Image Denoising Methods With Diffusion-wave Regulating Characteristic

In the information age, as an important information carrier, image has been deeply into almost all the ?elds, such as arti?cial intelligence, aerospace technology, biomedical technology, military technology, public security technology, and our daily life.However, images are often degraded because of imaging systems, environment or human factors, in the process of the image acquisition, compression, or transmission. This will affect not only the visual perception but also the subsequent higher level processing, such as identi?cation, classi?cation, decision and so on. Thus, image denoising is particularly important in the ?eld of image processing. Image denoising is an inverse problem in mathematics, i.e., to infer the unknown original image results only from the observed image(noisy image). It implies that the optimal solutions of models for image denoising problem are not unique or stable, which will bring great di?culties to design the algorithm of the image denoising. In addition, since the important details in the image will be lost inevitable when removing the image noise, how to preserve the important structures and textures as better as possible is another di?cult problem.The partial differential equation(PDE) based image denoising algorithms, which can better preserve the image structures when removing the noise, have became one of the most active research areas in the ?eld of image denoising. However, most of the partial differential equation models which are applied to image denoising are diffusion based parabolic equations. The diffusion effect of the parabolic equations is bene?-cial to remove the noise, but such ability is quite limited. A large number of research have demonstrated that hyperbolic equations with wave characteristic are superior to parabolic equations in the image structures detection and enhancement. Therefore, a class of diffusion-wave equations based models are proposed to deal with the trade-off between noise removal and structure preservation. All the proposed models have one trait in common: they can interpolate between the diffusion and the wave equation,and hence, they enjoy intermediate properties. On one hand, parabolic equation with diffusion characteristic can remove the noise effectively; on the other hand, hyperbolic equation with wave characteristic is able to preserve the edge structures and the textures in a highly oscillatory regions. Moreover, all the proposed models have their respective characteristics, and improve the performance of the image denoising from different perspectives, speci?cally as follows:(1) A spatial fractional PDE model with diffusion-wave regulating characteristic is proposed. It interpolates between the second and the fourth order PDE models by the use of spatial fractional derivatives. Thus, it can effectively avoid the “staircase effect” of the second order model and the “speckle effect” of the second order model.Moreover, in this paper, the existence and uniqueness of the solution of our model are proved, and the numerical computation scheme is also given.(2) A diffusion-wave equation model with variable nonlinearity is proposed. The diffusion modes of our model can be adaptively changed according to the image structures by means of the variable exponent functions, and hence, it can better preserve the image structures. In addition, a measure of the edge continuity is proposed. The noise and edges are distinguished by means of the continuity of the image gradient instead of the magnitude of image gradient, then it can give a better direction for the regulation of the diffusion mode. Finally, the existence and uniqueness of the solution of our model are proved in this paper.(3) A nonlocal diffusion tensor based diffusion-wave equation model is proposed and applied to denoising and enhancement of coherent ?ow-like structures. In order to cope with shortcoming of locality of the differential operator, we generalize the diffusion tensor to a nonlocal framework, and incorporate it into the parabolic-hyperbolic PDE model. Thus, it can also enhance the ?ow-like structures when denoising. Moreover, the existence and uniqueness of the solution of our model are proved in this paper.(4) A fractional diffusion-wave equation model with nonlocal regularization is proposed. The nonlocal term constructed by the fractional derivatives is incorporated into the diffusion-wave equation, and hence, the proposed model is based on the linear equation without introducing tricky nonlinear term. In this paper, the existence and uniqueness of the solution of our model are proved, and a stable numerical computation scheme is also given. The experimental results indicate that the proposed model is comparable to some nonlinear models.