Spatial-Aware Graph Completion Network (SAGCNet)

• The paper introduces a novel SAGCNet framework that performs full-volume reconstruction by jointly modeling local inter-slice correlations and global 3D spatial context.
• It employs a two-stream graph completion module (VSGC) and a volumetric spatial adapter (VSA) to integrate slice-level relational reasoning with volumetric priors in a transformer-based encoder.
• Empirical evaluations on UKBB, MESA, and ACDC datasets demonstrate that SAGCNet outperforms baseline models with improved PSNR, SSIM, and MAE across various missing-slice rates.

Searching arXiv for the specified paper and closely related context. Spatial-Aware Graph Completion Network (SAGCNet) is a unified framework for missing slice imputation in volumetric cardiac magnetic resonance (CMR) imaging. It is designed for settings in which a CMR volume contains missing or unusable through-plane slices and seeks to synthesize the full volume in one shot from the available slices. The method addresses two stated difficulties of volumetric MRI synthesis: modeling local inter-slice correlations and dependencies, and exploiting 3D spatial information together with global context. Its two principal components are a volumetric slice graph completion (VSGC) module, which performs slice-level relational reasoning under incomplete observations, and a volumetric spatial adapter (VSA), which injects 3D spatial awareness into a transformer-based visual encoder derived from UNETR (Liu et al., 9 Aug 2025).

1. Problem formulation and scope

SAGCNet is formulated for missing slice imputation in volumetric CMR, where a volume is represented as $V \in \mathbb{R}^{N \times H \times W}$, with $N$ through-plane slices of spatial size $H \times W$ (Liu et al., 9 Aug 2025). The volume is partitioned into available slices $V_a \in \mathbb{R}^{M \times H \times W}$ and missing slices $V_m \in \mathbb{R}^{P \times H \times W}$ such that $N = M + P$, and the missing rate is defined as $\eta = P/N$. A binary slice mask $M \in \{0,1\}^N$ indicates whether a slice is observed or missing, with $M_i = 1$ for observed slices and $M_i = 0$ otherwise.

The network receives a 3D volume with missing slices. For implementation, volumes are zero-padded along the through-plane to a fixed slice count; each 2D slice is resized to $N$0; and slice intensities are min-max normalized to $N$1 (Liu et al., 9 Aug 2025). After encoding, graph completion, and decoding, the model reconstructs $N$2 in one shot, thereby imputing all missing slices and optionally refining available ones.

During training, missing slice positions are randomly sampled per iteration at a fixed missing rate $N$3 to simulate arbitrary clinical missing scenarios and to enhance robustness. This training regime functions as a form of masked volumetric reconstruction. A plausible implication is that the model is intended not merely for deterministic interpolation between adjacent slices, but for more general missingness patterns distributed throughout a volume.

2. Network composition and architectural flow

SAGCNet combines a modified UNETR backbone, multiple VSGC insertion points, and a CNN-based U-shaped decoder with skip connections (Liu et al., 9 Aug 2025). The visual encoder is a transformer-based encoder in which two VSA modules are inserted per transformer block, one before and one after multi-head self-attention (MHSA). VSGC modules are inserted after the 3rd, 6th, and 12th transformer blocks to progressively refine slice-level graph representations through multi-view graph completion.

Channel adapters play two roles in this design. First, they map encoded feature tensors to per-slice node features suitable for graph construction. Second, they fuse graph-derived embeddings back into the encoder’s feature space through residual connections. This coupling makes the graph reasoning pathway and the volumetric transformer pathway jointly operative rather than sequentially isolated.

The architectural flow is therefore explicitly hybrid. Transformer blocks provide high-capacity visual encoding; VSA adds volumetric priors; VSGC imposes slice-level relational structure; and the decoder reconstructs the full 3D volume slice-wise. The paper’s concise forward-pass description specifies the following sequence: encoding with VSA-enhanced transformer blocks, channel-adapter projection into node features, kNN graph construction, two-view graph completion with contrastive alignment, fusion back into the encoder stream, and final U-Net-style decoding to $N$4 (Liu et al., 9 Aug 2025).

