Advertisement

Journal of Mathematical Imaging and Vision

, Volume 39, Issue 1, pp 28–44 | Cite as

Virtual Super Resolution of Scale Invariant Textured Images Using Multifractal Stochastic Processes

  • Pierre ChainaisEmail author
  • Émilie Kœnig
  • Véronique Delouille
  • Jean-François Hochedez
Article

Abstract

We present a new method of magnification for textured images featuring scale invariance properties. This work is originally motivated by an application to astronomical images. One goal is to propose a method to quantitatively predict statistical and visual properties of images taken by a forthcoming higher resolution telescope from older images at lower resolution. This is done by performing a virtual super resolution using a family of scale invariant stochastic processes, namely compound Poisson cascades, and fractional integration. The procedure preserves the visual aspect as well as the statistical properties of the initial image. An augmentation of information is performed by locally adding random small scale details below the initial pixel size. This extrapolation procedure yields a potentially infinite number of magnified versions of an image. It allows for large magnification factors (virtually infinite) and is physically conservative: zooming out to the initial resolution yields the initial image back. The (virtually) super resolved images can be used to predict the quality of future observations as well as to develop and test compression or denoising techniques.

Keywords

Natural images Scale invariance Multifractal analysis Extrapolation Enhancement Infinitely divisible cascades 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bacry, E., Muzy, J.: Log-infinitely divisible multifractal processes. Commun. Math. Phys. 236, 449–475 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barnsley, M.: Fractal functions and interpolation. Constr. Approx. 2(1), 303–329 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barnsley, M.: Fractals Everywhere, 2nd edn. Academic Press, San Diego (1993) zbMATHGoogle Scholar
  4. 4.
    Barral, J., Mandelbrot, B.: Multiplicative products of cylindrical pulses. Probab. Theory Relat. Fields 124, 409–430 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Benzi, R., Biferale, L., Crisanti, A., Paladin, G., Vergassola, M., Vulpiani, A.: A random process for the construction of multiaffine fields. Physica D 65, 352–358 (1993) zbMATHCrossRefGoogle Scholar
  6. 6.
    Benzi, R., Ciliberto, S., Tripicione, R., Baudet, C., Massaioli, F.: Extended self similarity in turbulent flows. Phys. Rev. E 48, R29–R32 (1993) CrossRefGoogle Scholar
  7. 7.
    Biermé, H., Meerschaert, M.M., Scheffler, H.P.: Operator scaling stable random fields. Stoch. Process. Appl. 117(3), 312–332 (2007) zbMATHCrossRefGoogle Scholar
  8. 8.
    Carey, W., Chuang, D., Hemami, S.: Regularity-preserving image interpolation. Image Process. IEEE Trans. 8(9), 1293–1297 (1999). DOI: 10.1109/83.784441 CrossRefGoogle Scholar
  9. 9.
    Castaing, B., Dubrulle, B.: Fully developed turbulence: a unifying point of view. J. Phys. II France 5, 895–899 (1995) CrossRefGoogle Scholar
  10. 10.
    Castaing, B., Gagne, Y., Hopfinger, E.: Velocity probability density functions of high Reynolds number turbulence. Physica D 46, 177–200 (1990) zbMATHCrossRefGoogle Scholar
  11. 11.
    Chainais, P.: Multidimensional infinitely divisible cascades. application to the modelling of intermittency in turbulence. Eur. J. Phys. B 51, 229–243 (2006) CrossRefGoogle Scholar
  12. 12.
    Chainais, P.: Infinitely divisible cascades to model the statistics of natural images. IEEE Trans. Pattern Mach. Intell. (2007). DOI: 10.1109/TPAMI.2007.1113 (ISSN: 0162-8828) Google Scholar
  13. 13.
    Chainais, P., Riedi, R., Abry, P.: Scale invariant infinitely divisible cascades. In: Int. Symp. on Physics in Signal and Image Processing, Grenoble, France (2003) Google Scholar
  14. 14.
    Chainais, P., Riedi, R., Abry, P.: On non scale invariant infinitely divisible cascades. IEEE Trans. Inf. Theory 51(3), 1063–1083 (2005) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Chainais, P., Riedi, R., Abry, P.: Warped infinitely divisible cascades: beyond scale invariance. Trait. Signal 22(1) (2005) Google Scholar
