Piecewise smooth (PWS) images (e.g., depth maps or animation images) contain unique signal characteristics such as sharp object boundaries and slowly varying interior surfaces. Leveraging on recent advances in graph signal processing, in this paper, we propose to compress the PWS images using suitable graph Fourier transforms (GFTs) to minimize the total signal representation cost of each pixel block, considering both the sparsity of the signal's transform coefficients and the compactness of transform description. Unlike fixed transforms, such as the discrete cosine transform, we can adapt GFT to a particular class of pixel blocks. In particular, we select one among a defined search space of GFTs to minimize total representation cost via our proposed algorithms, leveraging on graph optimization techniques, such as spectral clustering and minimum graph cuts. Furthermore, for practical implementation of GFT, we introduce two techniques to reduce computation complexity. First, at the encoder, we low-pass filter and downsample a high-resolution (HR) pixel block to obtain a low-resolution (LR) one, so that a LR-GFT can be employed. At the decoder, upsampling and interpolation are performed adaptively along HR boundaries coded using arithmetic edge coding, so that sharp object boundaries can be well preserved. Second, instead of computing GFT from a graph in real-time via eigen-decomposition, the most popular LR-GFTs are pre-computed and stored in a table for lookup during encoding and decoding. Using depth maps and computer-graphics images as examples of the PWS images, experimental results show that our proposed multiresolution-GFT scheme outperforms H.264 intra by 6.8 dB on average in peak signal-to-noise ratio at the same bit rate.