Abstract
Recent works show that the determination of singularity exponents in images can be useful to assess their information content, and in some cases they can cast additional information about underlying physical processes. However, the concept of singularity exponent is associated to differential calculus and thus cannot be easily translated to a digital context, even using wavelets. In this work we show that a recently patented algorithm allows obtaining precise, meaningful values of singularity exponents at every point in the image by the use of a discretized combinatorial mask, which is an extension of a particular wavelet basis. This mask is defined under the hypothesis that singularity exponents are a measure not only of the degree of regularity of the image, but also of the reconstructibility of a signal from their points.
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References
Aurell, E., Boffetta, G., Crisanti, A., Palading, G., Vulpiani, A.: Predictability in the large: an extension of the concept of lyapunov exponent. Journal of Physics A 30, 1–26 (1997)
Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Physics reports 356(6), 367–474 (2001)
Daubechies, I.: Ten lectures on wavelets. CBMS-NSF Series in App. Math. Capital City Press, Montpelier (1992)
Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. John Wiley and Sons, Chichester (1990)
Frisch, U.: Turbulence: The legacy of A.N. Kolmogorov. Cambridge Univ. Press, Cambridge (1995)
Isern-Fontanet, J., Turiel, A., Garcia-Ladona, E., Font, J.: Microcanonical multifractal formalism: application to the estimation of ocean surface velocities. Journal of Geophysical Research 112, C05024 (2007)
Jaffard, S.: Multifractal formalism for functions. I. Results valid for all functions. SIAM Journal of Mathematical Analysis 28(4), 944–970 (1997)
Jones, C.L., Lonergan, G.T., Mainwaring, D.E.: Wavelet packet computation of the hurst exponent. J. Phys. A: Math. Gen. 29(10), 2509 (1996)
Mallat, S., Huang, W.L.: Singularity detection and processing with wavelets. IEEE Trans. in Inf. Th. 38, 617–643 (1992)
Mallat, S., Zhong, S.: Wavelet transform maxima and multiscale edges. In: Ruskai, M.B., et al. (eds.) Wavelets and their Applications. Jones and Bartlett, Boston (1991)
Mallat, S., Zhong, S.: Characterization of signals from multiscale edges. IEEE Trans. on Pattern Analysis and Machine Intelligence 14, 710–732 (1992)
Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, London (1999)
Muzy, J.F., Bacry, E., Arneodo, A.: Wavelets and multifractal formalism for singular signals: Application to turbulence data. Physical Review Letters 67, 3515–3518 (1991)
Parisi, G., Frisch, U.: On the singularity structure of fully developed turbulence. In: Ghil, M., Benzi, R., Parisi, G. (eds.) Turbulence and Predictability in Geophysical Fluid Dynamics. Proc. Intl. School of Physics E. Fermi, pp. 84–87. North Holland, Amsterdam (1985)
Pont, O., Turiel, A., Pérez-Vicente, C.: Application of the microcanonical multifractal formalism to monofractal systems. Physical Review E 74, 61110 (2006)
Pont, O., Turiel, A., Pérez-Vicente, C.: Description, modeling and forecasting of data with optimal wavelets. Journal of Economic Interaction and Coordination 4, 39–54 (2009)
Pont, O., Turiel, A., Pérez-Vicente, C.: Empirical evidences of a common multifractal signature in economic, biological and physical systems. Physica A 388, 2025–2035 (2009)
She, Z.S., Leveque, E.: Universal scaling laws in fully developed turbulence. Physical Review Letters 72, 336–339 (1994)
Simonsen, I., Hansen, A., Magnar, O.: Determination of the hurst exponent by use of wavelet transforms. Phys. Rev. E 58(3), 2779–2787 (1998)
Struzik, Z.R.: Determining local singularity strengths and their spectra with the wavelet transform. Fractals 8(2), 163–179 (2000)
Turiel, A.: Relevance of multifractal textures in static images. Electronic Letters on Computer Vision and Image Analysis 1(1), 35–49 (2003)
Turiel, A.: Method and system for the singularity analysis of digital signals, patent registered under number PCT/ES2008/070195 (2008)
Turiel, A., Grazzini, J., Yahia, H.: Multiscale techniques for the detection of precipitation using thermal IR satellite images. IEEE Geoscience and Remote Sensing Letters 2(4), 447–450 (2005), doi:10.1109/LGRS.2005.852712
Turiel, A., Isern-Fontanet, J., García-Ladona, E., Font, J.: Multifractal method for the instantaneous evaluation of the stream function in geophysical flows. Physical Review Letters 95(10), 104502 (2005), doi:10.1103/PhysRevLett.95.104502
Turiel, A., Mato, G., Parga, N., Nadal, J.P.: The self-similarity properties of natural images resemble those of turbulent flows. Physical Review Letters 80, 1098–1101 (1998)
Turiel, A., Nieves, V., García-Ladona, E., Font, J., Rio, M.H., Larnicol, G.: The multifractal structure of satellite temperature images can be used to obtain global maps of ocean currents. Ocean Science 5, 447–460 (2009)
Turiel, A., Parga, N.: The multi-fractal structure of contrast changes in natural images: from sharp edges to textures. Neural Computation 12, 763–793 (2000)
Turiel, A., Pérez-Vicente, C.: Role of multifractal sources in the analysis of stock market time series. Physica A 355, 475–496 (2005)
Turiel, A., Pérez-Vicente, C., Grazzini, J.: Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study. Journal of Computational Physics 216(1), 362–390 (2006)
Turiel, A., del Pozo, A.: Reconstructing images from their most singular fractal manifold. IEEE Trans. on Im. Proc. 11, 345–350 (2002)
Turiel, A., Solé, J., Nieves, V., Ballabrera-Poy, J., García-Ladona, E.: Tracking oceanic currents by singularity analysis of micro-wave sea surface temperature images. Remote Sensing of Environment 112, 2246–2260 (2008)
Turiel, A., Yahia, H., Pérez-Vicente, C.: Microcanonical multifractal formalism: a geometrical approach to multifractal systems. Part I: Singularity analysis. Journal of Physics A 41, 15501 (2008)
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Pont, O., Turiel, A., Yahia, H. (2011). An Optimized Algorithm for the Evaluation of Local Singularity Exponents in Digital Signals. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds) Combinatorial Image Analysis. IWCIA 2011. Lecture Notes in Computer Science, vol 6636. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21073-0_31
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DOI: https://doi.org/10.1007/978-3-642-21073-0_31
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