Image Denoising Using Generalized Anisotropic Diffusion

Motivated by some recent works in fractional-order anisotropic diffusions, we introduce a class of generalized anisotropic diffusion equations for image denoising. We first define a new type of derivative (called G-derivative), which contains the fractional derivative as a special case, using the Fourier transform, then the generalized anisotropic diffusion equations are Euler–Lagrange equations of a cost functional which is an increasing function of the absolute value of the G-derivative of the image intensity function. All the G-derivative operators constitute a ring, and the semigroup property of the G-derivative consists with the semigroup property of the fractional derivative, so the resulting generalized anisotropic diffusions can be seen as generalizations of the fractional-order anisotropic diffusions. We also discuss the generalized Sobolev space described by the G-derivative and some variants of generalized anisotropic diffusions. The discretization of the G-derivative is computed in the frequency domain, and the stability analysis of the difference scheme is given. We list some generalized anisotropic diffusions and apply them to image denoising. Numerical results show that new models have great potentials in image denoising.