P(x) Harmonic Surface and Its Projection for Noise Removal

Abstract

Over the last two decades, many image processing problems have been modeled by partial differential equations (PDEs), such as restoration and segmentation. Due to good performance at controlling the trade-off between noise removal and edge-preserving, second-order PDEs play leading role in image restoration. The dilemma for second-order PDEs is the so- called staircasing effect. One remedy for this issue is the fourth- order PDEs, but the fourth-order PDEs suffer from the problem of oversmoothing the edges. In this paper, a novel PDE is proposed based on minimal surface and p(x) harmonic maps. The proposed model behaves like TV model at edge regions while like heat transfer equation within homogeneous regions. The proposed model is further projected to the normal direction of a smoothed version of the original image for the purpose of edge preserving. Several experiments have been conducted and promising results are observed. promising results have been observed, the second-order PDEs suffer from the so-called staircasing effect as well. During the last few years, alleviating this staircasing effect has been the interest of many works. One solution is to introduce high-order PDEs (specifically, fourth-order PDEs), for example, the model proposed by You and Kaveh (5) and the LLT model (6). Fourth-order PDEs can effectively prevent the staircases; meanwhile, they damp high frequency components of images much faster than second-order PDEs, this would result in over- smoothed step edges. Another approach to impede the staircasing effect is to make use of the p(x) harmonic maps (7), in which p(x) ranges from 1 to 2, such that the diffusion combines the advantages of TV model and Gaussian process. The demerit of this p(x) harmonic model is the difficulties for theoretical analysis.