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Remotely sensed images often suffer from the common problems of stripe noise and random dead pixels. The techniques to recover a good image from the contaminated one are called image destriping (for stripes) and image inpainting (for dead pixels). This paper presents a maximum a posteriori (MAP)-based algorithm for both destriping and inpainting problems. The main advantage of this algorithm is that it can constrain the solution space according to a priori knowledge during the destriping and inpainting processes. In the MAP framework, the likelihood probability density function (PDF) is constructed based on a linear image observation model, and a robust Huber-Markov model is used as the prior PDF. The gradient descent optimization method is employed to produce the desired image. The proposed algorithm has been tested using moderate resolution imaging spectrometer images for destriping and China-Brazil Earth Resource Satellite and QuickBird images for simulated inpainting. The experiment results and quantitative analyses verify the efficacy of this algorithm.