Estimation of fractional Brownian motion embedded in a noisy environment using nonorthogonal wavelets

We show that nonorthogonal wavelets can characterize the fractional Brownian motion (fBm) that is in white noise. We demonstrate the point that discriminating the parameter of fBm from that of noise is equivalent to discriminating the composite singularity formed by superimposing a peak singularity on a Dirac singularity. We characterize the composite singularity by formalizing this problem as a nonlinear optimization problem. This yields our parameter estimation algorithm. For fractal signal estimation, Wiener filtering is explicitly formulated as a function of the signal and noise parameters and the wavelets. We show that the estimated signal is a 1/f process. Comparative studies through numerical simulations of our methods with those of Wornell and Oppenheim (1992) are presented.

[1]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[2]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[3]  Richard Kronland-Martinet,et al.  Asymptotic wavelet and Gabor analysis: Extraction of instantaneous frequencies , 1992, IEEE Trans. Inf. Theory.

[4]  Patrick Flandrin,et al.  Wavelet analysis and synthesis of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[5]  C.-C. Jay Kuo,et al.  Fractal estimation from noisy data via discrete fractional Gaussian noise (DFGN) and the Haar basis , 1993, IEEE Trans. Signal Process..

[6]  Patrick Flandrin,et al.  On the spectrum of fractional Brownian motions , 1989, IEEE Trans. Inf. Theory.

[7]  Mohamed A. Deriche,et al.  Signal modeling with filtered discrete fractional noise processes , 1993, IEEE Trans. Signal Process..

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

[9]  M. S. Keshner.  1/f noise , 1982, Proceedings of the IEEE.

[10]  Stéphane Mallat,et al.  Characterization of Self-Similar Multifractals with Wavelet Maxima , 1994 .

[11]  Mohamed A. Deriche,et al.  Maximum likelihood estimation of the parameters of discrete fractionally differenced Gaussian noise process , 1993, IEEE Trans. Signal Process..

[12]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[13]  Gregory W. Wornell,et al.  Estimation of fractal signals from noisy measurements using wavelets , 1992, IEEE Trans. Signal Process..

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

[15]  Sally Floyd,et al.  Wide area traffic: the failure of Poisson modeling , 1995, TNET.

[16]  John E. Howland,et al.  Computer graphics , 1990, IEEE Potentials.

[17]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[18]  C.-C. Jay Kuo,et al.  An improved method for 2-D self-similar image synthesis , 1996, IEEE Trans. Image Process..

[19]  A.H. Tewfik,et al.  Correlation structure of the discrete wavelet coefficients of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[20]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[21]  S. Jaffard.  Pointwise smoothness, two-microlocalization and wavelet coefficients , 1991 .

[22]  Alex Pentland,et al.  Fractal-Based Description of Natural Scenes , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[23]  Gregory W. Wornell,et al.  A Karhunen-Loève-like expansion for 1/f processes via wavelets , 1990, IEEE Trans. Inf. Theory.

[24]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[25]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[26]  Walter Willinger,et al.  Long-range dependence in variable-bit-rate video traffic , 1995, IEEE Trans. Commun..

[27]  Anthony V. Fiacco,et al.  Computational Algorithm for the Sequential Unconstrained Minimization Technique for Nonlinear Programming , 1964 .

[28]  Bruno Torrésani,et al.  Local frequency analysis with two-dimensional wavelet transform , 1994, Signal Process..

[29]  J. R. Wallis,et al.  Some long‐run properties of geophysical records , 1969 .

[30]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[31]  M. Fox,et al.  Fractal feature analysis and classification in medical imaging. , 1989, IEEE transactions on medical imaging.