Original: Structure-Aware Image Segmentation with Homotopy Warping
Revised: Structure-Aware Image Segmentation with Homotopy Warping

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AffiliationAddress email
Revised: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Besides per-pixel accuracy, topological correctness is also crucial for the segmentation of images with fine-scale structures, e.g., satellite images and biomedicalimages.
Revised: Besides per-pixel accuracy, topological correctness is also crucial for the segmentation of images with fine-scale structures, e.g., satellite images and biomedicalimages.

Original: In this paper, by leveraging the theory of digital topology, we identifylocations in an image that are critical for topology.
Revised: In this paper, by leveraging the theory of digital topology, we identifylocations in an image that are critical for topology.

Original: By focusing on these criticallocations, we propose a new homotopy warping loss to train deep image segmentation networks for better topological accuracy.
Revised: By focusing on these criticallocations, we propose a new homotopy warping loss to train deep image segmentation networks for better topological accuracy.

Original: To efficiently identify thesetopologically critical locations, we propose a new algorithm exploiting the distance transform.
Revised: To efficiently identity thesetopologically critical locations, we propose a new algorithm exploiting the distance transform.

Original: The proposed algorithm, as well as the loss function, naturallygeneralize to different topological structures in both 2D and 3D settings.
Revised: The proposed algorithm, as well as the loss function, naturallygeneralize to different topological structures in both 2D and 3D settings.

Original: Theproposed loss function helps deep nets achieve better performance in terms oftopology-aware metrics, outperforming state-of-the-art structure/topology-awaresegmentation methods.
Revised: Theproposed loss function helps deep nets achieve better performance in terms oftopology-aware metrics, outperforming state-of-the-art structure/topology-awaresegmentation methods.

Original:  Introduction
Revised:  Introduction

Original: Image segmentation with topological correctness is a challenging problem, especially for imageswith fine-scale structures, e.g., satellite images, neuron images and vessel images.
Revised: Image segmentation with topological correctness is a challenging problem, especially for imageswith fine-scale structures, e.g., satellite images, neuron images and vessel images.

Original: Deep learningmethods have delivered strong performance in image segmentation task [34, 23, 7, 8, 9].
Revised: Deep learningmethods have delivered strong performance in image segmentation task [33, 22, 7, 8, 9].

Original: However,even with satisfying per-pixel accuracy, most existing methods are still prone to topological errors,i.e., broken connections, holes in 2D membranes, missing connected components, etc.
Revised: However,even with satisfying per-pixel accuracy, most existing methods are still prone to topological errors,i.e., broken connections, holes in 2D membranes, missing connected components, etc.

Original: Theseerrors may significantly impact downstream tasks.
Revised: Theseerrors may significantly impact downstream tasks.

Original: For example, the reconstructed road maps fromsatellite images can be used for navigation [3, 5].
Revised: For example, the reconstructed road maps fromsatellite images can be used for navigation [3, 5].

Original: A small amount of pixel errors will result inbroken connections, causing incorrect navigation route.
Revised: A small amount of pixel errors will result inbroken connections, causing incorrect navigation route.

Original: See Fig.
Revised: See Fig.

Original: 1 for an illustration.
Revised: 1 for an illustration.

Original: In neuronreconstruction [19, 28, 51, 56, 55], incorrect topology of the neuron membrane will result in erroneousmerge or split of neurons, and thus errors in morphology and connectivity analysis of neuronal circuits.
Revised: In neuronreconstruction [18, 27, 49, 54, 53], incorrect topology of the neuron membrane will result in erroneousmerge or split of neurons, and thus errors in morphology and connectivity analysis of neuronal circuits.

Original: Topological errors usually happen at challenging locations, e.g., weak connections or blurred locations.
Revised: Topological errors usually happen at challenging locations, e.g., weak connections or blurred locations.

Original: But not all challenging locations are topologically relevant; for example, pixels at the boundary ofthe object of interest can generally be challenging, but not relevant to topology.
Revised: But not all challenging locations are topologically relevant; for example, pixels near the peripheral ofthe object of interest can generally be challenging, but not relevant to topology.

Original: To truly suppresstopological errors, we need to focus on topologically critical locations , i.e., challenging locationsthat are topologically relevant.
Revised: To truly suppresstopological errors, we need to focus on topologically critical locations , i.e., challenging locationsthat are topologically relevant.

