Preservation of Fine Structures in PDE-Based Image Denoising

Image denoising processes often lead to significant loss of fine structures such as edges and textures. This paper studies various innovative mathematical and numerical methods applicable for conventional PDE-based denoising models. The method of diffusion modulation is considered to effectively minimize regions of undesired excessive dissipation. Then we introduce a novel numerical technique for residual-driven constraint parameterization, in order for the resulting algorithm to produce clear images whose corresponding residual is as free of image textures as possible. A linearized Crank-Nicolson alternating direction implicit time-stepping procedure is adopted to simulate the resulting model efficiently. Various examples are presented to show efficiency and reliability of the suggested methods in image denoising. 1. Introduction During the last two decades, as the field of image processing requires higher reliability and efficiency, mathematical techniques have become important components of image processing. Various mathematical frameworks employing powerful tools of partial differential equations (PDEs) and functional analysis have emerged and successfully applied to various image processing tasks, particularly for image denoising and restoration [1–9], see also [10, 11]. Those PDE-based denoising techniques have allowed researchers and practitioners not only to introduce effective new models but also to improve traditional algorithms. However, most PDE-based models tend to either converge to a piecewise constant image or introduce image blur (undesired dissipation), partially because the models are derived by minimizing a functional of the image gradient. As a consequence, the conventional PDE-based models may lose interesting image fine structures during the denoising. In order to reduce the artifact, researchers have studied various mathematical and numerical techniques which either incorporate more effective constraint terms and iterative refinement [5, 12–14] or minimize a functional of second derivatives of the image [15–18]. These new mathematical models may preserve fine structures better than conventional ones; however, more advanced models and appropriate numerical procedures are yet to be developed. Most image denoising models incorporate parameters which are closely related to the noise level. Since it is often the case that the noise level is unknown, the problem of choosing parameters occasionally becomes a difficult task and, as a result, the resulting algorithm may produce unsatisfactory images. This paper suggests the method of