🔬 materials science

Fitting the topology of synthetic particle systems with a novel graph representation

This paper introduces a novel graph-based representation that enables the computationally efficient fitting of particle system topology via persistent homology, thereby improving the structural and topological agreement of synthetic 3D images with real materials while preserving key morphological characteristics.

Original authors: Martin Alexander Memmesheimer, Claudia Redenbach

Published 2026-07-21

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CC BY 4.0

Original authors: Martin Alexander Memmesheimer, Claudia Redenbach

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a materials scientist trying to understand why a block of concrete is strong, or why a layer of paint protects a car from rust. The secret isn't just in the ingredients; it's in how those ingredients are arranged. Think of a material like a crowded dance floor. If everyone is standing in a neat grid, the crowd moves differently than if they are jumbled in a chaotic huddle. Scientists use special X-ray cameras to take 3D pictures of these microscopic dance floors, but the cameras are expensive, and the pictures are huge and hard to analyze. So, instead of taking more photos, researchers try to build "fake" versions of these materials on computers. They want to create digital sand, gravel, or pigment particles that look and act exactly like the real thing.

To do this, they need to understand "topology." In everyday language, topology is the study of shapes and connections that don't change when you stretch or squish them. It asks questions like: "Are these particles all touching in one big clump?" or "Are there any tunnels or holes running through the material?" A powerful tool called "persistent homology" helps scientists map these connections, acting like a topological fingerprint that counts the loops and voids in the data. The challenge is that these 3D X-ray images are so massive that trying to calculate these fingerprints directly is like trying to count every grain of sand on a beach by hand—it takes too long and crashes computers.

This is where a new study by Martin Alexander Memmesheimer and Claudia Redenbach comes in. They realized that instead of fighting the giant 3D images, they could translate the problem into a simpler language: a graph. Imagine taking a photo of a crowded room and replacing every person with a single dot, and drawing a line between two dots only if those two people can see each other without anyone blocking the view. This "particle graph" is a tiny, lightweight sketch of the massive 3D image. The researchers built a clever system to turn a real X-ray scan into this dot-and-line sketch, and then to turn a fake, computer-generated sketch back into a 3D image.

The team's main discovery is that by doing their math on these simple graphs instead of the heavy 3D images, they can "tune" the fake materials to match the real ones much more closely. They used a method called "topological optimization," which is like a video game where the computer keeps adjusting the fake particles until their "topological fingerprint" aligns with the real one. They found that this method works incredibly well. When they tested it on four different samples of recycled concrete, the fake images they created matched the real ones significantly better in terms of how connected the particles were and how many holes existed in the structure. The results showed a high degree of overlap (around 87%) and a strong preservation of the material's topological features, though not a perfect, pixel-for-pixel match.

However, the paper is careful to note what this method doesn't do. It doesn't create new particles out of thin air, nor does it drastically alter the fundamental size of the particles. The fake images started with the exact same number and size of particles as the real ones; the computer just moved them around to fix the connections. The study does find, however, that while the method is great at fixing the "skeleton" of the material (the connections and holes), it does slightly affect the particle geometry. The reconstruction process tends to make the surfaces of the particles smoother than the real jagged edges, and the volume of smaller particles can decrease slightly while larger ones appear to grow. To fix this, the researchers added a little bit of "noise" (random bumps) to the surface, which helped the fake particles look more realistic, though it didn't perfectly restore the original jaggedness.

Ultimately, the paper suggests that this graph-based approach is a powerful new tool. It allows scientists to take a rough, computer-generated pile of particles and polish it until its internal structure is much closer to reality, all without needing supercomputers to crunch the numbers. It's a bit like taking a rough draft of a story and using a smart editor to fix the plot holes and character connections, ensuring the final story feels just as real as the one that actually happened, even if the words on the page were written by a machine.

Technical Summary: Fitting the Topology of Synthetic Particle Systems with a Novel Graph Representation

Problem Statement
The macroscopic properties of particulate materials (e.g., concrete, pigments, iron ore) are determined by the shape, arrangement, and topology of their constituent particles. While X-ray micro-computed tomography (µCT) can capture 3D particle structures, the scarcity of real data limits data-driven analysis. Consequently, researchers rely on synthetic data generation. However, existing generation methods primarily focus on single-particle characteristics (size, shape) and often fail to reproduce the complex topological structure of the particle system, such as connectivity and higher-dimensional features.

