MATCH: Multi-faceted Adaptive Topo-Consistency for Semi-Supervised Histopathology Segmentation
Authors: Meilong Xu
Xiaoling Hu
Shahira Abousamra
Chen Li
Chao Chen
Venue: NeurIPS 2025 poster ()
Keywords: Digital pathology
Medical image segmentaton
Semi-supervised Learning
Primary area: applications

Abstract: In semi-supervised segmentation, capturing meaningful semantic structures from unlabeled data is essential. This is particularly challenging in histopathology image analysis, where objects are densely distributed. To address this issue, we propose a semi-supervised segmentation framework designed to robustly identify and preserve relevant topological features. Our method leverages multiple perturbed predictions obtained through stochastic dropouts and temporal training snapshots, enforcing topological consistency across these varied outputs. This consistency mechanism helps distinguish biologically meaningful structures from transient and noisy artifacts. A key challenge in this process is to accurately match the corresponding topological features across the predictions in the absence of ground truth. To overcome this, we introduce a novel matching strategy that integrates spatial overlap with global structural alignment, minimizing discrepancies among predictions. Extensive experiments demonstrate that our approach effectively reduces topological errors, resulting in more robust and accurate segmentations essential for reliable downstream analysis. Code is available at https://github.com/Melon-Xu/MATCH.

MATCH: Multi-faceted Adaptive Topo-Consistency for Semi-Supervised Histopathology Segmentation
Meilong Xu
Xiaoling Hu 2. Shahira Abousamra
Chen Li
Chao Chen'
Stony Brook University, NY, USA
2 Massachusetts General Hospital and Harvard Medical School, MA, USA
3 Department of Biomedical Data Science, Stanford University, CA, USA
Abstract
In semi-supervised segmentation, capturing meaningful semantic structures from unlabeled data is essential. This is particularly challenging in histopathology image analysis, where objects are densely distributed. To address this issue, we pro- pose semi-supervised segmentation framework designed to robustly identify and preserve relevant topological features. Our method leverages multiple perturbed predictions obtained through stochastic dropouts and temporal training snapshots, enforcing topological consistency across these varied outputs. This consistency mechanism helps distinguish biologically meaningful structures from transient and noisy artifacts. A key challenge in this process is to accurately match the corre- sponding topological features across the predictions in the absence of ground truth. To overcome this, we introduce a novel matching strategy that integrates spatial overlap with global structural alignment, minimizing discrepancies among predic- tions. Extensive experiments demonstrate that our approach effectively reduces topological errors, resulting in more robust and accurate segmentations essential for reliable downstream analysis. Code is available at https://iitub.coo/MMeoor Xu/MATCH
Introduction
Accurate segmentation of glands and nuclei in histopathology images critical for digital pathology. significantly influencing diagnosis, prognosis, and treatment planning by enabling precise quan- tification of morphological and structural tissue features (8] 43 27). Numerous fully-supervised segmentation methods [49 33 12
have demonstrated substantial success. How- ever, densely distributed cellular structures in histopathology images often induce topological errors, including false merges or splits, severely impacting clinical reliability. Additionally, fully super- vised methods demand extensive annotated datasets, which are costly, time-comseee cone-consuming, and not scalable 553 32. This limitation motivates exploring semi-supervised learning (SSL) strategies capable of leveraging abundant unlabeled data alongside limited annotations.
25 [7
Recent SSL approaches have significantly enhanced segmentation accuracy in contexts of limited 30
Nevertheless, these methods typically do not explicitly target topological errors, resulting seemingly small segmentation eπrors with consequential significant topological inaccuracies that affect segmentation robustness. To explicitly address topological errors, persistent homology 7 offers a rigorous mathematical framework that captures and characterizes topological features, such as connected components and loops in data across multiple scales. The output, persistence diagram, summarizes these structures as dots in 2D diagram. For each dot, the coordinate difference (y - xx) captures the persistence of the
supervision 779 (55 37
39 (64, 788, 81 777 82
33
76
84
73
74] 45)
Equal contribution. Email: Meilong Xu (meixu@cs..cc mmeixu@esstonmybooo.eeuu.)
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

0.7
Consistently Survive!
Death
Death
Birth
Birth
(a) lh, lh4
(b) TopoSemiScg
(c) Ours
Figure/ 1: Intuition of the proposed framework. (a) Colored likelihood maps are coming from the MC dropout. Connected components consistently matched in at least three predictions retain identical colors across instances, indicating topological stability; components shown in grey fail to reach this consensus and are therefore treated as topologically transient. (b) Limitation of ToppoSemiee [69], which relies on a fixed persistence threshold (ø 0.7, red dashed line) and therefore overlooks less-persistent yet meaningful structures (e.g. the violet point). (c) Our method adaptively identifies relevant topological structures without the need for human-selected thresholds.
topological structure across scales. Building upon this mathematical foundation, TopoSemiSeg (69 introduces topology-aware constraints into SSL frameworks, utilizing persistent homology to enforce topological consistency between teacher and student model predictions. Despite its effectiveness, TopoSemiSeg mainly relies predefined, hand-picked persistence threshold to identify meaningful topological structures. Such fixed thresholds are not data-driven, potentially biased, and can exclude relevant structures or retain irrelevant ones, shown in Figure T
To address this issue, we investigate how to identify reliable topological structures from model predictions in a robust and adaptive manner, and enforce model consistency over these structures. We first revisit the fundamental principles of semi-supervised learning robustness against perturbations. For an image without a training label, to identify reliable information, semi-supervised approaches typically add perturbations at the input level (i.e., augmentation) and at the model level (i.e., Monte Carlo Dropout). Pixel-level predictions that persist across these perturbations considered reliable and used to self-supervise the model.
Our main idea is to tightly couple this SSL reasoming. Moving beyond pixel-level, we identify topological structures that persist across different perturbations. These structures are considered reliable and used to self- the model. This idea avoids a hand- sicked threshold to determine reliable topological structures, and adaptively identifies truly relevant structures to enhance the model's topological reasoning power in an SSL setting.
principle with topological
Building on this idea, we propose novel SSL segmentation framework employing dual-level topo- logical consistency. Our method identifies significant topological features by examining predictions generated with different model perturbations. We formulate the structure correspondence task as contrastive learning problem, distinguishing stable features, i.c., those consistently detected across multiple predictions, from transient or noisy structures. To identify the stable topological structures, we introduce an advanced matching algorithm that integrates spatial overlap, topological persistence, and spatial proximity criteria to associate topological structures across diverse predictions reliably.
As for perturbations, we propose to employ Monte Carlo (MC) dropout perturbations 9
Mcanwhile,
stress the importance of temporal view of SSL. Previous works, such 33
36
35
51],
demonstrate that evaluating the predictions different training snapshots can reveal informative signals for robust prediction. Inspired by this, we propose also compare topological structures across model snapshots at different training epochs. By explicitly optimizing for dual-level topological consistency, our framework enhances structural coherence within the student model without relying on extensive pixel-wise annotations. Our key contributions can be summarized as follows:
We propose novel integration of topological reasoning into the semi- supervipe segmentation framework to robustly identify and preserve meaningful topological structures.
We introduce dual-level topological consistency, measuring structural stability from intra-perturbed predictions (MC dropout) and temporal training snapshots, effectively utilize unlabeled data.

---

Image
lh
lhz
hh
2h
th
l2
Wasserstein Matching
Betti Matching
Our Matching
Pabe Pusliwr
Figure 2: Comparison of our matching with Betti Matching [56] and Wasserstein Matching (21). We match two likelihood maps obtained from the same input histopathology patch. The birth critical points of the matched pairs are highlighted in the same color. Note that Wasserstein Matching gets most matches wrong, and Betti Matching also gets two matches wrong while pairing biologically unrelated features when lacking the guidance of the ground truth.
We develop a novel matching algorithm that integrates spatial overlap, topological persistence, and spatial proximity to accurately match topological structures across predictions.
Through extensive experiments on three widely used histopathology image datasets, our method significantly improves the topological accuracy while achieving comparable pixel-wise performance with limited annotations.
Related Works
Segmentation with Limited Supervision. Semi-supervised learning enhances medical image segmentation by effectively utilizing limited labeled data together with abundant unlabeled data. Con- sistency regularization approaches, such as the Mean Teacher model [57], ensure stable segmentation despite input variations [57 37
46 69). Pseudo-labeling progressively improves accuracy by leveraging confident model predictions on unlabeled data 775 50, 82). Adversarial training aligns segmentation outputs from labeled and unlabeled datasets using discriminator networks 24 34).
Additionally, uncertainty estimation methods such as MC dropout and Bayesian neural networks en- hance reliability by effectively handling uncertainty during pseudo-label generation
79 39 70),
while entropy minimization is used to reduce prediction uncertainty [13 66]. Contrastive learning strengthens segmentation robustness by training models differentiate similarities and distinctions
among data pairs, thereby boosting overall segmentation quality 77
76
Yopology-Awart Image Segmentation. Topology-awar methods have been proposed enforce
correct topology, like connectivity or correct counts in segmentation tasks 21 23 52
5
71
72
20
65
56
60 68 80 40). These methods typically use differentiable loss functions derived from
topological data analysis tools, including persistent homology [21] [56), discrete Morse theory 22
22
[6
topological interactions [15] 2 homotopy warping 20) centerline-based comparisons 52
61
These methods generally rely heavily on precisely annotated labels. Xu el al. (69 propose
TepoSe to combine SSL with topological constraints. Classical persistent homology-
segmentation methods rely on Wasserstein matching (21 69], which compares persistence diagrams based solely on feature lifespans. However, this approach may produce ambiguous or incorrect correspondences, illustrated in Figure 2 To alleviate spatial inconsistencies, several methods were proposed [56, 63). Betti Matching [66] embeds predictions and ground truth into a shared super-level filtration, ensuring alignment only among overlapping topological features. However, as shown Figure 2 it cannot ensure fully correct matching when the ground truth is missing and too sensitive to preserve some transient structures. Our proposed MATCH-Pair could achieve almost completely accurate matching without ground truth.
3 Methodology
The motivation our proposed framework is to identify meaningful topological structures directly from perturbed predictions without the ground truth. The main challenge is to accurately match corresponding topological structures across multi-facet predictions that often contain substantial noise and variability. To overcome this challenge, introduce MATCH-PP Pair, a pairwise matching

---

algorithm, and MATCH-Global, an extended global matching algorithm, to robustly identify stable structures across multiple predictions. Building upon these matching algorithms, we propose dual- level topological consistency constraints: intra-topological consistency, enforcing consistency across multiple stochastic predictions, and temporal-topological consistency, ensuring stability across consecutive training snapshots. These consistency constraints directly optimize the student model, enabling it to learn robust segmentation representations from limited labeled data.
The proposed MATCH framework leverages labeled data via supervised loss and unlabeled data through pixel-wise and dual-level topological consistency.
Our method overview is shown in Figure 3
In this section, we will start by introducing the preliminaries of classic SSL. Next, we will use 3 subsections to introduce MATCH-Pair. MATCH- G-oballl.,, and the dual-Ievel topological consistency.
Preliminaries: SSL training. We address the semi- sunn--sueevised image segmentation problem by leveraging a teacher-student framework, widely adopted paradigm in semi- semi-supervised learning No,
[57] Let D = {(o ) denote the labeled dataset, where x/ represents the input image and vf E {0, 1 is the corresponding pixel-wise annotation. Let Dy = {ay Ne, denote the unlabeled dataset. In our setting, the number of labeled samples is significantly smaller than the number of unlabeled samples, i.c., N1 < Ny. Our objective is to train a segmentation model fo. parameterized by 0, that accurately predicts segmentation masks using labeled and unlabeled data.
In this framework, the student model fo, is trained using both supervised and unsupervised losses, while the teacher model fo, provides stable targets for the student by being updated as an exponential
moving average (EMA) of the student's parameters: 0 ad((T) + (1 a))o++1) where controls the update rate. For the supervised loss on labeled data, we employ a combination of
Dice loss and cross-entropy loss to capture both overlap and pixel-wise discrepancies, Lrup
Diice (Je, (x=())))))) + LCE (x), (x*),
To leverage the unlabeled data, we enforce consistency between the student and teacher predictions. Specifically, the student receives a strongly augmented version of an unlabeled image x while the teacher processes a weakly augmented version. The pixel-wise consistency loss is defined as the cross-entropy between the student and teacher outputs, Leons = [fe(fe, (A.(z")) )), fo, (xw(x"))) where A, and A., denote strong and weak augmentations, respectively.
3.1 MATCH-Pair: Spatial-Aware Pairwise Matching
Accurate identification of corresponding topological structures between the likelihood maps is crucial for robust histopathology image segmentation. We employ persistent homology with super-level set filtration to extract O-D topological features from likelihood maps, producing persistence diagrams that characterize each component by its persistence and critical points. To find correspondence between different persistence diagrams, traditional methods based on Wasserstein distance emphasize topological persistence without considering spatial relationships, often leading to incorrect associations between spatially distant yet similarly persistent features. In contrast, approaches based solely on spatial overlap tend to match transient structures of minimal significance incorrectly. To address these limitations, we propose MATCH-Pair, Hungarian overlap-matching algorithm that integrates spatial overlap, topological persistence, and spatial proximity. The overall pipeline is depicted in Figure
Given two likelihood maps 1h, lh2 [0, 1] which are the softmax-activater outputs of the final UNet layer, we compute the persistence diagrams: Dgm(Ihx) = {1h, de), k E [1,2] with the persistence pers; = Id: - b,]. Each persistence pair (b,, yields a connected spatial region M,, defined by flood-fill algorithm [54). This algorithm generates a binary mask M, starting from the birth pixel b; The region is expanded iteratively to neighboring pixels, provided that their likelihood exceeds the threshold 1- d,
To compute the relative significance of each structure, the persistence values are normalized to derive weighting factor: Wp.s max
pers e (1, Here, i and index the topological features 3
pers
from the stt and 2nd persistence diagrams respectively, where i 1...., ni and j E 1,..., 12 with ni and ni being the number of features in each diagram. & distinguishes between the two likelihood maps being compared. The notation Wa,i refers to the normalized persistence weight of the i-th topological feature in the k-th likelihood map.

