Fourth Order Partial Differential Equations for Image Processing. (Paperback)

It has long been known that nonlinear diffusion partial differential equations (PDEs) can be useful in the problem of noise removal in image processing. Most PDEs considered for this purpose are second order equations, but higher order equations have been considered to overcome some problems attendant to second order methods. Some background and history of the development of such fourth order equations is given to motivate their study. One equation in particular, the well-known You-Kaveh model, is selected to study further. The impact of various parameters upon the denoising performance of this model is studied, along with several other practical concerns regarding of the model, and a simple smoothing technique involving parameter manipulation is proposed. This technique is shown to improve noise removal and decrease processing time. The analytical behavior of fourth order diffusion PDEs is also studied. While the You-Kaveh equation is found to possess a structure known to exhibit unstable behavior, insights are gained into what could contribute to a well-posed diffusion equation. Results from the field of maximal regularity are culled to find sufficient conditions for the existence of unique solutions to fourth order nonlinear diffusion PDEs. Motivated by this study of the numerical and the analytical aspects of fourth order PDEs for denoising, two new denoising models, variations on the You-Kaveh equation, are proposed. Both of these new equations utilize fractional derivatives, and are inspired by similar work on second order diffusions. Extensive numerical experiments are performed on natural images to demonstrate the performance of these models, and to compare them to You-Kaveh as well as to second order methods. Additionally, a detailed proof is given, showing that both of these equations satisfy conditions sufficient to yield the existence of unique solutions locally in time.