  16. 16.
    Chang, S., Cvetkovic, Z., Vetterli, M.: Resolution enhancement of images using wavelet transform extrema extrapolation. In: Acoustics, Speech, and Signal Processing, 1995. ICASSP-95, 1995 International Conference on, vol. 4, pp. 2379–2382 (1995). DOI: 10.1109/ICASSP.1995.479971
  17. 17.
    Decoster, N., Roux, S., Arneodo, A.: A wavelet-based method for multifractal image analysis. II. Applications to synthetic multifractal rough surfaces. Eur. Phys. J. B 15, 739–764 (2000) CrossRefGoogle Scholar
  18. 18.
    Delouille, V., Chainais, P., Hochedez, J.F.: Quantifying and containing the curse of high resolution coronal imaging. Ann. Geophys. 26(10), 3169–3184 (2008) CrossRefGoogle Scholar
  19. 19.
    Ebert, D., Musgrave, F., Peachy, D., Perlin, K., Worley, S.: Texturing and Modeling: A Procedural Approach, 3rd edn. Morgan Kaufmann, San Mateo (2003) Google Scholar
  20. 20.
    Fattal, R.: Image upsampling via imposed edge statistics. In: SIGGRAPH ’07: ACM SIGGRAPH 2007 papers, p. 95. ACM, New York (2007). DOI: 10.1145/1275808.1276496 CrossRefGoogle Scholar
  21. 21.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, New-York (1966) zbMATHGoogle Scholar
  22. 22.
    Freeman, W.T., Jones, T.R., Pasztor, E.C.: Example-based super-resolution. IEEE Comput. Graph. Appl. 22(2), 56–65 (2002) CrossRefGoogle Scholar
  23. 23.
    Frisch, U.: Turbulence. The Legacy of A. Kolmogorov. Cambridge University Press, Cambridge (1995) Google Scholar
  24. 24.
    Glasner, D., Bagon, S., Irani, M.: Super-resolution from a single image. In: Computer Vision, 2009 IEEE 12th International Conference on, pp. 349–356 (2009). DOI: 10.1109/ICCV.2009.5459271
  25. 25.
    Grenander, U., Srivastava, A.: Probability models for clutter in natural images. IEEE Trans. Pattern Anal. Mach. Intell. 23(4), 424–429 (2001) CrossRefGoogle Scholar
  26. 26.
    Guofang, T., Zhang, C., Wu, J., Liu, X.: Remote sensing image processing using wavelet fractal interpolation. In: Proceedings of the International Conference on Communications, Circuits and Systems, vol. 2, p. 706 (2005). DOI: 10.1109/ICCCAS.2005.1495209
  27. 27.
    HaCohen, Y., Fattal, R., Lischinski, D.: Image upsampling via texture hallucination. In: Proceedings of IEEE Int. Conf. on Computational Photography (2010) Google Scholar
  28. 28.
    Han, Z., Denney, T.J.: Incremental Fourier interpolation of 2-d fractional Brownian motion. Ind. Electron. IEEE Trans. 48(5), 920–925 (2001). DOI: 10.1109/41.954556 CrossRefGoogle Scholar
  29. 29.
    Hochedez, J.F., et al.: EUI, the ultraviolet imaging telescopes of solar orbiter. In: Proceedings of the 2nd Solar Orbiter Workshop, vol. 641. ESA-SP, Athens (2006) Google Scholar
  30. 30.
    Jaffard, S.: Multifractal formalism for functions, Part 1 & 2. SIAM J. Math. Anal. 28(4), 944–998 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Jaffard, S.: Beyond Besov spaces Part 1: Distributions of wavelet coefficients. J. Fourier Anal. Appl. 10(3), 221–246 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Jaffard, S.: Wavelet techniques in multifractal analysis. In: Lapidus, M., van Frankenhuijsen, M. (eds.) Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Proceedings of Symposia in Pure Mathematics, vol. 72(2), pp. 91–152. AMS, Providence (2004) Google Scholar
  33. 33.
    Kirshner, H., Porat, M.: On the role of exponential splines in image interpolation. Image Process., IEEE Trans. 18(10), 2198–2208 (2009). DOI: 10.1109/TIP.2009.2025008 CrossRefGoogle Scholar
  34. 34.
    Kœnig, E., Chainais, P.: Virtual resolution enhancement of scale invariant textured images using stochastic processes. In: Proceedings of IEEE-ICIP 2009, Cairo (2009) Google Scholar
  35. 35.
    Kœnig, E., Chainais, P., Delouille, V., Hochedez, J.F.: Amélioration virtuelle de la résolution d’images du soleil par augmentation d’information invariante d’échelle. In: Proceedings of the 22nd Colloquium GRETSI, Dijon (2009) Google Scholar
  36. 36.
    Lashermes, B., Jaffard, S., Abry, P.: Wavelet leaders based multifractal analysis. In: Proc. of Int. Conf. on Acoustics, Speech and Signal Proc. Philadelphia, USA (2005) Google Scholar
  37. 37.
