A Nonlinear Entropic Variational Model for Image Filtering
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A Nonlinear Entropic Variational Model for Image Filtering A. Ben Hamza Concordia Institute for Information Systems Engineering, Concordia University, Montr´eal, Quebec H3G 1T7, Canada Email: [email protected]
Hamid Krim Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695-7911, USA Email: [email protected]
Josiane Zerubia Ariana Research Group, INRIA/I3S, BP 93, 06902 Sophia Antipolis Cedex, France Email: [email protected] Received 12 August 2003; Revised 8 June 2004 We propose an information-theoretic variational filter for image denoising. It is a result of minimizing a functional subject to some noise constraints, and takes a hybrid form of a negentropy variational integral for small gradient magnitudes and a total variational integral for large gradient magnitudes. The core idea behind this approach is to use geometric insight in helping to construct regularizing functionals and avoiding a subjective choice of a prior in maximum a posteriori estimation. Illustrative experimental results demonstrate a much improved performance of the approach in the presence of Gaussian and heavy-tailed noise. Keywords and phrases: MAP estimation, variational methods, robust statistics, differential entropy, gradient descent flows, image denoising.
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INTRODUCTION
In recent years, variational methods and partial differential equations-(PDE) based methods [1, 2, 3, 4, 5, 6] have been introduced to explicitly account for intrinsic geometry to address a variety of problems including image segmentation, mathematical morphology, motion estimation, image classification, and image denoising [7, 8, 9, 10, 11, 12]. The latter will be the focus of the present paper. The problem of signal/image denoising has been addressed using a number of different techniques including wavelets [13], order statisticsbased filters [14], PDE-based algorithms [9, 15], and variational approaches [16, 17, 18]. In particular, a large number of PDE-based methods have particularly been proposed to tackle the problem of image denoising [12, 19, 20] with a good preservation of edges. Much of the appeal of PDE-based methods lies in the availability of a vast arsenal of mathematical tools which at the very least act as a key guide in achieving numerical accuracy as well as stability. PDEs or gradient descent flows are generally a result of variational problems using the Euler-Lagrange principle [21]. One popular
variational technique used in image denoising is the total variation-based approach. It was developed in [4] to overcome the basic limitations of all smooth regularization algorithms, and a variety of numerical methods have also recently been developed for solving total variation minimization problems [22, 23]. In this paper, we present a variational approach to maximum a posteriori (MAP) estimation. The core idea behind this approach is to use geometric insight in helping construct regularizing functionals and avoiding a subjective choice of a prior in MAP estimation. Using tools from robust statis
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