Original: Without identifying and targeting these locations, neural networksthat are optimized for standard pixel-wise losses (e.g., cross-entropy loss or mean-square-error loss)cannot avoid topological errors, even if we increase the training set size.
Revised: Without identifying and targeting these locations, neural networksthat are optimized for standard pixel-wise losses (e.g., cross-entropy loss or mean-square-error loss)cannot avoid topological errors, even if we increase the training set size.

Original: Existing works have targeted these topologically critical locations.
Revised: Existing works have targeted these topologically critical locations.

Original: The closest method to our workis TopoNet [25], which is based on persistent homology [16, 15].
Revised: The closest method to our workis TopoNet [24], which is based on persistent homology [15, 14].

Original: The main idea is to identifytopologically critical locations corresponding to critical points of the likelihood map predicted bythe neural network.
Revised: The main idea is to identifytopologically critical locations corresponding to critical points of the likelihood map predicted bythe neural network.

Original: The selected critical points are reweighed in the training loss to force the neuralnetwork to focus on them, and thus to avoid topological errors.
Revised: The selected critical points are reweighed in the training loss to force the neuralnetwork to memorize them, and thus to avoid topological errors.

Original: But there are two main issues withthis approach: 1) the method is based on the likelihood map, which can be noisy with a large amount of irrelevant critical points.
Revised: But there are two main issues withthis approach: 1) the method is based on the likelihood map, which can be noisy with a large amount of irrelevant critical points.

Original: This leads to inefficient optimization during training.
Revised: This leads to inefficient optimization during training.

Original: The computationfor persistent homology is cubic to the image size.
Revised: The computationfor persistent homology is cubic to the image size.

Original: It is too expensive to recompute at every iteration.
Revised: It is too expensive to recompute at every iteration.

Original: [Figure]
Revised: [Figure]

Original: In this paper, we propose a novel approach to identify topologically critical locations in a more efficient andaccurate manner.
Revised: In this paper, we propose a novel approach to identify topologically critical locations in a more efficient andaccurate manner.

Original: These locations arepenalized in the proposed homotopywarping loss to achieve better topological accuracy.
Revised: These locations arepenalized in the proposed homotopywarping loss to achieve better topological accuracy.

Original: Our method is partiallyinspired by the warping error previously proposed to evaluate the topological accuracy [27].
Revised: Our method is partiallyinspired by the warping error previously proposed to evaluate the topological accuracy [26].

Original: Given a binarypredicted mask f B and a ground truthmask g , we “warp” one towards another without changing its topology.
Revised: Given a binarypredicted mask f B and a ground truthmask g , we “warp” one towards another without changing its topology.

Original: From the view of topology, to warpmask f B towards g , we find a maskf ∗ B that is homotopy equivalent to f Band is as close to g as possible [22].
Revised: In the language of topology, to warpmask f B towards g , we find a maskf ∗ B that is homotopy equivalent to f Band is as close to g as possible [21].

Original: The difference between the warped mask f ∗ B and g constitutes the topologically critical locations.
Revised: The difference between the warped mask f ∗ B and g constitutes the topologically critical locations.

Original: We can also warp g towards f B and find another set of topologically critical locations.
Revised: We can also warp g towards f B and find another set of topologically critical locations.

Original: See Fig.
Revised: See Fig.

Original: 4and Fig.
Revised: 4and Fig.

Original: 5 for illustrations.
Revised: 5 for illustrations.

Original: These locations directly correspond to topological difference betweenthe prediction and the ground truth, tolerating geometric deformations.
Revised: These locations directly correspond to topological difference betweenthe prediction and the ground truth, tolerating geometric deformations.

Original: Our homotopy warping losstargets them to fix topological errors of the model.
Revised: Our homotopy warping losstargets them to fix topological errors of the model.

Original: The warping of a mask is achieved by iteratively flipping labels at pixels without changing thetopology of the mask.
Revised: The warping of a mask is achieved by iteratively flipping labels at pixels without changing thetopology of the mask.

Original: These flippable pixels are called simple points/pixels in the classic theory ofdigital topology [31].
Revised: These flippable pixels are called simple points/pixels in the classic theory ofdigital topology [30].

