An Optimized Algorithm for the Evaluation of Local Singularity Exponents in Digital Signals

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.

[1]  C L Jonesyxk.  Wavelet packet computation of the Hurst exponent , 1996 .

[2]  Antonio Turiel,et al.  Multiscale techniques for the detection of precipitation using thermal IR satellite images , 2005, IEEE Geoscience and Remote Sensing Letters.

[3]  Antonio Turiel,et al.  Microcanonical multifractal formalism: Application to the estimation of ocean surface velocities , 2007 .

[4]  E. Bacry,et al.  Wavelets and multifractal formalism for singular signals: Application to turbulence data. , 1991, Physical review letters.

[5]  Stéphane Mallat,et al.  Singularity detection and processing with wavelets , 1992, IEEE Trans. Inf. Theory.

[6]  Antonio Turiel,et al.  Multifractal method for the instantaneous evaluation of the stream function in geophysical flows. , 2005, Physical review letters.

[7]  J. Nadal,et al.  Self-Similarity Properties of Natural Images Resemble Those of Turbulent Flows , 1998, cond-mat/0107314.

[8]  Leon G. Higley,et al.  Forensic Entomology: An Introduction , 2009 .

[9]  A. Vulpiani,et al.  Predictability: a way to characterize complexity , 2001, nlin/0101029.

[10]  Antonio Turiel,et al.  Application of the microcanonical multifractal formalism to monofractal systems. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  A. Crisanti,et al.  Predictability in the large: an extension of the concept of Lyapunov exponent , 1996, chao-dyn/9606014.

[12]  Antonio Turiel,et al.  Description, modelling and forecasting of data with optimal wavelets , 2008 .

[13]  Stéphane Mallat,et al.  Characterization of Signals from Multiscale Edges , 2011, IEEE Trans. Pattern Anal. Mach. Intell..

[14]  Antonio Turiel,et al.  Numerical methods for the estimation of multifractal singularity spectra on sampled data: A comparative study , 2006, J. Comput. Phys..

[15]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[16]  Antonio Turiel,et al.  The multifractal structure of satellite sea surface temperature maps can be used to obtain global maps of streamlines , 2009 .

[17]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[18]  Antonio Turiel,et al.  Empirical evidences of a common multifractal signature in economic, biological and physical systems , 2009 .

[19]  N. Parga,et al.  Ju l 2 00 1 The multi-fractal structure of contrast changes in natural images : from sharp edges to textures , 2008 .

[20]  Stéphane Jaffard,et al.  Multifractal formalism for functions part II: self-similar functions , 1997 .

[21]  Z. Struzik.  Determining Local Singularity Strengths and their Spectra with the Wavelet Transform , 2000 .

[22]  U. Frisch.  Turbulence: The Legacy of A. N. Kolmogorov , 1996 .

[23]  Antonio Turiel,et al.  Microcanonical multifractal formalism—a geometrical approach to multifractal systems: Part I. Singularity analysis , 2008 .

[24]  Antonio Turiel.  Relevance of multifractal textures in static images , 2003 .

[25]  Antonio Turiel,et al.  Reconstructing images from their most singular fractal manifold , 2002, IEEE Trans. Image Process..

[26]  Jean-Michel Poggi,et al.  Wavelets and their applications , 2007 .

[27]  Antonio Turiel,et al.  Role of multifractal sources in the analysis of stock market time series , 2005 .

[28]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[29]  Joaquim Ballabrera-Poy,et al.  Tracking oceanic currents by singularity analysis of Microwave Sea Surface Temperature images , 2008 .

[30]  Stéphane Jaffard,et al.  Multifractal formalism for functions part I: results valid for all functions , 1997 .

[31]  She,et al.  Universal scaling laws in fully developed turbulence. , 1994, Physical review letters.

[32]  I. Simonsen,et al.  Determination of the Hurst exponent by use of wavelet transforms , 1997, cond-mat/9707153.