This organization is notable because the graph operates at slice level rather than voxel level. The stated rationale is efficiency: the graph contains $N$5 nodes, one per slice, making it lightweight relative to full 3D CNN or transformer processing over the entire volume.

3. Volumetric slice graph completion (VSGC)

VSGC constructs a graph $N$6 for each volume, where each node $N$7 corresponds to slice $N$8 and has feature $N$9 derived from the encoder through a channel adapter (Liu et al., 9 Aug 2025). The node attribute matrix is $H \times W$0 with $H \times W$1, and the number of nodes is $H \times W$2. For missing slices, node attributes are treated as missing in the attribute-view and initialized with learnable parameters.

Edges are defined by k-nearest neighbors in slice feature space with $H \times W$3. The adjacency matrix is $H \times W$4, where $H \times W$5 if $H \times W$6 and $H \times W$7 otherwise. When normalization is required, the symmetric normalized adjacency is

$H \times W$8

VSGC models incompleteness at both the attribute and structure levels through a two-view graph completion strategy. In the attribute-view, the augmented graph is constructed as

$H \times W$9

where $V_a \in \mathbb{R}^{M \times H \times W}$0 is broadcast to shape $V_a \in \mathbb{R}^{M \times H \times W}$1, $V_a \in \mathbb{R}^{M \times H \times W}$2 denotes element-wise product, and $V_a \in \mathbb{R}^{M \times H \times W}$3 are learnable parameters that initialize missing attributes. In the structure-view, personalized PageRank (PPR) propagation is used to enrich node features:

$V_a \in \mathbb{R}^{M \times H \times W}$4

The resulting structure-view graph is denoted $V_a \in \mathbb{R}^{M \times H \times W}$5, with the paper describing PPR as a mechanism to propagate information and enhance diversity.

Both views are encoded with a Graph Attention Network (GAT). The paper specifies two GAT layers and gives the single-head formulation as

$V_a \in \mathbb{R}^{M \times H \times W}$6

$V_a \in \mathbb{R}^{M \times H \times W}$7

where $V_a \in \mathbb{R}^{M \times H \times W}$8 is the input node feature, $V_a \in \mathbb{R}^{M \times H \times W}$9 and $V_m \in \mathbb{R}^{P \times H \times W}$0 are trainable parameters, and $V_m \in \mathbb{R}^{P \times H \times W}$1 is the neighbor set of node $V_m \in \mathbb{R}^{P \times H \times W}$2 (Liu et al., 9 Aug 2025).

The two views are aligned through a symmetric cross-view InfoNCE objective:

$V_m \in \mathbb{R}^{P \times H \times W}$3

where $V_m \in \mathbb{R}^{P \times H \times W}$4 is cosine similarity and $V_m \in \mathbb{R}^{P \times H \times W}$5. The explicit purpose of this term is to maximize mutual information between the attribute-view and structure-view embeddings, enforce one-to-one alignment of corresponding nodes, and separate non-corresponding nodes. The fused graph embeddings are then returned to the encoder via a channel adapter with a residual connection.

The design justification given for VSGC is that slice-wise graphs can encode local through-plane correlations as well as dependencies between non-adjacent slices, while the two-view formulation mitigates the incompleteness of both node attributes and graph structure. This suggests that VSGC is not simply a graph regularizer, but a completion mechanism embedded within the backbone.

4. Volumetric spatial adapter (VSA) and 3D context modeling

The VSA is the component responsible for injecting 3D spatial awareness and global volumetric context into the transformer blocks while remaining lightweight (Liu et al., 9 Aug 2025). Each VSA consists of normalization, a down-projection that reduces channel dimensionality, a depth-wise 3D convolution, and an up-projection back to the original dimensionality, all under a residual formulation:

$V_m \in \mathbb{R}^{P \times H \times W}$6

where $V_m \in \mathbb{R}^{P \times H \times W}$7 is the incoming feature tensor, $V_m \in \mathbb{R}^{P \times H \times W}$8 denotes layer normalization, and $V_m \in \mathbb{R}^{P \times H \times W}$9 and $N = M + P$0 are linear projections analogous to $N = M + P$1 pointwise convolutions.

