ed.) Encyclopedia of Computer Science and Engineering, pp. 2495–2504. Wiley, Hoboken (2008). doi:10.1002/9780470050118.ecse609. Also available from http://www.nada.kth.se/~tony/abstracts/Lin08-EncCompSci.html

    Google Scholar 

  48. Lindeberg, T.: Generalized Gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space. Technical report TRITA-CSC-CV 2010:3, ISSN 1653-7092, School of Computer Science and Communication, KTH (Royal Institute of Technology), Stockholm, Sweden, 2010. Also available from http://www.csc.kth.se/~tony/abstracts/Lin10-GenGaussScSp.html

  49. Lindeberg, T., Fagerström, D.: Scale-space with causal time direction. In: Proc. 4th European Conf. on Computer Vision, Cambridge, UK, Apr, 1996, vol. 1064, pp. 229–240. Springer, Berlin (1996)

    Google Scholar 

  50. Lindeberg, T., Gårding, J.: Shape-adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D structure. In: Eklundh, J.-O. (ed.) Proc. 3rd European Conf. on Computer Vision, Stockholm, Sweden, May 1994. Lecture Notes in Computer Science, vol. 800, pp. 389–400. Springer, Berlin (1994)

    Google Scholar 

  51. Lindeberg, T., Gårding, J.: Shape-adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D structure. Image Vis. Comput. 15, 415–434 (1997)

    Article  Google Scholar 

  52. Lindeberg, T., Akbarzadeh, A., Laptev, I.: Galilean-corrected spatio-temporal interest operators. In: International Conference on Pattern Recognition, Cambridge, pp. I:57–62. (2004). Extended version available as technical report ISRN KTH NA/P–04/05–S from KTH

    Google Scholar 

  53. Lindeberg, T., Akbarzadeh, A., Laptev, I.: Galilean-corrected spatio-temporal interest operators. Technical Report ISRN KTH/NA/P–04/05–SE, Dept. of Numerical Analysis and Computing Science, KTH, Mar. 2004

  54. Lukas, B.D., Kanade, T.: An iterative image registration technique with an application to stereo vision. In: Image Understanding Workshop (1981)

    Google Scholar 

  55. Mikolajczyk, K., Schmid, C.: Scale and affine invariant interest point detectors. Int. J. Comput. Vis. 60(1), 63–86 (2004)

    Article  Google Scholar 

  56. Nagel, H., Gehrke, A.: Spatiotemporal adaptive filtering for estimation and segmentation of optical flow fields. In: Proc. 5th European Conf. on Computer Vision, Freiburg, Germany, Jun. 1998, pp. 86–102. Springer, Berlin (1998)

    Google Scholar 

  57. Pauwels, E.J., Fiddelaers, P., Moons, T., van Gool, L.J.: An extended class of scale-invariant and recursive scale-space filters. IEEE Trans. Pattern Anal. Mach. Intell. 17(7), 691–701 (1995)

    Article  Google Scholar 

  58. Pazy, A.: Semi-groups of Linear Operators and Applications to Partical Differential Equations. Applied Mathematical Sciences. Springer, Berlin (1983)

    Google Scholar 

  59. Perona, P.: Steerable-scalable kernels for edge detection and junction analysis. Image Vis. Comput. 10, 663–672 (1992)

    Article  Google Scholar 

  60. Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  61. Schaffalitzky, F., Zisserman, A.: Viewpoint invariant texture matching and wide baseline stereo. In: Proc. 8th Int. Conf. on Computer Vision, Vancouver, Canada, July 2001, pp. II:636–643. (2001)

    Google Scholar 

  62. Schoenberg, I.J.: On Pòlya frequency functions. II. Variation-diminishing integral operators of the convolution type. Acta Sci. Math. (Szeged) 12, 97–106 (1950)

    MathSciNet  Google Scholar 

  63. Schoenberg, I.J.: On smoothing operations and their generating functions. Bull. Am. Math. Soc. 59, 199–230 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  64. Simoncelli, E.P., Freeman, W.T., Adelson, E.H., Heeger, D.J.: Shiftable multi-scale transforms. IEEE Trans. Information Theory 38(2), 587–607 (1992)

