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Notes on Discrete Gaussian Scale Space

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Abstract

Gaussian scale space is a well-known linear multi-scale representation for continuous signals. The exploration of its so-called deep structure by tracing critical points over scale has various theoretical applications and allows for the construction of a scale space hierarchy tree. However, implementation issues arise, caused by discretization and quantization errors. In order to develop more robust scale space based algorithms, the discrete nature of computer processed signals has to be taken into account. We propose suitable neighborhoods, boundary conditions, and sampling methods. In analogy to prevalent approaches and inspired by Lindeberg’s scale space primal sketch, a discretized diffusion equation is derived, including requirements imposed by the chosen neighborhood and boundary condition. The resulting discrete scale space respects important topological invariants such as the Euler number, a key criterion for the successful implementation of algorithms operating on critical points in its deep structure. Relevant properties of the discrete diffusion equation and the Eigenvalue decomposition of its Laplacian kernel are discussed and a fast and robust sampling method is proposed. We finally discuss properties of topological graphs under the influence of smoothing, setting the stage for more robust deep structure extraction algorithms.

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References

  1. Becciu, A., Duits, R., Janssen, B., Florack, L., van Assen, H.C.: Cardiac motion estimation using covariant derivatives and helmholtz decomposition. In: Statistical Atlases and Computational Models of the Heart. Imaging and Modelling Challenges - Second International Workshop, STACOM 2011, Held in Conjunction with MICCAI 2011, Toronto, Sept 22, 2011, Revised Selected Papers, LNCS 7085, pp. 263–273, 2011.

  2. Biasotti, S., Falcidieno, B., Spagnuolo M.: Extended Reeb graphs for surface understanding and description. In Borgefors G., Nystrm, I., Sanniti di Baja G. (eds.) Discrete Geometry for Computer Imagery 2000. Lecture Notes in Computer Science 1953, pp. 185–197. (2000)

  3. Binotto, A., Weber, D., Daniel, C., Stork, A., Eduardo Pereira, C., Kuijper, A., Fellner, D.: Iterative sle solvers over a CPU–GPU platform. In: 12th IEEE International Conference on High Performance Computing and Communications, IEEE HPCC-10, pp. 305–313. Melbourne, Sep 1–3 2010, IEEE (2010)

  4. Brimkov, Valentin E., Barneva, Reneta P.: Connectivity of discrete planes. Theor. Comput. Sci. 319(1–3), 203–227 (June 2004)

  5. Duits, R., Florack, L.M.J., de Graaf, J., ter Haar, B.M.: Romeny. On the axioms of scale space theory. J. Math. Imaging. Vis. 20(3), 267–298 (2004)

    Article  Google Scholar 

  6. Duits, Remco, Franken, Erik: Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of hardi images. Int. J. Comput. Vis. 92(3), 231–264 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Duits, Remco, Tom, C.J., Dela, Haije, Creusen, Eric J., Arpan, Ghosh: Morphological and linear scale spaces for fiber enhancement in dw-mri. J. Math. Imaging Vis. 46(3), 326–368 (2013)

    Article  Google Scholar 

  8. Felsberg, Michael, Duits, Remco, Florack, Luc: The monogenic scale space on a bounded domain and its applications. Scale-Space, LNCS 2695, 209–224 (2003)

    Google Scholar 

  9. Florack, Luc: A spatio-frequency trade-off scale for scale-space filtering. IEEE Trans. Pattern Anal. Mach. Intell. 22(9), 1050–1055 (2000)

    Article  Google Scholar 

  10. Griffin, L.D., Colchester, A.: Superficial and deep structure in linear diffusion scale space: isophotes, critical points and separatrices. Image Vis. Comput. 13(7), 543–557 (September 1995)

  11. Griffin, L.D., Colchester, A., Robinson, G.: Scale and segmentation of grey-level images using maximum gradient paths. Image Vis. Comput. 10(5), 389–402 (1992)

    Article  Google Scholar 

  12. Hancock, Edwin R., Wilson, Richard C.: Pattern analysis with graphs: parallel work at Bern and York. Pattern Recognit. Lett. 33(7), 833–841 (2012)

    Article  Google Scholar 

  13. Iijima, T.: Basic theory on normalization of a pattern (in case of typical one-dimensional pattern). Bull. Electr. Lab. 26, 368–388 (1962)