The modified UNETR attention mechanism follows the standard expression

$N = M + P$2

with $N = M + P$3, $N = M + P$4, and $N = M + P$5 denoting queries, keys, and values, and $N = M + P$6 the head dimension. VSA is positioned before and after MHSA in each transformer block for what the paper describes as enhanced performance and stability.

The role of VSA is explicitly complementary to VSGC. The depth-wise 3D convolution aggregates features across slices, embedding local 3D spatial patterns and global volumetric context into transformer features. Combined with slice-level graph reasoning, the method integrates two forms of context: local inter-slice correlations captured by the graph and broader volumetric priors captured by VSA. The paper frames this as a response to the limitations of purely 2D slice-based processing and to the cost of full 3D models.

The ablation findings support the importance of this component. On UKBB and ACDC, removing VSA reduces performance from full SAGCNet’s $N = M + P$7 and $N = M + P$8 PSNR/SSIM to $N = M + P$9 and $\eta = P/N$0, respectively (Liu et al., 9 Aug 2025). Within the paper’s experimental scope, this establishes that volumetric spatial adaptation makes a measurable contribution independent of the graph module.

5. Training objective, datasets, and empirical results

The total training loss is

$\eta = P/N$1

with $\eta = P/N$2, $\eta = P/N$3, and $\eta = P/N$4 (Liu et al., 9 Aug 2025). The masked reconstruction loss is

$\eta = P/N$5

and the perceptual synthesis loss is

$\eta = P/N$6

where $\eta = P/N$7 is a fixed feature extractor. The contrastive loss is the cross-view objective defined within VSGC.

Training uses the Adam optimizer with initial learning rate $\eta = P/N$8, cosine annealing to $\eta = P/N$9, batch size $M \in \{0,1\}^N$0, and $M \in \{0,1\}^N$1 epochs on an NVIDIA A100 GPU (Liu et al., 9 Aug 2025). Hyperparameters include $M \in \{0,1\}^N$2 kNN neighbors, two GAT layers, and contrastive temperature $M \in \{0,1\}^N$3. Data augmentation consists of random flipping and rotation.

The reported datasets are UK Biobank (UKBB), with $M \in \{0,1\}^N$4 patients and $M \in \{0,1\}^N$5 temporal phases per patient; MESA, with $M \in \{0,1\}^N$6 CMR volumes and $M \in \{0,1\}^N$7–$M \in \{0,1\}^N$8 slices per volume; and ACDC, with $M \in \{0,1\}^N$9 patients and $M_i = 1$0–$M_i = 1$1 slices per volume. All datasets use an $M_i = 1$2 train and $M_i = 1$3 test split. Missing rates are evaluated at $M_i = 1$4.

The principal quantitative results are summarized below.

Dataset	Missing rate and SAGCNet result	Comparator summary
UKBB	$M_i = 1$5: $M_i = 1$6 dB / $M_i = 1$7	best baseline UNETR: $M_i = 1$8 / $M_i = 1$9
UKBB	$M_i = 0$0: $M_i = 0$1 / $M_i = 0$2	baselines $M_i = 0$3–$M_i = 0$4 / $M_i = 0$5–$M_i = 0$6
UKBB	$M_i = 0$7: $M_i = 0$8 / $M_i = 0$9	baselines $N$00–$N$01 / $N$02–$N$03
MESA	$N$04: $N$05 / $N$06	UNETR: $N$07 / $N$08; others $N$09–$N$10 / $N$11–$N$12
MESA	$N$13: $N$14 / $N$15	UNETR: $N$16 / $N$17; others $N$18–$N$19 / $N$20–$N$21
MESA	$N$22: $N$23 / $N$24	UNETR: $N$25 / $N$26; others $N$27–$N$28 / $N$29–$N$30
ACDC	$N$31: $N$32 / $N$33	UNETR: $N$34 / $N$35; others $N$36–$N$37 / $N$38–$N$39
ACDC	$N$40: $N$41 / $N$42	baselines $N$43–$N$44 / $N$45–$N$46
ACDC	$N$47: $N$48 / $N$49	baselines $N$50–$N$51 / $N$52–$N$53