---

[3]
Likelhhodd
intra
Student Model
fuzz
fa.
[b]
Lemp
Student Model
[o,
Teacher
fa,
Iterations
Figure Overview of the proposed MATCH framework with dual-Ievel topological consistency. Note that the Listra and Lump are used to directly optimize the parameters of the student model.
Flood
max persy
Hungart
S(M
Uh.
Dooth
Flood
perss pers
-FIII
Birth
Figure 4: Pipeline the MATCH-Pair algorithm between two persistence diagrams.
To evaluate the similarity between spatial masks M1,5 and M2.j. a combined metric that integrates spatial overlap, normalized persistence weights, and spatial proximity is defined as:
[M M2.il
(1-
déj
Si = W1,1 W2.5
[M U M222
dmax
where di is the Euclidean distance between birth critical points of the corresponding masks, and dmax denotes the maximum distance among all mask pairs. This similarity metrie ensures the prioritization of spatially close, persistent, and well-overlapping structures.
global one-to-one assignment between features from the two maps is obtained via the Hungarian algorithm [3I, minimizing the cost (defined as the complement of similarity):
min (1-
Tiz {0, 1}
Pairs achieving scores above a predefined threshold Tprimary constitute valid matches.
3.2 MATCH-Global: Multi-faceted Global Matching
While MATCH-Pass addresses an optimal correspondence between two persistence diagrams, many practical scenarios often involve multiple stochastic predictions (facets). Finding the corresponding topological structures among multiple facets a challenge. Thus, we extend MATCH- to MATCH- Global, global multi-facet matching approach to link homologous O-dimensional components consistently across all facets, assigning global indices to anatomical or topological structures.
Given a series of likelihood maps L [lhe )T lhe E 10 each generates a persistence diagram: Dgm, = [(br,, de Each pair (br,i, de,s) corresponds to spatial mask Mr,s, the normalized persistence weight Wa,s |de, be,l//mxxx
//maxx à Idv,, 3 by l and birth-critical point Ct
5

---

Matching is performed sequentially across facets, For each adjacent pair of facets (t, t + 1) we form the weighted overlap matrix:
1 Wet WcH,,,oU(MMM. M,41,)) (1 -
with d max;,j |cc. - via the proposed MATCH-Pair algorithm.
1.,+,,ll -e-+1, introduces a soft spatial penalty. Optimal assignments are solved
These matches form an undirected graph G = (V.8) with vertices V {tt, i S ) representing the structures and edges & = U g(t) indicating matches. Connected K components {Cx k= k=] of G are identified by preadth-first search, providing globally consistent identities: C₁ ((1,)) mask M,,, belongs identity k).
1
Thus, the global multi-facet matching framework integrates pairwise correspondences into globally coherent tracks, robustly accommodating missing detections, splits, and merges, thereby ensuring topological consistency across multiple facets.
3.3 Dual-Level Topological Consistency
After identifying consistent topologiaal structures across multiple facets, propose dual-level topological consistency losses to enhance segmentation reliability and coherence. Specifically, we introduce two complementary loss terms: toraa-ooplloical consistency, which ensures consistency among stochastic predictions from MC dropout realizations [9] and temporal-topological consistency, which maintains consistency across consecutive training iterations.
In both scenarios, topological features are extracted from multiple prediction facets. We then apply proposed MATCH-Global algorithm (see Section 3.2) to classify these topological structures into two distinct categories: matched (Cmmmrhh, Cumpp)) representing features consistently identified across multiple predictions, and unmatched (Cuummatch, Cumpuichh) ), denoting features that are inconsistent or unstable across predictions. Specifically, matched structures are encouraged toward optimal probabil- ity distributions at their birth and death critical points, whereas unmatched structures, indicative of prediction uncertainty instability, are driven toward shorter topological lifespans. Formally, we define the associated losses as:
Cmarch (t, i) (P!".) 2 (1))
(t,
(P(C)) -P(!))
where P!f and ((t) represent the predicted probability values at the birth (br,;) and death (d.,) critical points, respectively, of the i-th topological feature extracted from the I-th prediction.
The intra-topological consistency loss aggregates these penalties over multiple stochastic predictions through MC dropout within each iteration:
Banza
Cintra
Σ
Cmatch (t,
Σ
Caiag (t,
Bintra
C
(B)) intra
Cuummachh, (b)
(1.6)
(
where Binra indicates the number MC dropout predictions within each iteration. Similarly, the temporal-topological consistency enforces the constraints across predictions from consecutive training snapshots:
Breop Σ 6=1
Cremp
Σ
Cmatch (t,
Laiag (t,
Bremp
Clemp
Crmp
Cumppacch.
(1,6)
(t.i)Ecmmzat..
where Bremp presents the number of temporal training snapshots. Finally, our dual-level topological consistency losses are integrated into the overall training objective alongside the supervised and pixel-wise consistency terms:
Lintal Luup + Acons + Xintra Cintra Xnemp
where hyperparameters Acous Aimma and Atemp balance their respective contributions, ensuring the model jointly meets pixel-level accuracy and robust topological coherence.

---

Table Quantitative results on three histopathology image datasets. We compare our method with several state-of-the-ee semi- supervisee medical image segmentation methods on two settings of 10%
and 20% labeled data. The statistically significant best results are highlighted in bold, while the second-best are marked with underline.
Pixel-wise
Topology-wise
Dataset
Label Ratio
Method
Dice Obj
BE
BME
DIU !
MT
S7
0.821 0.006
2.238
0.153
62,250
3.127
74,630 +: 2.967
80.100
3.809
UA-MT
79
0.837 0.005
1,703 0.112
66.450
3.218
65,420 2.847
1,732 0.118
74.600
3,407
68,300 3.004
10
XNet
83
0.872 0.004
0,578 0.053
15.050
1.118
55,880 2.516
PMT
0.876 0.004
0.520 0.051
14.200
1.013
$7,100 2.638
SepoSemiSeg
69
0.884 0.002
0,227 0,014
10.475
0.458
49.690 1,947
Ours
0,888 0.002
0.197 = 0.012
9.175 + 0.580
45,950 + 1,880
CRAG
MT 57
0.858 0.008
2,603 0.161
99.025
3.912
95.215 3.487
EM 59
0.869 0.006
1,933 0.136
75.225 3.772
63,823 3.139
MT
0.859 0.006
1.822 +: 0.129
3.586 70.850
61.138 2.918
URPC 39
0.849 0 0.007
2,489 0.152
99,500 4.085
87,681 3.276
20
XNet
83
0.883 + 0.005
0.422 = 0.055
10,900 1.127
50.537 2.547
TopoSemiSog
69
0.898 0.004
0,226 0.019
8,575 + 0.736
43,712 + 1.842
Ours
0.909 0.005
0.188 0.018
7,425 1 0.570
40.250 1.720
(Full)
Fully-Sure Superviee
0.223
0.790 0.005
2.392 0.162
31,125 3.274
76.130 2.965
EM
59
0.819 : 0.006
1.431 + 0.143
3,846 19.188
61.245 + 3.302
URPC
79
0.849 0.004
1.155 0.123
19.588 3,408
54,832 3.017
10
XNel
S3
0.874 0.003
0.843 0.051
14.238 1.154
40,912 2.422
PMT
10
0.872 0.004
0.798 0.052
13.920
1.097
39,850 2.487
TopoSemiSeg
69
0.878 0.003
0.551 + 0.014
8,300 + 0.478
35.845 + 1.965
Ours
0,501
0.391 7,850
30.525
GlaS
MT S7
0.863 0.005
2.126 0.171
29.963 3,987
64,275 3,496
EM 39
0.865 0.006
1.255 + 0.138
17.275 3.783
58.673 3.255
UA-MT
0.866 0.005
1.123 0.132
18.038 3.599
53.014 3.069
URPC 39
0.878 : 0.004
0.759 = 0.067
14,350 1.212
42,587 + 2.601
20
XNel
83
2.328
PMT
10
0.887 0.003
0.698 + 0.062
9,980 + 1.118
34,805 2.271
TopoSemiSeg
0.895
0.510 0.053
9,825 0.813
30,462
Ours
0.894 0,004
0,392 0.056
7,925 + 0.725
26.175 1.633
100 (Full)
Pully- Supervidd
0.917 0.006
0,273 0.026
6,875 + 0.276
19.620 + 0.712
MT
S7
0,748 0.006
10.210 = 0.486
292.857 = 6.542
1526.079 = 35.842
UA-MT
79
0.741 + 0.007
10:227 + 0.497
255 428 + 5.983
1316.272 + 30.216
79
6,829 0.319
214 = 5.327
1098 372 24.392
10
XNet
83
0.762 0.005
7.152 0.338
220.405 = 4.611
1122. 799 25.116
PMT
0.764 0.004
7.515 0.352
227.650 1: 4.805
1210.400
26.954
TopoSemiScg
0.003
3.067
Ours
0.785 0.003
5,594 0.361
192.863 :: 1.137
1011.857
12,648
MoNuSeg
MT
0.767 0.005
12.522 0.547
246. 786 + 8.018
1350, + 32.407
EM
0.777 0.006
0.335
198.571 6.731
1142.661
27.581
UA-MT 79
0.772 1 0.007
9.406 1: 0.444
246.857 = 7.944
1336 684 = 31.268
39
20
XNet
33
0.776 0.003
6,750 + 0.316
198.525 + 5.421
1117.406 + 26.014
PMT
6,500 0,308
195, 125 - 6.289
1080.476 + 25,145
TopoSemiSog
69
0.793 0.004
5,150 0.145
188.642 = 3.255
1105 946 18.486
Ours
0.790 0.006
0.156 4,300
179.325 3: 2.383
982.286 14,953
(Full)
Fully- Supervied
142.429
4.674
729.017 17.662
4 Experiments
We conduct comprehensive evaluations on three publicly available histopathology image datasets on both pixel-wise and topology-wiss metrics. We benchmark our method against classic and recent
Mate-of-the-art semi-supervised segmentation methods, including MT [57 EM 59 UA-MT 79
URPC 39 XNet 83 PMT [0]. and TopoSemiSeg 669
Implementation Details. The implementation details will be provided in the Supplementary.
7

---

(a) Original Img GT
(b) /errs
1h2/er22
(d) tha/erra
(c) tha/erra
(1) Result Uncertainty
Figure 5: Qualitative illustration MC dropout predictions (after the model convergence). Top row: original patch, the four likelihood maps, and the final segmentation. Bottom row: ground-truth mask, corresponding error maps, and the pixel-wise variance (uncertainty) map.
Datasets. We evaluate our proposed method on Colorectal Adenocarcinoma Gland (CRAG) M. Gland Segmentation in Colon Histology Images Challenge (GlaS) 53 and Multi-Organ Nuclei Segmentation (MoNuSeg)
32 More details are provided in the Supplementary.
Evaluation Metrics. To better evaluate our proposed method, we use pixel-wise metrics including
Object-level Dice Score (Dice (Dice_obj) [67]: topology-wise metries including Betti Error (21), Betti
Matching Error [56). and Discrepancy between Intersection and Union (DIU) 40 More details are provided in the Supplementary.
4.1 Results
Uncertainty Throughout the Topological Consistency. As illustrated in Figure 5 our proposed MATCH not only produces robust segmentation result (top, (f)) but also furnishes an informative pixel-wise uncertainty map without any mucertainty-speciffe training objective or doing post hoc calibration. Visually, the variance map (bottom, (f)) concentrates along the gland boundaries where the four likelihood maps disagree, and these regions coincide almost perfectly with the binary error maps (bottom, (b) (e)). Quantitatively, the Pearson correlation coefficients (PCC) 48 between the uncertainty and the error maps are 0. 16768, 0.728, 0.757, and 0.753 for the four facets, respectively. This confirms that the uncertainty tightly coupled with prediction errors. Hence, reliable uncertainty estimation and the attendant suppression of spurious structures emerge naturally as a by-product of the proposed consistency mechanism, with no additional supervision or model modification required.
Quantitative Results. As shown in Table I across the three histopathology image datasets, our proposed method consistently achieves superior performance compared state-of-the-aar semi- supervised segmentation methods, under both 10% and 20% labeled data settings. Specifically, our method yields higher topology-wist accuracy with comparable pixel-wise performance. These results collectively illustrate that our framework effectively leverages limited annotations to achieve robust segmentation accuracy and enhanced topological fidelity.
Qualitative Results. We provide the qualitative results Figure 6 The qualitative comparison highlights that our proposed method consistently outperforms other semi-supervised methods in preserving accurate glandular structures and topology across various histopathology samples. The comparative methods exhibit notable topological errors, including fragmentation, merging, and boundary leakage, as indicated by the red boxes. In contrast, our method effectively mitigates these errors, demonstrating superior robustness in maintaining topological integrity and accurate boundary delincation, thereby underscoring its effectiveness for precise medical image analysis tasks.
4.2 Ablation Study
To comprehensively explore the robustness and efficacy of our proposed strategy, hyperparameter- selection, and experimental settings, we conduct the ablation experiments on the CRAG dataset using 20% labeled data.

---

(a) Image
(b) (c) MT
(e) UA-MT
(f) URPC
(e) XNet
(h) PMT
() TopoSemi
(i Ours
Figure 6: Qualitative results for semi-supervised methods on 10% and 20% labeled data. Rows 1-2 correspond to CRAG dataset, rows 3-4 correspond to the GlaS dataset. From left to right: (a) raw image, (b) ground-truth mask, (c) to (i) present the 7 baselines. ()) indicates the results of our method. The regions prone to topological errors are highlighted in red boxes.
Table Ablation study on matching algorithms.
Table 3: Effect of IoU ppatial-proximitg (SP).
Matching
Pixel-wise wise
Topology wise
IoU SP
Pixel-wis
Topulugy-wise
Dijc_ 1
BE 4
BME
DNU 4
Dicc_obj
BB
BME
DIU J
Wasser. 21
08844 0.007
0,423
0.026
9:647 0.866
58.592
2.594
0,890
0.005
0.233
0.020
8,300
0,650
43.750 4: 2.100
Beui (5h
0,889 1 0.005
0,237
1 0.01 8.216 = 0,717 44.157 1: 2.146
X
0.882 1: 0.006
0,247
- 0.022
9,600 E. 0.650
46 66.000
1: 2.250
Ours
0.909 0.005
0.188
0,088
7,455 = 0,570 40.250
1.720
0.909 0.005
0,188
0.018
7,425
0.570
40.250 :: 1.720
Ablation Study on Matching Algorithm. To validate the effectiveness of our proposed matching algorithm, we compare it against established alternatives, including Wasserstein Matching [2] and Betti-Matchin [56]. As shown in Table 2 our algorithm consistently achieves superior performance in both pixel- and topology-we -ise metrics. Specifically, Wasserstein Matching, relying exclusively on persistence values without spatial information, exhibits the worst results. Although Betti-Mo Matching incorporates spatial context, it still performs suboptimally compared to our method.
Ablation Study on IoU and Spatial Proximity (SP). To validate the effectiveness of the individual items of our matching cost, we conduct an ablation study on loU and spatial proximity. The results in Table 3 quantitatively substantiate the complementary roles of the IoU and the spatial proximity factor in our Hungarian assignment cost. Removing either the proximity or the overlap item could degrade the performance. The overlap itself cannot fully distinguish spatially adjacent structures. These results demonstrate that both items are necessary to achieve topologically accurate matching.
Sensitivity Analysis on Bintra and Bremp- We further analyzed the sensitivity of our method to the number of MC dropout samples Biua and temporal training snapshots Beemp. Table shows that employing too few facets yields unreliable estimation of topological consistency, resulting in suboptimal segmentation performance. Conversely, increasing the number of facets beyond an optimal point introduces redundant information and additional variability, degrading model performance. Therefore, the optimal number that strikes a practical balance, ensuring the best performance while remaining computationally efficient.
Table 4: Influence of B and Bremp-
Table 5: Efficacy of Cintra and Lump-
Brut Buem
Pixel-wise
Topelogy-wse
Lut Cump
-wise
Topology-wise
RE
1
BF
RMF
DIU
0.778 + 0.010 0,255
9.350
0.620
48.600 : 2,300
0.862
0,400 + 0,02 11,680 + 59,930 + 150
0.892 +: 0.007 0214 0,000
105
0.00
105
2,050
0.006
0.215
0,000
7,920
0.590
44.750
:: 1.970
0.872 + 0.111
0.275 0.023
10,050
+ 0,600
54.250 +: 2.251
X
0,882 + 0.008
0.238
1 0031
8.540
2: 0.450
45.310
1 2.400
4 0.909 - 0,08
0,188 IMIIS 7,425 0.570
40.250 - 1.730
/
0.909 + 0.0S5
0.188
+ 0J18
7.425
0.570 40.250 + 1.720
Ablation Study on Loss Components. To evaluate the contributions of individual loss terms in
our dual-level topological consistency framework, we conduct experiments selectively enabling or disabling the Limra and Cemp- As presented in Table 5 each loss individually improves the pixel- and

