Graph-based rotation of the DCT basis for motion-adaptive transforms

In this paper, we consider motion-adaptive transforms that are based on vertex-weighted graphs. The graphs are constructed by motion vector information and the weights of the vertices are given by scale factors, where the scale factors are used to control the energy compaction of the transform. The vertex-weighted graph defines a one dimensional linear subspace. Thus, our transform basis is subspace constrained. To find a full transform matrix that satisfies our subspace constraint, we rotate the discrete cosine transform (DCT) basis such that the first basis vector matches the subspace constraint. Since rotation is not unique in high dimensions, we choose a simple rotation that only rotates the DCT basis in the plane which is spanned by the first basis vector of the DCT and the subspace constraint. Experimental results on energy compaction show that the motion-adaptive transform based on this rotation is better than the motion-compensated orthogonal transform based on hierarchical decomposition while sharing the same first basis vector.