Graph-Based Separable Transforms (GBSTs)

Updated 29 November 2025

GBSTs are a class of linear, data-adaptive transforms defined on product graphs that generalize the DCT/DST to non-Euclidean domains.
They achieve efficient computation by independently applying one-dimensional graph-based transforms along each product graph dimension to enhance energy compaction and decorrelation.
GBSTs are practically applied in video coding, image compression, graph neural networks, and spatio-temporal analysis, offering improved coding gains and parameter efficiency.

Graph-Based Separable Transforms (GBSTs) are a class of linear transforms defined for signals or data arrays indexed by product graphs, in which the transformation is constructed by independently applying one-dimensional graph-based transforms (GBTs) along each factor graph dimension. This framework generalizes classical separable transforms such as the 2D Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) to settings dictated by data-adaptive, non-Euclidean, or product-structured domains. GBSTs optimize for energy compaction, decorrelation, or spectral sparsification, with applications spanning video coding, image compression, graph neural networks, spatio-temporal analysis, and high-dimensional signal processing.

1. Mathematical Construction of GBSTs

Given a signal block $X \in \mathbb{R}^{N \times N}$ , GBSTs are derived from two product graphs—typically modeled as weighted line or path graphs for video/image blocks—with their respective graph Laplacians $L_\text{row}$ and $L_\text{col}$ of the form: $L(w,v) = D - A + V$ where $A$ is the adjacency matrix (with edge weights $w$ ), $D$ is the degree matrix, and $V$ is a possible self-loop of weight $v$ at a terminal vertex. The eigenvectors of each Laplacian yield orthonormal bases $U_\text{row}, U_\text{col}$ .

The separable 2D GBST is then given by

$\widehat{X} = U_\text{col}^\top X U_\text{row}$

or equivalently, for vectorized $x = \text{vec}(X)$ ,

$\widehat{x} = (U_\text{col} \otimes U_\text{row}) x$

This separable structure allows for efficient computation, as transforms can be implemented by two cascade one-dimensional multiplications, mirroring the algorithmic legacy of DCT/DST (Egilmez et al., 2019).

2. Linkages to Classical Transforms and Interpretability

Many standard transforms are special cases of GBSTs with appropriate graph parameters:

DCT-2: Path graph with no self-loop, $v=0$ .
DST-7: Self-loop present at one end, $v=w$ .
DCT-8: Self-loop at opposite end, $v=w$ .

The eigenvectors of $L(w,v)$ , via a second-difference recurrence, reproduce the analytical forms of DCT/DST for $v/w \in \{0,1\}$ : $u^{(k)}_n = \sqrt{\frac{2}{N}} \cos \left( \frac{\pi k (2n+1)}{2N} \right)$ for DCT-2, and sin/cos-shifted variants for DST-7/DCT-8 (Egilmez et al., 2019). This correspondence aligns GBSTs with coding standards and ensures backwards compatibility in codec augmentations.

3. Graph Learning and Data Adaptivity

GBSTs are distinguished by their capacity for data-dependent optimization. Learning the graph parameters $(w,v)$ is performed by maximizing a graphical-model log-likelihood, given empirical residual block statistics. The optimization problem: $\min_{w \ge 0, v \ge 0} \; \text{Tr}[L(w,v) S] - \log \det L(w,v)$ is convex, with rapid convergence due to the low parameter count (often two per graph) (Egilmez et al., 2019, Egilmez et al., 2019). This process may be generalized to more flexible path graphs, supporting closed-form edge weight estimation or iterative maximum likelihood updating in product-graph settings (Pakiyarajah et al., 21 May 2025, Lu et al., 26 Feb 2024).

For practical deployment, graphs can be learned offline (no decoder-side computational cost), online via adaptive clustering (for local block adaptation), or jointly optimized with secondary transforms under rate-distortion objectives (Pakiyarajah et al., 21 May 2025, Lu et al., 26 Feb 2024).

4. Generalizations and Extensions

GBSTs extend beyond DCT/DST replacements:

DTT+ Family: Augment base discrete trigonometric transform graphs with rank-one updates (self-loop + scaling), yielding DTT+ kernels that interpolate between DCT/DST and KLT. These corrections permit integer approximations (INT-DTT+) with structured sparsification of the Cauchy correction matrix, formalizing computationally efficient, data-adaptive separable transforms that approach KLT coding gain with DCT-level complexity (Fernández-Menduiña et al., 22 Nov 2025).
Bi-Fractional GBSTs: For product graphs with heterogeneous dimensions, the 2D Graph Bi-Fractional Fourier Transform (GBFRFT) applies fractional orders independently to each dimension, improving adaptability and spectral selectivity compared to single-order graph FRFTs (Wang et al., 13 Oct 2025).
Irregularity-aware GBSTs: The IAGFT extends GBSTs by incorporating pixel/perceptual weighting in the inner product and solving for the Q-Laplacian eigenproblem, optionally with block clustering for perceptual class adaptation (Fernández-Menduiña et al., 2023).
Spatio-temporal GBSTs: GBST formalism encompasses separable compositions of spatial and temporal graph wavelets, supporting provably stable scattering transform networks optimized for generalization and computational cost (Pan et al., 2020).

