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In this paper, we introduce a new decoding algorithm for DCT-based video encoders, such as Motion JPEG (M-JPEG), H26x, or MPEG. This algorithm considers not only the compression artifacts but also the ones due to transmission, acquisition or storage of the video. The novelty of our approach is to jointly tackle these two problems, using a variational approach. The resulting decoder is object-based, allowing independent and adaptive processing of objects and backgrounds, and considers available information provided by the bitstream, such as quantization steps, and motion vectors. Several experiments demonstrate the efficiency of the proposed method. Objective and subjective quality assessment methods are used to evaluate the improvement upon standard algorithms, such as the deblocking and deringing filters included in MPEG-4 postprocessing.