Quantization of KLT Matrices via GMRF Modeling of Image Blocks for Adaptive Transform Coding

TL;DR

This work tackles quantizing local KLTs for forward adaptive transform coding by modeling image-block KLTs with a finite-lattice non-causal homogeneous Gauss-Markov random field (GMRF) parameterized by a small vector $\boldsymbol{\theta}$. By choosing asymmetric Neumann boundary conditions, the authors show that the KLT of the block corresponds to the eigenvectors of the precision matrix $\mathbf{Q}(\boldsymbol{\theta})$, enabling a low-dimensional, scalable transform design via $4$ GMRF parameters. They introduce a coding-optimized parameter estimation approach that maximizes transform coding gain under high-rate assumptions and develop a codebook design workflow that yields transform matrices from GMRF eigenvectors. Experimental results demonstrate that GMRFTs can outperform the 2D-DCT in natural images, especially in variable block-size coding, and offer competitive performance against recent model-free methods while maintaining scalability across block sizes. This approach provides a practical path toward texture-aware, shape-adaptive, and scalable transform coding for image and video compression.

Abstract

Forward adaptive transform coding of images requires a codebook of transform matrices from which the best transform can be chosen for each macroblock. Codebook construction is a problem of designing a quantizer for Karhunen-Lóeve transform (KLT) matrices estimated from sample image blocks. We present a novel method for KLT matrix quantization based on a finite-lattice non-causal homogeneous Gauss-Markov random field (GMRF) model with asymmetric Neumann boundary conditions for blocks in natural images. The matrix quantization problem is solved in the GMRF parameter space, simplifying the harder problem of quantizing a large matrix subject to an orthonormality constraint to a low-dimensional vector quantization problem. Typically used GMRF parameter estimation methods such as maximum-likelihood (ML) do not necessarily maximize the coding performance of the resulting transform matrices. To this end we propose a method for GMRF parameter estimation from sample image data, which maximizes the high-rate transform coding gain. We also investigate the application of GMRF-based transforms to variable block-size adaptive transform coding.

Figures (10)

• Figure 1: (a) Non-causal GMRF neighborhood structures of orders 1 to 6 for the pixel $s$. All pixels labeled by values $t \leq M$ belong to the M-th order neighborhood of $s$. (b) 1st-order and (c) 2nd order GMRF models with homogeneous spatial interactions.
• Figure 2: Effect of boundary conditions on $8 \times 8$ blocks (taken from Lena image): assymetric Neumann (left), periodic (center), and Dirichlet.
• Figure 3: (a) A finite lattice homogeneous field (gray square) satisfying conditions C1 and C2. (b) modified field with diagonally symmetric spatial interactions along the boundary such that precision matrix is symmetric.
• Figure 4: Precision matrix for the 2nd order model shown in Fig. \ref{['order2_model']}(b), where $q_1 = 1-(\theta_v+\theta_h+\theta_{b})$, $q_2 = 1-\theta_h$, $q_3 = 1-\theta_v$, $q_4 = -(\theta_v+\theta_{b})$, and $q_5 = -(\theta_h+\theta_{b})$.
• Figure 5: Basis images of $64 \times 64$ transform matrices: (a) is 2D-DCT, and (b), (c) and (d) are 3 examples of GMRFTs taken from a codebook containing 7 matrices.