5. Algorithmic Complexity and Practical Integration

Separable GBSTs maintain the computational advantages of classical separable transforms. For block size $N$ :

Matrix multiplies: Two $N \times N$ multiplications per block.
Graph learning updates: Convex, typically two or $O(N)$ edge/self-loop weights to estimate.
Transform signaling: As few as one bit per block (DCT/GBT switch), or additional bits in codecs that select among multiple transform candidates (Lu et al., 26 Feb 2024).

Compared to full separable KLTs ( $N(N+1)/2$ parameters per direction, $O(N^3)$ multiplies per block), GBSTs are more parameter-efficient and robust in data-limited regimes (Pakiyarajah et al., 21 May 2025, Egilmez et al., 2019).

The progressive decomposition in the DTT+ framework allows sparse Cauchy corrections, enabling integer implementations with low additional cost over standard integer DCT (Fernández-Menduiña et al., 22 Nov 2025).

6. Performance Evaluation and Empirical Outcomes

GBSTs deliver coding gains over fixed transforms, particularly on structured or texture-rich residual data:

VVC reference (MTS): Replacing standard DST-7/DCT-8 with learned GBSTs yielded average coding gain of $\sim$ 0.4% BD-rate over DCT-2/DST-7/DCT-8 (Egilmez et al., 2019).
AVM intra-coding: Joint rate-distortion optimized path-graph GBSTs achieved up to $2$ percentage point absolute improvement over separable KLTs in block-wise encoding (Pakiyarajah et al., 21 May 2025).
Low-complexity INT-DTT+: More than 3% BD-rate savings over VVC MTS baseline, with complexity on par with integer DCT (Fernández-Menduiña et al., 22 Nov 2025).
Texture-rich intra frames: Adaptive online path-graph GBSTs outperformed DCT by $-7.8\%$ BD-rate on H.264/AVC intra-predicted texture data, with stronger results for uniform image blocks (Lu et al., 26 Feb 2024).
Light field coding: Optimized spatio-angular GBSTs on local super-rays outperformed HEVC and JPEG Pleno by up to 1 dB PSNR at high bit-rates (Rizkallah et al., 2019).
Perceptually-inspired graph learning: Separable IAGFTs achieve $\sim$ 6% BD-rate savings in MS-SSIM over JPEG and DCT-based approaches (Fernández-Menduiña et al., 2023).

Energy compaction and decorrelation are often closer to KLT, while computational cost remains competitive or favorable.

7. Applications and Theoretical Foundations

GBSTs have found utility in:

Video coding: Explicit multiple-transform selection frameworks in VVC, AVM, and H.264/AVC, integrating GBSTs as primary or mode-dependent candidates (Egilmez et al., 2019, Fernández-Menduiña et al., 22 Nov 2025, Lu et al., 26 Feb 2024, Pakiyarajah et al., 21 May 2025).
Image compression: Block-wise IAGFT for perceptual coding, light field energy compaction (Fernández-Menduiña et al., 2023, Rizkallah et al., 2019).
Graph neural networks: Depthwise/separable graph convolution architectures (DSGC) for efficient learning and expressivity on arbitrary graph domains (Lai et al., 2017).
Spatio-temporal analysis: Separable spatio-temporal graph scattering transforms for stable feature extraction and robust classification under low-data regimes (Pan et al., 2020).
General graph signal processing: Bi-fractional GBSTs for filtering and deblurring on heterogeneous graph products (Wang et al., 13 Oct 2025).

Under Gaussian Markov random field (GMRF) models, GBSTs are provably optimal in the MSE sense for separable covariance structures (Egilmez et al., 2019); learning graphs from data imposes regularization that improves robustness and generalization relative to sample-KLT estimates.

In summary, GBSTs unify a wide spectrum of separable transforms through the lens of graph signal processing, yielding parameter-efficient, data-adaptive, and computationally tractable solutions applicable to a diverse array of high-dimensional signal domains.