Digital color images are compressed using common standards such as JPEG. The JPEG compression is a lossy process, which means that most of the compression is obtained by loss of data, and the original image cannot be restored completely from the compressed object. The compression technique is used in various applications such as imaging and equalization of channels. But the decompression technique is not having enough sources to handle various problems. The process of sampling and sub sampling is done by both compression and decompression. This paper proposed JPEG decompression based on the MAP (Maximum A Posteriori probability) formulation with spare priors by ADMM (Alternating Direction Method of Multipliers). The main contribution is the observation that using a Gaussian approximation of the quantization noise and a tight frame in the sparse prior based on l1-norm allows for fast computation of the inverse critical for the ADMM method. In addition, it shows that a minor modification of the proposed algorithm solves simultaneously the problem of image denoising. The experimental section analyzes the behavior of the proposed decompression algorithm in a small number of iterations with an interesting conclusion that this mode outperforms full convergence.
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