Original: Note that in this paper, we focus on the topology of binary masks, simplepoints and simple pixels can be used interchangeably.
Revised: Note that in this paper, we focus on the topology of binary masks, simplepoints and simple pixels can be used interchangeably.

Original: To find the optimal warping of a mask towardsanother mask is challenging due to the huge search space.
Revised: To find the optimal warping of a mask towardsanother mask is challenging due to the huge search space.

Original: To this end, we propose a new heuristicmethod that is computationally efficient.
Revised: To this end, we propose a new heuristicmethod that is computationally efficient.

Original: We filter the image domain with the distance transformand flip simple pixels based on their distance from the mask.
Revised: We filter the image domain with the distance transformand flip simple pixels based on their distance from the mask.

Original: This algorithm is proven efficient anddelivers high quality locally optimal warping results.
Revised: This algorithm is proven efficient anddelivers high quality locally optimal warping results.

Original: Overall, our contributions can be summarized as follows:
Revised: Overall, our contributions can be summarized as follows:

Original: • We propose a novel homotopy warping loss , which penalizes errors on topologically criticallocations.
Revised: • We propose a novel homotopy warping loss , which penalizes errors on topologically criticallocations.

Original: These locations are defined by homotopic warping of predicted and groundtruth masks.
Revised: These locations are defined by homotopic warping of predicted and groundtruth masks.

Original: The loss can be incorporated into the training of topology-preserving deepsegmentation networks.• By exploiting distance transforms of binary masks, we propose a novel homotopic warpingalgorithm to identify topologically critical locations in an efficient manner.
Revised: The loss can be incorporated into the training of topology-preserving deepsegmentation networks.• By exploiting distance transforms of binary masks, we propose a novel homotopic warpingalgorithm to identify topologically critical locations in an efficient manner.

Original: This is essentialin incorporating the homotopy warping loss into the training of deep nets.
Revised: This is essentialin incorporating the homotopy warping loss into the training of deep nets.

Original: Our loss is a plug-and-play loss function.
Revised: Our loss is a plug-and-play loss function.

Original: It can be used to train any segmentation network to achievebetter performance in terms of topology.
Revised: It can be used to train any segmentation network to achievebetter performance in terms of topology.

Original: We conduct experiments on both 2D and 3D benchmarksto demonstrate the efficacy of the proposed method.
Revised: We conduct experiments on both 2D and 3D benchmarksto demonstrate the efficacy of the proposed method.

Original: Our method performs strongly in multipletopology-relevant metrics (e.g., ARI, Warping Error and Betti Error).
Revised: Our method performs strongly in multipletopology-relevant metrics (e.g., ARI, Warping Error and Betti Error).

Original: We also conduct severalablation studies to further demonstrate the efficiency and effectiveness of the technical contributions.
Revised: We also conduct severalablation studies to further demonstrate the efficiency and effectiveness of the technical contributions.

Original:  Related Works
Revised:  Related Works

Original: Deep Image Segmentation.
Revised: Deep Image Segmentation.

Original: Deep learning methods (CNNs) have achieved satisfying performancesfor image segmentation [34, 7, 8, 9, 37, 41].
Revised: Deep learning methods (CNNs) have achieved satisfying performancesfor image segmentation [33, 7, 8, 9, 36, 39].

Original: By replacing fully connected layer with fully convolutional layers, FCN [34] transforms a classification CNNs (e.g., AlexNet [32], VGG [45], or
Revised: By replacing fully connected layer with fully convolutional layers, FCN [33] transforms a classification CNNs (e.g., AlexNet [31], VGG [43], or

Original: ResNet [24]) to fully-convolutional neural networks.
Revised: ResNet [23]) to fully-convolutional neural networks.

Original: In this way, FCN successfully transfers thesuccess of image classification [32, 45, 49] to dense prediction/image segmentation.
Revised: In this way, FCN successfully transfers thesuccess of image classification [31, 43, 47] to dense prediction/image segmentation.

Original: Instead of using
Revised: Instead of using

Original: Conditional Random Field (CRF) as post-processing, Deeplab (v1-v2) [7, 8] methods add anotherfully connected CRF after the last CNN layer to make use of global information.
Revised: Conditional Random Field (CRF) as post-processing, Deeplab (v1-v2) [7, 8] methods add anotherfully connected CRF after the last CNN layer to make use of global information.