---

Table 6: The results the Roads dataset.
Table 7: Density- quantitative results.
Labeled Ratio Method
RF4
RMF
DIU 4
Setting
Dice ségj
RF..
RMF
10%
Ours
TapoSamiSeg 8.324 + 9.681
7,892 +: 0,64 8.147 + 0.521
0.647
9.576 + 0.583
0.671
Sparse (Chur. < 30 cells)
Crowded (02) 2 100 cells)
0,002 0.004
0.75 0:009
4,620 + 0.140
6,890
+ 0.340
163 132 + 2,156
198.525
20%
TopoSemiSeg
Outs
7.467 +: 0.582 8.213
6,983 1: 0,507 7,024 : 0.436 8,149 1: 0.492
0,514
9.387
0.598
Ours (whole test image)
Crowded (Ours, 2 100 cells)
0,774 0.007 5,610
0.790 0:006
4,930 + 0.156
+ 0.198
186313 + 2.715
179.225
2,383
topology-wise performance compared to the baseline without these constraints. Combining both losses achieves the strongest overall performance, confirming that Litra and Ciemp complement each other by addressing different sources of topological inaccuracies stochastic noise within single facets and structural inconsistencies across training iterations.
Ablation Study on 1-D Topological Features. We mainly focus on 0-D topological features due to the following factors: For the primary application in our study (gland and nuclei segmentation). the most critical topological errors involve incorrect splitting or merging of individual structures, which are well-captured by O-D persistent homology. For the validation on I-dimensional structures. we conducted additional experiments on the Roads dataset [42]. The results are shown in Table O verifies that our method could learn good topological representations from unlabeled data on 1 d-dimensional topological features.
Crowding-Awwer Aware Ablation Study. To quantify the influence nuclei density on model performance, randomly cut the test images into patches of size 256 x 256. For every patch, count nuclei in the ground-truth truth instance map. Patches with <= 30 nuclei are labeled Sparse; those with >=100 nuclei are labeled Crowded. We sampled 14 samples for fair comparison and show the results below in Table 7 The experiments above verify that our approach is density-aware. It achieves state-of-the arrt performance typical tissue, excels in sparse fields, and maintains clear advantage over the strongest baseline under extreme nuclear crowding.
Ablation Study on the Alternatives of MC-dropout. We choose two alternative perturbation methods: Variational Inference (VI) (26). which generates multiple predictions by sampling from the learned variational posterior distribution, and Temperature Scaling [14), which produces diverse predictions through multiple sampling from temperature modulltee probability distributions. The experiments are conducted on CRAG 20% labeled data, and the results are shown Table g
Table 8: Ablation of perturbation methods.
Table Comparison with foundation models.
Method
Dice_obj f
Bl
BME L
Method
Dice_obj 1
BB J
BML
Variational Inference
0.895
0,006
0.212 d: 0.022
9.125 1: 0.685
LoRA SAM
0.882 0.006
0.660 4: 0.012
27,300
2.937
Temperauuee Scaling
0.891 = 0.007
0.258
: 0.025
9,850 1: 0.795
LoRA- MedSAM
0.896 1 0.005
0,268
:: 0.025
11.275 1: 1.899
MC- Dropout
0.909 0.005
0.188
0,018
7.425
0.570
0,909 0.005
0,188
0.018
7,425 0.570
Comparison with Self-Supervised Methods Finetuned Limited Labeled Data. To comprehen- sively evaluate the effectiveness of our method, we compare our method against some foundation models, like SAM (29] and MedSAM [41]. We use LoRA [19] finetune these two models using 20% labeled data on the CRAG dataset and report the performance in Table 9 The results show that even with powerful foundation models, like SAM or MedSAM, topological errors can still exist without explicit topological modeling.
Conclusion
We present semi- supervised segmentation framework that preserves significant topological structures in histopathology with limited annotations. Dual-level topological consistency across Monte Carlo dropout predictions and temporal training snapshots separates stable biological patterns from noise. For alignment, MATCH-PAR -air achieves spatially accurate matching between noisy persistence diagrams by combining spatial overlap, persistence, and proximity, and MATCH- scales to multiple facets. Experiments show consistent gains in robustness and substantial reductions in topological errors, enabling more reliable downstream analyses in digital pathology.
10

---

Acknowledgement. This work was partially supported by grants NSF CCF-2144901, NIH ROINS143143, and NIH RO1CA297843.
References
[1] Hritam Basak and Zhaozheng Yin. Pseudo-label guided contrastive learning for semi-supervised medical image segmentation. In CVPR, 2023.
[2] Alexander H Berger, Laurin Lux, Nico Stucki, Vincent Bürgin, Suprosanna Shit, Anna Banaszak, Daniel Rueckert, Ulrich Bauer, and Johannes C Pactzold. Topologically faithful multi-class segmentation in medical images. In MICCAI, 2024.
[3] David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin A Raffel. Mixmatch: A holistic approach to semi-supervised learning. In NeurIPS, 2019.
[4] Liang Chich Chen, Yukun Zhu, George Papandreou, Florian Schroff, and Hartwig Adam. Encoder-decoder with atrous separable convolution for semantic image segmentation. In ECCV, 2018.
[5] James R Clough, Nicholas Byrne, Ilkay Oksuz, Veronika Zimmer, Julia A Schnabel, and Andrew King. topological loss function for deep-learning based image segmentation using persistent homology. TPAMI, 2020.
[6] David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko. Lipschitz functions have I p-stable persistence. Foundations of Computiaioonal Mathematics, 2010.
[7] Herbert Edelsbrunner and John Harer. Computational topology: an introduction. American Mathematical Soc., 2010.
[8] Matthew Fleming, Sreelakshmi Ravula, Sergei F Tatishhhev, and Hanlin L. Wang. Colorectal carcinoma: Pathologic aspects. Journal of gastrointestinal oncology, 2012.
[9] Yarin Gal and Zoubin Ghahramani. Dropout bayesian approximation: Representing model uncertainty
in deep learning. JCML, 2016.
[10] Ning Gao, Sanping Zhou, Le Wang, and Nanning Zheng. Pmt: Progressive mean teacher via exploring temporal consistency for semi-supervised medical image segmentation. In ECCV, 2024.
[11] Simon Graham, Hao Chen, Jevgenij Gamper, Qi Dou, Pheng-Ann Heng, David Snead, Yee Wah Tsang, and Nasir Rajpoot. Mild-net: Minimal information loss dilated network for gland instance segmentation in colon histology images. MedIA, 2019.
[12] Simon Graham, Quoe Dang Vu, Shan E Ahmed Raza, Ayesha Azam, Yee Wah Tsang, Jin Tae Kwak, and Nasir Rajpoot. Hever-net: Simultancous segmentation and classification of nuclei in multi-tissue histology images. MedIA, 2019.
[13] Yves Grandvalet and Yoshua Bengio. Semi-supervised learning by entropy minimization. In NeurlPS, 2004.
[14] Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modemn neural networks. In ICML, 2017.
[15] Saumya Gupta, Xiaoling Hu, James Kaan, Michael Jin, Mutshipay Mpoy, Katherine Chung, Gagandeep Singh, Mary Saltz, Tahsin Kure, Joel Saltz, et al. Leamingg topological interactions for multi-class medical image segmentation. In ECCV, 2022.
[16] Saumya Gupta, Yikai Zhang, Xiaoling Hu, Prateek Prasanna, and Chao Chen. Topology-awan uncertainty for image segmentation. In NeurlPS, 2023.
[17] Hongliang He, Jun Wang, Pengxu Wei, Fan Xu, Xiangyang Ji, Chang Liu, and Jie Chen. Toposeg: Topology- aware nuclear instance segmentation. In ICCV, 2023.
[18] Fabian Hörst, Moritz Rempe, Lukas Heine, Constantin Seibold, Julius Keyl, Giulia Baldini, Selma Ugurel, Jens Siveke, Barbara Grünwald, Jan Egger, al. Cellvit: Vision transformers for precise cell segmentation and classification. MedlA, 2024.
[19] Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang. Lu Wang. Weizhu Chen, al. Lora: Low-rank adaptation large language models. In ICLR, 2022.
[20] Xiaoling Hu. Structure-aware image segmentation with homotopy warping. In NewrIPS, 2022.
11

---

[21] Xiaoling Hu, Fuxin Li, Dimitris Samaras, and Chao Chen. Topology-press preserving deep image segmentation. In NeurlPS, 2019.
[22] Xiaoling Hu, Dimitris Samaras, and Chao Chen. Learning probabilistic topologiaal representations using discrete morse theory. ICLR, 2023.
[23] Xiaoling Hu, Yusu Wang, Li Fuxin, Dimitris Samaras, and Chao Chen. Topology-awee aware segmentation using discrete morse theory. In ICLR, 2021.
[24] Wei-Chih Hung. Yi-Hsuan Tsai, Yan-Ting Liou, Yen-Yu Lin, and Ming-Hsuan Hsuan Yang. Adversarial learning for semi-supervised semantic segmentation. arXiv preprint arXiv: 1802.09934 2018.
[25] Fabian Isensee, Paul F Jacger, Simon AA Kohl, Jens Petersen, and Klaus H Maier-Hein. nnu-net: a self-configuring method for deep learning-basee based biomedical image segmentation. Nature methods, 2021.
[26] Michael I Jordan, Zoubin Ghahramani, Tommi S Jaakkola, and Lawrence K Saul. An introduction to variational methods for graphical models. Machine learning, 1999.
[27] Takahiro Karasaki, David A Moore, Selvaraju Veeriah, Cristina Naceur-Lombardell, Antonia Toncheva, Neil Magno, Sophia Ward, Maise Al Bakir, Thomas BK Watkins, Kristiana Grigoriadis, et al. Evolutionary characterization of lung adenocarcinoma morphology in tracerx. Nature medicine, 2023.
[28] Diederik Kingma. Adam: A method for stochastic optimization. arXiv preprint
2014.
[29] Alexander Kirillov, Eric Mintun, Nikhila Ravi, Hanzi Mao, Chloc Rolland, Laura Gustafson, Tete Xiao, Spencer Whitchead, Alexander C Berg, Wan-Yen Lo, et al. Segment anything. In ICCV, 2023.
[30] Aishik Konwer, Zhijian Yang, Erhan Bas, Cao Xiao, Prateek Prasanna, Parminder Bhatia, and Taha Kass-Hout. Enhancing sam with efficient prompting and preference optimization for semi-supervised medical image segmentation. In CVPR, 2025.
[31] Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly. 1955.
[32] Neeraj Kumar, Ruchika Verma, Deepak Anand, Yanning Zhou, Omer Fahri Onder, Efstratios Tsougenis, Hao Chen, Pheng-Ann Ann Heng. Jiahui Li, Zhiqiang Hu, et al. A multi-organ nucleus segmentation challenge. TMI, 2019.
[33] Samuli Laine and Timo Aila. Temporal ensembling for semi-supervised lcarning. arXiv preprint arXiv:1610.022. 1610.022222 2016.
[34] Tao Lei, Dong Zhang, Xiaogang Du, Xuan Wang, Yong Wan, and Asoke K Nandi. Semi-supervised medical image segmentation using adversarial consistency learning and dynamic convolution network. TMI, 2022.
[35] Chen Li, Xiaoling Hu, Shahira Abousamra, and Chao Chen. Calibrating uncertainty for semi-supervised crowd counting. In ICCV, 2023.
[36] Chen Li, Xiaoling Hu, and Chao Chen. Confidence estimation using unlabeled data, In ICLR, 2023.
[37] Xiaomeng Li, Lequan Yu, Hao Chen, Chi-Wing Fu, Lei Xing, and Pheng-Ann Heng. Transformation- consistent self-ensembling model for semi-supervised medical image segmentation. TNNLS, 2020.
[38] Xiangde Luo, Jieneng Chen, Tao Song, and Guotai Wang. Semi-supervised medical image segmentation through dual-task consistency. In AAAI, 2021.
[39] Xiangdc Luo, Guotai Wang, Wenjun Liao, Jieneng Chen, Tao Song. Yinan Chen, Shichuan Zhang, Dimitris N Metaxas, and Shaoting Zhang. Semi-supervised medical image segmentation via uncertainty rectified pyramid consistency, MedlA, 2022.
[40] Laurin Lux, Alexander Berger, Alexander Weers, Nico Stucki, Daniel Rueckert, Urrich Bauer, and Johannes Paetzold. Topograph: An efficient graph- framework for strictly topology preserving image segmentation. In ICLR, 2025.
[41] Jun Ma, Yuting Hc, Feifci Li, Lin Han, Chenyu You, and Bo Wang. Segment anything in medical images. Nature Communications, 2024.
[42] Volodymyr Mnih. Machine learning for aerial image labeling. University of Toronto (Canada), 2013.
12

