Variable-size Symmetry-based Graph Fourier Transforms for image compression

TL;DR

Experiments show that SBGFTs outperform the primary transforms integrated in the explicit Multiple Transform Selection used in the latest VVC intra-coding, providing a bit rate saving percentage of $\mathbf {6.23\%}$ , with only a marginal increase in average complexity.

Abstract

Modern compression systems use linear transformations in their encoding and decoding processes, with transforms providing compact signal representations. While multiple data-dependent transforms for image/video coding can adapt to diverse statistical characteristics, assembling large datasets to learn each transform is challenging. Also, the resulting transforms typically lack fast implementation, leading to significant computational costs. Thus, despite many papers proposing new transform families, the most recent compression standards predominantly use traditional separable sinusoidal transforms. This paper proposes integrating a new family of Symmetry-based Graph Fourier Transforms (SBGFTs) of variable sizes into a coding framework, focusing on the extension from our previously introduced 8x8 SBGFTs to the general case of NxN grids. SBGFTs are non-separable transforms that achieve sparse signal representation while maintaining low computational complexity thanks to their symmetry properties. Their design is based on our proposed algorithm, which generates symmetric graphs on the grid by adding specific symmetrical connections between nodes and does not require any data-dependent adaptation. Furthermore, for video intra-frame coding, we exploit the correlations between optimal graphs and prediction modes to reduce the cardinality of the transform sets, thus proposing a low-complexity framework. Experiments show that SBGFTs outperform the primary transforms integrated in the explicit Multiple Transform Selection (MTS) used in the latest VVC intra-coding, providing a bit rate saving percentage of 6.23%, with only a marginal increase in average complexity. A MATLAB implementation of the proposed algorithm is available online at [1].

Figures (11)

• Figure 1: Grid axes and their corresponding edge symmetries for a 2DG. The vertical, diagonal, horizontal, and anti-diagonal axes are highlighted in blue, orange, red, and green, respectively. Example pairs of additional edges that would satisfy the ES property shown in parenthesis for the associated axis are drawn in matching colors.
• Figure 2: Example of an NS with respect to the dashed blue reflection axis. The red nodes for $\mathcal{V}_s$ and the black connections for $\mathcal{E}_s$ indicate the associated support. More edges may be present without influencing the NS.
• Figure 3: Position of all the reflection axes for a $N{\times}N$ grid. The reflection axes in each direction lead to SBGs with ES with respect to the main grid axis in the perpendicular direction, as shown by the maching color in Fig. \ref{['fig:symmetries_on_grid']}.
• Figure 4: Example of SBGs constructed on a $16{\times}16$ 2DGG, putting $k=2$ in \ref{['eq:axesLR']}--\ref{['eq:axesAD']}. Each graph has ES with respect to one of the grid axes highlighted by the continuous blue lines. Black edges are the baseline grid edges, while red edges are added edges that induce the NS property with respect to the reflection axis drawn as a dashed blue line.
• Figure 5: Symmetry ratios of all eigenvectors for $8{\times} 8$ SBGs built with reflection axes different than main grid axes. Only the first half of the axes are depicted since the other half enjoys the same symmetry ratio. The SBGs are grouped by the reflection axes direction: (a) horizontal, (b) vertical, (c) diagonal, and (d) anti-diagonal.

Theorems & Definitions (12)

• Definition 2.1: 2D grid graph
• Definition 2.2: Grid axes
• Definition 2.3: Edge symmetric graphs
• Remark 2.1
• Lemma 2.1
• Definition 2.4: Reflection axis
• Definition 2.5: Support
• Definition 3.1: Symmetry-based graphs
• Remark 3.1
• Definition 3.2: Mirroring operation with respect to $s$