Traveling wave solutions of fourth order PDEs for image processing

The authors introduce two nonlinear advection-diffusion equations, each of which combines Burgers's convection with a fourth order nonlinear diffusion previously designed for image denoising. One equation uses the L-2-curvature diminishing diffusion of You and Kaveh [IEE Trans. Image Process., 9 (2000), pp. 1723-1730], and the other uses the "low curvature image simplifiers" diffusion of Tumblin and Turk [Proceedings of the 26th Annual Conference on Computer Graphics, ACM Press/Addison-Wesley, New York, 1999, pp. 83-90]. The new PDEs are compared with a third advection-diffusion equation that combines Burgers's convection with a second order diffusion recommended by Perona and Malik for denoising and edge detection [IEEE Trans. Pattern Anal. Machine Intell., 12 (1990), pp. 629-639]. We prove results regarding the existence and nonexistence of traveling wave solutions of each PDE. Visualizations of each ODE's phase space show qualitative differences between the two fourth order problems. The combined work gives insight into the existence of finite time singularities in solutions of the diffusion equations.