---

[43] Rodolfo Montironi, Roberta Mazzuccheli, Marina Scarpelli, Antonio Lopez-Beltran, Giovanni Fellegara, and Ferran Algaba. Gleason grading of prostate cancer in needle biopsies radical prostatectomy specimens: contemporary approach, current clinical significance and sources of pathology discrepancies. BJU international, 2005.
[44] Tanya Nair, Doina Precup, Douglas I. Arnold, and Tal Arbel. Exploring uncertainty measures in deep networks for multiple sclerosis lesion detection and segmentation. MedlA, 2020.
[45] Thanh-Huy Nguyen, Nguyen Lan Vi Vu, Hoang-Thien Nguyen, Quang-Ving Vinh Dinh, Xingjian Li, and Min Xu. Semi-supervised histopathology image segmentation with feature diversified collaborative learning. In AAAI Bridge P Program on Al for Medicine and Healthcare, 2025.
[46] Yassine Ouali, Céline Hudelot, and Myriam Tami. Semi-super semantic segmentation with cross- consistency training. In CVPR, 2020.
[47] A Paszke. Pytorch: An imperative style, high-performance deep learning library. arXiv preprint arXiv: 191.01091,.01003. 2019.
[48] Karl Pearson. Vii. note on regression and inheritance in the case of parents. proceedings of the royal society of London, 1895.
[49] Olaf Ronneherger, Philipp Fischer, and Thomas Brox, U-net: Convolutional networks for biomedical image segmentation. In MICCAI, 2015.
[50] Constantin Mare Seibold, Simon ReiB, Jens Kleesick, and Rainer Stiefelhagen. Reference-guidee pscudo- label generation for medical semantic segmentation. In AAAI, 2022.
[51] Wooscok Shin, Hyun Joon Park, Jin Sob Kim, and Sung Won Han. Revisiting and maximizing temporal knowledge semi-supervised semantic segmentation. arXiv preprint arXiv:2405.20610, 2024.
[52] Suprosanna Shit, Johannes C Pactzold, Anjany Sekuboyina, Ivan Ezhov, Alexander Unger, Andrey Zhylka, Josien PW Pluim, Ulrich Bauer, and Bjoern II Menze. cldice-a novel copology-preservingg loss function for tubular structure segmentation. In CVPR, 2021.
[53] Korsuk Sirinukunwattana, Josien PW Pluim, Hao Chen, Xiaojuan Qi, Pheng- Ann Heng. Yun Bo Guo, Li Yang Wang, Bogdan J Matuszewski, Elia Bruni, Urko Sanchez, et al. Gland segmentation in colon histology images: The glas challenge contest. MedlA, 2017.
[54] Alvy Ray Smith. Tint fill. In Proceedings of the 6th annual conference on Computer graphics and interactive techniques, 1979.
[55] Kihyuk Sohn, David Berthelot, Nicholas Carlini, Zizhao Zhang, Han Zhang, Colin A Raffel, Ekin Dogus Cubuk, Alexey Kurakin, and Chun-Liang Li. Fixmatch: Simplifying semi-supervised learning with consistency and confidence. NeurlPS, 2020.
[56] Nico Stucki, Johannes Paetzold, Suprosanna Shit. Bjoern Menze, and UIrich Bauer. Topologically faithful image segmentation via induced matching of persistence barcodes. In ICML, 2023.
[57] Antti Tarvainen and Harri Valpola. Mcan teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. In NeurIPS, 2017.
[58] Ruchika Verma. Neeraj Kumar, Abhijeet Patil, Nikhil Cherian Kurian, Swapnil Rane, Simon Graham, Quoe Dang Vu, Mieke Zwager, Shan E Ahmed Raza, Nasir Rajpoot, et al. Monusac2020: A multi-organ nuclei segmentation and classification challenge. TMI, 2021.
[59] Tuan-Hung Vu, Himalaya Jain, Maxime Bucher, Matthieu Cord, and Patrick Pérez. Advent: Adversarial entropy minimization for domain adaptation in semantic segmentation. In CVPR, 2019.
[60] Fan Wang, Huidong Liu, Dimitris Samaras, and Chao Chen. Topogan: A topalogy-aware generative adversarial network. In ECCV, 2020.
[61] Haotian Wang, Min Xian, and Aleksandar Vakanski. Ta-net: Topology-awee aware network for gland segmenta- tion. In WACV, 2022.
[62] Kaiping Wang. Bo Zhan, Chen Zu, Xi Wu, Jiliu Zhou, Luping Zhou, and Yan Wang. Semi-supervised medical image segmentation via a cnntretunyerttiitt guided mean teacher model with contrastive learning. MedIA, 2022.
13

---

[63] Bo Wen, Haochen Zhang, Dirk-Uwe G Bartsch. William R Freeman, Truong Q Nguyen. and Cheolhong An. Topology-preservin image segmentation with spatial-aware persistent feature matching. arXiv preprint arXiv:2412.0201 2024.
[64] Huisi Wu, Zhaoze Wang, Youyi Song. Lin Yang, and Jing Qin. Cross-patch dense contrastive learning for semi-supervised segmentation of cellular nuclei in histopathologic images. In CVPR, 2022.
[65] Kesheng Wu, Ekow J. Otoo, and Kenji Suzuki. Optimizing two-pass connected-component labeling algorithms. Pattern Analysis and Applications, 2009.
[66] Junsong Xie, Qian Wu, and Renju Zhu. Entropy-gun guided contrastive learning for semi-supervised medical image segmentation. IET Image Processing, 2024.
[67] Yutong Xie, Hao La, Jianpeng Zhang, Chunhua Shen, and Yong Xia. Deep segmentationn-mmnndtioo model for gland instance segmentation. In MICCAI, 2019.
[68] Mcilong Xu, Saumya Gupta, Xiaoling Hu, Chen Li, Shahira Abousamra, Dimitris Samaras, Prateck Prasanna, and Chao Chen. Topocellgen: Generating histopathology cell topology with diffusion model. In CVPR, 2025.
[69] Meilong Xu, Xiaoling Hu, Saumya Gupta, Shahira Abousamra, Chao Chen. Semi-supervised segmentation of histopathology images with noise-aware topological consistency. ECCV, 2024.
[70] Zhe Xu, Yixin Wang, Donghuan Lu, Xiangde Luo, Jiangpeng Yan, Yefeng Zheng, and Raymond Kai-yu Tong, ambiguity-sclective consistency regularization for mean-teacher supervisd medical image segmentation. MedlA, 2023.
[71] Jiaqi Yang, Xiaoling Hu, Chao Chen, and Chialing Tsai. 3d aypology-preseringg segmentation with compound multi-slice representation. ISBI, 2021.
[72] Jiaqi Yang, Xiaoling Hu, Chao Chen, and Chialing Tsai. A topological-attentnnn convistm network and its application to em images. In MICCAI, 2021.
[73] Jiaqi Yang, Nitish Mchta, Gozde Demirci, Xiaoling Hu, Meera S Ramakrishnan, Mina Naguib, Chao Chen, and Chia-Ling Tsai. Anomaly-guidee weakly supervised lesion segmentation retinal oct images. MedlA, 2024.
[74] Jiaqi Yang, Nitish Mehta. Xiaoling Hu, Chao Chen, and Chia-Ling Tsai. A multimodal approach combining structural and cross-domain textual guidance for weakly supervised oct segmentation. arXiv preprint 1rXiv:2411.1265,, 2024.
[75] Huifeng Yao, Xiaowei Hu, and Xiaomeng Li. Enhancing pseudo label quality for semi-supervised domain-generalizee medical image segmentation. In AAAI, 2022.
[76] Chenyu You, Weicheng Dai, Fenglin Liu, Yifei Min, Nicha C Dvomek, Xiaoxiao Li, David Clifton, Lawrence Staib, and James S Duncan. Mine your own anatomy: Revisiting medical image segmentation with extremely limited labels. TPAMI, 2024.
[77] Chenyu You, Weicheng Dai, Yifci Min, Fenglin Liu, David Clifton, S Kevin Zhou, Lawrence Staib, and James Duncan. Rethinking semi-supervised medical image segmentation: A variance-reduuctoon perspective. In NeurIPS, 2023.
[78] Chenyu You, Yuan Zhou, Ruihan Zhao, Lawrence Staih, and James S Duncan. Simevd: Simple contrastive voxel-wise representation distillation for semi-supervised medical image segmentation. TMI, 2022.
[79] Lequan Yu, Shujun Wang, Xiaomeng Li, Chi- Chi-Wing Fu, and Pheng-Ann Heng. Uncertainty-aware self- ensembling model for semi- semi-supervised 3d left atrium segmentation. In MICCAI, 2019.
[80] Guoqing Zhang, Caixia Dong, and Yang Li. segmentation. In BMVC, 2023.
hard pixel mining for tubular structure
[81] Wenqiao Zhang, Lei Zhu, James Hallinan, Shengyu Zhang, Andrew Makmur, Qingpeng Cai, and Beng Chin Ooi. Boostmis: Boosting medical image semi-supervised learning with adaptive pscudo labeling and informative active annotation. In CVPR, 2022.
[82] Zhenxi Zhang, Chumna Tian, Harrison X Bai, Zhicheng Jiao, and Xilan Tian. Discriminative error prediction network for semi- supervise colon gland segmentation. MedIA, 2022.
14

---

[83] Yanfeng Zhou, Jiaxing Huang, Chenlong Wang. Le Song, and Ge Yang. Xnet: Wavelet-based low and high frequency fusion networks for fully-and semi-supervised semantic segmentation of biomedical images. In ICCV, 2023.
[84] Yanfeng Zhou, Lingrui Li, Zichen Wang, Guole Liu, Ziwen Liu, and Ge Yang. Xnet v2: Fewer limitations, better results greater universality. BIBM, 2024.
[85] Zongwei Zhou, Md Mahfuzur Rahman Siddiquee, Nima Tajbakhsh, and Jianming Liang. Unet++:: A nested u-net architecture medical image segmentation. In Deep learning in medical image analysis and multimodal learning for clinical decision support: 4th international workshop. DLMIA 2018, and 8th international workshop, ML-CDD LLSCDS 2018, held in conjunction with MICCAI 2018, 2018
15

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification:
The main claims of our paper are well reflected in the abstract and introduction.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results
be expected to generalize other settings.
is fine to include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
The limitation of this work is discussed in the Supplementary.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification. asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low images are taken in low lighting. Or speech-to-text system might be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [NA]
16

---

Justification:
This work does not include theoretical results.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless whether the code and data are provided or not)?
Answer: [Yes]
Justification:
The implementation details provided in the Supplementary.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make clear how to reproduce that algorithm. (b) the contribution is primarily a new model architecture, the paper should describe the architccture clearly and fully.
(c) If the contribution is new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. the case of closed-source models, may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
17

---

Answer: [Yes]
Justification:
We have provided the code in the abstract.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.co cc/
for more details.
While we encourage the release of code and data, we understand that this might not be possible, "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (c.g., for a new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
//nips.com wttpcccpppbccccceec guwdes/dde/eeee Cooeeeoodeesssseeeeeee http:p//iiiitiieeeeeeoo/eaaasses
ssiniiiiccyy for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Does the paper specify all the training and test details (c.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary understand the results?
Question:
Answer: [Yes]
Justification: Some ablation studies are provided in the main paper and the rest of training and test details is provided in the Supplementary.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [Yes]
Justification: This paper reports error bars suitably and correctly defined.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the enror bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
18

---

It should be clear whether the error bar is the standard deviation or the standard error of the mean.
is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out range (e.g. negative еπхо rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification:
We include the GPU usage, learning rate, batch size, etc., in the Supplementary.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics
Answer: [Yes]
Justification: The research conducted in the paper conforms with the NeurIPS Code of Ethics in every respect.
Guidelines:
The answer NA means that the authors have reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g.. if there is special consid- cration due to laws or regulations in their jurisdiction..
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification; The discussion of both potential positive societal impacts and negative societal impacts of the work performed is provided in the Supplementary.
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
19

---

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: This work doesn't 't pose such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Question: Are the creators or original owners of assets (e.g.. code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: The creators or original owners of assets used in the paper are properly credited and the license and terms of use explicitly mentioned and properly respected.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (c.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the batasets.papersstt paperswi thcode. cww/ datasets package should be provided. For popular datasets,
has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
20

---

If this information is not available online, the authors are encouraged to reach out to the asset's creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: This work does not release new assets.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/mmoel as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsoureing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification: This work does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question:
Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification:
This work does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and expect authors to adhere to the NeurlPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such the institution conducting the review.
16. Declaration of LLM usage
21

---

Question: Does the paper describe the usage of LLMs if it is important, original, or non-standard component of the core methods this research? Note that if the LLM used only for writing, editing, or formatting purposes and does not impact the core methodology. scientific rigorousness, or originality the research, declaration is not required.
Answer: [NA]
Justification: The core method development in this research does not involve LLMs any important, original, or non-standard components.
Guidelines:
The answer NA
that the method development in this research does involve LLMs as any important, original, or non-standard components.
Please refer to our LLM policy for what should or should be described.
22

---

MATCH: Multi-faceted Adaptive Topo-Consistency for Semi-Supervised Histopathology Segmentation
Supplementary Material-
6 Overview
In the supplementary, we begin with a brief introduction to the persistent homology in Section 7 followed by detailed introductions of the datasets in Section and the evaluation metrics in Section Then, we provide the implementation details in Section [0] followed by the references of our baselines in Section []] We also provide additional ablation studies in Section [2] further illustrate the efficacy and robustness of our method and hyper-paramedee selections. The limitations are provided in Section
1 followed by analysis on the broader impact in Section
7 Brief Introduction to Persistent Homology
Persistent homology 6 7, a fundamental concept in topological data analysis (TDA), offers a robust framework for capturing and quantifying the topological features of data across multiple scales. In the context of image segmentation, particularly when dealing with likelihood maps that represent the probability of each pixel belonging specific class, persistent homology provides means analyze the underlying topological structures inherent in these probabilistic representations.
Given likelihood map Ω2 + [0, 1]. where Q R² represents the image domain, construct
filtration of super-level sets:
Fg = fo E f() 2 a), a [0,
As a decreases from to O, the super-level set Fa transitions from empty regions to encompass the entire domain Q, revealing the sequential emergence, merging, and disappearance of connected components and loops. Persistent homology tracks these topological changes across the filtration, recording the corresponding birth and death thresholds of each feature in persistence diagram.
A persistence diagram is a multiset of points {(b,, d in the extended plane IR,, where each point corresponds to topological feature that appears (birth b,) and disappears (death d,) during the filtration process. Features that persist across a wide range a values (i.e., with large Id; 6,) are considered topologically signifieant, while those with short lifespans often attributed to noise.
8 Dataset Details
Colorectal Adenocarcinoma Gland (CRAG) consists of 213 hematoxylin and cosin (H&E)- stained colorectal adenocareinoma image tiles acquired at 20 magnification, each with detailed annotations at the instance level. Most images are in approximately 1512 x 1516 pixels. Officially, the dataset is partitioned into 173 training samples and 40 testing samples. For experiments, the training subset is further divided into 153 images for model training and 20 images for validation. For supi-supe scenarios with 10% and 20% labeled data, randomly select 16 and 31 labeled images, respectively, for training.
Gland Segmentation in Colon Histology Images Challenge (GlaS) 53 comprises 165 images sourced from 16 H&E-stained histological slides colorectal adenocarcinoma at stages T3 or T4. The official split includes 85 training images and 80 testing images. In experimental setup, the training set divided into 68 images for model training and 17 for validation. We randomly select 7 and 14 labeled images to represent 10% and 20% of labeled training data scenarios, respectively.
Multi-Organ Nuclei Segmentation (MoNuSeg)
dataset contains 44 H&E-stained histology images of dimensions 1000 x 1000 pixels, encompassing nuclei annotations from seven distinct organs. Officially, it consists of 30 training images with a total of 21, 623 annotated nuclei and 14 images designated for testing. For our experiments, we reserve 20% (6 images) of the training set for validation. experiments involving 10% and 20% labeled data splits, we randomly select and labeled images, respectively, for training.
23

