An Algorithm for Learning Orthonormal Matrix Codebooks for Adaptive Transform Coding

This paper proposes a novel data-driven approach to designing orthonormal transform matrix codebooks for adaptive transform coding of any non-stationary vector processes which can be considered locally stationary. Our algorithm, which belongs to the class of block-coordinate descent algorithms, relies on simple probability models such as Gaussian or Laplacian for transform coefficients to directly minimize with respect to the orthonormal transform matrix the mean square error (MSE) of scalar quantization and entropy coding of transform coefficients. A difficulty commonly encountered in such minimization problems is imposing the orthonormality constraint on the matrix solution. We get around this difficulty by mapping the constrained problem in Euclidean space to an unconstrained problem on the Stiefel manifold and leveraging known algorithms for unconstrained optimization on manifolds. While the basic design algorithm directly applies to non-separable transforms, an extension to separable transforms is also proposed. We present experimental results showing that adaptive coding with our transform codebook designs outperform the commonly used discrete cosine transform (DCT) in many regions of natural images and motion-compensated prediction error images of video frames. According to these results, the proposed codebook designs slightly outperform those found by a common alternative approach, the sparsity-based optimization.