Original: Moreover, Deeplabv3 [9] introduces dilated/atrous convolution to increase the receptive field and make better use ofcontext information to achieve better performance.
Revised: Moreover, Deeplabv3 [9] introduces dilated/atrous convolution to increase the receptive field and make better use ofcontext information to achieve better performance.

Original: Besides the methods mentioned above, UNet [41] has also been one of the most popular methodsfor image segmentation, especially for images with fine structures.
Revised: Except for the methods mentioned above, UNet [39] has also been one of the most popular methodsfor image segmentation, especially for images with fine structures.

Original: UNet architecture is based on
Revised: UNet architecture is based on

Original: FCN with two major modifications: 1) Similar to encoder-decoder, UNet is symmetric.
Revised: FCN with two major modifications: 1) Similar to encoder-decoder, UNet is symmetric.

Original: The output iswith the same size as input images, thus suitable for dense prediction/image segmentation, and 2)
Revised: The output iswith the same size as input images, thus suitable for dense prediction/image segmentation, and 2)

Original: Skip connections between downsampling and upsampling paths.
Revised: Skip connections between downsampling and upsampling paths.

Original: The skip connections of UNet areable to combine low level/local information with high level/global information, resulting in bettersegmentation performance.
Revised: The skip connections of UNet areable to combine low level/local information with high level/global information, resulting in bettersegmentation performance.

Original: Though obtaining satisfying pixel performances, these methods are still prone to structural/topologicalerrors, as they are usually optimized via pixel-wise loss functions, such as mean-square-error loss(MSE) and cross-entropy loss.
Revised: Though obtaining satisfying pixel performances, these methods are still prone to structural/topologicalerrors, as they are usually optimized via pixel-wise loss functions, such as mean-square-error loss(MSE) and cross-entropy loss.

Original: As illustrated in Fig.1, a small amount of pixel errors will affect oreven damage the downstream tasks.
Revised: As illustrated in Fig.1, a small amount of pixel errors will affect oreven damage the downstream tasks.

Original: Topology-Aware Segmentation.
Revised: Topology-Aware Segmentation.

Original: Topology-aware segmentation methods have been proposed tosegment with correct structure/topology.
Revised: Topology-aware segmentation methods have been proposed tosegment with correct structure/topology.

Original: By identifying critical points of the predicted likelihoodmaps, persistent-homology-based losses [25, 11] penalize topologically critical locations.
Revised: By identifying critical points of the predicted likelihoodmaps, persistent-homology-based losses [24, 11] penalize topologically critical locations.

Original: However,the identified critical points can be very noisy and often are not relevant to the topological errors.
Revised: However,the identified critical points can be very noisy and often are not relevant to the topological errors.

Original: Illustrations are included in Supplementary Material.
Revised: Illustrations are included in Supplementary Material.

Original: Moreover, the computation of persistenthomology is expensive, making it difficult to evaluate the loss and gradient at every training iteration.
Revised: Moreover, the computation of persistenthomology is expensive, making it difficult to evaluate the loss and gradient at every training iteration.

Original: Other methods indirectly preserve topology by enhancing the curvilinear structures.
Revised: Other methods indirectly preserve topology by enhancing the curvilinear structures.

Original: VGG-UNet
Revised: VGG-UNet

Original: [36]uses the response of pretrained filters to enhance structures locally.
Revised: [35]uses the response of pretrained filters to enhance structures locally.

Original: But it does not truly preserve thetopology, and cannot generalize to higher dimensional topological structures, such as voids.
Revised: But it does not truly preserve thetopology, and cannot generalize to higher dimensional topological structures, such as voids.

Original: Severalmethods extract skeletons of the masks and penalize heavily on pixels of the skeletons.
Revised: Severalmethods extract skeletons of the masks and penalize heavily on pixels of the skeletons.

Original: This ensuresthe prediction to be correct along the skeletons, and thus are likely correct in topology.
Revised: This ensuresthe prediction to be correct along the skeletons, and thus are likely correct in topology.

Original: clDice
Revised: clDice

Original: [44]extracts the skeleton through min/max-pooling operations over the likelihood map.
Revised: [42]extracts the skeleton through min/max-pooling operations over the likelihood map.