---

Evaluation Metrics
We evaluate the segmentation quality from both pixel- and topology-wise. Object-level Dice coefficient (Dice_ (ioe__bj [67 is selected to measure pixel-wise performance, which measures instance- wise overlap between predicted and ground-truth masks and is thus well suited to the precise delineation of individual structures required in digital pathology.
To evaluate the topologiaal accuracy, we select three topological evaluation metrics, Betti Error Betti Matching Error [56], and Discrepancy between Intersection and Union (DIU) [40] Betti Error (BE) mainly computes the mean absolute difference in O-dimensional Betti numbers over 256 x 256 dldding-windww patches. Betti Matching Error (BME), which extends BE by enforcing spatial correspondence when pairing topological features, thereby penalizing misplaced components even when counts preserved. Introduced in [40), DIU quantifies how faithfully the topology of the common and combined foreground regions agrees.
10 Implementation Details
Our model trained in two distinct stages. In the initial stage, we perform pretraining using only supervised loss and pixel-wise consistency loss. For all three datasets, the pretraining stage proceeds for 12, 000 iterations, The second stage involves fine-tuning the model by integrating our proposed dual-level topological consistency constraints, which last for additional 1, 000 iterations. We use UNet [49] as the backbone for both the student and teacher models.
All training is implemented using PyTorch [47] and optimized using the Adam optimizer [28 Training hyperparameters are set as follows: the batch size is 16 and the learning rate is 5 10 Both labeled and unlabeled data undergo pre-processing through random cropping (with cropping size of 256 x 256), followed by data augmentation procedures including random rotation and flipping as weak augmentations, and color adjustments and morphological shifts for stronger augmentations.
In particular, we adopt a random cropping strategy for enforcing intra-topological consistency, while fixed patch cropping strategy is used for temporal-topological consistency. The inputs to the student model estimate the intra- and cmpporal-topologcca consistency are all original patches, without any transformations. The EMA decay rate is set to 0.999. Within the supervised loss, the weights assigned to the cross-entropy loss and Dice loss are equally set to 0.5. The weight of the pixel-wise consistency loss is calculated by the Gaussian ramp-up function Acons = k x 5-(1 7)² where k = 0.1 and I' is the total number of iterations.
Additionally. Aunua and Asemp are both set 0.001. This balanced configuration ensures effective integration of topological constraints while maintaining stable training dynamics. Note that dual- level topological consistency is used to optimize the student model directly, and we use the student model to do the inference. The experiments are conducted on an NVIDIA RTX A6000 GPU (48 GB), using 24-core Intel Xeon@ Gold 6248R CPU 3.00 GHz and 192 GB RAM. The training time of one iteration is 1020.04 ms, and GPU memory consumption is 25.726 GB using UNet with batch size 16. The training time of TopoSemiSeg for one iteration is 610.80 ms, and the GPU memory consumption is 15.235 GB. For the non-PH baseline, like PMT [10] the training time per iteration is 582.34 ms,
11 Baseline Reference
We select 7 classical and recent state-of-the-ard methods as comparatives. The implementations of some of them are based on publicly available repositories. Here, we provide the source of our baselines for reference and greatly appreciate their efforts in building the open-source community:
MT 57 EM 59,, UA-MT 79 and URPC 39 are based on the implementations from:
httss//github.com/Hl..oo http://htwb...........aalaaatatd LAMSSSLAMISS
XNet 83 is based on the implementations from:
Yanfeng-Zhou
PMT [10 is based on the implementations from: http://gilub.cco//Arr44PPPPPPP
24

---

TopoSemiSeg 669
is based on the implementations from:
12
Additional Ablation Study
Here, we provide additional ablation studies illustrate the efficacy and robustness of our selected backbone and hyperparameters.
Ablation Study on Aimtra and Atemp- The results shown in Table T0 demonstrate the impact of varing weights of intra- and temporal-topological consistency (Ainta and \uemp) When two weights are both 0.001, the performance is the best across both pixel-wise and topology-wise metries. As these weights increase from 0.001 to 0.01, there's a clear degradation in performance, indicating that excessively large consistency constraints may introduce unnecessary regularization, thus impairing the segmentation quality. When both weights are reduced to 0. 0.0005, the dual-level consistency regularization becomes too weak to meaningfully optimize the student model, leading to diminished topological guidance and corresponding drop in both pixel-wise and topology -wioe performance.
Ablation Study on EMA Decay a. Table 11 investtigates the influence of the EMA decay parameter o. a 0.999 yields the best performance. When decreasing a from 0.999 to 0.996, the results remain competitive but slightly deteriorate, highlighting that a higher EMA decay value effectively leverages historical model parameters for improved topological and segmentation robustness. In contrast, very high values (e.g. 0.9999) excessively rely on historical information, marginally weakening the adaptability and performance of the model.
Table 10: Influence of Xintr and semp.
Table 11: Impact of the EMA decay
Auaa
Aamp
Pisel wise
Topology wise
Pixel wise
Topology wise
Dicc_obj f
BE
BME
DIU 4
Diec_obj
BME
DIU
0.01
0.012
0.275
2,300
0,012
9,500
2.100
0.005
0,005
0.892
0,007
0.214
1: 0.000
8,105
0.600
44.105
2.050
0.999
0,909 1: 0,005
0.188
0,018
7,425 0,570
40.250
1.720
0.001
0.001
0,909
0,085
0,188
1: 0.018
7.425
1: 0.570
40.250
1: 1,720
0.996
0.902
1: 0.006
0,205
+ 0,020
8,250 =
0.610
42,500
+: 1,900
0.0005
0.0005
0.895
0.007
0.235
0.000
8,950
0.590
44.225 1.850
0.99
0.882
0,260
0,025 11,000
0,700
50,000
2.300
Ablation Study on Different Backbones. To further verify the robustness of our proposed method, we conduct ablation experiments on different backbones. The results are shown in Table [2] Specifi- cally, D3+pLabV3+ [4] and UNet++ (85] show modest but clear improvements in both pixel-wise and topology-wise metrics. The UNet [49 backbone achieves the most substantial gains, particu- larly in topology-wise metrics. These results demonstrate that integrating our MATCH framework consistently improves performance across multiple backbones.
Table 12: Performance compprriooccomppiioon of different backbones w or w/o our MATCH.
Backhone
Piacl Wise
Topelogy Wise
Dice_Obj
BE
BME J
VOL
DespLabV3+
0,889 0.010
+
11,782 +
50887 +
Dbs5 40m
0.892 : 0.008
0.245 = 0:022
10.129 = 0.638
47.412 = 2.047
UNeea+
0,886 0.008
0.245 + 0.023
9:210
0.603
45.517 =: 2.041
UNee++
HOurs
0,890 0.006
0,238 d: 0,020
9:021
0,580
45.073 1.995
UNer
0.894
0.006
0,232 :: 0.019
8,872
2: 0.579
44.281 :: 1.881
UNel
+Ours
0,909 0.005
0,188 :: 0,018
7.425
0.570
40.250 =: 1.730
Ablation Study on Applying Dual-Level Topo-Consistency between Teacher and Student Models. We conduct an ablation study to assess the impact of enforcing dual-level topological consistency in a sttueet-ttudent framework. Specifically, the dual-level topological consistency is estimated from the teacher model's 's multiple predictions, and consistency constraints are applied between the student output and the most recent prediction from the teacher. We compare this teacher-student configuration with a student-only model, both trained under identical consistency constraints. The results in Table [3 reveal that the student-only model consistently achieves superior performance. The relatively poorer performance of the teacher-student configuration suggests that leveraging the
25

---

teacher's predictions, potentially noisy or outdated, introduces additional uncertainty and adversely affects the student's ability to effectively capture stable topologiaa structures.
Table 13: Ablation study on applying dual-level topo-consistency between teacher and student models.
Mode
Pixel-wise
wopelogy-wise
Dice_ Diee_Obj
BE
BME
DIU 4
Teacher-Studeen
0.885 + 0.007
0:217
0.021
8.102 + 0.620
42.520 + 1.880
Student Only
0.909 0.005
0.188 0.018
7,425 1 0.570
40.250 11 1.720
Extension from Binary to Multi-Class Segmentation. To extend our method to the multi-class setting, we choose a multi-class nuclei segmentation dataset, MoNuSAC (58), to conduct experiments. This dataset contains four cell types: Epithelial, Lymphocyte, Macrophage, and Neutrophil. We conducted experiments using 20% labeled data and report the class-wise performance of TopoSemiSeg and our method in Table [4] As demonstrated in our class-specific results on MoNuSAC (Epithelial, Lymphocyte, Macrophage, and Neutrophil), our approach consistently outperforms TopoSemiSeg across all cell types, with particularly notable improvements in topological metrics (BE and BME) that are crucial for distinguishing overlapping structures.
Table 14: The multi-class segmentation results on the MoNuSAC dataset.
Class
Method
Dice_obj
BEI
BME L
TopoSemiSey
0.778 1 0.009
5,342 + 0.187
195.158 +: 4.627
Epithelial
Ours
1: 0.008
5.128
d: 0.189
186.847
Lymphocyte
TopoSemiSe
0.751 +: 0.013
6.089
+ 0.223
218.394
5.841
Ours
0.756 1: 0.012
5.794
1: 0.235
207.693
4.672
TopoSemiSeg
0.765 + 0.011
5.687 + 0.201
206.732
4.985
Macrophage
Ours
0.769 + 0.010
5,423 + 0.208
195.381
4.127
Neutrophil
TopoSemiSeg
0.738 + 0.016
6.521 =: 0.267
234.576
6.123
Ours
0.742 1: 0.015
6.187 J. 0.281
221.459 1: 5.894
Ablation Study on the Sensitivity of Tprimary- We provide the ablation study on the sensitivity of Tprimory in Table [5] The results have shown that our method is robust to selecting Tprimary Moreover, the low threshold of 0.1 was chosen to be inclusive rather than restrictive: it allows more potential matches to be considered valid while letting the Hungarian algorithm determine optimal assignments based on our comprehensive similarity metric (combining spatial overlap, persistence weights, and proximity).
Table 15: Effect of Tprimary-
Table 16: Effect of dropout rates.
Tpinay
Dice_
BF J BME
DIU 4
Dropout Rate Dice obj
BF
BMF
0.05
0.906 = 0.0060. 195
0.0197.850 + 0.62041.750
1.850
10%
0.898 + 0 0.0060.210 0.0208. + 0.650
0.1 00,rret)0,9099 = 0.0050.188.
0.0187.425 = 0,570440.250
1.720
20% (current) 0,909 + 0. + 0.0187.425 + 0.570
0.2
0.908 1 0.0050.191
0.0207.680 1 0.59041.100
1.780
30%
0,910 1: 0.0050.185 0.0177.350 1 0,560
0.3
0.905
0.0060.201 0.00218.50
0.65042.850 1.920
50%
0,890 1: 0.0070.220 0.0228.800 1: 0.720
The Impact of Different Dropout Rates. We also add complementary ablation studies on the dropout rate of the MC-dropout. Other settings are kept unchanged. We conduct the ablation experiments on CRAG 20% labeled data and report the performance in Table The results reveal an optimal dropout rate range of 20%-30% for our framework, where performance plateaus with minimal differences between these rates. Lower dropout rates provide insufficient perturbation diversity for reliable topological matching. In contrast, excessive dropout introduces detrimental noise that degrades both pixel- and hopology-wise performance, confirming that moderate stochasticity is essential for effective topological consistency estimation.
Alltion Study on Lcons. We conducted the ablation study on the Luons and the results are shown in Table 17 Based on the results of the ablation study and the principles of semi-supervised learning, removing the pixel-wise consistency term in the training stages would result in significant performance degradation across all metrics.
26

---

Table 17: Ablation Study on Lcons
Method
Dice ohj 1
BF +
BMF
wHo Coces
0.875 + 0.008 0,909 0.005
0.285 = 0.025
9,850 = 0.680
Ours
0.188 = 0.018
7.425 0.570
Downstream Analysis: Cell counting To further analyze the impact of our method on downstream analysis, we conducted cell counting study the same MoNuSeg test cohort. We used the connected component analysis [65] to identify the cells and calculate the total cell count, the predicted total cell count, and the absolute counting eπror (mean + std). The results
shown in Table [8] Note that the Total GT cell count and the predicted cell count are reported for the entire test cohort, while the absolute count error is reported per image (with total of 14 test images).
We observed that our method yields noticeably smaller counting errors than both baseline approaches (one topo method and one non-topo method). This confirms that although the pixel-wise segmentation performances are comparable, fixing the topological errors on a few pixels leads to more accurate biological readouts.
Table 18: Downstream Analysis Cell Counting.
Method
Total GT Cell Count
Predicted Cell Count
Absolute Counting Error (Mean + Std)
Dice_obe Dije_obj
PMT [8]
6024
8106
148.71 1: 99.41
0.778 1. 0.006
TopoSemiSeg [57]
7877
56.09
MATCH
6024
7511
106.21 = 49.30
0,790 + 0.006
13
Limitations
potential limitation of our MATCH framework arises from its reliance on stable feature extraction from persistence diagrams, which can be challenged when predictions exhibit extreme noise or minimal structural differences. In addition, the framework introduces nontrivial computational overhead: computing persistence diagrams and performing AATCH--Gloal/MMTTCH across Monte Carlo dropout samples and temporal snapshots require multiple forward passes and matching steps, resulting in longer training times and increased memory usage.
alignments
14 Broader Impact
Our method significantly contributes to enhancing segmentation robustness by effectively leveraging unlabeled data, reducing reliance extensive annotations, and ensuring topological accuracy crucial for clinical and biomedical analysis. This approach not only facilitates efficient utilization of limited labeled data but also provides insightful uncertainty estimates beneficial for downstream diagnostic applications.
A negative broader impact could include inadvertent propagation of segmentation inaccuracies if poorly matched topological structures influence model learning, potentially affecting reliability in critical medical decisions.
27


