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In this paper, we propose a total variation (TV) based filter adapted to the statistics of quantum noise which follows Poisson distribution, for the enhancement and restoration of the digital mammographic images. The proposed method is developed in a variational framework which reduces to a minimization problem. The proposed model consists of two terms viz. data fidelity term and regularization function and to make a proper balance between these two terms during the filtering process a regularization parameter has been introduced. For digital implementations, the proposed model has been discretized using finite difference schemes. A comparative study of the proposed scheme has also been performed with the other existing techniques in terms of mean square error (MSE), peak signal-to-noise ratio (PSNR), correlation parameter (CP) and mean structure similarity index map (MSSIM).