SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Details of the network. Structures interact with each other in the image space and are not independent. During uncertainty estimation, it is therefore crucial to consider their spatial context, i.e., interstructural uncertainty. Hence, we use Graph Neural Networks (GNN) [59], specifically Graph Convolution Networks (GCN) [31], to jointly reason about the structures and capture the high-order spatial interactions. In the graph, each node represents a structure, and edges between nodes exist when corresponding structures have non-zero overlaps (typically at endpoints). The input feature vector for each node is constructed as shown in Fig. 6. For every structure, we first concatenate  $[x^c, f^c, m]$ , where  $x^c$  comes from the original input  $x$ ;  $f^c$  from the likelihood map  $f$  (not  $f_n$ ); and  $m$  is a binary map indicating the presence of the structure. These  $x^c, f^c, m$  are smaller crops/bounding boxes centered on the structure. After passing them through convolution blocks, we apply channelwise pooling to obtain a fixed-length feature vector for training. We further concatenate the persistence value of the saddle point associated with the original DMT structure (aka stable manifold). Persistence value (from persistent homology [13]) is defined as the difference of function (likelihood) values of 2 critical cells (saddle-maxima pair). It captures the importance of a structure, thus making it a valuable feature in our framework. Note that we do not use the perturbed  $f_n$  from the Prob. DMT method when constructing the feature vector.
Training the network. We train the regression network using the attenuation loss proposed in [30]. As there are no labels to learn uncertainty, it is implicitly learned during regression optimization. We fix a Gaussian likelihood, and so variance  $\hat{\delta}^2$  is used as a measure of uncertainty. The network's head is split into two — to predict  $\hat{p}(e)$  of being a true positive structure and its associated uncertainty  $\hat{\delta}_e^2$ . For numerical stability, we actually predict the log variance  $s_e = \log \hat{\delta}_e^2$ . The training loss is given in Eq. 1. The structures that we obtain from Prob. DMT may not always fully overlap with the true GT structures, that is, some structures may only have partial overlap. We thus do not impose any hard
2For neighborhood, we use 8-connectivity for 2D, and 26-connectivity for 3D in this work.
<sup>3</sup>If the path does not reach  $c_{m}$  , we impose a maximum limit on the update steps to prevent an infinite loop.
Figure 6: Construction of the input feature vector for each node (structure) in the GCN.
constraints in Eq. 1, instead,  $z_{e}$  is a soft label, and is given by:  $z_{e} = (\sum y\odot m) / (\sum m)$ , where  $y$  is the GT and  $\odot$  is the Hadamard product. This value simply represents the proportion of the structure that overlaps with the GT, i.e., the fraction of the structure that is a true positive.
$$
L _ {U Q} (\phi) = \frac {1}{| E |} \sum_ {\forall e \in E} \left(\frac {1}{2} \frac {\left\| \hat {p} (e) - z _ {e} \right\| ^ {2}}{\exp \left(s _ {e}\right)} + \frac {1}{2} s _ {e}\right) \tag {1}
$$
In [30],  $\hat{\delta}^2$  denoted the pixel-wise uncertainty of the framework's input. In our setting, the input to our framework is  $f = F_{\theta}(x)$ , and so  $\hat{\delta}^2$  is modeled to capture the structure-wise uncertainty inherent in data  $x$  and model  $F_{\theta}$ . Training  $\hat{\delta}^2$  in this manner ensures that the network does not trivially predict high or low uncertainty, rather, predicts an uncertainty estimate that is dependent on the input.
3.3 Proposed module  $M_{\phi}$
For Eq. 1 to hold, we require  $M_{\phi}$  to be a probabilistic network. We already show in Sec. 3.1 our formulation for Prob. DMT. Additionally, the regression network is also probabilistic as we use MC dropout [16].
