Dealing with Topological Information within a Fully Convolutional Neural Network

Bruno La\"y; Etienne Decenci\`ere; Fu Min; Gervais Gauthier; H\'el\`ene Burdin; Juanjuan Chen; Santiago Velasco-Forero; Th\'er\`ese Baldeweck; Thomas Bornschloegl

arxiv: 1906.11600 · v1 · pith:CA4DCXZPnew · submitted 2019-06-27 · 💻 cs.CV · eess.IV

Dealing with Topological Information within a Fully Convolutional Neural Network

Etienne Decenci\`ere , Santiago Velasco-Forero , Fu Min , Juanjuan Chen , H\'el\`ene Burdin , Gervais Gauthier , Bruno La\"y , Thomas Bornschloegl

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Th\'er\`ese Baldeweck

This is my paper

Pith reviewed 2026-05-25 14:46 UTC · model grok-4.3

classification 💻 cs.CV eess.IV

keywords fully convolutional neural networkgeodesic operatortopological informationimage segmentationhistological imageswhole slide imagingreceptive field

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The pith

Pre-processing with a geodesic operator lets fully convolutional networks exploit global topological information despite limited receptive fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Fully convolutional neural networks process images through local filters whose receptive fields are small, so they cannot directly access global properties such as topology. The paper shows that a geodesic operator applied as pre-processing can encode the needed topological cues into the input channels that the network does see. The method is demonstrated on the segmentation of histological images of pigmented reconstructed epidermis captured by whole-slide imaging. If correct, the same network architecture can now be used on tasks where global connectivity matters without enlarging receptive fields or altering the model. A reader would care because many medical and scientific imaging problems require both local detail and global structure.

Core claim

A geodesic operator is used to pre-process the input so that a fully convolutional neural network can exploit global topological information despite its limited receptive field size. The operator is applied to the segmentation of histological images of pigmented reconstructed epidermis acquired via Whole Slide Imaging.

What carries the argument

The geodesic operator, which injects global topological information into the local input features seen by the network.

If this is right

The network can now segment images correctly when global connectivity or topology determines the correct labels.
No change to the convolutional architecture is required to incorporate the missing global information.
The same pre-processing step can be reused for other fully convolutional tasks that need topology.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach could be tested on non-histological images where topology is critical, such as road networks or vessel segmentation.
It might reduce the need for dilated convolutions or attention layers in some topology-sensitive tasks.
If the operator can be made differentiable, end-to-end training that includes the pre-processing step becomes possible.

Load-bearing premise

The geodesic operator can be defined and applied so that it adds the relevant topological information without distorting other image features or creating artifacts the network cannot learn around.

What would settle it

Segmentation accuracy on the histological test images would remain unchanged or decrease when the geodesic pre-processing step is removed, compared with the identical network trained on raw inputs.

Figures

Figures reproduced from arXiv: 1906.11600 by Bruno La\"y, Etienne Decenci\`ere, Fu Min, Gervais Gauthier, H\'el\`ene Burdin, Juanjuan Chen, Santiago Velasco-Forero, Th\'er\`ese Baldeweck, Thomas Bornschloegl.

**Figure 1.** Figure 1: Examples of original images with overlayed contours of the reference segmentation (best viewed in colour). Red/top contour: frontier between background and SC. Note that its position can be completely shifted if the detached layer is unbroken (top) or broken (bottom). Green/middle contour: frontier between SC and living epidermis. Cyan/bottom contour: frontier between living epidermis and collagen scaffol… view at source ↗

**Figure 2.** Figure 2: Top: original image, showing the selected crops. Middle: results without using global information. Bottom: result using global information, thanks to the presented method. These segmentation results have not been postprocessed. They are overlayed on the original data using the following colour code: SC (magenta), living epidermis (orange) and other regions (cyan) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Top: original image. Bottom: image after pre-processing based on the geodesic reconstruction. Differences are mainly visible on the holes within the tissue sample. 3.4 Post-processing The current results are already satisfactory. There are however a few defects in the resulting segmentation (as can be seen in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Zoom-in on some test images to illustrate the results, as well as its robustness to acquisition artifacts. The contours of the segmentation computed with the final model are overlayed on the original images. be large enough and neural network architectures that use downsampling layers impose that the dimensions be multiple of some 2n (where n is the number of such layers, supposing that the sampling steps … view at source ↗

**Figure 5.** Figure 5: Zoom-in onto the two more significant errors found on the 23 images of the test database. Processing times are as follows. The standard U-Net takes 171 seconds to process the full 23 test images on a conventional laptop with a NVidia GeForce GTX 980M graphics card. The improved method, including the geodesic reconstruction, takes 407 seconds. We think that the pre-processing could be optimized, but the c… view at source ↗

read the original abstract

A fully convolutional neural network has a receptive field of limited size and therefore cannot exploit global information, such as topological information. A solution is proposed in this paper to solve this problem, based on pre-processing with a geodesic operator. It is applied to the segmentation of histological images of pigmented reconstructed epidermis acquired via Whole Slide Imaging.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's main move is a geodesic pre-processing step to feed global topology into an FCN for histology segmentation, but the abstract supplies no equations, results, or ablations so the claim stays untested.

