Traditionally, bit-depth expansion is an image processing technique to display a low bit-depth image on a high bit-depth monitor. In this paper, we study a variational method for expanding the bit-depth of low contrast images. Our idea is to develop a variational approach containing an energy functional to determine a local mapping function f(r,x) for bit-depth expansion via a smoothing technique, such that each pixel can be adjusted locally to a high bit-depth value. In order to enhance low contrast images, we make use of the histogram equalization technique for such local mapping function. Both bit-depth expansion and equalization terms can be combined together into the resulting objective function. In order to minimize the differences among the local mapping function at the nearby pixel locations, the spatial regularization of the mapping is incorporated in the objective function. Experimental results are reported to show that the performance of the proposed method is competitive with the other compared methods for several testing low contrast images.