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.

In this work a discrete fractional Brownian motion (FBM) model is applied to xray images as a measure of the regional texture.FBM is a generalization of ordinary Wiener -Levy Brownian motion.A Parameter H is introduced which describes the roughness of the realizations. Using generated realizations, a Cramer -Rao bound for the variance of an estimate of H was evaluated using asymptotic statistics.The results show that the accuracy of the estimate is independent of the true H. A maximum likelihood estimator is derived for H and applied to the data sets The results were close to the C -R bound.The MLE is then applied to sequences of digital coronary angiograms. The results show that the H parameter is a useful index by which to segment vessels from background noise.

In previous work, the concept of estimation theory has been applied to provide a basis for determining the fractal dimension of a given data set when fractional Brownian motion (FBIvi) is assumed to be a suitable model. However, the generated FIM functions used to test the approach were only one dimensional, even though the estimation was applied to images.In the current work, the generation process is extended to two dimensions so that FBM images are generated directly.A series of 32 x 32 FIl(4 images were formed for fractal dimensions of 2.2 to 2.8. The texture in these images is seen to be representative of textures observed in xray images of bone. Furthermore by combining the FBM realization with a deterministic function such as a sinusoid, it is found that complex appearing images can be broken down into two basic components: fractal, and determinisitic.Thus this methodology may prove useful in the analysis and presentation of medical images.