논문 12-37A-11-07 한국통신학회논문지 '12-11 Vol.37A No.11 http://dx.doi.org/10.7840/kics.2012.37A.11.961 961 THE METHOD OF NONFLAT TIME EVOLUTION (MONTE) IN PDE-BASED IMAGE RESTORATION Youngjoon Cha  , Seongjai Kim * ABSTRACT This article is concerned with effective numerical techniques for partial differential equation (PDE)-based image restoration. Numerical realizations of most PDE-based denoising models show a common drawback: loss of fine structures. In order to overcome the drawback, the article introduces a new time-stepping procedure, called the method of nonflat time evolution (MONTE), in which the timestep size is determined based on local image characteristics such as the curvature or the diffusion magnitude. The MONTE provides PDE-based restoration models with an effective mechanism for the equalization of the net diffusion over a wide range of image frequency components. It can be easily applied to diverse evolutionary PDE-based restoration models and their spatial and temporal discretizations. It has been numerically verified that the MONTE results in a significant reduction in numerical dissipation and preserves fine structures such as edges and textures satisfactorily, while it removes the noise with an improved efficiency. Various numerical results are shown to confirm the claim. Key Words : Method of nonflat time evolution (MONTE), net diffusion (ND) function, diffusion equalization, fine structures, total variation (TV) model, numerical dissipation ※ The work of the first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 20110179).  주저자：Department of Applied Mathematics, Sejong University, Gunja-Dong 98, Seoul 143-747, Korea, yjcha@sejong.ac.kr, 정회원 * Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762-5921, USA, skim@math.msstate.edu 논문번호：KICS2012-09-459, 접수일자：2012년 9월 26일, 최종논문접수일자：2012년 11월 8일 Ⅰ . INTRODUCTION Image restoration is an important image processing (IP) step for various image-related applications and is often necessary as a pre-processing for other imaging techniques such as segmentation, registration, and compression and visualization. Thus image restoration methods have been considered as an important process in IP, computer graphics, and their applications [1,2,3,4] . There have been lots of partial differential equation (PDE)-based models in image restoration such as the Perona-Malik model [5] , the total variation (TV) minimization [6,7] , and color restoration models [8, 9,10] . These PDE-based models have been extensively studied to answer fundamental questions in image restoration and have allowed researchers and practitioners not only to introduce new mathematical models but also to analyze and improve traditional algorithms [11, 12, 13] . Good references to work on them are e.g. Aubert- Kornprobst [14] , Osher-Fedkiw [15] , and Sapiro [4] . However, most of these PDE-based restoration models and their numerical realizations show a common drawback: loss of fine structures. In particular, they often introduce an excessive and undesirable numerical dissipation on regions where the image content changes rapidly such as on edges and textures. Therefore it is very important and challenging to develop mathematical models and numerical techniques which can effectively preserve fine structures during the restoration. Although there have been developed new mathematical models for a better preservation of fine structures [2, 16] , more