    Article  MathSciNet  Google Scholar 

  65. Sporring, J., Nielsen, M., Florack, L., Johansen, P. (eds.): Gaussian Scale-Space Theory: Proc. PhD School on Scale-Space Theory. Series in Mathematical Imaging and Vision. Springer, Copenhagen (1996)

    Google Scholar 

  66. ter Haar Romeny, B. (ed.): Geometry-Driven Diffusion in Computer Vision. Series in Mathematical Imaging and Vision. Springer, Dordrecht (1994)

    Google Scholar 

  67. ter Haar Romeny, B.: Front-End Vision and Multi-Scale Image Analysis. Springer, Dordrecht (2003)

    Google Scholar 

  68. ter Haar Romeny, B., Florack, L., Nielsen, M.: Scale-time kernels and models. In: Scale-Space and Morphology: Proc. Scale-Space’01, Vancouver, Canada, July 2001. Lecture Notes in Computer Science. Springer, Berlin (2001)

    Google Scholar 

  69. Tuytelaars, T., van Gool, L.: Matching widely separated views based on affine invariant regions. Int. J. Comput. Vis. 59(1), 61–85 (2004)

    Article  Google Scholar 

  70. Valois, R.L.D., Cottaris, N.P., Mahon, L.E., Elfer, S.D., Wilson, J.A.: Spatial and temporal receptive fields of geniculate and cortical cells and directional selectivity. Vis. Res. 40(2), 3685–3702 (2000)

    Article  Google Scholar 

  71. Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner-Verlag, Stuttgart (1998)

    MATH  Google Scholar 

  72. Weickert, J., Ishikawa, S., Imiya, A.: Linear scale-space has first been proposed in Japan. J. Math. Imaging Vis. 10(3), 237–252 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  73. Witkin, A.P.: Scale-space filtering. In: Proc. 8th Int. Joint Conf. Art. Intell., Karlsruhe, Germany, Aug. 1983, pp. 1019–1022 (1983)

    Google Scholar 

  74. Young, R.A.: The Gaussian derivative theory of spatial vision: Analysis of cortical cell receptive field line-weighting profiles. Technical Report GMR-4920, Computer Science Department, General Motors Research Lab., Warren, Michigan (1985)

  75. Young, R.A.: The Gaussian derivative model for spatial vision: I. Retinal mechanisms. Spat. Vis. 2, 273–293 (1987)

    Article  Google Scholar 

  76. Young, R.A., Lesperance, R.M.: The Gaussian derivative model for spatio-temporal vision: II. Cortical data. Spat. Vis. 14(3, 4), 321–389 (2001)

    Article  Google Scholar 

  77. Young, R.A., Lesperance, R.M., Meyer, W.W.: The Gaussian derivative model for spatio-temporal vision: I. Cortical model. Spat. Vis. 14(3, 4), 261–319 (2001)

    Article  Google Scholar 

  78. Yuille, A.L., Poggio, T.A.: Scaling theorems for zero-crossings. IEEE Trans. Pattern Anal. Mach. Intell. 8, 15–25 (1986)

    Article  MATH  Google Scholar 

  79. Zhang, H., Zhang, W.: Maximum principles and bounds in a class of fourth-order uniformly elliptic equations. J. Phys. A: Math. Gen. 35, 9245–9250 (2002)

    Article  MATH  Google Scholar 

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    Tony Lindeberg

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The support from the Swedish Research Council, Vetenskapsrådet (contract 2004-4680), the Royal Swedish Academy of Sciences as well as the Knut and Alice Wallenberg Foundation is gratefully acknowledged.

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Lindeberg, T. Generalized Gaussian Scale-Space Axiomatics Comprising Linear Scale-Space, Affine Scale-Space and Spatio-Temporal Scale-Space. J Math Imaging Vis 40, 36–81 (2011). https://doi.org/10.1007/s10851-010-0242-2

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