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Scalar Function Topology Divergence: Comparing Topology of 3D Objects

This paper introduces Scalar Function Topology Divergence (SFTD), a novel topological tool that measures multi-scale dissimilarity between scalar functions while preserving feature localization, demonstrating its effectiveness as a loss function for improving 3D shape reconstruction and segmentation accuracy in computer vision tasks.

Original authors: Ilya Trofimov, Daria Voronkova, Eduard Tulchinskii, Evgeny Burnaev, Serguei Barannikov

Published 2026-09-10

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CC BY 4.0

Original authors: Ilya Trofimov, Daria Voronkova, Eduard Tulchinskii, Evgeny Burnaev, Serguei Barannikov

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

In the world of computer vision, machines are constantly asked to see the world not just as a collection of pixels, but as a collection of shapes. When a computer analyzes a medical scan or a photograph, it often builds a mathematical map, a landscape of hills and valleys, to decide where one object ends and another begins. For decades, scientists have measured how well a computer's guess matches reality by counting errors pixel by pixel, much like checking a spelling test letter by letter. But this approach misses the forest for the trees. It cannot easily tell the difference between a shape that has a hole in the middle and one that is solid, or a shape with two separate parts versus one that is connected. These features—cavities, loops, and connections—are the true skeleton of an object, and preserving them is crucial for tasks like reconstructing 3D models of cells or identifying tumors in the brain.

A team of researchers from the Skolkovo Institute of Science and Technology and several international partners has developed a new way to measure these shapes that pays attention to their structure and, just as importantly, where that structure is located. They call their tool Scalar Function Topology Divergence, or SFTD. Unlike previous methods that could only tell if two shapes had the same number of holes or loops, SFTD can pinpoint exactly where those features are and whether they appear in the right place. This distinction is vital because a computer might correctly identify that a cell has a hole, but if it places that hole in the wrong spot, the model is still fundamentally flawed. By focusing on the location of these topological features, the researchers have created a tool that not only detects errors more accurately but also helps machines learn to build better, more truthful 3D models.

The core of this work lies in a new method for comparing two mathematical landscapes. Imagine two maps of the same terrain, one drawn by a human and one by a machine. Traditional tools might look at the maps and say they are identical because they both show three hills and two valleys. However, if the machine's map has shifted those hills to different coordinates, the traditional tools might miss the mistake. The researchers introduced a concept they call an F-Cross-Barcode, which acts like a detailed ledger of differences. Instead of just listing the features, this ledger tracks exactly where a feature exists in one map but is missing or misplaced in the other. It is like having a map that not only lists the cities but highlights the specific streets where the city names have been swapped. This allows the system to see that while the number of features is the same, their arrangement is wrong.

To turn this insight into a practical tool for training computers, the team created a single number, the SFTD score, which measures the total distance between where features should be and where they actually are. If the score is zero, the shapes are topologically identical in both structure and location. If the score is high, it means the shapes differ significantly. Crucially, this score can be used as a guide during the learning process. When a computer is trying to learn how to reconstruct a 3D object from a 2D image, it makes mistakes. By calculating the SFTD score after each attempt, the computer receives a specific signal telling it to move the misplaced features back to their correct locations, rather than just adjusting the overall brightness or contrast of the image.

The researchers tested this approach on several challenging problems. In one experiment, they compared two simple shapes that looked identical to standard analysis tools because they had the same number of holes and bumps. The new tool, however, immediately spotted that the holes were in different places. In a more complex test involving the reconstruction of red blood cells and cell nuclei from 2D microscope images, the team added their new tool to the computer's learning process. The results showed that the computer, when guided by this new method, produced 3D models that were more accurate in their volume, surface texture, and overall shape than models trained with previous methods. In some cases, the new method reduced the error in matching the shape of the cell by a significant margin, proving that paying attention to the location of topological features leads to better results.

The utility of this method extends beyond just building shapes; it also helps in finding errors in existing ones. In the field of medical imaging, specifically in segmenting brain tumors, doctors need to know exactly where the tumor tissue, the swelling, and the dead tissue are located. The researchers applied their tool to compare a computer's prediction of a tumor against the ground truth. They found that while standard tools could not tell the difference between a correct prediction and one with misplaced voids, their method highlighted the exact location of the error. This capability is a powerful addition to the medical toolkit, offering a way to verify that a computer's diagnosis is not just statistically similar to the truth, but structurally and spatially correct.

The researchers also demonstrated that their method works efficiently on graphs, which are networks of connected points used to model everything from social networks to molecular structures. They showed that the tool could distinguish between different patterns in these networks that other methods missed. Furthermore, in 2D image segmentation tasks, their approach was found to be not only more accurate at preserving the correct topology but also significantly faster than existing alternatives, running roughly thirty-five times quicker than a competing method. This speed and accuracy combination suggests that the tool could be easily integrated into existing systems without slowing them down.

Ultimately, this work represents a shift in how we ask computers to understand shape. It moves the focus from a simple count of features to a precise understanding of their placement. By ensuring that the holes, loops, and connections in a digital model appear exactly where they should, the researchers have provided a way to build more reliable and realistic representations of the physical world. The code for this new tool is now available for others to use, opening the door for further applications in fields where the integrity of a shape is as important as its appearance. The findings suggest that when we teach machines to see, we must also teach them to understand where the parts of an object belong, a lesson that this new topological tool helps them learn.

Technical Summary: Scalar Function Topology Divergence (SFTD)

Problem Statement

In computer vision tasks such as image segmentation and 3D shape reconstruction, predictions are often defined as scalar functions $f: \mathbb{R}^n \to \mathbb{R}$ , where the final object shape is determined by thresholding these functions. Standard evaluation metrics (e.g., Dice score, cross-entropy) operate pixel-wise or voxel-wise. These metrics fail to capture topological correctness, specifically the preservation of multi-scale shape patterns such as cavities, holes, and connectivity.