Inference procedure. We take  $T$  runs of  $M_{\phi}$  and compute the uncertainty as the mean  $\bar{\delta}_e^2 = \frac{1}{T}\sum_{t=1}^{T}(\hat{\delta}_e^2)_t$ . We similarly obtain  $\bar{p}(e)$  from  $\hat{p}(e)$ . In Fig. 7, we illustrate the post-processing steps to obtain the structure-wise uncertainty heatmap. First, we obtain maps  $\bar{p} = \cup \bar{p}_e$  and  $\bar{\delta}^2 = \cup \bar{\delta}_e^2$  having the same spatial resolution as the input  $x$ . We then binarize  $\bar{p}$ , and overlay it onto the segmentation map obtained from  $F_\theta$ . We do this because Prob. DMT
Figure 7: Post-processing procedure.
gives us one-pixel wide skeleton structures but we need to recover the structure thickness. Next, we use shortest distance to assign uncertainty values from  $\bar{\delta}^2$  to the pixels in the overlaid map. The shortest distance uses paths only along the foreground pixels. In Fig. 7 we show how we obtain the final uncertainty heatmap from the skeleton heatmap. We also note that the overlaid map is an additional output of our method: it is an improved segmentation map that can be used instead of the one obtained by  $F_{\theta}$ . We provide more details in Appendix D.
Datasets. We evaluate our method on four datasets: DRIVE [65], ROSE [40], ROADS [44] and PARSE 2022 Grand Challenge [39, 70]. The DRIVE dataset contains 2D retinal vasculature; ROSE is a 2D retinal OCTA (Optical Coherence Tomography Angiography) segmentation dataset, ROADs is a large non-medical dataset containing aerial images, and PARSE contains 3D CT scans of pulmonary arteries. We further describe the datasets and the data splits in Appendix E.
Baselines. We broadly split our comparison baselines into three types: a) Standard vessel segmentation methods: UNet [57], DeepVesselNet [68], and  $\mathbf{CS}^2$ -Net [48]; b) Pixel-wise uncertainty estimation methods: Prob.-UNet [32], and PHiSeg [4]; c) Structure-wise uncertainty estimation method: Hu et al. [28]. Implementation details are provided in Appendix F.
Evaluation metrics. To evaluate the quality of uncertainty quantification, we use Expected Calibration Error (ECE) [50] and Reliability Diagrams (RD) [8]. Furthermore, we also evaluate the segmentation on metrics such as DICE [73], clDice [62], ARI [2], VOI [42], Betti Number error [27] and Betti Matching error [66]. We include clDice, ARI, VOI, Betti Number and Betti
Input image
GT
UNet
PHiSeg
Hu et al.
GT
Figure 8: Qualitative results compared to the uncertainty baselines. We show uncertainty estimates in the form of a heatmap. Green highlights false negatives and yellow highlights false positives. Row 1: DRIVE; Row 2: ROSE; Row 3: ROADs; Row 4: PARSE (3D render).
Figure 9: Reliability diagrams of samples from each dataset.
Matching as they are topology-based metrics and hence are sensitive to the performance on thin structures. Detailed definitions are present in Appendix G.
Tab. 1 shows the quantitative results against uncertainty methods, and Tab. 2 shows the quantitative results on different backbone architectures. We show the respective qualitative results in Fig. 8 and Fig. 10. In Fig. 9, we plot the Reliability Diagrams. We also perform the unpaired t-test [67] (95% confidence interval) to determine the statistical significance. Each table reports the mean and standard deviations for every metric, with statistically significant better performances in bold and numerically better (but not significant) performances in italics. For all the probabilistic methods, the average of five runs was used. For our method, we generated the structure-wise uncertainty estimates and the segmentation map by following the steps outlined in the 'Inference procedure' in Sec. 3.3. Due to space constraints, results on two metrics Betti Number and Betti Matching are reported in Appendix H. We discuss the remaining performances below.
Performance of uncertainty estimation. Tab. 1 shows that our method outperforms others on both ECE and segmentation metrics. Fig. 9 displays RDs, with our method following the ideal line much closely compared to others. This is because we explicitly model the distribution of the structures, thereby quantifying the uncertainty of the segmentation network. In Fig. 8, we also see that our method generates better fidelity structure-wise uncertainty maps compared to Hu et al. Our heatmaps
Figure 10: Qualitative results over different segmentation backbones. Green highlights false negatives. Row 1: DRIVE; Row 2: ROSE; Row 3: ROADs; Row 4: PARSE.
