Variable exponent functionals in image restoration

We study a functional with variable exponent, 1 < p (x) ≤ 2, which provides a model for image denoising and restoration. Here p (x) is defined by the gradient information in the observed image. The diffusion derived from the proposed model is between total variation based regularization and Gaussian smoothing. The diffusion speed of the corresponding heat equation is tuned by the variable exponent p (x). The minimization problem and its associated flow in a weakened formulation are discussed. The existence, uniqueness, stability and long-time behavior of the proposed model are established in the variable exponent functional space W^{1, p (x)}. Experimental results illustrate the effectiveness of the model in image restoration.