Reviews:
[{"review_id": "6aX64zH0M8", "reviewer": "Reviewer_t5XP", "summary": "This paper presents a semi-supervised segmentation method for histopathology images to resolve the problem of topological errors. It enforces topological consistency across different outputs, including outputs from stochastic dropouts and temporal training snapshots. The experiment shows that it reduces topological errors compared with other methods.", "questions": "1. Beyond the three datasets (CRAG, GlaS, MoNuSeg), how does the model generalize to other pathology tasks?\n2. The paper only extracts 0-D topological features. Do you plan to extend the framework to 1-D or higher-order topological features?\n3. Have you evaluated the risk of error propagation in pseudo-labels?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. This paper proposes a innovative matching algorithm between a pair of perturbed predictions, which solves topological errors.\n2. This paper provides a new view of the reliability of the segmentation predictions.\n\nWeaknesses:\n1. The concept of temporal topological consistency has been explored in semi-supervised learning, so I think as a standalone module it lacks novelty.\n2. The paper only extracts 0-D topological features for matching, without considering structures with holes or annular structures, which commonly exist in histopathology images.\n3. The method’s performance is only validated on three datasets restricted to gland and nucleus segmentation tasks, with no validation on other pathological tasks.\n4. The method relies solely on MC dropout for generating perturbed predictions without comparing it with other perturbation methods, which may limit the model's performance.\n5. The method generates multiple perturbations for each input image through Monte Carlo dropout, increase computational time and memory consumption, especially when dealing with whole-slide images (WSI).", "was_revised": true, "final_justification": "Thank authors for their response, which has addressed some of my concerns. However, the validation of 1D topological structure capability was conducted solely on the Roads dataset, with no evidence on medical datasets for critical histopathological structures(e.g., holes or annular configurations). Additionally, there was no lightweight strategies to mitigate the computational cost. While authors assert that \"the MATCH algorithm and dual-level topological consistency are domain-agnostic\", there was no cross-task validation evidence. I will maintain my original rating.", "rebuttal": "We sincerely thank the reviewer for the thoughtful and positive feedback on our method. We greatly appreciate the recognition of the innovative matching algorithm we proposed to resolve topological errors between perturbed predictions and the acknowledgment of our contribution to a new perspective on assessing the reliability of segmentation predictions. Below, we address your concerns one by one. \n\n**Q1: The standalone module of temporal topological consistency lacks novelty.**\n\n**A1**: While we acknowledge that temporal consistency has been explored in semi-supervised learning, our contribution represents a novel and substantial extension beyond existing approaches in several key aspects. \n- First, unlike previous temporal consistency methods that operate at the pixel or feature level, our work introduces the first application of temporal topological consistency specifically for preserving topological structures across training snapshots. \n- Second, our dual-level consistency framework uniquely combines both intra-topological consistency (across MC-dropout predictions) and temporal consistency within a unified topological reasoning paradigm, rather than treating them as separate mechanisms. \n- Third, integrating temporal consistency with our novel MATCH-Global matching algorithm creates a synergistic effect that enables robust identification of stable topological structures without requiring hand-picked persistence thresholds—a critical limitation in existing topology-aware methods like TopoSemiSeg.\n\n**Q2: Only 0-D topological features are extracted, without considering structures with holes or annular structures.**\n\n**A2**: Our decision to concentrate on 0-D topological features was motivated by the following factor: For the primary applications in our study (gland and nuclei segmentation), the most critical topological errors involve incorrect splitting or merging of individual structures, which are well-captured by 0-D persistent homology. \n\nFor validation on 1-dimensional structures, we conducted additional experiments on the Roads dataset [r1], which contains aerial images of road networks with complex branching and connectivity patterns. While this dataset is not histopathological, it verifies that our method could also learn good topological representations from the unlabeled data regarding 1-dimensional topological features. The results are shown below:\n\n| Labeled Ratio | Method       | BE ↓               | BME ↓               | DIU ↓               |\n|--------------|--------------|-------------------|-------------------|-------------------|\n| **10%**      | TopoSemiSeg  | 8.324 ± 0.729     | 9.681 ± 0.647     | 10.952 ± 0.671    |\n|              | **Ours**     | **7.892 ± 0.634** | **8.147 ± 0.521** | **9.376 ± 0.583** |\n| **20%**      | TopoSemiSeg  | 7.467 ± 0.582     | 8.213 ± 0.514     | 9.387 ± 0.538     |\n|              | **Ours**     | **6.983 ± 0.507** | **7.024 ± 0.436** | **8.149 ± 0.492** |\n\n\nThe complete extension results of our method to the 1-dimensional topological feature will be provided in the final version.\n\n**Q3: No validation on other pathological tasks.**\n\n**A3**: As suggested by our title, we focused on gland and nuclei segmentation as these represent fundamental histopathology tasks where topological errors are most critical and prevalent due to dense, overlapping structures. Our core contributions, the MATCH algorithm and dual-level topological consistency, are domain-agnostic. They rely on general topological properties rather than task-specific features. Accurate segmentation of the glands and nuclei could be the foundation of the downstream analysis. We will leave exploring other pathological tasks as future work.\n\n**Q4: The alternatives to generate perturbed predictions other than MC-dropout.**\n\n**A4**: We choose two alternative perturbation methods: Variational Inference (VI) [r2], which generates multiple predictions by sampling from the learned variational posterior distribution, and Temperature Scaling [r3], which produces diverse predictions through multiple sampling from temperature-modulated probability distributions. The experiments are conducted on CRAG 20% labeled data, and the results are shown below:\n\n| Method                 | Dice_obj ↑       | BE ↓            | BME ↓          |\n|------------------------|-----------------|---------------|---------------|\n| Variational Inference  | 0.895 ± 0.006   | 0.242 ± 0.022 | 9.125 ± 0.685 |\n| Temperature Scaling    | 0.891 ± 0.007   | 0.258 ± 0.025 | 9.850 ± 0.795 |\n| **MC-Dropout**         | **0.909 ± 0.005** | **0.188 ± 0.018** | **7.425 ± 0.570** |\n\nMC-dropout is chosen as a widely used and easily implementable method for generating perturbed predictions. However, our framework can readily accommodate other perturbation approaches, as this is not our primary contribution. We will provide complete results in the final version.\n\n**Q5: Increasing computational time and memory consumption, especially when dealing with whole-slide images (WSI).**\n\n**A5**: We acknowledge the reviewer’s concern regarding computational overhead, which was introduced by generating multiple perturbations through Monte Carlo dropout. Although our method does incur increased computational time and memory usage compared to TopoSemiSeg, specifically an iteration training time of 1020.04 ms and GPU memory consumption of 25.726 GB using UNet with batch size 16, this overhead is justified by the performance gains demonstrated in our experiments. The training time of TopoSemiSeg for one iteration is 610.80 ms. The GPU memory consumption is 15.235 GB. To address the reviewer’s specific concern, we emphasize that our approach processes patches cropped from whole-slide images rather than entire whole slide images themselves. The largest image resolution processed is approximately 1516×1516 pixels, significantly smaller than a whole-slide image.\n\n**Q6: The evaluation of the risk of error propagation in pseudo-labels.**\n\n**A6**: Our method does not employ pseudo-labeling strategies that risk error propagation. Instead, we enforce topological consistency constraints directly during training through our dual-level framework, without generating or incorporating pseudo-labels into the training set.\n\n[r1] Mnih V. Machine learning for aerial image labeling[M]. University of Toronto (Canada), 2013.\n\n[r2] Jordan, Michael I., et al. \"An introduction to variational methods for graphical models.\" Machine learning 37.2 (1999): 183-233.\n\n[r3] Guo C, Pleiss G, Sun Y, et al. On calibration of modern neural networks[C]//International conference on machine learning. PMLR, 2017: 1321-1330."}, {"review_id": "H2e64m2VTg", "reviewer": "Reviewer_gBzP", "summary": "The authors aim at improving histopathology image segmentation when unlabeled data are predominant during training. For this reason, they propose a Semi-Supervised Learning (SSL) image segmentation framework and enforce image segmentation prediction robustness against perturbations adopting topological reasoning. In particular, the authors model perturbations with Monte Carlo dropout mechanism and by considering the predictions in different training epochs. As for the topological consistency, they formulate the structure correspondence task as a contrastive learning problem to distinguish features considered stable when matching per spatial overlap, topological persistence and spatial proximity criteria across themselves (via MATCH-Pair) and across multiple predictions (via MATCH-Global). The method is validated in the task of binary gland and cell segmentation across 3 different datasets and with different percentages of label data, ablation study correlates the results analysis.", "questions": "1. Could the authors clarify whether the teacher model used in the SSL framework was pretrained using only the same limited set of labeled data available to the student? If additional labeled data were used during pretraining, how do the authors justify this in the context of a semi-supervised learning setup, where strict limitations on label availability are assumed?\n2. Given the authors’ emphasis on the challenges posed by densely packed structures, have they evaluated how the model’s performance varies with object density (e.g., comparing sparse vs. crowded regions in MoNuSeg)? Would the authors consider including a stratified analysis or crowding-aware ablation to validate their method's robustness under varying tissue densities?\n3. While simplifying all tasks to binary segmentation may ease implementation, have the authors considered extending or evaluating their method in the multiclass setting, particularly for GlaS and MoNuSeg? More critically, given the importance of separating overlapping nuclei in histopathology, why was instance-level structure not considered, either during training or evaluation? Could the topological matching mechanism benefit from incorporating explicit instance constraints to better resolve ambiguous or overlapping objects?\n4. Could the authors define the variables $d_i$ and $b_i​$ more clearly when first introduced (l. 147), especially for readers unfamiliar with persistent homology?\n5. What is the precise definition and origin of the likelihood maps $lh1$​ and $lh2$? Do these correspond to the sigmoid-activated outputs of the final UNet layer, or are they derived differently?\n6. At line 153, is the use of the index notation $i,j $ consistent with the dimensions and semantics of the features being described? Furthermore, could the authors clarify the role of the indices $i,j,k$, as they are used without explicit definition and may confuse readers interpreting the formulation?", "limitations": "The authors have adequately addressed the potential negative societal impact of their work", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n- The proposed method appears technically sound and even though it does not introduce an entirely new method (SSL, contrastive learning paradigm, persistent diagrams, Monte Carlo dropout perturbations, consistency between different training temporal views, Hungarian matching logic are already known technique and widely used in the state of the art), their usage context and combination appears in this work original and more efficient than other state-of-the-art methods (i.e. exploiting the persistent structures between perturbations allow avoiding hand-picking threshold to determine reliable structures as happening in TopoSemiSeg).  Notably, the MATCH-Pair and MATCH-Global can be considered a smart glue to define valid matches between 2 persistent diagrams and between multiple facets, respectively. \n\nWeaknesses: \n1. The authors state that they use a pretrained model to initialize the SSL training. However, it is unclear whether this pretraining was performed using only the limited labeled data available to the student model, or if additional labeled data were used. If the teacher model is pretrained on a larger labeled dataset than what is available during the SSL phase, this would violate the core assumption of semi-supervised learning, namely, that only a small fraction of labeled data is accessible. In such a case, the observed performance gains may stem from the additional supervision during pretraining rather than from the proposed SSL strategy itself.\n2. The authors state that capturing meaningful semantic structures from unlabeled data is essential, particularly in scenarios where objects are densely distributed (l. 3). They also acknowledge that such dense distributions often lead to topological errors (l. 21). This is indeed a critical aspect in histopathology image segmentation. However, despite choosing the MoNuSeg dataset, which contains patches with densely packed nuclei, the authors do not analyze how the model's performance varies with respect to object crowding.\n3. The authors evaluate their method on a combination of binary (CRAG) and inherently multiclass (GlaS and MoNuSeg) datasets, but convert all tasks into binary segmentation. While the decision to reduce the problem to a binary setting is understandable given the technical challenges of extending semi-supervised learning to multiclass segmentation, it nonetheless limits the scope of the evaluation and the relevance of the results to real-world histopathology applications. More critically, the method does not account for instance-level structure, despite the authors’ emphasis on the challenges posed by densely packed cellular objects. Unlike the multiclass case, instance-level information is essential for properly resolving overlapping nuclei, and omitting this consideration weakens the effectiveness of the proposed topological reasoning. Topological consistency alone may struggle to disambiguate closely spaced structures without explicit instance-level constraints.\n4. The variables $d_i$ and $b_i​$, first introduced at line 147, are never explicitly defined in the manuscript. While their meaning may be inferred by readers familiar with SSL literature (the birth and depth points in the persistence diagrams), this lack of definition may hinder understanding for a broader audience. For clarity and completeness, all variables, especially those central to the method, should be explicitly defined when first introduced\n5. The authors do not clearly specify the source of the likelihood maps $lh_{1}$ $lh_{2}$ (l. 147) specifically, whether these correspond to the output of a sigmoid activation applied to the final layer of the UNet. Clarifying this detail would improve both the clarity and reproducibility of the method, as the exact definition of these maps is critical for understanding how predictions are calibrated and how consistency is enforced during training.\n6. It is unclear whether the index notation $i,j$,  used at line 153 is correct in the given context. Additionally, the indices $i,j,k,$ are never explicitly defined in the manuscript, which may cause confusion, especially for readers trying to follow the mathematical formulation in detail. Providing a clear definition of the indexing convention would improve the rigor and readability of the method description.\n\nMinor Weaknesses: \n1. line 62 typo topolgoical\n2. since Figure 3 is referenced before $L_{intra}$ and $L_{temp}$ it is better to repeat their name definition in the caption\n3. line 177 typo: missing {\n4. line 177 fragmented sentence: G identified…\n5. line 196 format typo inconsistency with Eq. at line 195: $P^{(t)}bt, i$ and $P^{(t)}dt, i$", "was_revised": true, "final_justification": "After reading the authors’ responses, I find that five of the six main weaknesses I previously identified have been satisfactorily addressed:\n\n(W1) has been verified through the authors' clarification;\n(W2) is now supported by new experiments;\n(W4–W6) are resolved by clearer notation and a commitment to update the final manuscript accordingly.\n\nOne point of improvement remains: (W3) pertains to the scope of the method, which is currently explicitly developed for binary segmentation. While binary segmentation is indeed a relevant and useful task in computational pathology, expanding to multiclass, instance (or even panoptic) segmentation would enable explicit counting, differentiation and classification between various tissue structures, support quantitative analysis, and facilitate tissue phenotyping, key tasks in digital pathology. The authors have provided appreciated multiclass segmentation results in their response and evidence of implicit instance segmentation effects provided by their strategy, but the model could be further improved by incorporating explicit instance segmentation strategies, which they acknowledge (with the proposal of potential strategies) as future work.\n\nI have also read all other reviews and the authors’ responses. Considering that most concerns have been addressed, aside from the other two remaining weaknesses highlighted by reviewer ZP1X, I would slightly raise my score to 4 (Borderline accept)", "rebuttal": "We sincerely thank the reviewer for the thorough technical evaluation and appreciate the recognition that our novel combination of established techniques represents an original and more efficient approach than existing methods. We are particularly grateful for acknowledging that our method successfully avoids hand-picked threshold limitations and for characterizing MATCH-Pair and MATCH-Global as \"smart glue\" for defining valid matches across persistence diagrams, which accurately captures our core innovation. We will carefully modify the typos and polish the script in the final version.\n\n**Q1: SSL training initialization problem.**\n\n**A1**: Thanks for pointing out the unclear parts. Both teacher and student models start from the same backbone initialization. No additional labels are involved in the pretraining stage. The models see the same labeled data during the two stages. Using the supervised loss on the labeled subset together with a pixel-wise consistency loss on unlabeled patches, we obtain roughly correct probability maps that capture most objects but may contain topological defects (splits, merges, spurious holes). Then, we incorporate our dual-level topological consistency losses in the second stage to explicitly penalize missing/overlapping components and fix the topological errors.\n\n**Q2: Crowding-aware ablation.**\n\n**A2**: We appreciate the reviewer’s suggestion and have now quantified the influence of nuclei density on model performance.\nWe randomly cut the test images into patches of size 256×256. For every patch, we count nuclei in the ground-truth instance map. Patches with <= 30 nuclei are labelled Sparse; those with >=100 nuclei are labelled Crowded. As reported in the main paper, the whole test image indicates that we did the inference on the entire test image. We sampled 14 samples to achieve a fair comparison and show the results below:\n\n| Setting                                              | Dice_obj ↑      | BE ↓             | BME ↓              |\n|------------------------------------------------------|---------------|----------------|------------------|\n| Sparse (Ours, $\\leq 30$ cells)                          | 0.804 ± 0.004 | 4.620 ± 0.140  | 163.132 ± 2.136  |\n| Crowded (TopoSemiSeg, $\\geq 100$ cells)           | 0.756 ± 0.009 | 6.890 ± 0.240  | 198.525 ± 3.125  |\n| Crowded (Ours, $\\geq 100$ cells)                  | 0.774 ± 0.007 | 5.610 ± 0.198  | 186.313 ± 2.715  |\n| Ours (whole test image)                           | 0.790 ± 0.006 | 4.930 ± 0.156  | 179.225 ± 2.383  |\n\nThe experiments above verify that our approach is density-aware. It achieves state-of-the-art accuracy on typical tissue, excels in sparse fields, and maintains a clear advantage over the strongest baseline under extreme nuclear crowding. The revised final version will include the full stratified and the crowding-aware ablation studies.\n\n**Q3: Extend from binary to multi-class segmentation. Lack of explicit instance-level constraints.**\n\n**A3**: To extend our method to the multi-class setting, we choose a multi-class nuclei segmentation dataset, MoNuSAC, to conduct experiments. This dataset contains four cell types: Epithelial, Lymphocyte, Macrophage, and Neutrophil. We conducted experiments using 20% labeled data and report the class-wise performance of TopoSemiSeg and our method below:\n\n| Class        | Method        | Dice_obj ↑       | BE ↓             | BME ↓             |\n|--------------|---------------|-----------------|-----------------|------------------|\n| Epithelial   | TopoSemiSeg   | 0.778 ± 0.009   | 5.342 ± 0.187   | 195.158 ± 4.627  |\n|              | **Ours**      | **0.781 ± 0.008** | **5.128 ± 0.189** | **186.847 ± 3.958** |\n| Lymphocyte   | TopoSemiSeg   | 0.751 ± 0.013   | 6.089 ± 0.223   | 218.394 ± 5.841  |\n|              | **Ours**      | **0.756 ± 0.012** | **5.794 ± 0.235** | **207.693 ± 4.672** |\n| Macrophage   | TopoSemiSeg   | 0.765 ± 0.011   | 5.687 ± 0.201   | 206.732 ± 4.985  |\n|              | **Ours**      | **0.769 ± 0.010** | **5.423 ± 0.208** | **195.381 ± 4.127** |\n| Neutrophil   | TopoSemiSeg   | 0.738 ± 0.016   | 6.521 ± 0.267   | 234.576 ± 6.123  |\n|              | **Ours**      | **0.742 ± 0.015** | **6.187 ± 0.281** | **221.459 ± 5.894** |\n\nAs demonstrated in our class-specific results on MoNuSAC (Epithelial, Lymphocyte, Macrophage, and Neutrophil), our approach consistently outperforms TopoSemiSeg across all cell types, with particularly notable improvements in topological metrics (BE and BME) that are crucial for distinguishing overlapping structures.\nRegarding instance-level segmentation, our topological matching mechanism could benefit from incorporating explicit instance constraints, as the spatial overlap and proximity components in our MATCH-Pair algorithm already capture some instance-level information by distinguishing spatially separated objects with similar topological persistence. Future work will explore extending our dual-level consistency framework to explicit instance segmentation tasks, where the topological stability across perturbations could provide robust supervision for resolving ambiguous boundaries between overlapping nuclei, potentially incorporating instance-aware loss terms that leverage our adaptive matching capabilities.\n\n**Q4: The unclear definition of the $b_i$ and $d_i$.**\n\n**A4**: We will add the introduction before using it in the final version. $b_i$ is the threshold at which a connected component first appears (birth), while $d_i$ is the threshold at which that component merges with an order one or vanishes. Their values are the pixel values of the critical points. A primer on persistent homology already appears in the Supplementary, and we will move an abbreviated version of this introduction to the main paper in the final version.\n\n**Q5: The lack of a precise definition of the likelihood maps.**\n\n**A5**: We thank the reviewer for this important clarification request regarding the likelihood maps $lh_1$ and $lh_2$. To clarify, the likelihood maps correspond to the softmax-activated outputs of the final UNet layer, representing normalized probability distributions over segmentation classes for each pixel. In the final version, we will add this specification to improve methodological clarity and reproducibility.\n\n**Q6: Undefined indices, $i, j, k$.**\n\n**A6**: We acknowledge this critical concern regarding notation and will clarify the index definitions for improved readability. At line 153, the indices $i, j, k$ represent specific roles in our formulation: $i$ and $j$ index the topological features from the first and second persistence diagrams respectively (where $i \\in \\{1, ..., n_1\\}$ and $j \\in \\{1, ..., n_2\\}$ with $n_1, n_2$ being the number of features in each diagram), while $k$ distinguishes between the two likelihood maps being compared (i.e., $k \\in \\{1, 2\\}$). The notation $w_{k, i}$ thus refers to the normalized persistence weight of the $i$-th topological feature in the $k$-th likelihood map, ensuring dimensional consistency throughout our similarity metric computation. In the final version, we will add explicit definitions of all indices and their ranges to eliminate potential confusion and enhance mathematical clarity."}, {"review_id": "ZcmCGMuZjS", "reviewer": "Reviewer_iCZE", "summary": "The authors introduce MATCH, a self-supervised framework for image segmentation in the medical field based on a teacher-student network architecture, where the student learns from both supervised loss on labeled data (standard Dice + cross-entropy) and on unlabeled data (through consistency at the topological level). In particular, it uses dual-level topological consistency via stochastic predictions from Monte Carlo dropout and temporal training snapshots. This process allows the framework to learn useful representations.", "questions": "What is the computational cost of the topological loss in terms of speed and memory consumption, particularly compared to other competitive methods like TopoSemiSeg, which uses Betti matching?\n\nThe main results in Table 1 do not show where your method saturates (there is still quite a difference between the 20% labeled and fully supervised results). How much can the unsupervised loss replace the contribution of supervised loss in training?\n\nHow is the generated uncertainty map correlated to real-world segmentation uncertainty, e.g., segmentation annotations by two experts?", "limitations": "One limitation I can think of is that the choice of hyperparameters in the model is somewhat equivalent to manually setting up thresholds for TopoSemiSeg, which the authors criticise and present as one primary motivation for their method. How stable are those hypoparameters?", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\nThe topological consistency is enforced by MATCH-pair, a Hungarian overlap-matching algorithm that integrates spatial overlap, topological persistence, and spatial proximity, which is rather interesting and well-motivated.\n\nThe contribution grounded in geometric intuition is supported by good uncertainty maps (Figure 5) and improved segmentation results (Figure 6).\n\nThe evaluation on three datasets is also quite thorough, and the proposed model seems to consistently outperform others.\n\nWeaknesses:\nThe results are only marginally better than the best current method in terms of segmentation quality. The Dice score is often just 0.001–0.006 higher than TopoSemiSeg, which is the main weakness in my opinion.