Technical Summary: GeoTopoDiff

Problem Statement

The reconstruction of large-scale 3D porous microstructures from sparse Micro-Computed Tomography (µCT) slices presents a significant challenge for diffusion-based generative models. While diffusion models have shown promise in solving ill-posed inverse problems, existing voxel-based approaches face a fundamental trade-off in industrial applications: achieving high topological fidelity typically requires fully observed, high-resolution µCT scans, which limits field of view and throughput. Conversely, conditional diffusion pipelines based on sparse boundary observations (e.g., only top and bottom slices) are more efficient but suffer from the inherent ill-posedness of the inverse problem.

The core limitation identified is the latent space representation. Current methods perform reverse denoising in voxel occupancy space, which captures local pore morphology but fails to explicitly model the discrete pore-throat topology. Consequently, sparse boundary conditions can lead to multiple visually plausible reconstructions that possess contradictory pore-throat graph topologies, resulting in poor consistency in transport properties like permeability.

Methodology: GeoTopoDiff

The authors propose GeoTopoDiff, a graph diffusion framework designed to reconstruct 3D porous microstructures from sparse top and bottom µCT slices. The method shifts the learning paradigm from a voxel-based space to a mixed graph state space that simultaneously encompasses continuous pore geometry and discrete pore-throat topology.

1. Mixed Graph State Representation

Instead of operating on voxel grids, GeoTopoDiff represents the porous medium as a graph $S = (B, C, E)$:

• $B$ (Continuous Geometry): Stores pore-body centroids $(x, y, z)$ and equivalent radii.
• $C$ (Discrete Node States): Categorical labels indicating active pore bodies or null nodes.
• $E$ (Discrete Topology): Categorical labels indicating throat connections or null edges between pore bodies.

The framework employs a mixed forward diffusion process where continuous geometry variables are corrupted via Gaussian diffusion, while discrete node and edge states are corrupted via categorical diffusion.

2. Topology-Aware Partial Graph Prior

To constrain the reverse denoising process, GeoTopoDiff extracts a partial boundary graph ($G_B$) from the observed top and bottom slices. This graph includes:

• Directly observed boundary geometry and node states.
• Soft topological evidence derived from top-bottom consistency (matching pore components across slices).
• Masks ($M_B, M_C, M_{E,obs}$) that identify variables which can be "hard-clamped" during sampling to enforce boundary consistency.

3. Reverse Sampling and Decoding

The reverse process is a boundary-conditioned denoising step where a neural network ($D_\theta$) predicts geometry noise and logits for node/edge states.

• Geo-Topo Interaction: The model utilizes a specialized attention mechanism where geometry-aware node updates aggregate edge-neighbour messages, and edge updates use incident node features. This allows topology to influence geometry denoising and vice versa.
• Clamping: After each reverse step, directly observed boundary variables are projected (clamped) to match the observed slices, reducing posterior uncertainty in the unobserved interior.
• Graph-to-Volume Decoder: The final denoised graph $S_0$ is decoded into a 3D voxel probability map using an analytic soft occupancy field combined with residual refinement, ensuring the output remains coupled to the sampled pore graph.

Key Contributions

1. Formulation of a Mixed Inverse Problem: The authors formulate sparse-boundary 3D porous reconstruction as a mixed continuous-discrete diffusion inverse problem, explicitly preserving both pore geometry and pore-throat topology.
2. GeoTopoDiff Framework: They introduce a framework that transfers diffusion prior learning into a mixed graph state space, utilizing a topology-aware partial graph prior to constrain the reverse denoising of unobserved interior regions.
3. Empirical Validation: The method is evaluated on anisotropic PTFE and isotropic Fontainebleau sandstone, demonstrating improvements over representative baselines in morphology, graph topology fidelity, and transport-oriented consistency.

Experimental Results

The model was tested on two datasets (PTFE and Fontainebleau sandstone) using only top and bottom slices as input, with internal slices reserved for ground truth evaluation. GeoTopoDiff was compared against six baselines, including statistical methods, 2D-to-3D GANs, and voxel-based diffusion models.

Key Performance Metrics:

• Morphology: GeoTopoDiff reduced morphology-related errors (measured by Two-Point Correlation Functions and Pore-Size Distribution) by 19.8% on average compared to baselines.
• Topology & Transport: The method reduced topology-sensitive transport errors (permeability mean relative error) by 36.5% on average. Specifically, it reduced permeability error by 42.3% for PTFE and 27.9% for sandstone compared to the strongest baseline (LDM-3DG).
• Connectivity: The disconnected rate was reduced by 44.9% (PTFE) and 31.0% (sandstone), indicating superior preservation of through-plane connectivity.
• Ablation Studies: Removing the boundary graph prior or the discrete topological diffusion component led to the most significant performance degradation, confirming that topology must be generated as a first-class variable rather than inferred post-hoc.

Significance and Claims

The paper claims that the shift from voxel-based latent spaces to a mixed graph state space is critical for scientific inverse problems where the target phenomenon is governed by high-level structure. By explicitly constraining the solution space with a boundary partial graph prior, GeoTopoDiff enables diffusion sampling to operate over the structural degrees of freedom that determine whether an interior is not only visually plausible but also connectivity- and transport-consistent.

The authors posit that for porous media, where functional properties depend on the pore network structure (porosity, connectivity, permeability), designing the diffusion state space around the underlying structure is as important as improving the denoising network itself. The work suggests a broader lesson for scientific inverse problems: when the target is governed by high-level structure, explicit structural modeling within the generative process can significantly reduce posterior uncertainty under sparse observations.

The authors note limitations regarding computational scalability (quadratic scaling with the number of pore bodies due to dense edge sets) and dependence on the quality of boundary segmentation, but they assert that the current approach successfully bridges the gap between high-throughput industrial imaging needs and topological fidelity. All models and code are made publicly available.