read the letter

The central idea is to run a geodesic operator on the input images so that a standard FCN can pick up topological cues that its limited receptive field would otherwise miss. They apply this to segmenting pigmented reconstructed epidermis from whole-slide histology images. That is the whole contribution in one sentence. The approach is a direct engineering response to a known constraint rather than a new architecture or loss term. It keeps the network unchanged and moves the topology handling into the data preparation stage. That choice is reasonable when the goal is to stay within existing FCN pipelines used in medical imaging labs. The paper does identify a concrete setting where local context fails and offers one concrete workaround. The application domain is narrow but real: whole-slide images of reconstructed skin tissue where topology matters for the biological structures being segmented. On the positive side, the proposal is simple enough that a reader could try the same pre-processing on their own data without needing new code for the network itself. The stress-test note is right that nothing in the given material shows an internal contradiction. The weakest assumption is that the geodesic operator can be tuned so it adds the right long-range signal without creating new artifacts that the network then has to ignore. The abstract gives no equations for the operator, no parameter settings, and no before-after numbers on Dice scores or topology metrics. Without those, it is impossible to tell whether the method actually improves results or merely changes the input distribution. The citation pattern cannot be checked from what is supplied, but the claim itself does not rest on circular fitting. This work is aimed at practitioners who already run FCNs on histology and have noticed topology errors in their outputs. A reader facing the same receptive-field problem might borrow the pre-processing idea and test it quickly. It does not contain enough detail or evidence to serve as a reference for new theoretical work. I would send the manuscript to peer review because the idea is stated clearly enough that referees can ask for the missing implementation details and quantitative results; a desk reject would be too quick given the practical framing.

Referee Report

0 major / 3 minor

Summary. The paper proposes pre-processing input images with a geodesic operator to enable fully convolutional neural networks (FCNs) to exploit global topological information despite their limited receptive field size. The method is applied to semantic segmentation of histological images of pigmented reconstructed epidermis acquired via Whole Slide Imaging.

Significance. If validated, the approach provides a practical way to embed non-local topological cues into pixel-wise FCN inputs without modifying network architecture or receptive field. This is a standard technique for injecting global context and could benefit medical image segmentation tasks where topology is clinically relevant. The manuscript does not report machine-checked proofs or parameter-free derivations.

minor comments (3)

The abstract supplies no equations, implementation details, or quantitative results; the full manuscript should include the precise definition of the geodesic operator (e.g., distance metric and stopping criterion) and how it is normalized before being fed to the FCN.
Add an ablation study comparing the geodesic-preprocessed input against standard distance transforms or multi-scale inputs to isolate the contribution of the topological encoding.
Clarify whether the geodesic operator is applied to the raw image, to a preliminary segmentation, or to a distance map derived from annotations, and report any sensitivity to parameter choices in the operator.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of minor revision. The positive assessment of the geodesic pre-processing approach for embedding topological cues into FCN inputs is appreciated. Since no specific major comments were raised, we provide a brief response below.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper proposes an external pre-processing geodesic operator to embed global topological information into the input of an FCN, addressing the limited receptive field. This is presented as a standard image-processing step applied before network training, with no equations, fitted parameters, or predictions that reduce to the method's own outputs by construction. No self-citation chains or ansatzes are invoked to justify the core claim, and the approach remains independent of the network's learned weights or internal derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5608 in / 1072 out tokens · 36409 ms · 2026-05-25T14:46:57.455988+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A new method is proposed here to cope with non local information within convolutional neural networks. It is based on a geodesic reconstruction of the input image from the top and bottom of the image, channel-wise... Jn+1 = δ(Jn) ⋀ I
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the definition of this frontier is based on non local information

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 3 internal anchors

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SegNet: A Deep Convolutional Encoder-Decoder Architecture for Image Segmentation

Badrinarayanan, V., Kendall, A., Cipolla, R.: SegNet: A Deep Convolutional Encoder-Decoder Architecture for Image Segmentation. arXiv:1511.00561 [cs] (Nov 2015), http://arxiv.org/abs/1511.00561, arXiv: 1511.00561

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work page arXiv 2017

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ELECTRON

Fukushima, K.: Neural Network Model for a Mechanism of Pattern Recognition Unaﬀected by Shift in Position- Neocognitron. ELECTRON. & COMMUN. JAPAN 62(10), 11–18 (1979)

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Backpropagation applied to handwritten zip code recognition,

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IEEE Transactions on Image Processing 2(2), 176–201 (Apr 1993)

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Wang, D., Khosla, A., Gargeya, R., Irshad, H., Beck, A.H.: Deep Learning for Identifying Metastatic Breast Cancer. arXiv:1606.05718 [cs, q-bio] (Jun 2016), http://arxiv.org/abs/1606.05718, arXiv: 1606.05718

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ADADELTA: An Adaptive Learning Rate Method

Zeiler, M.D.: ADADELTA: An Adaptive Learning Rate Method. CoRR abs/1212.5701 (2012), http://arxiv.org/abs/1212.5701

work page internal anchor Pith review Pith/arXiv arXiv 2012