Research Article A Total Variation Model Based on the Strictly Convex Modification for Image Denoising Boying Wu, 1 Elisha Achieng Ogada, 1,2 Jiebao Sun, 1 and Zhichang Guo 1 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 Department of Mathematics, Egerton University, P.O. Box 536-20115, Egerton, Kenya Correspondence should be addressed to Jiebao Sun; sunjiebao@126.com Received 18 April 2014; Revised 16 May 2014; Accepted 26 May 2014; Published 18 June 2014 Academic Editor: Adrian Petrusel Copyright © 2014 Boying Wu et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose a strictly convex functional in which the regular term consists of the total variation term and an adaptive logarithm based convex modifcation term. We prove the existence and uniqueness of the minimizer for the proposed variational problem. Te existence, uniqueness, and long-time behavior of the solution of the associated evolution system is also established. Finally, we present experimental results to illustrate the efectiveness of the model in noise reduction, and a comparison is made in relation to the more classical methods of the traditional total variation (TV), the Perona-Malik (PM), and the more recent D--PM method. Additional distinction from the other methods is that the parameters, for manual manipulation, in the proposed algorithm are reduced to basically only one. 1. Introduction Noise removal, edge detection, contrast enhancement, in- painting, and segmentation have been the subject of intense mathematical image analysis and processing research for nearly three decades. Several methods have been pursued over the passage of time. Tese include wavelet transform [1, 2], curvelet shrinkage methods [3–6], and variational partial diferential equation (PDE) based methods [7, 8]. Tese methods generate processes that can easily be divided into either linear and nonlinear processes or isotropic and nonisotropic processes [9]. Due its ability to preserve crucial image features, such as edges, nonlinear anisotropic difusion is favored over isotropic difusion [10]. Much interest, therefore, has focused on understanding operations and mathematical properties of the nonlinear anisotropic difusion and associated variational formulations [11, 12], formulation of well-posed and stable equations [8, 11], extending and modifying anisotropic dif- fusion [11, 13, 14], and studying the relationships that exist between the various image processing techniques [11, 15, 16]. Te objective of any image denoising process should not focus only on the removal of noise, but it should also ensure that no spurious details are created on the restored image and that the edges are preserved or sharpened [7, 17, 18]. It is, therefore, necessary to develop formulations which are sen- sitive to the local image structure, especially edges/contours [19]. Consequently, a number of edge indicators have been proposed and logically grafed into the partial diferential equation (PDE) based evolution equations [7, 11, 20]. Some of these PDEs originate from variational problems. For instance, Rudin et al. [8] proposed a minimization functional, widely referred to as the total variation (TV) functional, of the form min  {()=∫ Ω |∇|+∫ Ω (−) 2 }, (1) where  is the fdelity parameter,  = () denotes the noise image, and Ω is an open bounded subset of R 2 . TV functionals are defned in the space of functions of bounded variation (BV) and, therefore, do not necessarily require image functions to be continuous and smooth. Tis fact makes them allow jumps or discontinuities, and hence they are able to preserve edges. Te original TV formulation has certain weaknesses. Firstly, the formulation is susceptible to backward difusion since it is not strictly convex. Secondly, the numerical imple- mentation cannot be accomplished without additional small Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 948392, 16 pages http://dx.doi.org/10.1155/2014/948392