Table 1: Comparison against uncertainty baselines (all use UNet [57] as the backbone)
Dataset	Method	ECE (%)↓	Dice↑	clDice↑	ARI↑	VOI↓
DRIVE	Prob.-UNet [32]	8.3316 ± 0.0043	0.7779 ± 0.0219	0.7663 ± 0.0492	0.7759 ± 0.0532	0.3560 ± 0.0203
PHiSeg [4]	7.9316 ± 0.0032	0.7851 ± 0.0295	0.7712 ± 0.0497	0.7767 ± 0.0497	0.3527 ± 0.0308
Hu et al. [28]	8.0883 ± 0.0036	0.7866 ± 0.0141	0.7725 ± 0.0392	0.7768 ± 0.0403	0.3489 ± 0.0286
Ours	4.1633 ± 0.0043	0.7976 ± 0.0195	0.7974 ± 0.0372	0.7996 ± 0.0301	0.3322 ± 0.0229
ROSE	Prob.-UNet [32]	7.2795 ± 0.0022	0.7378 ± 0.0284	0.6485 ± 0.0258	0.7219 ± 0.0538	0.7769 ± 0.0146
PHiSeg [4]	7.0875 ± 0.0036	0.7415 ± 0.0267	0.6552 ± 0.0236	0.7309 ± 0.0425	0.7638 ± 0.0128
Hu et al. [28]	6.9243 ± 0.0033	0.7429 ± 0.0132	0.6598 ± 0.0172	0.7506 ± 0.0302	0.7616 ± 0.0123
Ours	3.9904 ± 0.0041	0.7593 ± 0.0171	0.6782 ± 0.0119	0.7837 ± 0.0314	0.7403 ± 0.0239
ROADS	Prob.-UNet [32]	8.4318 ± 0.0042	0.7194 ± 0.0418	0.8058 ± 0.0615	0.7350 ± 0.0494	0.5602 ± 0.0308
PHiSeg [4]	7.9331 ± 0.0038	0.7203 ± 0.0366	0.8113 ± 0.0521	0.7392 ± 0.0416	0.5559 ± 0.0295
Hu et al. [28]	7.8034 ± 0.0029	0.7275 ± 0.0361	0.8282 ± 0.0493	0.7314 ± 0.0391	0.5644 ± 0.0239
Ours	4.1442 ± 0.0031	0.7461 ± 0.0364	0.8496 ± 0.0455	0.7601 ± 0.0349	0.5463 ± 0.0218
PARSE	Prob.-UNet [32]	9.9918 ± 0.0069	0.6002 ± 5.7751	0.6179 ± 0.0804	0.6523 ± 0.0654	0.8923 ± 0.0417
PHiSeg [4]	9.9280 ± 0.0077	0.5910 ± 3.0858	0.6080 ± 0.0743	0.6512 ± 0.0521	0.8839 ± 0.0297
Hu et al. [28]	7.7891 ± 0.0075	0.6044 ± 2.3583	0.6153 ± 0.0724	0.6537 ± 0.0363	0.8803 ± 0.0318
Ours	4.0289 ± 0.0073	0.6190 ± 3.0826	0.6221 ± 0.0613	0.6658 ± 0.0461	0.8701 ± 0.0332
assign non-zero uncertainty to several false positives/negatives in the backbone UNet's outputs. This is because we reason about every structure while Hu et al. limits the structure space via pruning.
Performance over different backbones. Tab. 2 and Fig. 10 show that our method is backbone-agnostic. It consistently improves the segmentation quality and produces high fidelity uncertainty maps for each of the underlying networks. This validates the practical applicability of our method.
Performance of proofreading. One of the motivations of this work is to streamline the proofreading process. Structure-wise uncertainty can be used as a guide, with a user having to simply accept/reject a structure. We conduct experiments on the ROSE dataset and simulate user interaction with our method and Hu et al.'s. The user is given each method's final segmentation map, and inspects structures in decreasing order of uncertainty (till 0.5). Each uncertain structure is subjected to a yes/no decision, which is denoted as one 'click'. The results are in Fig. 11. Our findings are consistent with the observation that Hu et al. assigns zero uncertainty to many structures; thus their margin of improvement is limited and saturates quickly.
Figure 11: Proofreading.
