Fractional brownian motion: a maximum likelihood estimator and its application to image texture

Area of Science:

• Digital Signal Processing
• Image Analysis
• Texture Quantification

Background:

• Fractals are valuable for texture characterization.
• Estimating fractal dimension parameter H is crucial.
• Extending fractional Brownian motion to discrete cases is needed.

Purpose of the Study:

• To extend fractional Brownian motion theory to discrete processes.
• To develop and evaluate a maximum likelihood estimator (MLE) for fractal parameter H.
• To assess the performance of the MLE in noisy conditions and its application to 2D images.

Main Methods:

• Theoretical extension of fractional Brownian motion to discrete processes.
• Derivation of an asymptotic Cramer-Rao bound for H estimation variance.
• Development and application of a maximum likelihood estimator (MLE) for H.
• Generation of discrete fractional Brownian motion for simulations.
• Application of MLE to 2D X-ray images of the human calcaneus.

Main Results:

• The power spectral density of discrete fractional Brownian motion is approximately proportional to |f|a.
• The developed MLE for H nearly achieves the asymptotic Cramer-Rao bound.
• Additive Gaussian noise significantly impacts H estimation accuracy, even at high signal-to-noise ratios (30 dB).
• The MLE demonstrates strong potential for texture quantification in 2D X-ray images.

Conclusions:

• The developed MLE is effective for estimating fractal parameter H in discrete processes.
• Noise robustness of the MLE requires further investigation for practical applications.
• Fractal analysis using this methodology shows significant promise for quantitative texture analysis in medical imaging.