For the single-missing-slice setting ($N$54), the comparison against I2GAN is also explicit. On UKBB, I2GAN obtains MAE $N$55, PSNR $N$56, and SSIM $N$57, whereas SAGCNet obtains MAE $N$58, PSNR $N$59, and SSIM $N$60. On ACDC, I2GAN obtains MAE $N$61, PSNR $N$62, and SSIM $N$63, whereas SAGCNet obtains MAE $N$64, PSNR $N$65, and SSIM $N$66 (Liu et al., 9 Aug 2025).

Evaluation uses MAE, PSNR, and SSIM. The PSNR expression is

$N$67

where $N$68 is the peak intensity after normalization. SSIM uses the standard luminance-contrast-structure formulation:

$N$69

Qualitatively, SAGCNet is reported to reduce error regions around cardiac anatomical boundaries and structures and to produce visually realistic slices with better continuity than Pix2Pix, MMGAN, ResViT, UNETR, and MTT-Net (Liu et al., 9 Aug 2025). This suggests that the model’s gains are not limited to average intensity fidelity but extend to structural continuity across slices.

6. Ablation, efficiency, limitations, and reproducibility

The ablation study isolates the contributions of VSA, VSGC, the two graph views, and the contrastive objective (Liu et al., 9 Aug 2025). On UKBB and ACDC, the reported PSNR/SSIM values are:

• w/o VSA: $N$70 and $N$71
• w/o VSGC: $N$72 and $N$73
• w/o attribute-view: $N$74 and $N$75
• w/o structure-view: $N$76 and $N$77
• w/o $N$78: $N$79 and $N$80
• Full SAGCNet: $N$81 and $N$82

These results are interpreted in the paper as demonstrating that both VSA and VSGC are critical and that contrastive alignment contributes measurable gains. Within the paper’s experimental frame, the two-view graph completion strategy is therefore empirically justified rather than only architecturally motivated.

The efficiency argument rests on two points. First, the graph operates at slice level, so with $N$83 the number of edges is approximately $N$84. Second, a GAT layer has computational complexity $N$85 per head, plus $N$86 for linear projections. With two GAT layers and small $N$87, the paper characterizes this cost as modest (Liu et al., 9 Aug 2025). Figure 1 is described as showing favorable PSNR-versus-FLOPs behavior and faster convergence relative to baselines, while VSA uses down/up projections and depth-wise 3D convolutions to control parameter count.

The stated limitations are also specific. Fixed kNN graph construction with $N$88 may under-express long-range slice relations in highly anisotropic or irregular volumes; the method assumes consistent slice ordering and orientation; severe acquisition variability or artifacts may challenge both kNN-based adjacency and depth-wise convolution assumptions; and the model operates per volume without explicit temporal modeling across cardiac phases (Liu et al., 9 Aug 2025). The paper also notes that generalizability beyond CMR and to other modalities or anatomies warrants further validation, and that favorable runtime and FLOPs characteristics are not fully tabulated. These remarks constrain the scope of the reported results and identify likely directions for extension, including dynamic or learned adjacency and 4D cine modeling.

Reproducibility materials are provided through the project repository at https://github.com/JK-Liu7/SAGCNet, which includes training scripts, model definitions for VSA and VSGC, and checkpoints (Liu et al., 9 Aug 2025). The key implementation details identified for reproduction are the UNETR backbone with VSA modules before and after MHSA, VSGC after the 3rd, 6th, and 12th transformer blocks, $N$89 neighbors, two GAT layers, fixed $N$90 per experiment, random missing slice sampling per iteration, and the preprocessing and optimization settings listed above. Within the stated setup, reproducing the method involves choosing $N$91, training for $N$92 epochs, and evaluating PSNR, SSIM, and MAE on the held-out test split.