An efficient maximum likelihood estimator for two-dimensional fractional brownian motion

For pattern recognition, natural scenes and medical images are often modeled as two-dimensional fractional Brownian motion (2D FBM), which can be easily described by the Hurst exponent (H), a real number between 0 and 1. The Hurst exponent is directly related to the fractal dimension (D) by D = 3-H, and hence it very suitably serves as a characteristic index. Therefore, how to estimate the Hurst exponent effectively and efficiently is very important in pattern recognition. In this paper, a partially iterative algorithm, simply called an iterative maximum likelihood estimator (MLE) for 2D DFBM is first proposed and then a more sufficiently iterative algorithm, simply called an efficient MLE for 2D DFBM, is further proposed via a perfect structure of the log-likelihood function related to the Hurst exponent. Except for theoretical knowledge, two practical algorithms are also correspondingly provided for easy applications. Experimental results show that the MLE for 2D DFBM is effective and workable; its accuracy gets higher as the image size increases, and hence its recognition resolution is much finer to identify small difference among patterns. Furthermore, the efficient MLE is much quicker than the iterative MLE, especially at larger image sizes.