Existing topological comparison tools, primarily based on the Wasserstein distance between independently computed persistence barcodes, have two critical limitations:

Lack of Localization: They measure global topological similarity but cannot identify where topological features differ between a prediction and a ground truth.
Dimensional Constraints: Many existing methods are limited to 2D spaces or do not account for the spatial arrangement of features.

Methodology

The authors propose a new framework for comparing scalar functions that incorporates the localization of topological features. The core components are:

1. F-Cross-Barcodes

To compare two scalar functions $f$ and $g$ defined on a common domain (an undirected graph or a Euclidean lattice), the authors construct a doubled graph $\tilde{G}$ (or an extended lattice $\tilde{L}$ ).

Construction: The domain is expanded to include two copies of the original vertices (or grid points) plus a special vertex $O$ .
Filtration: A new vertex function $\tilde{f}$ is defined on this extended structure. For the original vertices, values correspond to $f$ ; for the "primed" vertices, values correspond to $\min(f, g)$ ; and the special vertex $O$ takes the global minimum.
Definition: The F-Cross-Barcode is the persistence barcode derived from the lower-star filtration of this extended structure.
Interpretation: Unlike standard barcodes which describe a single function, intervals in an F-Cross-Barcode represent ranges of sublevel sets where a topological feature exists in one function but not the other (or exists in different locations).

2. Scalar Function Topology Divergence (SFTD)

To utilize these barcodes in machine learning pipelines (e.g., as a loss function), the authors define a scalar metric:
$\text{SFTD}_k(f, g) = \sum_{(b, d) \in \text{F-Cross-Barcode}_k(f, g)} |d - b|^p$
where $k$ is the homology dimension and $p$ is typically 1 or 2.

Properties: SFTD is not symmetric; the authors often use the average of $\text{SFTD}(f, g)$ and $\text{SFTD}(g, f)$ .
Differentiability: The paper provides an algorithm for differentiating SFTD with respect to network parameters, enabling its use as a loss term in deep learning.
Stability: The method is proven to be stable with respect to noise (bounded by the bottleneck distance).

3. Visualization

The authors introduce a visualization technique where the birth and death events of intervals in the F-Cross-Barcode are mapped back to the original spatial domain. This highlights specific regions (points or voxels) where topological discrepancies occur.

Key Contributions

New Tools: Introduction of F-Cross-Barcodes and SFTD for multi-scale, topology-aware comparison of scalar functions.
Localization: Unlike Wasserstein distance between independent barcodes, SFTD accounts for the spatial localization of topological features.
Visualization: A technique to visualize the specific areas of topological dissimilarity.
Differentiation: An algorithm for computing gradients of SFTD for use in optimization.
Empirical Validation: Demonstration that SFTD improves 3D shape reconstruction and identifies topological errors in segmentation where standard metrics fail.

Experimental Results

Synthetic Data

Indistinguishable Barcodes: The authors demonstrated cases where two functions have identical persistence barcodes (and thus zero Wasserstein distance) but differ in the location of local minima or defects.
SFTD Sensitivity: In these cases, SFTD was non-zero, and the visualization correctly pinpointed the locations of the topological differences (e.g., different locations of local minima in 2D, defects in lattices, and bridge positions in concentric spheres).

Graph Eigenvectors

The method was applied to compare eigenvectors of the graph Laplacian. Standard Euclidean or cosine distances could not distinguish these vectors effectively, but SFTD revealed topological similarities and differences, showing block structures and diagonal patterns in similarity heatmaps.

3D Shape Reconstruction (SHAPR)

Dataset: "Red blood cell" and "Cell nuclei" datasets reconstructed from 2D fluorescence microscopy images.
Comparison: The authors replaced the topological loss term in the SHAPR model (previously using Wasserstein-2 distance between independent barcodes) with a symmetrized SFTD.
Outcome:
- Red Blood Cells: The model augmented with SFTD (dimensions 0–3) achieved the lowest errors in IoU, volume, surface, and roughness. It also yielded the lowest Wasserstein-2 distances between ground truth and predicted barcodes.
- Cell Nuclei: SFTD improved IoU error and Wasserstein-2 distance for $H_2$ homology.
- Visual Inspection: Reconstructions using SFTD showed fewer topological defects (cavities, holes, surface breaks) compared to models using only Wasserstein loss.

3D Segmentation (Brain Tumors)

Dataset: BraTS 21 dataset using a pretrained Swin UNETR model.
Finding: Two segmentations (ground truth and prediction) with voids in different locations had identical persistence barcodes. Standard metrics could not distinguish the error.
Result: SFTD successfully identified the difference, and the visualization highlighted the specific voids that were mislocalized.

2D Segmentation

The authors claim SFTD outperforms Betti matching loss in 2D segmentation problems in terms of topological accuracy while being approximately 35 times faster.

Significance and Claims

The paper claims that SFTD provides a necessary advancement over existing topological comparison methods by integrating localization into the metric.

Loss Function Utility: SFTD serves as an effective additional loss term that enforces topological similarity at multiple scales during training, leading to better 3D reconstructions and segmentation.
Diagnostic Tool: It acts as a diagnostic tool to identify and visualize specific topological errors (e.g., misplaced holes or cavities) that are invisible to standard pixel-wise metrics or global barcode distances.
Generality: The method is applicable to scalar functions on undirected graphs and Euclidean spaces of any dimensionality.

The authors conclude that these tools offer a robust, differentiable, and visualizable approach to topological data analysis in computer vision, with potential for broader applications in graph learning and 3D vision.