    Google Scholar 

  14. Janssen, B., Duits, R., ter Haar Romeny, B.M.: Linear image reconstruction by Sobolev norms on the bounded domain. In: SSVM, LNCS 4485, pp. 55–67 (2007)

  15. Kanters, F., Florack, L., Duits, R., Platel, B., ter Haar Romeny, B.: Scalespaceviz: alpha-scale spaces in practice. Pattern Recognit. Image Anal. 17, 106–116 (2007)

    Article  Google Scholar 

  16. Kanters, F., Lillholm, M., Duits, R., Janssen, B., Platel, B., Florack, L. M. J., ter Haar Romeny, B.M.: On image reconstruction from multiscale top points. In: Scale Space and PDE Methods in Computer Vision, LNCS 3459, pp. 431–442 (2005)

  17. Kanters, F.M.W., Florack, L.M.J.: Deep structure, singularities, and computer vision. Eindhoven University of Technology, Technical report, Sept 2003

  18. Kenyon, R.: The Laplacian on planar graphs and graphs on surfaces. ArXiv e-prints, Mar 2012

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Kovalevsky, V.A.: Discrete topology and contour definition. Pattern Recognit. Lett. 2(5), 281–288 (1984)

    Article  Google Scholar 

  21. Kuijper, A.: The Deep Structure of Gaussian Scale Space Images. PhD thesis, Utrecht University, 2002. ISBN 90-393-3061-1

  22. Kuijper, A.: On detecting all saddle points in 2D images. Pattern Recognit. Lett. 25(15), 1665–1672 (2004)

    Article  Google Scholar 

  23. Kuijper, A., Florack, L.M.J.: Calculations on critical points under Gaussian blurring. In: Nielsen, M., Johansen, P., Fogh Olsen, O., Weickert, J. (eds.) Scale-Space Theories in Computer Vision, volume 1682 of Lecture Notes in Computer Science. Springer, Berlin Heidelberg (1999)

    Google Scholar 

  24. Kuijper, A., Florack, L.M.J.: The relevance of non-generic events in scale space. In ECCV 2002, LNCS 2350, pp. 190–204 (2002)

  25. Kuijper, A., Florack, L.M.J.: Understanding and modeling the evolution of critical points under Gaussian blurring. In ECCV 2002, LNCS 2350, pp. 143–157 (2002)

  26. Kuijper, A., Florack, L.M.J.: The relevance of non-generic events in scale space models. Int. J. Comput. Vis. 1(57), 67–84 (2004)

    Article  Google Scholar 

  27. Kuijper, Arjan: Exploring and exploiting the structure of saddle points in gaussian scale space. Comput. Vis. Image Underst. 112(3), 337–349 (2008)

    Article  Google Scholar 

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

    Book  Google Scholar 

  29. Lindeberg, T.: Edge detection and ridge detection with automatic scale selection. Int. J. Comput. Vis. 30(2), 117–154 (1998)

    Article  Google Scholar 

  30. Lindeberg, T.: Feature detection with automatic scale selection. Int. J. Comput. Vis. 30(2), 79–116 (1998)

    Article  Google Scholar 

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

    Article  Google Scholar 

  32. Lindeberg, T.: Discrete Scale-Space Theory and the Scale-Space Primal Sketch. PhD thesis, Royal Institute of Technology, 1991

  33. Lindeberg, Tony: Detecting salient blob-like image structures and their scales with a scale-space primal sketch: a method for focus-of-attention. Int. J. Comput. Vis. 11(3), 283–318 (1993)

    Article  Google Scholar 

  34. Lindeberg, T.: Generalized gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space. J. Math. Imaging Vis. 22, 1–46 (2010). doi:10.1007/s10851-010-0242-2

  35. Lindeberg, T.: Image matching using generalized scale-space interest points. In: Kuijper, A., Bredies, K., Pock, T., Bischof, H. (eds.) SSVM, volume 7893 of Lecture Notes in Computer Science, pp. 355–367. Springer, New York (2013)

    Google Scholar 

  36. Lizhi, Cheng, Zengrong, Jiang: An efficient algorithm for cyclic convolution based on fast-polynomial and fast-w transforms. Circuits Syst. Signal Process. 20, 77–88 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Loog, M., Duistermaat, J.J., Florack, L.M.J.: On the behavior of spatial critical points under Gaussian blurring, a folklore theorem and scale-space constraints. Scale-Space and Morphology in Computer Vision, LNCS 2106, 183–192 (2001)