Table 2: Comparison against different segmentation backbones
Dataset	Method	Dice↑	cIDice↑	ARI↑	VOI↓
DRIVE	UNet [57]	0.7728 ± 0.0336	0.7586 ± 0.0405	0.7530 ± 0.0519	0.3697 ± 0.0329
UNet [57] + Ours	0.7976 ± 0.0195	0.7974 ± 0.0372	0.7996 ± 0.0301	0.3322 ± 0.0229
DeepVesselNet [68]	0.8015 ± 0.0260	0.7997 ± 0.0431	0.7729 ± 0.0457	0.3413 ± 0.0256
DeepVesselNet [68] + Ours	0.8173 ± 0.0190	0.8285 ± 0.0361	0.8037 ± 0.0361	0.3238 ± 0.0192
CS2-Net [48]	0.8189 ± 0.0176	0.8125 ± 0.0413	0.8204 ± 0.0495	0.3417 ± 0.0203
CS2-Net [48] + Ours	0.8301 ± 0.0172	0.8367 ± 0.0305	0.8495 ± 0.0301	0.3243 ± 0.0258
ROSE	UNet [57]	0.7375 ± 0.0197	0.6453 ± 0.0165	0.7206 ± 0.0426	0.8488 ± 0.0126
UNet [57] + Ours	0.7593 ± 0.0171	0.6782 ± 0.0119	0.7837 ± 0.0314	0.7403 ± 0.0239
DeepVesselNet [68]	0.7653 ± 0.0101	0.6634 ± 0.0192	0.7622 ± 0.0302	0.7426 ± 0.0163
DeepVesselNet [68] + Ours	0.7795 ± 0.0205	0.6873 ± 0.0195	0.7936 ± 0.0282	0.7164 ± 0.0226
CS2-Net [48]	0.7623 ± 0.0285	0.6799 ± 0.0127	0.7702 ± 0.0322	0.7236 ± 0.0157
CS2-Net [48] + Ours	0.7886 ± 0.0208	0.6968 ± 0.0149	0.7981 ± 0.0211	0.7072 ± 0.0168
ROADS	UNet [57]	0.7011 ± 0.0426	0.7918 ± 0.0679	0.7143 ± 0.0526	0.5832 ± 0.0345
UNet [57] + Ours	0.7461 ± 0.0364	0.8496 ± 0.0455	0.7601 ± 0.0349	0.5463 ± 0.0218
DeepVesselNet [68]	0.7518 ± 0.0345	0.8248 ± 0.0574	0.7923 ± 0.0441	0.5641 ± 0.0321
DeepVesselNet [68] + Ours	0.7673 ± 0.0324	0.8513 ± 0.0519	0.8139 ± 0.0464	0.5357 ± 0.0329
CS2-Net [48]	0.7539 ± 0.0366	0.8341 ± 0.0511	0.8197 ± 0.0426	0.5475 ± 0.0468
CS2-Net [48] + Ours	0.7692 ± 0.0372	0.8559 ± 0.0528	0.8368 ± 0.0419	0.5261 ± 0.0411
PARSE	UNet [57]	0.5905 ± 3.0661	0.6104 ± 0.0727	0.6509 ± 0.0852	1.9738 ± 0.0414
UNet [57] + Ours	0.6190 ± 3.0826	0.6221 ± 0.0613	0.6658 ± 0.0461	0.8701 ± 0.0332
DeepVesselNet [68]	0.7208 ± 3.0452	0.6801 ± 0.0554	0.6923 ± 0.0524	0.4907 ± 0.0701
DeepVesselNet [68] + Ours	0.7376 ± 3.1863	0.6983 ± 0.0622	0.7098 ± 0.0613	0.4711 ± 0.0613
CS2-Net [48]	0.7630 ± 3.9415	0.6918 ± 0.0695	0.7138 ± 0.0695	0.4273 ± 0.0521
CS2-Net [48] + Ours	0.7720 ± 2.8109	0.7113 ± 0.0689	0.7343 ± 0.0733	0.4078 ± 0.0642
To demonstrate the efficacy of the proposed method, we conduct ablation studies of the different components in our pipeline, as well as check the effect of changing hyperparameter values.
We also include ablation studies on the dimensionality of the input feature vector, and size of the crops/bounding boxes (which we report in Appendix I). All analyses are on the DRIVE dataset using UNet [57] as the backbone.
Table 3: Ablation of different modules
DMT	Reg. Net	ECE (%)↓	clDice↑
DMT	GNN	6.3481 ± 0.0082	0.7729 ± 0.0304
Prob. DMT	MLP	4.8202 ± 0.0046	0.7745 ± 0.0305
Prob. DMT	GNN	4.1633 ± 0.0043	0.7974 ± 0.0372
Ablation of different modules. We conduct ablation studies on both parts of our framework: structure generation (DMT vs Prob. DMT), and regression network (GNN vs Multi-layer Perceptron (MLP)). The results are in Tab. 3. Prob. DMT results in a sharp improvement in ECE compared to the original DMT; this supports our assertion that Prob. DMT models intra-structural uncertainty. Similarly, using GNN over MLP results in improvement. The message-passing in GNNs accounts for inter-structural uncertainty, thus yielding higher fidelity uncertainty estimates.
