Globally Convergent Algorithms for Estimating Generalized Gamma Distributions in Fast Signal and Image Processing

Many applications in real-time signal, image, and video processing require automatic algorithms for rapid characterizations of signals and images through fast estimation of their underlying statistical distributions. We present fast and globally convergent algorithms for estimating the three-parameter generalized gamma distribution (G Gamma D). The proposed method is based on novel scale-independent shape estimation (SISE) equations. We show that the SISE equations have a unique global root in their semi-infinite domains and the probability that the sample SISE equations have a unique global root tends to one. The consistency of the global root, its scale, and index shape estimators is obtained. Furthermore, we establish that, with probability tending to one, Newton-Raphson (NR) algorithms for solving the sample SISE equations converge globally to the unique root from any initial value in its given domain. In contrast to existing methods, another remarkable novelty is that the sample SISE equations are completely independent of gamma and polygamma functions and involve only elementary mathematical operations, making the algorithms well suited for real-time both hardware and software implementations. The SISE estimators also allow the maximum likelihood (ML) ratio procedure to be carried out for testing the generalized Gaussian distribution (GGD) versus the G Gamma D. Finally, the fast global convergence and accuracy of our algorithms for finite samples are demonstrated by both simulation studies and real image analysis.