While modern displays offer high dynamic range (HDR) with large bit-depth for each rendered pixel, the bulk of legacy image and video contents were captured using cameras with shallower bit-depth. In this paper, we study the bit-depth enhancement problem for images, so that a high bit-depth (HBD) image can be reconstructed from an input low bit-depth (LBD) image. The key idea is to apply appropriate smoothing given the constraints that reconstructed signal must lie within the per-pixel quantization bins. Specifically, we first define smoothness via a signal-dependent graph Laplacian, so that natural image gradients can nonetheless be interpreted as low frequencies. Given defined smoothness prior and observed LBD image, we then demonstrate that computing the most probable signal via maximum a posteriori (MAP) estimation can lead to large expected distortion. However, we argue that MAP can still be used to efficiently estimate the AC component of the desired HBD signal, which along with a distortion-minimizing DC component, can result in a good approximate solution that minimizes the expected distortion. Experimental results show that our proposed method outperforms existing bit-depth enhancement methods in terms of reconstruction error.