Symmetry-Based Graph Fourier Transforms for Image Representation

It is well-known that the application of the Discrete Cosine Trans- form (DCT) in transform coding schemes is justified by the fact that it belongs to a family of transforms asymptotically equivalent to the Karhunen-Loe`ve Transform (KLT) of a first order Markov process. However, when the pixel-to-pixel correlation is low the DCT does not provide a compression performance comparable with the KLT. In this paper, we propose a set of symmetry-based Graph Fourier Transforms (GFT) whose associated graphs present a totally or par- tially symmetric grid. We show that this family of transforms well represents both natural images and residual signals outperforming the DCT in terms of energy compaction. We also investigate how to reduce the cardinality of the set of transforms through an analysis that studies the relation between efficient symmetry-based GFTs and the directional modes used in H.265 standard. Experimental results indicate that coding efficiency is high.