What if we take all the overlapping patches from a given image and organize them to create the shortest path by using their mutual Euclidean distances? This suggests a reordering of the image pixels in a way that creates a maximal 1D regularity. What could we do with such a construction? In this talk we consider a wider perspective of the above, and introduce a wavelet transform for graph-structured data. The proposed transform is based on a 1D wavelet decomposition coupled with a pre-reordering of the input so as to best sparsify the given data. We adopt this transform to image processing tasks by considering the image as a graph, where every patch is a node, and edges are obtained by Euclidean distances between corresponding patches. We show several ways to use the above ideas in practice, leading to state-of-the-art image denoising, deblurring, inpainting, and face-image compression results.