  38. Nowak, R.: 2D DFT, July 2005. http://cnx.org/content/m10987/2.4/

  39. Olsen, O.F.: Multi-scale watershed segmentation. In: Sporring, J., Nielsen, M., Florack, L.M.J., Johansen, P. (eds.) Gaussian Scale-Space Theory, volume 8 of Computational Imaging, 2nd edn. Kluwer Academic, Dordrecht (1997)

    Google Scholar 

  40. Perez, A.: Determining the genus of a graph abstract. Harvard College, Cambridge (2009)

    Google Scholar 

  41. Pfaltz, J.L.: Surface networks. Geogr. Anal. 8(1), 77–93 (1976)

    Article  Google Scholar 

  42. Raz, R.: On the complexity of matrix product. In: Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, STOC ’02, pp. 144–151, New York, 2002. ACM

  43. Schneider, Bernhard: Extraction of hierarchical surface networks from bilinear surface patches. Geogr. Anal. 37(2), 244–263 (2005)

    Article  Google Scholar 

  44. Schwarzkopf, A., Kalbe, T., Goesele, M., Kuijper, A., Bajaj, C.: Volumetric nonlinear anisotropic diffusion on GPUs. In: Third International Conference on Scale Space Methods and Variational Methods in Computer Vision (SSVM) (May 29th–June 2nd, 2011, Ein Gedi, Israel), LNCS 6667, pp. 62–73, 2012

  45. Scott, Paul J.: An algorithm to extract critical points from lattice height data. Int. J. Mach. Tools Manuf. 41(13–14), 1889–1897 (2001)

    Article  Google Scholar 

  46. Shinagawa, Y., Kunii, T.L.: Constructing a Reeb graph automatically from cross sections. IEEE Comput. Gr. Appl. 11, 44–51 (November 1991)

  47. Takahashi, Shigeo, Ikeda, Tetsuya, Shinagawa, Yoshihisa, Kunii, Tosiyasu L., Ueda, Minoru: Algorithms for extracting correct critical points and constructing topological graphs from discrete geographical elevation data. Comput. Gr. Forum 14(3), 181–192 (1995)

  48. Tschirsich, M.: The Discrete Scale Space as a Base for Robust Scale Space Algorithms. Department of Computer Science, Technical University of Darmstadt, Darmstadt. Technical report (June 2012)

  49. Tschirsich, M., Kuijper, A.: A discrete scale space neighborhood for robust deep structure extraction. In S+SSPR 2012, LNCS 7626, pp. 126–134. Springer, New York (2012)

  50. Tschirsich, Martin, Kuijper, Arjan: Laplacian eigenimages in discrete scale space. Struct. Synt. Stat. Pattern Recognit. LNCS 7626, 162–170 (2012)

    Article  Google Scholar 

  51. Tschirsich, M., Kuijper, A.: Discrete deep structure. In Scale Space and Variational Methods in Computer Vision-4th International Conference, SSVM 2013, (Schloss Seggau, Leibnitz, June 2–6, 2013), LNCS 7893, pp. 343–354, 2013

  52. Viola, P., Jones, M.: Robust real-time object detection. Int. J. Comput. Vis. 24(2), 137–154 (2001)

    Article  Google Scholar 

  53. Weickert, J.A.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  55. Witkin, A.P.: Scale-space filtering. In; Proceedings of the 8th International Joint Conference Artificial Intelligence, pp. 1019–1022. Karlsruhe, August 1983

  56. Xiao, Bai, Hancock, Edwin R., Wilson, Richard C.: Graph characteristics from the heat kernel trace. Pattern Recognit. 42(11), 2589–2606 (2009)

    Article  MATH  Google Scholar 

  57. Hancock, E.R., Wilson, R.C.: Geometric characterization and clustering of graphs using heat kernel embeddings. Image Vis. Comput. 28(6), 1003–1021 (2010)

    Article  Google Scholar 

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Authors and Affiliations

  1. Technische Universität Darmstadt, Darmstadt, Germany

    Martin Tschirsich

  2. Technische Universität Darmstadt and Fraunhofer IGD, Darmstadt, Germany

    Arjan Kuijper

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  1. Martin Tschirsich
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  2. Arjan Kuijper
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Correspondence to Arjan Kuijper.

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Tschirsich, M., Kuijper, A. Notes on Discrete Gaussian Scale Space. J Math Imaging Vis 51, 106–123 (2015). https://doi.org/10.1007/s10851-014-0509-0

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