Often an image g(x,y) is regularized and even restored by minimizing the Mumford-Shah functional. Properties of the regularized image u(x,y) depends critically on the numerical value of the two parameters alpha and gamma controlling smoothness and fidelity. When alpha and gamma are constant over the image, small details are lost when an extensive filtering is used in order to remove noise. In this paper, it is shown how the two parameters alpha and gamma can be made self-adaptive. In fact, alpha and gamma are not constant but automatically adapt to the local scale and contrast of features in the image. In this way, edges at all scales are detected and boundaries are well-localized and preserved. In order to preserve trihedral junctions alpha and gamma become locally small and the regularized image u(x,y) maintains sharp and well-defined trihedral junctions. Images regularized by the proposed procedure are well-suited for further processing, such as image segmentation and object recognition.