Whittle maximum likelihood estimator for isotropic fractional Brownian images

Fractional Brownian motion (fBm) of H parameter is a stochastic fractal process that can be used to create virtual landscapes or to model 2D physical phenomenon. In this communication, we explore the Whittle maximum likelihood estimator (WMLE) to assess the H parameter of isotropic fBm images. We have compared a 1D estimator assessed from the lines increments of the image, to a 2D estimator of the 2D increments of the image. These 2 estimators were tested on 2D fBm generated using the exact Stein method. Results are of high quality. The mean H values are very close to the true ones for both estimators. The standard deviations of the 2D estimates are 2 times smaller than in 1D and should be preferred for practical application as for example texture analysis.

[1]  Rachid Harba,et al.  Fast and exact synthesis for 1-D fractional Brownian motion and fractional Gaussian noises , 2002, IEEE Signal Processing Letters.

[2]  Patrice Abry,et al.  Wavelet Analysis of Long-Range-Dependent Traffic , 1998, IEEE Trans. Inf. Theory.

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

[4]  R. Kumaresan,et al.  Isotropic two-dimensional Fractional Brownian Motion and its application in Ultrasonic analysis , 1992, 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[5]  R. Dahlhaus.  Efficient parameter estimation for self-similar processes , 1989, math/0607078.

[6]  Hashem Dezhbakhsh,et al.  On the typical spectral shape of an economic variable , 2003 .

[7]  R. Harba,et al.  Fractal organization of trabecular bone images on calcaneus radiographs , 1994, Journal of bone and mineral research : the official journal of the American Society for Bone and Mineral Research.

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

[9]  Trieu-Kien Truong,et al.  Spectral representation of fractional Brownian motion in n dimensions and its properties , 1995, IEEE Trans. Inf. Theory.

[10]  Michael L. Stein,et al.  Fast and Exact Simulation of Fractional Brownian Surfaces , 2002 .

[11]  S. Kay,et al.  Fractional Brownian Motion: A Maximum Likelihood Estimator and Its Application to Image Texture , 1986, IEEE Transactions on Medical Imaging.

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

[13]  Steven Kay,et al.  Fundamentals Of Statistical Signal Processing , 2001 .

[14]  Heinz-Otto Peitgen,et al.  The science of fractal images , 2011 .

[15]  Benoit B. Mandelbrot,et al.  Some noises with I/f spectrum, a bridge between direct current and white noise , 1967, IEEE Trans. Inf. Theory.

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

[17]  C. R. Dietrich,et al.  Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix , 1997, SIAM J. Sci. Comput..

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

[19]  Rachid Harba,et al.  Detection of diabetic foot hyperthermia by infrared imaging , 2014, 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[20]  P. Whittle,et al.  Estimation and information in stationary time series , 1953 .

[21]  R. Harba,et al.  A practical comparison of fBm estimators , 2010, IEEE 10th INTERNATIONAL CONFERENCE ON SIGNAL PROCESSING PROCEEDINGS.