\n\nThe computational cost of the topological loss in terms of model training and inference is not clearly described. How does the cost-value ratio look, particularly considering the marginal improvement in performance?\n\nThe authors do not compare their method with self-supervised approaches like SAM and its variants such as CellVit. Fine-tuning an SSL method on limited labeled data is also considered semi-supervision.\n\nThe novelty of the proposed method is not very high, as it is a combination of modules that are based on existing methods.", "was_revised": true, "final_justification": "Authors did a strong rebuttal and I would incline to raise my score to accept.", "rebuttal": "We sincerely thank the reviewer for recognizing our proposed MATCH-pair algorithm, highlighting its integration of spatial overlap, topological persistence, and spatial proximity. The reviewer also acknowledged our method's geometric intuition, supported by clear uncertainty visualizations and improved segmentation performance.\n\n**Q1: Marginal improvement on pixel-level performance.**\n\n**A1**: We acknowledge the reviewer’s observation that the improvement in Dice scores over TopoSemiSeg is modest; however, as shown in Table 1, our primary objective was not simply to enhance an already-saturated pixel-wise metric but rather to restore topological fidelity without compromising segmentation accuracy. Even minor topological errors involving only a few pixels can significantly impact downstream analyses, thus emphasizing the importance and practical relevance of the improvements achieved by our approach.\n\n**Q2: The computational cost of the topological loss regarding speed and memory consumption.**\n\n**A2**: We appreciate the reviewer's question regarding the computational cost of our topological loss. Our proposed method requires 1020.04 ms per training iteration and a GPU memory usage of 25.726 GB, while TopoSemiSeg consumes 610.80 ms per iteration and 15.235 GB GPU memory under comparable experimental settings. \n\nHowever, we respectfully clarify that TopoSemiSeg employs optimal matching based on the Wasserstein distance between persistence diagrams instead of Betti Matching, which does not account for spatial correspondences. In contrast, our dual-level topological consistency explicitly incorporates spatial correspondence, justifying the additional computational overhead by achieving better preservation of topological structures, which is critical for accurate downstream analyses.\n\n**Q3: Comparison with self-supervised methods finetuned on limited labeled data.**\n\n**A3**: We use LoRA to fine-tune the SAM and MedSAM using 20% labeled data on the CRAG dataset and report the performance below:\n\n| Method                 | Dice_obj ↑     | BE ↓             | BME ↓              |\n|------------------------|---------------|----------------|-----------------|\n| LoRA-SAM-CRAG-20%      | 0.882 ± 0.006 | 0.440 ± 0.042  | 27.300 ± 2.937  |\n| LoRA-MedSAM-CRAG-20%   | 0.898 ± 0.005 | 0.268 ± 0.025  | 11.275 ± 1.899  |\n| **Ours**                                        | **0.909 ± 0.005**  |  **0.188 ± 0.018**  |  **7.425 ± 0.570**  |\n\nThe results show that even with powerful foundation models, like SAM or MedSAM, topological errors can still exist without explicit topological modeling. \n\n**Q4: The novelty of the proposed method is not very high.**\n\n**A4**: Our work introduces several key innovations that advance semi-supervised histopathology segmentation beyond existing approaches. While prior topology-aware methods like TopoSemiSeg rely on fixed, hand-picked persistence thresholds that may exclude relevant structures or retain irrelevant ones, we propose the first adaptive topological consistency framework that automatically identifies meaningful structures without human-selected parameters. We introduce dual-level topological consistency by uniquely combining intra-prediction consistency (across Monte Carlo dropout realizations) with temporal consistency (across training snapshots), providing a more comprehensive approach to structural stability than single-level methods. Furthermore, we develop MATCH-Pair and MATCH-Global algorithms that integrate spatial overlap, topological persistence, and spatial proximity for robust feature matching across multiple predictions, addressing a critical limitation where existing methods like Wasserstein matching produce ambiguous correspondences due to the lack of spatial awareness. This multi-faceted adaptive approach represents a significant departure from threshold-dependent methods, enabling more robust identification of biologically meaningful topological structures while naturally providing uncertainty estimation as a byproduct of the consistency mechanism.\n\n**Q5: The main results in Table 1 do not show where your method saturates (there is still quite a difference between the 20% labeled and fully supervised results). How much can the unsupervised loss replace the contribution of the supervised loss in training?**\n\n**A5**: The observed performance gap between our method (trained with only 20% labeled data) and the fully supervised baseline reflects inherent constraints in scenarios typical to digital pathology, where extensive, high-quality annotations are rarely feasible. Given this practical limitation, our method effectively leverages unlabeled data through dual-level topological consistency losses, demonstrating substantial capability in extracting meaningful and robust representations. Although the results indicate potential for further performance gains with increased supervision, the significant improvement achieved underscores the efficacy of the unsupervised components in partially substituting supervised signals and enhancing generalizability, particularly when abundant annotations remain inaccessible.\n\n**Q6: How is the generated uncertainty map correlated to real-world segmentation uncertainty, e.g., segmentation annotations by two experts?**\n\n**A6**: While direct validation of uncertainty maps against ground truth segmentation uncertainty from multiple expert annotations would be ideal, such multi-annotator datasets are incredibly challenging to obtain in pathology due to the inherent difficulty and cost of pathological labeling. Consequently, we employ indirect validation methods to assess the correlation between our generated uncertainty maps and real-world segmentation uncertainty. These indirect approaches include computing Pearson correlation coefficients between predicted uncertainty and segmentation performance metrics and demonstrating the utility of uncertainty estimates through downstream applications such as uncertainty-guided active learning or segmentation refinement, where improved task performance is a proxy measure for uncertainty map reliability.\n\n**Q7: Sensitivity to hyperparameter tuning.**\n\n**A7**: In our evaluations, sensitivity analysis (Tables 3a and 3b) reveals that performance degrades gracefully across a wide parameter range, demonstrating robustness. Additional experiments, such as the ablation studies on the dropout rate and $\\tau_{primary}$ (Reviewer Hseg, Q1 & A1, Q7 & A7), also show the robustness of our method."}, {"review_id": "VQ0KKXja2r", "reviewer": "Reviewer_Hseg", "summary": "This paper proposes MATCH, a semi-supervised segmentation framework specifically designed for histopathology images.  It effectively mitigates topological errors by imposing adaptive topological consistency constraints. A central and novel aspect of this approach is the ability to discern reliable topological structures from multiple perturbed predictions, eliminating the reliance on fixed persistence thresholds, a requirement in prior methods such as TopoSemiSeg. The framework innovatively introduces a dual-level topological consistency mechanism. This includes intra-topological consistency across Monte Carlo dropout predictions and temporal-topological consistency across training snapshots. Moreover, the method utilizes novel matching algorithms, namely MATCH-Pair and MATCH-Global. These algorithms ingeniously integrate spatial overlap, topological persistence, and spatial proximity to accurately align topological structures across different predictions. Comprehensive experiments conducted on three histopathology datasets, namely CRAG, GlaS, and MoNuSeg, vividly demonstrate the superiority of MATCH. The results show notable improvements in both pixel-wise and topology-wise metrics when compared to existing state-of-the-art semi-supervised methods.", "questions": "1. Under what conditions does the matching algorithm fail? Can you provide examples where MATCH-Global produces incorrect correspondences and discuss potential mitigation strategies?\n2. It is stated that a novel integration of topological reasoning into SSL is one of the key contributions (Line 66). Nevertheless, the approach presented in this paper appears to be an enhancement of the TopoSemiSeg [57] approach. More evidence should be provided to demonstrate the innovativeness of the integration of topological reasoning and SSL in this paper.\n3. Given that there are some matching algorithms in this paper, could you please provide a detailed computational complexity analysis along with runtime comparisons against well-known or relevant baseline methods?\n4. Could you elaborate on whether and how the dropout rate of Monte Carlo (MC) affects the algorithm proposed in this paper?", "limitations": "The limitations section needs expansion in the main paper to include computational and methodological constraints.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\n1. The paper effectively overcomes a crucial shortcoming of existing topology-aware semi-supervised approaches by eliminating the need for hand-picked persistence thresholds. The adaptive identification of meaningful topological structures is well-motivated and solid from a technical perspective.\n\n2. The dual-level consistency framework is well-designed, combining intra-prediction consistency (MC dropout) with temporal consistency across training epochs. The MATCH-Pair and MATCH-Global algorithms effectively integrate multiple criteria (spatial overlap, persistence, proximity) for robust structure matching.\n\n3. The evaluation is comprehensive, covering three datasets with both pixel-wise and topology-specific metrics. The consistent improvements across different label ratios (10%, 20%) and datasets demonstrate the method's effectiveness.\n\n### Weaknesses\n\n1. Although the method purports to eschew hand-selected thresholds, it incorporates several hyperparameters ( $\\tau_{primary}$) for which the sensitivity has not been comprehensively analyzed. The ablation studies are restricted and do not encompass all crucial parameters.\n2. The paper fails to sufficiently discuss the computational cost of the matching algorithms. In particular, MATCH-Global, which necessitates solving multiple Hungarian assignment problems, might impose a substantial constraint on practical implementation.\n3. The elaboration and pre-explanation of the theory are inadequate. Specifically, the significance of \"Birth\" and \"Death\" depicted in Figure 1 has not been expounded upon. It is further recommended to specify what exactly \"dual-level\" refers to in Figure 3.\n4. The paper has multiple grammatical mistakes and ambiguous expressions. For instance, \"topolgoical\" is misspelled in line 62, and there are issues with inconsistent notation. Also, in certain sections, the mathematical notation could be made more accurate.", "was_revised": true, "final_justification": "The author has effectively resolved my concerns in the rebuttal.", "rebuttal": "We sincerely thank the reviewer for recognizing the key contributions of our work, including the adaptive topological consistency that removes the need for fixed thresholds, the well-designed dual-level consistency framework, and the effectiveness of our MATCH-Pair and MATCH-Global algorithms. We also appreciate acknowledging our comprehensive evaluation and consistent performance gains across datasets. We address your concerns point by point below.\n\n**Q1: Ablation study on the sensitivity of  $\\tau_{primary}$.**\n\n**A1**: We provide the ablation study on the sensitivity of $\\tau_{primary}$. The results have shown that our method is really robust to the selection of $\\tau_{primary}$. Moreover, the low threshold of 0.1 was chosen to be inclusive rather than restrictive: it allows more potential matches to be considered valid while letting the Hungarian algorithm determine optimal assignments based on our comprehensive similarity metric (combining spatial overlap, persistence weights, and proximity). This design philosophy aligns with our adaptive approach. Rather than using a high threshold to filter matches aggressively, we use a permissive threshold and rely on our sophisticated matching algorithm to identify the truly meaningful correspondences. This approach ensures we don't exclude potentially relevant topological structures prematurely, which would contradict our core contribution of avoiding hand-picked filtering thresholds.\n\n| $\\tau_{primary}$     | Dice_obj ↑     | BE ↓           | BME ↓          | DIU ↓            |\n|---------------|----------------|----------------|----------------|------------------|\n| 0.05          | 0.906 ± 0.006  | 0.195 ± 0.019  | 7.850 ± 0.620  | 41.750 ± 1.850   |\n| 0.1 (current) | **0.909 ± 0.005**  | **0.188 ± 0.018**  | **7.425 ± 0.570**  | **40.250 ± 1.720**   |\n| 0.2           | 0.908 ± 0.005  | 0.191 ± 0.020  | 7.680 ± 0.590  | 41.100 ± 1.780   |\n| 0.3           | 0.905 ± 0.006  | 0.201 ± 0.021  | 8.150 ± 0.650  | 42.850 ± 1.920   |\n\n\n**Q2: Computational cost of the matching algorithms.**\n\n**A2**: While MATCH-Global requires solving multiple Hungarian assignment problems, our method remains computationally feasible with a training time of 1020.04 ms per iteration compared to 610.80 ms for TopoSemiSeg. The peak GPU memory usage is 25.726 GB during training (batch size 16) and 8.49 GB during inference, enabling deployment on single-GPU systems.\n\n**Q3: Inadequate elaboration, pre-explanation of the theory, and has multiple grammatical mistakes.**\n\n**A3**: We acknowledge these critical concerns and will address them in the final version. Regarding theoretical exposition, we provide detailed explanations of the \"Birth\" and \"Death\" critical points in persistent homology in the Supplementary materials. We will move this foundational content to the main paper before introducing these concepts, while also enhancing Figure 3 to explicitly clarify that \"dual-level\" refers to our intra-topological consistency (across Monte Carlo dropout predictions) and temporal-topological consistency (across training snapshots). We will also carefully correct all grammatical errors and ensure consistent mathematical notation throughout the manuscript.\n\n**Q4: The failure cases of the matching algorithm.**\n\n**A4**: Our proposed algorithm may fail in the following cases: First, when dealing with highly fragmented or merged structures due to poor image quality or preprocessing artifacts, the flood-fill algorithm used to generate spatial masks may produce unreliable region boundaries, leading to inaccurate IoU calculations and subsequent mismatches. Second, in severe topological noise where numerous spurious connected components with similar persistence values are generated, the algorithm might struggle to distinguish between meaningful biological structures and artifacts, particularly when spatial proximity alone is insufficient for disambiguation. Additional analysis and visualizations will be provided in the final version.\n\n**Q5: More evidence should be provided to demonstrate the innovativeness of integrating topological reasoning and SSL in this paper.**\n\n**A5**: While we acknowledge that TopoSemiSeg [57] pioneered the integration of topological reasoning into SSL frameworks, our approach introduces several fundamental innovations that distinguish it from existing work. As reviewer ZP1X noted, our method presents novel contributions in addressing the core limitations of TopoSemiSeg's reliance on fixed, hand-picked persistence thresholds. Our key innovations include: (1) a data-driven approach to identify meaningful topological structures through multi-faceted predictions rather than predetermined thresholds, (2) the introduction of dual-level topological consistency that combines both intra-predictions (MC dropout) and temporal consistency across training snapshots, and (3) the development of MATCH-Pair and MATCH-Global algorithms that integrate spatial overlap, topological persistence, and spatial proximity for robust structure matching without ground truth. These contributions represent a paradigm shift from threshold-based topological filtering to adaptive, perturbation-based identification of stable topological features, fundamentally advancing how topological reasoning is integrated into SSL frameworks. The experimental results demonstrate that our approach significantly outperforms TopoSemiSeg across all topology-wise metrics while maintaining comparable pixel-wise performance, validating the effectiveness of our novel integration approach.\n\n**Q6: Computational complexity analysis along with the runtime comparisons of the matching algorithms.**\n\n**A6**: The computational complexity of our MATCH-Pair algorithm consists of several components: persistence diagram computation requires $O(n^3)$ time complexity using the reduction algorithm, where $n=MN$ pixels, spatial mask generation via flood-fill takes $O(n)$ time, and similarity matrix computation between $k_1$ and $k_2$ topological features in two persistence diagrams requires $O(k_1 \\times k_2)$ operations for IoU and Euclidean distance calculations, and the Hungarian algorithm for optimal assignment has $O(min(k_1, k_2)^3)$ complexity, yielding an overall complexity of $O(n^3+k_1k_2+min(k_1, k_2)^3)$. In comparison, Wasserstein matching (used in TopoSemiSeg) requires $O(n^3+k^3)$ for calculating the persistence diagrams and the matching. Betti matching has $O(n^3+n^2+k^2)$ complexity from super-level filtration embedding and overlapping feature identification. Since $k \\ll n$ in practice, the dominant $O(n^3)$ term from barcode computation is shared across all methods, while MATCH-Pair achieves superior matching accuracy with only marginal additional cost in the matching phase. \n\n**Q7: How does the dropout rate of MC-dropout affect the algorithm proposed in this paper?**\n\n**A7**: We add complementary ablation studies on the dropout rate of the MC-dropout. Other settings are kept unchanged. We conduct the ablation experiments on CRAG 20% labeled data and report the performance below:\n\n| Dropout Rate | Dice_obj ↑     | BE ↓            | BME ↓           |\n|--------------|---------------|----------------|----------------|\n| 10%          | 0.898 ± 0.006 | 0.210 ± 0.020  | 8.200 ± 0.650  |\n| 20% (current)          | 0.909 ± 0.005 | 0.188 ± 0.018  | 7.425 ± 0.570  |\n| 30%          | 0.910 ± 0.005 | 0.185 ± 0.017  | 7.350 ± 0.560  |\n| 50%          | 0.890 ± 0.007 | 0.220 ± 0.022  | 8.800 ± 0.720  |\n\nThe predicted results reveal an optimal dropout rate range of 20%-30% for our framework, where performance plateaus with minimal differences between these rates. Lower dropout rates provide insufficient perturbation diversity for reliable topological matching. In contrast, excessive dropout introduces detrimental noise that degrades both pixel- and topology-wise performance, confirming that moderate stochasticity is essential for effective topological consistency estimation."}, {"review_id": "Ct7wqDHrss", "reviewer": "Reviewer_ZP1X", "summary": "This paper introduces a method with three modules for the segmentation of histopathological images: the first module aims at increasing the consistency across predictions from slightly augmented images (semi-supervised learning); the second module aims at increasing the consistency across multiple stochastic predictions (MC dropout); the third module aims at increasing the topological consistency across different snapshots of the model during training. The method is compared with prior related work, and ablation studies are conducted.", "questions": "- What's the exact running time of this method? And what about the running time of a baseline that does not include the persistence-homology method?\n- There are two algorithms to compute the Matches (Section 3.1 and 3.2). Which one is actually used for the final method?\n- How/which statistical significance tests were used?", "limitations": "A limitation is mentioned in the supplementary material. I think that there are more obvious limitations with this method. First, relying on persistence homology typically translates to extremely large training time (see one of my questions). Second, since this method relies on different \"modules\" (see the three extra terms in the total loss function, above Section 4), the underperformance on one of them due to dataset particularities could lead to a general underperformance. Additionally, the sensitivity analysis on the hyper-parameters show that an optimal hyper-parameter value depends on the dataset, and there is no easy/intuitive way to tune it.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n- Novelty: consistency matching strategies are common in semi-supervised learning, but they are typically focused on pixel-level consistency and not topology consistency, which is novel.\n- Number of experiments: experiments include comparisons with several related modern prior work on three datasets, ablation and sensitivity studies.\n- Importance in microscopy imaging:Since there exist fully-annotated microscopy datasets and annotating new ones is time-consuming, new methods in semi-supervised learning strategies impact scientific areas utilizing microscopy imaging.\n- Clarity: the method is well described.\n\n# Weaknesses\n- Experiments were conducted only once (one random seed).\n- The ablation studies were only conducted on two out of the three components. Specifically, there is an ablation study for the \"intra\" and for the \"temp\" terms, but not for the \"const\" term in the equation above Section 4.\n- The ablation studies show that the method is sensitive to the hyper-parameters (see Table A, B_intra and B_temp value, whose best value was 4; not 3 or 5). It also doesn't seem to be an easy/intuitive way to tune them. Furthermore, it hyper-parameter values probably depend on the dataset.\n- Running time, which is crucial especially in Persistence-Homology-based methods, is unreported.\n\nMinor:\n- Typo in 211: \"EM [48]. UA-MT\" it should be a comma.\n- Typo in L62: \"topolgoical\"\n- L261: This is not an ablation study but a sensitivity study, i.e., the goal is to measure how sensitive the method is to a hyper-parameter choice.", "was_revised": true, "final_justification": "My final recommendation is the same as my initial recommendation: Acceptance.\n\nAfter engaging with the authors during the discussion period, two of the weaknesses that I highlighted were addressed. I have also followed the comments from the other reviewers and the authors' answers, and I believe that they were addressed to a good extent as well. I still believe that there are some limitations in the method: 1) the method being too slow (which makes it unusable for 3D images), 2) the method being sensitive to the hyper-parameter values. However, with these two weaknesses, acknowledged by the authors, I find that this work has sufficient technical novelty to be accepted in NeurIPS.", "rebuttal": "We sincerely thank the reviewer for the comprehensive comments and appreciate the recognition of our key contributions: the novelty of extending consistency matching from pixel-level to topological consistency in semi-supervised learning, the thoroughness of our experimental validation across three datasets with extensive baselines and ablation studies, and the practical importance for microscopy imaging, where annotation is time-consuming. We are also grateful for the positive feedback on methodological clarity. We will modify the typos and make clearer statements in the final version.\n\n**Q1: Experiments were conducted only once (one random seed); How/which statistical significance tests were used?**\n\n**A1**: Our experiments were conducted with multiple random seeds (as evidenced by the error bars reported in Table 1), and we employed standard deviation calculations across these runs to assess variability. For statistical significance testing, we used unpaired t-tests to compare our method against baseline approaches, with significance levels set at p < 0.05.  The error bars in our results represent standard deviations across $5$ independent runs with different random initializations, ensuring reproducibility and statistical validity.\n\n**Q2: Ablation study on \"const\" term.**\n\n**A2**: We conducted the ablation study on the “const” item and report the performance below:\n\n| Method    | Dice_obj ↑      | BE ↓            | BME ↓           |\n|-----------|---------------|----------------|----------------|\n| w/o const | 0.875 ± 0.008 | 0.285 ± 0.025  | 9.850 ± 0.680  |\n| **Ours**\t  | **0.909 ± 0.005**  |  **0.188 ± 0.018**  | **7.425 ± 0.570** |\n\nBased on the ablation study results and the principles of semi-supervised learning, removing the pixel-wise consistency term in the training stages would result in significant performance degradation across all metrics. Without this foundational constraint, the model would rely solely on limited labeled data and the sparse topological consistency signals, which only constrain specific critical points rather than the complete segmentation boundary.\n\n**Q3: The way to tune the hyperparameters is not that easy and intuitive.**\n\n**A3**: The hyperparameter sensitivity observed in our ablation studies (Table 3 (a)) reflects careful optimization rather than a limitation. The optimal values ($B_{intra} = B_{temp} = 4$) were systematically determined through grid search on the CRAG dataset and demonstrate robust transferability across domains. Importantly, these hyperparameters were tuned exclusively on CRAG and directly transferred to the two additional evaluation datasets without further adjustment, achieving consistent performance improvements.\n\n**Q4: Exact running time of this method and a baseline that does not include the PH-based method.**\n\n**A4**: We provide detailed runtime analysis comparing our method against non-persistent homology baselines. Our MATCH framework requires 1020.04 ms per training iteration with dual-level topological consistency, while PMT [8] (a non-PH baseline) at 582.34 ms per iteration, with total training time of 4h36m and peak GPU memory usage of 25.726 GB (batch size 16, UNet backbone). While this computational cost is substantial, it is justified by the significant improvements in topological accuracy essential for reliable histopathology analysis, where the clinical cost of topological errors far outweighs the additional computational expense, and our method remains practically feasible on standard single-GPU systems.\n\n**Q5: Which one is actually used for the final method?**\n\n**A5**: Sorry for the misleading. Our final method employs both algorithms in a hierarchical manner: MATCH-Pair (Section 3.1) serves as the foundational pairwise matching component, while MATCH-Global (Section 3.2) extends this approach to handle multiple facets by sequentially applying MATCH-Pair between adjacent prediction pairs and then constructing a global correspondence graph. Specifically, MATCH-Global utilizes MATCH-Pair's Hungarian algorithm with our spatial-aware similarity metric to establish pairwise correspondences, then performs breadth-first search on the resulting graph to identify globally consistent topological structures across all MC dropout predictions and temporal snapshots. Therefore, MATCH-Global represents the complete matching framework used in our dual-level topological consistency approach, with MATCH-Pair functioning as its core algorithmic building block."}]