Original: DMT Loss [26]uses the Morse complex of the likelihood map as the skeleton.
Revised: DMT Loss [25]uses the Morse complex of the likelihood map as the skeleton.

Original: However, these skeletons are notnecessarily topologically critical.
Revised: However, these skeletons are notnecessarily topologically critical.

Original: The penalization on them may not be relevant to topology.
Revised: The penalization on them may not be relevant to topology.

Original: We also note that many deep learning techniques have been proposed to ensure the segmentationoutput preserves details, and thus preserves topology implicitly [41, 34, 2, 14, 30, 29].
Revised: We also note that many deep learning techniques have been proposed to ensure the segmentationoutput preserves details, and thus preserves topology implicitly [39, 33, 2, 13, 29, 28].

Original: One mayalso use topological constraints as postprocessing steps once we have the predicted likelihoodmaps [21, 33, 48, 43, 53, 20, 52, 38, 57, 6, 1, 47, 39, 17].
Revised: One mayalso use topological constraints as postprocessing steps once we have the predicted likelihoodmaps [20, 32, 46, 41, 51, 19, 50, 37, 55, 6, 1, 45, 38, 16].

Original: Compared to end-to-end methods,postprocessing methods usually contain self-defined parameters or hand-crafted features, making itdifficult to generalize to different situations.
Revised: Compared to end-to-end methods,postprocessing methods usually contain self-defined parameters or hand-crafted features, making itdifficult to generalize to different situations.

Original: Instead of relying on the noisy likelihood maps [25, 11, 44, 26], we propose to use the warping ofbinary masks to identify the topologically critical locations.
Revised: Instead of relying on the noisy likelihood maps [24, 11, 42, 25], we propose to use the warping ofbinary masks to identify the topologically critical locations.

Original: The identified locations are more likelyto be relevant to topological errors.
Revised: The identified locations are more likelyto be relevant to topological errors.

Original: Penalizing on these locations ensures the training efficiency andsegmentation quality of our method.
Revised: Penalizing on these locations ensures the training efficiency andsegmentation quality of our method.

Original: Another difference from previous methods is that our methodrely on purely local topological computation (i.e., checking whether a pixel is simple within a localpatch), whereas previous methods are mostly relying on global topological computation.
Revised: Another difference from previous methods is that our methodrely on purely local topological computation (i.e., checking whether a pixel is simple within a localpatch), whereas previous methods are mostly relying on global topological computation.

Original:  Method
Revised:  Method

Original: By warping the binary predicted mask to the ground truth or conversely, we can accurately andefficiently identify the critical locations.
Revised: By warping the binary predicted mask to the ground truth or conversely, we can accurately andefficiently identity the critical locations.

Original: And then we propose a novel homotopy warping loss whichtargets them to fix topological errors of the model.
Revised: And then we propose a novel homotopy warping loss whichtargets them to fix topological errors of the model.

Original: The overall framework is illustrated in Fig.
Revised: The overall framework is illustrated in Fig.

Original: This section is organized as follows.
Revised: This section is organized as follows.

Original: We will start with necessary definitions and notations.
Revised: We will start with necessary definitions and notations.

Original: In
Revised: In

Original: Sec. 3.1, we give a concise description of digital topology and simple points.
Revised: Sec. 3.1, we give a concise description of digital topology and simple points.

Original: Next, we analyzedifferent types of warping errors in Sec. 3.2.
Revised: Next, we analyzedifferent types of warping errors in Sec. 3.2.

Original: The proposed warping loss is introduced in Sec. 3.3.
Revised: The proposed warping loss is introduced in Sec. 3.3.

Original: And finally, we explain the proposed new warping algorithm in Sec. 3.4.
Revised: And finally, we explain the proposed new warping algorithm in Sec. 3.4.

Original: 3.1 Digital Topology and Simple Points
Revised: 3.1 Digital Topology and Simple Points

Original: In this section, we briefly introduce simple point definition from the classic digital topology [31].
Revised: In this section, we briefly introduce simple point definition from the classic digital topology [30].

Original: We focus on the 2D setting, whereas all definitions generalize to 3D. Details on 3D images will beprovided in the Supplementary Material.
Revised: We focus on the 2D setting, whereas all definitions generalize to 3D. Details on 3D images will beprovided in the Supplementary Material.

