This paper addresses problems appearing in restoration algorithms based on utilizing both Tikhonov and bilateral total variation (BTV) regularization. The former regularization assumes that prior information has Gaussian distribution which indeed fails at edges, while the later regularization highly depends on the selected bilateral filter's parameters. To overcome these problems, we propose a locally adaptive regularization. In the proposed algorithm, we use general directional regularization functions with adaptive weights. The adaptive weights are estimated from local patches based on the property of the partially restored image. Unlike Tikhonov regularization, it can avoid smoothness across edges by using adaptive weights. In addition, unlike BTV regularization, the proposed regularization function doesn't depend on parameters' selection. The convexity conditions as well as the convergence conditions are derived for the proposed algorithm.