Motion-Adaptive Transforms Based on the Laplacian of Vertex-Weighted Graphs

Abstract

We construct motion-adaptive transforms for image sequences by using the eigenvectors of Laplacian matrices defined on vertex-weighted graphs, where the weights of the vertices are defined by scale factors. The vertex weights determine only the first basis vector of the linear transform uniquely. Therefore, we use these weights to define two Laplacians of vertex-weighted graphs. The eigenvectors of each Laplacian share the first basis vector as defined by the scale factors only. As the first basis vector is common for all considered Laplacians, we refer to it as subspace constraint. The first Laplacian uses the inverse scale factors, whereas the second utilizes the scale factors directly. The scale factors result from the assumption of ideal motion. Hence, the ideal unscaled pixels are equally connected and we are free to form arbitrary graphs, such as complete graphs, ring graphs, or motion-inherited graphs. Experimental results on energy compaction show that the Laplacian which is based on the inverse scale factors outperforms the one which is based on the direct scale factors. Moreover, Laplacians of motion-inherited graphs are superior than that of complete or ring graphs, when assessing the energy compaction of the resulting motion-adaptive transforms.