Inferring Information Across Scales in Acquired Complex Signals

  • Suman Kumar MajiEmail author
  • Oriol Pont
  • Hussein Yahia
  • Joel Sudre
Part of the Springer Proceedings in Complexity book series (SPCOM)


Transmission of information across the scales of a complex signal has some interesting potential, notably in the derivation of sub-pixel information, cross-scale inference and data fusion. It follows the structure of complex signals themselves, when they are considered as acquisitions of complex systems. In this work we contemplate the problem of cross-scale information inference through the determination of appropriate multiscale decomposition. Our goal is to derive a generic methodology that can be applied to propagate information across the scales in a wide variety of complex signals. Consequently, we first focus on the determination of appropriate multiscale characteristics, and we show that singularity exponents computed in microcanonical formulations are much better candidates for the characterization of transitions in complex signals: they outperform the classical “linear filtering” approach of the state-of-the-art edge detectors (for the case of 2D signals). This is a fundamental topic as edges are usually considered as important multiscale features in an image. The comparison is done within the formalism of reconstructible systems. Critical exponents, naturally associated to phase transitions and used in complex systems methods in the framework of criticality are key notions in Statistical Physics that can lead to the complete determination of the geometrical cascade properties in complex signals. We study optimal multiresolution analysis associated to critical exponents through the concept of “optimal wavelet”. We demonstrate the usefulness of multiresolution analysis associated to critical exponents in two decisive examples: the reconstruction of perturbated optical phase in Adaptive Optics (AO) and the generation of high resolution ocean dynamics from low resolution altimetry data.


Multiresolution Analysis Adaptive Optic Optical Phase Singularity Spectrum High Resolution Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Suman Kumar Maji’s Ph.D. is funded by a CORDIS grant and Région Aquitaine OPTAD research project grant.


  1. 1.
    Turiel A, del Pozo A (2002) Reconstructing images from their most singular fractal manifold. IEEE Trans Image Process 11:345–350 MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Parisi G, Frisch U (1985) On the singularity structure of fully developed turbulence and predictability in geophysical fluid dynamics. In: Ghil M, Benzi R, Parisi G (eds) Proc intl school of physics E Fermi. North-Holland, Amsterdam, pp 84–87 Google Scholar
  3. 3.
    Frisch U (1995) Turbulence. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  4. 4.
    Turiel A, Parga N (2000) The multi-fractal structure of contrast changes in natural images: from sharp edges to textures. Neural Comput 12:763–793 CrossRefGoogle Scholar
  5. 5.
    Boffetta G, Cencini M, Falcioni M et al. (2002) Predictability: a way to characterize complexity. Phys Rep 356(6):367–474 MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Pont O, Turiel A, Perez-Vicente C (2009) Empirical evidences of a common multifractal signature in economic, biological and physical systems. Physica A 388:2025–2035 ADSCrossRefGoogle Scholar
  7. 7.
    Turiel A, Perez-Vicente C, Grazzini J (2006) Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study. J Comput Phys 216:362–390 MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Canny J (1986) A computational approach to edge detection. IEEE Trans Pattern Anal Mach Intell 8:679–698 CrossRefGoogle Scholar
  9. 9.
    Faugeras O (1993) Three-dimensional computer vision: a geometric viewpoint. MIT Press, Cambridge. ISBN 0-262-06158-9 Google Scholar
  10. 10.
    Turiel A, Yahia H, Perez-Vicente C (2008) Microcanonical multifractal formalism: a geometrical approach to multifractal systems. Part I: singularity analysis. J Phys A, Math Theor 41:015501. doi: 10.1088/1751-8113/41/1/015501 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lovejoy S, Schertzer D (1990) Multifractals, universality classes, satellite and radar measurements of clouds and rain. J Geophys Res 95:2021–2034 ADSCrossRefGoogle Scholar
  12. 12.
    Arneodo A, Bacry E, Muzy J (1995) The thermodynamics of fractals revisited with wavelets. Physica A 213:232–275 ADSCrossRefGoogle Scholar
  13. 13.
    Pont O, Turiel A, Perez-Vicente C (2009) Description, modelling and forecasting of data with optimal wavelets. J Econ Interact Coord 4:39–54. doi: 10.1007/s11403-009-0046-x CrossRefGoogle Scholar
  14. 14.
    Pont O, Turiel A, Yahia H (2011) An optimized algorithm for the evaluation of local singularity exponents in digital signals. In: IWCIA, vol 6636, pp 346–357 Google Scholar
  15. 15.
    Sobel I (1978) Neighbourhood coding of binary images fast contour following and general array binary processing. Comput Graph Image Process 8:127–135 CrossRefGoogle Scholar
  16. 16.
    Prewitt J (1970) Object enhancement and extraction. In: Picture process psychopict, pp 75–149 Google Scholar
  17. 17.
    Roberts LG (1965) Machine perception of three dimensional solids. In: Tippett JT et al. (eds) Optical and electro-optical information processing. MIT Press, Cambridge Google Scholar
  18. 18.
    Rosenfeld A (1969) Picture processing by computer. Academic Press, New York zbMATHGoogle Scholar
  19. 19.
    Marr D, Hildreth E (1980) Theory of edge detection. Proc R Soc Lond B, Biol Sci 207:187–217 ADSCrossRefGoogle Scholar
  20. 20.
    Haralick RM (1984) Digital step edges from zero crossing of second directional derivatives. IEEE Trans Pattern Anal Mach Intell 6:58–68 CrossRefGoogle Scholar
  21. 21.
    Torre V, Poggio TA (1986) On edge detection. IEEE Trans Pattern Anal Mach Intell 8:147–163 CrossRefGoogle Scholar
  22. 22.
    Laligant O, Truchetet F (2010) A nonlinear derivative scheme applied to edge detection. IEEE Trans Pattern Anal Mach Intell 32:242–257 CrossRefGoogle Scholar
  23. 23.
    Pottier C, Turiel A, Garçon V (2008) Inferring missing data in satellite chlorophyll maps using turbulent cascading. Remote Sens Environ 112:4242–4260 CrossRefGoogle Scholar
  24. 24.
    Mallat S (1999) A wavelet tour of signal processing, 2nd edn. Academic Press, San Diego zbMATHGoogle Scholar
  25. 25.
    Benzi R, Biferale L, Crisanti A, Paladin G, Vergassola M, Vulpiani A (1993) A random process for the construction of multiaffine fields. Physica D 65(4):352–358 ADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Sudre J, Morrow R (2008) Global surface currents, a high resolution product for investigating ocean dynamics. Ocean Dyn 58:101–118. doi: 10.1007/s10236-008-0134-9 ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Suman Kumar Maji
    • 1
    Email author
  • Oriol Pont
    • 1
  • Hussein Yahia
    • 1
  • Joel Sudre
    • 2
  1. 1.Geostat TeamINRIA Bordeaux Sud-OuestBordeauxFrance
  2. 2.Dynbio TeamLEGOS, UMR CNRS 5556ToulouseFrance

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