Effect of hyperparameters. Our main hyperparameters are  $u, \alpha, \beta, \gamma$ , with  $u$  used in the Bernoulli distribution,  $\gamma$  in the path-generation algorithm, and (shape  $\alpha$ , scale  $\beta$ ) as prior hyperparameters of the Inverse Gamma distribution. We achieve the best ECE when  $u = 0.3, \gamma = 0.2, \alpha = 2.0, \beta = 0.01$ , however, a reasonable range always yields improvement, thus demonstrating the robustness of the method. We provide results of testing different hyperparameter values in Appendix I.
In this work, we propose to quantify the structure-wise uncertainty of a given segmentation network. Our framework explicitly models structures as samples from a probability distribution, thus helping to estimate intra-structural uncertainty. Furthermore, we incorporate inter-structural uncertainty by jointly reasoning over the structures, resulting in better fidelity uncertainty estimates. This structure-wise uncertainty quantification can streamline the proofreading process by reducing the time spent finding and correcting errors. Extensive experiments show the practical applicability of our method over different segmentation backbones and datasets. We further discuss the broader impact of our work and the limitations in Appendix J and Appendix K respectively.
Acknowledgements. This research was partially supported by grants NSF 2144901, NCI R21CA258493, and NIH 1R21CA258493-01A1. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
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Appendix A provides a pseudocode of our proposed Probabilistic DMT module.
Appendix B provides a detailed discussion on the hyperparameters used in this work.
Appendix C reports the computational complexity of the method.
Appendix D expands on the 'Inference procedure' outlined in Sec. 3.3 of the main paper.
Appendix E provides the details of the datasets used.
Appendix F gives implementation details of our method and the baseline methods.
Appendix G provides definitions of the evaluation metrics used.
Appendix H provides results on topological metrics.
Appendix I provides results of ablation studies mentioned in Sec. 4.2 of the main paper.
Appendix J discusses the broader impact of this work.
Appendix K discusses the limitations of this work.
A Probabilistic DMT
In Algo. 1, we provide a pseudocode of our Probabilistic DMT module proposed in Sec. 3.1. We set max_step as 50 in our implementation. The terminologies used are:  $f^c$  denotes the likelihood map from  $F_{\theta}$  centered on structure  $e$ ;  $(c_s, c_m)$  are critical points between which the path  $e$  is generated;  $c_s$  denotes the saddle point and  $c_m$  denotes the maxima.  $IG$  denotes the Inverse Gamma distribution, and  $\mathcal{N}$  denotes the Gaussian distribution.
Algorithm 1 Probabilistic DMT pseudocode
1: procedure PROB_DMT  $(f^c,e,c_s,c_m)$    
2:  $\hat{u}\sim$  Bernoulli(u)   
3: if  $\hat{u}$  is True then   
4:  $\hat{e}\gets e$    
5: else   
6:  $\hat{e}\gets$  GENERATE_PATH  $(f^{c},c_{s},c_{m})$    
7: end if   
8: return  $\hat{e}$    
9: end procedure   
10: procedure GENERATE_PATH  $(f,c_{s},c_{m})$    
11: initialize  $m\gets 0$  ▷  $m$  has same spatial dimension as  $f$    
12: initialize  $c\gets c_s$    
13: initialize  $m[c]\gets 1$    
14: initialize step  $\leftarrow 0$    
15:  $\sigma^2\sim IG(\alpha ,\beta)$    
16:  $f_{n}\sim f + \mathcal{N}(0,\sigma)$    
17: while  $c\neq c_m$  and step  $<  \max$  step do   
18: val  $\leftarrow 0$    
19: for  $c'\in$  Neighborhood(c) do   
20: val  $[c']\gets \gamma *\frac{1}{||c_m - c'_||_2} +(1 - \gamma)*f_n[c']$    
21: end for   
22:  $c\gets \mathrm{argmax}(val)$  ▷ Update current step   
23: m[c]  $\leftarrow 1$    
24: step  $\leftarrow$  step+1   
25: end while   
26: return m   
27: end procedure
B Discussion of hyperparameters
The main hyperparameters in this work are  $u, \alpha, \beta, \gamma$ . We describe the importance of each below: