<p>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. </p>
<p>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. </p>
A novel algorithm for designing optimized orthonormal transform-matrix codebooks for adaptive transform coding of a non-stationary vector process is proposed. This algorithm relies on a block-wise stationary model of a non-stationary process and finds a codebook of transform-matrices by minimizing the end-to-end mean square error of transform coding averaged over the distribution of stationary blocks of vectors. The algorithm, which belongs to the class of block-coordinate descent algorithms, solves an intermediate minimization problem involving matrix-orthonormality constraints in a computationally efficient manner by mapping the problem from the Euclidean space to the Stiefel manifold. As such, the algorithm can be broadly applied to any adaptive transform coding problem. Preliminary results obtained with inter-prediction residuals in an H265 video codec are presented to demonstrate the advantage of optimized adaptive transform codes over non-adaptive codes based on the standard DCT.
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