Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification

Abstract

Fractional Brownian motion (FBM) is a suitable description model for a large number of natural shapes and phenomena. In applications, it is imperative to estimate the fractal dimension from sampled data, namely, discrete-time FBM (DFBM). To this aim, the increment of DFBM, referred to as discrete-time fractional Gaussian noise (DFGN), is invoked as an auxiliary tool. The regular part of DFGN is first filtered out via Levinson's algorithm. The power spectral density of the regular process is found to satisfy a power law that its exponent can be well fitted by a quadratic function of fractal dimension. A new method is then proposed to estimate the fractal dimension of DFBM from the given data set. The computational complexity and statistical properties are investigated. Moreover, the proposed algorithm is robust with respect to amplitude scaling and shifting, as well as time shifting on the data. Finally, the effectiveness of this estimator is demonstrated via a classification problem of natural texture images.