Original: Connectivities of pixels.
Revised: Connectivities of pixels.

Original: To discuss the topology of a 2D binary image, we first define the connectivitybetween pixels.
Revised: To discuss the topology of a 2D binary image, we first define the connectivitybetween pixels.

Original: See Fig.
Revised: See Fig.

Original: 3 for an illustration.
Revised: 3 for an illustration.

Original: A pixel p has 8 pixels surrounding it.
Revised: A pixel p has 8 pixels surrounding it.

Original: We can either consider the 4 pixels that share an edge with p as p ’s neighbors (called 4-adjacency ), or consider all 8pixels as p ’s neighbors (called 8-adjacency ).
Revised: We can either consider the 4 pixels that share an edge with p as p ’s neighbors (called 4-adjacency ), or consider all 8pixels as p ’s neighbors (called 8-adjacency ).

Original: To ensure the Jordan closed curve theorem to hold, onehas to use one adjacency for foreground (FG) pixels, and the other adjacency for the background (BG)pixels.
Revised: To ensure the Jordan closed curve theorem to hold, onehas to use one adjacency for foreground (FG) pixels, and the other adjacency for the background (BG)pixels.

Original: In this paper, we use 4-adjacency for FG and 8-adjacency for BG.
Revised: In this paper, we use 4-adjacency for FG and 8-adjacency for BG.

Original: For 3D binary images, weuse 6-adjacency for FG and 26-adjacency for BG.
Revised: For 3D binary images, weuse 6-adjacency for FG and 26-adjacency for BG.

Original: Denote by N 4 ( p ) the set of 4-adjacency neighborsof p , and N 8 ( p ) the set of 8-adjacency neighbors of p .
Revised: Denote by N 4 ( p ) the set of 4-adjacency neighborsof p , and N 8 ( p ) the set of 8-adjacency neighbors of p .

Original: [Figure]
Revised: [Figure]

Original: points. For abinary image (2D/3D),a pixel/voxel is called a simple point if itcould be flipped from foreground (FG) tobackground (BG), or from BG to FG, withoutchanging the topology of the image [31].
Revised: a pixel/voxel is called a simple point if itcould be flipped from foreground (FG) tobackground (BG), or from BG to FG, withoutchanging the topology of the image [30].

Original: Thefollowing definition can be used to determinewhether a point is simple:
Revised: Thefollowing theorem can be used to determinewhether a point is simple:

Original: Definition 1 (Simple Point Condition [31])
Revised: Theorem 1 (Simple Point Condition [30]) .

Original: Let p be a point in a 2D binary image.
Revised: Let p be a point in a 2D binary image.

Original: Denote by F the set of FG pixels.
Revised: Denote by F the set of FG pixels.

Original: Assume 4adjacency for FG and 8-adjacent for BG.
Revised: Assume 4adjacency for FG and 8-adjacent for BG.

Original: pis a simple point if and only if both of the following conditions hold: 1) p is 4-adjacent tojust one FG connected component in N 8 ( p ) ;and 2) p is 8-adjacent to just one BG connected component in N 8 ( p ) .
Revised: pis a simple point if and only if both of the following conditions hold: 1) p is 4-adjacent tojust one FG connected component in N 8 ( p ) ;and 2) p is 8-adjacent to just one BG connected component in N 8 ( p ) .

Original: See Fig.
Revised: See Fig.

Original: 3 for an illustration of simple andnon-simple points in 2D case.
Revised: 3 for an illustration of simple andnon-simple points in 2D case.

Original: It is easy tocheck if a pixel p is simple or not by inspecting its 3 × 3 neighboring patch.
Revised: It is easy tocheck if a pixel p is simple or not by inspecting its 3 × 3 neighboring patch.

Original: The Definition 1 canalso generalize to 3D setting with 6- and 26-adjacencies for FG and BG, respectively.
Revised: The theorem can alsogeneralize to 3D setting with 6- and 26-adjacencies for FG and BG, respectively.

Original: 3.2 Homotopic Warping Error
Revised: 3.2 Homotopic Warping Error

Original: [Figure]
Revised: [Figure]

Original: In this section, we introduce the homotopic warping of one mask towardsanother.
Revised: In this section, we introduce the homotopic warping of one mask towardsanother.

