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Image Recovery via Diffusion Tensor and Time-Delay Regularization

We present a system of PDEs for image restoration, which consists of an anisotropic diffusion equation driven by a diffusion tensor, whose structure depends on the gradient of the image obtained from a coupled time-delay regularization equation, and governs the direction and the speed of the diffusion. The diffusion resulting from this model is isotropic inside a homogeneous region, anisotropic along its boundary, and is able to connect broken edges and enhance coherent structures. Experimental results are given to show its effectiveness in tracking edges and recovering images with high levels of noise. Moreover, the proposed model can be interpreted as a time continuous Hopfield neural network. This connection further illustrates how the proposed model enhances coherent structures. The existence, uniqueness, and stability for the solutions of the PDEs are proved.