Two main phenomena degrade the quality of a mammogram: the blur induced by X-rays scattering and a loss of contrast due to beam hardening. In this paper, we propose an original approach to restore the mammographies from these degradations. They are due to the physical phenomena occurring in the radiographic image formation process. Our objective has been to construct a physical model describing accurately the phenomena while being tractable to enable an inversion and therefore to construct a corrected mammography. In fact, we show that, in reason of the specific protocol for realizing a mammographic exam, and the composition of the breast in mainly two components (fat and glandular tissue), the observed values can be related to the thickness of glandular tissue crossed by the X-rays. This relation is nonlinear, and expresses the different phenomena taking place in the acquisition process. An adapted inversion scheme enables to build up a map of the thicknesses of glandular tissue. This representation enhances significantly the mammograms. As the approach only relies, on one hand on priors deduced from the acquisition geometry and from the breast composition, and on the other hand on a physical description of the acquisition process, it does not create artifacts that may alter the physician's diagnosis.