Author rebuttal:

Meta-review: Five experts reviewed this paper. Their post rebuttal recommendations are 3 Borderline Accepts and 2 Accepts. Reviewers appreciated the technical novelty of MATCH, the clear motivation, and the convincing empirical results on three challenging datasets (CRAG, GlaS, MoNuSeg). It can accurate align topological structures. This is significant because histopathology segmentation if histology images requires pixel-level accuracy but also preservation of fine-grained topological features. MATCH outperforms standard SSL baselines and SOTA topology-aware approaches. Reviewers also highlighted the detailed ablations, including studies on the intra- and temporal-topological losses, hyperparameter robustness, and generalization across segmentation backbones. Finally, the Reviewers found the paper to be clearly written and organized.

Overall, MATCH is well-motivated and makes a decent contribution to the SOTA in semi-supervised histopathology segmentation. Its novelty lies in introducing adaptive topological consistency through robust matching and dual-level constraints. The empirical improvements are convincing.

In the discussions and rebuttal, Reviewers raised concerns were about the (a) computational cost and scalability to larger or 3D datasets, (b) sensitivity to hyper-parameters, and (c) the method’s application to binary segmentation (it may require extensions for multi-class or instance-level settings). Some reviewers also requested clarifications on why Wasserstein-based matching underperformed, and on the incremental benefits of using both intra- and temporal-consistency terms. In the rebuttal, the authors addressed these issues with added experiments, ablations, and discussion. These concerns were well addressed in the rebuttal, and one of the reviewers raised their ratings. Although remaining concerns remain over the computational overhead, the strengths outweigh the weaknesses. I therefore recommend acceptance as a poster. The authors are encouraged to make the necessary changes to the best of their ability.
Decision: Accept (poster)