Fractional Brownian Motion: A Maximum Likelihood Estimator and Its Application to Image Texture

Abstract

Fractals have been shown to be useful in characterizing texture in a variety of contexts. Use of this methodology normally involves measurement of a parameter H, which is directly related to fractal dimension. In this work the basic theory of fractional Brownian motion is extended to the discrete case. It is shown that the power spectral density of such a discrete process is only approximately proportional to |f|a instead of in direct proportion as in the continuous case. An asymptotic Cramer-Rao bound is derived for the variance of an estimate of H. Subsequently, a maximum likelihood estimator (MLE) is developed to estimate H. It is shown that the variance of this estimator nearly achieves the minimum bound. A generation algorithm for discrete fractional motion is presented and used to demonstrate the capabilities of the MLE when the discrete fractional Brownian process is contaminated with additive Gaussian noise. The results show that even at signal-to-noise ratios of 30 dB, significant errors in estimation of H can result when noise is present. The MLE is then applied to X-ray images of the human calcaneus to demonstrate how the line-to-line formulation can be applied to the two-dimensional case. These results indicate that it has strong potential for quantifying texture.

Fractal estimation using models on multiscale trees

“…Sixty four fJ3m sample paths, each having a length of 2048 samples, were generated using the Cholesky decomposition method of [ 7 ] ; this is precisely the same approach as in Kaplan and Kuo [4], whose experimental results form the basis of comparison with ours.…”

Fractal Estimation using Models on Multiscale Trees

“…Sixty four fBm sample paths, each having a length of 2048 samples, were generated using the Cholesky decomposition method of [7], precisely the same approach as in Kaplan and Kuo [4] whose experimental results form the basis of comparison with ours.…”

IMAGING SYSTEMS USED FOR DIAGNOSIS OF PERIODONTAL PATHOLOGY Part 2: Alternative Imaging Systems and Image Processing Methods

“…Periodontal hastalığı ya da osteoporozu olan bireylerde trabeküllerde meydana gelen rezorpsiyon, fraktal boyutta değişikliklerle sonuçlanabilmektedir. 60,61 Kemik yapısına ilişkin hassas değerlendirmeler yapabilmemize olanak sağlayan ve kemiğin trabeküler yapısını objektif olarak belirleyebilen fraktal analiz yönteminden; alveol kemiğinde oluşan erken dönem periodontal değişikliklerin saptanmasında da yararlanılabilmektedir. Bu özellikleri ile fraktal analizin önemli bir tanı yöntemi olduğu gerek medikal gerekse dental alanda yapılan çalışmalarla vurgulanmaktadır.…”