Attempts to integrate topological data analysis, specifically persistent homology, into synthetic generation pipelines face a critical bottleneck: the direct application of these methods to large 3D µCT images is computationally prohibitive. Furthermore, computing persistent homology directly on image data requires binarization, which obscures explicit particle information and makes gradient-based optimization difficult due to the discrete nature of voxels.

Methodology
The authors propose a novel framework that bridges the gap between image data and topological optimization by introducing a particle graph representation. The methodology consists of three primary stages:

Graph Construction:
- Each particle in a labeled 3D image is represented as a node in a graph.
- An edge exists between two nodes if the corresponding particles are connected by an unobstructed line segment that does not intersect other particles.
- Node features ( $f_V$ ): Store the center of mass of each particle.
- Edge features ( $f_E$ ): Store the endpoints of the shortest unobstructed line segment between two connected particles.
- To reduce computational cost, a thresholded graph ( $G_\tau$ ) is used, retaining only edges where the distance between endpoints is below a threshold $\tau$ .
Topological Optimization in the Graph Domain:
- The goal is to align the topology of a synthetic particle graph with that of a real particle graph.
- Loss Function: The optimization minimizes the 2-Wasserstein distance between the persistence diagrams of the synthetic and real graphs.
- Differentiability: To enable gradient-based optimization (using the Adam optimizer), the authors employ an entropic regularized version of the Wasserstein distance (based on optimal transport theory). This ensures stable gradients.
- Differentiability of Persistence: Since persistence diagrams are derived from a Vietoris-Rips filtration on the graph, and the filtration weights depend on the edge features (which are continuous), the loss function is differentiable with respect to the edge features.
Image Reconstruction:
- The optimized graph is converted back to an image.
- For each particle (node), a set of points is aggregated, including its center of mass and the edge feature points associated with it.
- The particle shape is reconstructed as the convex hull of these points.
- Post-processing: A morphological closing is applied to fill voxel-wide gaps. Optionally, simplex noise can be added to the boundary to introduce irregularities, though the authors found this did not consistently improve topological metrics.

Key Contributions

Novel Graph Representation: The paper introduces a method to represent 3D particle systems as graphs where nodes are particles and edges encode geometric connectivity. This representation preserves explicit particle information while drastically reducing dimensionality compared to voxel grids.
Domain Transfer for Topological Optimization: The work successfully transfers the computation of persistent homology and its optimization from the image domain to the graph domain. This makes topological optimization computationally feasible for large 3D datasets.
Postprocessing Framework: The method acts as a postprocessing step for any synthetic generation method (demonstrated here with Random Sequential Adsorption), independent of the underlying generation algorithm.
Differentiable Topological Loss: The authors formulate a differentiable loss term based on entropic regularized Wasserstein distances, enabling gradient descent to adjust particle positions to match target topologies.

Results
The method was validated on four µCT images of recycled concrete (RC1–RC4) and synthetic images generated via Random Sequential Adsorption (RSA).

Reconstruction Quality: The reconstruction from the graph (without optimization) achieved a mean Dice score of approximately 0.87, indicating high geometric overlap with the original particles.
Topological Improvement:
- Optimizing the graph significantly reduced the Wasserstein distance between synthetic and real persistence diagrams.
- For 0-dimensional topology (connectivity), the error was reduced by over 95% for some datasets, achieving perfect agreement in others.
- For 1-dimensional (loops) and 2-dimensional (cavities) topologies, improvements ranged from 7% to 24% relative to unoptimized reconstructions.
Geometric Preservation: The optimization process preserved the particle size distribution and volume fraction. While the convex hull reconstruction slightly smoothed boundaries (reducing surface area), the overall morphological characteristics remained intact.
Geometric Tortuosity: The method improved the geometric tortuosity of the synthetic images, bringing them closer to the values observed in real µCT data. The improvement was driven primarily by the optimization of 1-dimensional topology.

Significance and Claims
The authors claim that their approach provides a robust, general-purpose tool for improving the topological fidelity of synthetic particle systems. By moving the optimization to the graph domain, they overcome the computational infeasibility of direct image-based topological optimization. The method allows for the generation of synthetic data that not only matches the statistical properties of particle size and shape but also accurately reproduces the complex topological connectivity of real materials.

The paper notes that while the method is effective, future work could address the computational scaling with the number of particles and the "domain gap" introduced by the convex hull reconstruction, potentially by exploring methods that better model concavities (e.g., spherical harmonics or alpha shapes) without sacrificing computational efficiency.

Technical Summary: Fitting the Topology of Synthetic Particle Systems with a Novel Graph Representation

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