    Levy-Vehel, J., Legrand, P.: Hölderian regularity-based image interpolation. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP, vol. 3, pp. III–III (2006). DOI: 10.1109/ICASSP.2006.1660788
  38. 38.
    Li, X.: Image resolution enhancement via data-driven parametric models in the wavelet space. EURASIP J. Image Video Process. 2007, 41516 (2007),12 p. DOI: 10.1155/2007/41516 Google Scholar
  39. 39.
    Li, X., Orchard, M.T.: New edge-directed interpolation. IEEE Trans. Image Process. 10, 1521–1527 (2001) CrossRefGoogle Scholar
  40. 40.
    Liu, Y., Fieguth, P.: Image resolution enhancement with hierarchical hidden fields. In: ICIAR ’09: Proceedings of the 6th International Conference on Image Analysis and Recognition, pp. 73–82. Springer, Berlin (2009). DOI: 10.1007/978-3-642-02611-9_8 Google Scholar
  41. 41.
    Mandelbrot, B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358 (1974) zbMATHCrossRefGoogle Scholar
  42. 42.
    Muzy, J., Bacry, E.: Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws. Phys. Rev. E 66, 056121 (2002) CrossRefGoogle Scholar
  43. 43.
    Muzy, J., Bacry, E., Arneodo, A.: Multifractal formalism for fractal signals: The structure function approach versus the wavelet transform modulus-maxima method. J. Stat. Phys. 70, 635–674 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Olshausen, B., Field, D.: Emergence of simple-cell receptive properties by learning a sparse code for natural images. Nature 381, 607–609 (1996) CrossRefGoogle Scholar
  45. 45.
    Portilla, J., Simoncelli, E.: A parametric texture model based on joint statistics of complex wavelet coefficients. Int. J. Comput. Vis. 40(1), 49–71 (2000) zbMATHCrossRefGoogle Scholar
  46. 46.
    Riedi, R.H.: Multifractal processes. Long-range dependence: theory and applications (2001) Google Scholar
  47. 47.
    Schertzer, D., Lovejoy, S.: Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes. J. Geophys. Res. 92, 9693 (1987) CrossRefGoogle Scholar
  48. 48.
    Schmitt, F., Marsan, D.: Stochastic equations generating continuous multiplicative cascades. Eur. Phys. J. B 20, 3–6 (2001) Google Scholar
  49. 49.
    Srivastava, A., Lee, A., Simoncelli, E., Zhu, S.C.: On advances in statistical modeling of natural images. J. Math. Imaging Vis. 18, 17–33 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Suetake, N., Sakano, M., Uchino, E.: Image super-resolution based on local self-similarity. Opt. Rev. 15, 26–30 (2008). DOI: 10.1007/s10043-008-0005-0 CrossRefGoogle Scholar
  51. 51.
    Thevenaz, P., Blu, T., Unser, M.: Interpolation revisited [medical images application]. Medical Imaging, IEEE Trans. 19(7), 739–758 (2000). DOI: 10.1109/42.875199 CrossRefGoogle Scholar
  52. 52.
    Turiel, A., Mato, G., Parga, N., Nadal, J.: Self-similarity properties of natural images resemble those of turbulent flows. Phys. Rev. Lett. 80(5), 1098–1101 (1998) CrossRefGoogle Scholar
  53. 53.
    Unser, M., Aldroubi, A., Eden, M.: Enlargement or reduction of digital images with minimum loss of information. IEEE Trans. Image Process. 4(3), 247–258 (1995) CrossRefGoogle Scholar
  54. 54.
    Unser, M., Zerubia, J.: A generalized sampling theory without band-limiting constraints. IEEE Trans. Circ. Syst. II 45(8), 959–969 (1998) zbMATHCrossRefGoogle Scholar
  55. 55.
    Wainwright, M., Simoncelli, E.: Scale mixtures of Gaussian and the statistics of natural images. Adv. Neural Inf. Process. Syst. 12, 855–861 (2000) NIPS’99 Google Scholar
  56. 56.
    Wainwright, M., Simoncelli, E., Willsky, A.: Random cascades on wavelet trees and their use in analyzing and modeling natural images. Appl. Comput. Harmon. Anal. 11, 89–123 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Wendt, H., Roux, S.G., Abry, P., Jaffard, S.: Wavelet leaders and bootstrap for multifractal analysis of images. Signal Process. 89, 1100–1114 (2009) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Pierre Chainais
    • 1
    Email author
  • Émilie Kœnig
    • 1
  • Véronique Delouille
    • 2
  • Jean-François Hochedez
    • 2
  1. 1.Université Blaise Pascal CNRS UMR 6158, LIMOSClermont UniversitéClermont-FerrandFrance
  2. 2.Royal Observatory of BelgiumBrusselsBelgium

Personalised recommendations