Temporal Signal Basis for Hierarchical Block Motion in Image Sequences

In classic data compression, the optimal transform for energy compaction is the Karhunen-Lòeve transform with the eigenvectors of the covariance matrix. In coding applications, neither the covariance matrix nor the eigenvectors can be easily transmitted to the decoder. In this letter, we introduce a covariance matrix model based on graphs determined by hierarchical block motion in image sequences and use its eigenvector matrix for compression. The covariance matrix model is defined using the graph distance matrix, where the graph is determined by block motion. As the proposed covariance matrix is closely related to the graph, the relation between the covariance matrix and the Laplacian matrix is studied and their eigenvector matrices are discussed. From our assumptions, we show that our covariance model can be viewed as a Gaussian graphical model where the signal is described by the second order statistics and the zeros in the precision matrix indicate missing edges in the graph. To assess the compression performance, we relate the coding gain due to the eigenbasis of the covariance model to that of the Laplacian eigenbasis. The experimental results show that the eigenbasis of our covariance model is advantageous for tree-structured block motion in image sequences.