Indian Journal of Economics and Development, Vol 5 (9), September 2017 ISSN (online): 2320-9836 ISSN (Print): 2320-9828 Boltzmann and Non-Boltzmann sampling for image processing T. Pramananda Perumal*, K.R. Srivaishnavi*, D.L. Asha Rani*, K.P.N. Murthy** *Presidency College, Kamarajar Salai, Triplicane, Chennai-600 005, Tamil Nadu, India **Chennai Mathematical Institute (CMI), Siruseri, Kelambakkam, Chennai-603 103, Tamil Nadu, India pramanandaperumal@yahoo.com, srivaishnavisuresh07@gmail.com, aancy35@gmail.com, k.p.n.murthy@gmail.com Abstract Objectives: We present two algorithms for image processing; the first is based on Boltzmann sampling and the second on entropic sampling. Methods: These algorithms come within the Bayesian framework which has three components: 1. Likelihood: a conditional density - the probability of a noisy image given a clean image, 2. A Prior and, 3. A Posterior: a conditional density - the probability of a clean image given a noisy image. The Likelihood provides a model for the degradation process; the Prior models what we consider as a clean image; it also provides a means of incorporating whatever data we have of the image; the Posterior combines the Prior and Likelihood and provides an estimate of the clean counterpart of the given noisy image. The algorithm sets a competition between: 1. The Likelihood that tries to anchor the image to the given noisy image so that the features present can be retained including perhaps the noisy ones and, 2. The Prior which tries to make the image smooth, even at the risk of eliminating some genuine features of the image other than the noise. Findings: A proper choice of the prior and the likelihood functions would lead to good image processing. We need also good estimators of the clean image. Application: The choice of estimators is somewhat straight forward for image processing employing Boltzmann algorithm. For non-Boltzmann algorithm we need efficient estimators that make full use of the entropic ensemble generated. Keywords: Image processing, Prior, Posterior, Boltzmann sampling, Entropic sampling, Bayesian. 1. Introduction We discuss in this paper two algorithms for image processing: one based on Boltzmann sampling (The application of Boltzmann sampling to image analysis was pioneered by Gemen and Gemen [1] and has since become an active field of research [2-6]) and the other on non-Boltzmann sampling (Non-Boltzmann sampling was pioneered by Torrie and Valleau [7]; their method, called Umbrella sampling, has since undergone a series of metamorphoses. We have multi-canonical Monte Carlo algorithm of Berg and Neuhaus [8], entropic sampling of Lee [9], and the algorithm of Wang and Landau [10]. A preliminary and incomplete work [11] on the application of non-Boltzmann sampling to image restoration, indicated that it has no great advantage over Boltzmann sampling). These algorithms are inspired by some recent and not-so-recent developments in Monte Carlo simulation of macroscopic systems. We are presently testing these algorithms on a few benchmark problems employing Monte Carlo simulation. The results shall be presented in a future communication. In this paper, we confine our attention to describing these algorithms and presenting some details on how to implement them. We begin with a mathematical description of an image, which forms the contents of section (2). This is followed by a brief description of the three basic ingredients of the Bayesian methodology for image processing: the Likelihood distribution (Likelihood function models the process of degradation of a clean image to a noisy image X; it is denoted by L(X| ); it is a conditional probability density function: the probability of X given ) [3], in section (3), the Prior (The prior is a probability density function; the prior models what we expect or what we know of the clean counterpart of the given noisy image [4]) in section (4) and the Posterior (Posterior is proportional to the product of the Likelihood and the prior as prescribed by Bayes’ theorem; it is a conditional density function P (Θ|X)) in section (5). 1 www.iseeadyar.org