Lifting scheme on graphs with application to image representation

We propose a new multiscale transform for scalar functions defined on the vertex set of a general undirected weighted graph. The transform is based on an adaption of the lifting scheme to graphs. One of the difficulties in applying directly the lifting scheme to graphs is the partitioning of the vertex set. We follow a recent greedy approach and extend it to a multilevel transform. We carefully examine each step of the algorithm, in particular its effect on the underlying basis. We finally investigate the use of the proposed transform to image representation by computing M-term nonlinear approximation errors. We provide a comparison with standard orthogonal and biorthogonal wavelet transforms.