Original: We warp a mask through asequence of flipping of simple points.
Revised: We warp a mask through asequence of flipping of simple points.

Original: Since we only flip simple points, bydefinition the warped mask will havethe same topology.
Revised: Since we only flip simple points, bydefinition the warped mask will havethe same topology.

Original: The operation iscalled a homotopic warping .
Revised: The operation iscalled a homotopic warping .

Original: It has beenproven that two binary images with thesame topology can always be warpedinto each other by flipping a sequenceof simple points [42].
Revised: It has beenproven that two binary images with thesame topology can always be warpedinto each other by flipping a sequenceof simple points [40].

Original: Consider two input masks, the prediction mask and the ground truth mask.
Revised: In our algorithm, we take two inputmasks, the prediction mask and theground truth mask.

Original: We can warp one of them (source mask) into another (target mask) in the best way possible, i.e.,the warped mask has the minimal number of difference with the target mask (formally, the minimal Hamming distance).
Revised: We can warp one of them (source mask) into another (target mask) in thebest way possible, i.e., the warped mask has the minimal number of difference with the targetmask (formally, the minimal Hamming distance).

Original: Once the warping is finished, the pixels at which the warped mask is differentfrom the target mask, called critical pixels , are a sparse set of pixels indicative of the topologicalerrors of the prediction mask.
Revised: Once the warping is finished, the pixels at whichthe warped mask is different from the target mask, called critical pixels , are a sparse set of pixelsindicative of the topological errors of the prediction mask.

Original: We will warp in both directions: from the prediction mask to the ground truth mask, and the opposite.
Revised: We will warp in both directions: from the prediction mask to the ground truth mask, and the opposite.

Original: They identify different sets of critical pixels for a same topological error.
Revised: They identify different sets of critical pixels for a same topological error.

Original: In Fig.
Revised: In Fig.

Original: 4, we show asynthetic example with red and white masks, as well warping in both directions.
Revised: 4, we show asynthetic example with red and white masks, as well warping in both directions.

Original: Warping the redmask towards the white mask ((a) and (b)) results in a single-pixel wide gap.
Revised: Warping the redmask towards the white mask ((a) and (b)) results in a single-pixel wide gap.

Original: The pixels in the gap(highlighted with red crosses) are critical pixels; flipping any of them will change the topology of the
Revised: The pixels in the gap(highlighted with red crosses) are critical pixels; flipping any of them will change the topology of the

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: warped red mask.
Revised: warped red mask.

Original: Warping the white mask towards the red mask ((d) and (
Revised: Warping the white mask towards the red mask ((d) and (

Original: )) results in a single pixelwide link connecting the warped white mask.
Revised: )) results in a single pixelwide link connecting the warped white mask.

Original: All pixels along the link (highlighted with red crosses)are critical; flipping any of them will change the topology of the warped white mask.
Revised: All pixels along the link (highlighted with red crosses)are critical; flipping any of them will change the topology of the warped white mask.

Original: Here if the red mask is the prediction mask, then this corresponds to a false negative connection, i.e.,a connection that is missed by the prediction.
Revised: Here if the red mask is the prediction mask, then this corresponds to a false negative connection, i.e.,a connection that is missed by the prediction.

Original: If the red mask is the ground truth mask, then thiscorresponds to a false positive connection.
Revised: If the red mask is the ground truth mask, then thiscorresponds to a false positive connection.

Original: Note the warping ensures that only topological errorsare represented by the critical pixels .
Revised: Note the warping ensures that only topological errorsare represented by the critical pixels .

Original: In the synthetic example (Fig. 4), the large area of error in thebottom left corner of the image is completely ignored as it is not topologically relevant.
Revised: In the synthetic example (Fig. 4), the large area of error in thebottom left corner of the image is completely ignored as it is not topologically relevant.

Original: In Fig.
Revised: In Fig.

Original: 5, we show a real example from the satellite image dataset, focusing on the errors related to1D topological structures (connection).
Revised: 5, we show a real example from the satellite image dataset, focusing on the errors related to1D topological structures (connection).

Original: In the figure we illustrate both a false negative connectionerror (highlighted with a red box) and false positive connection error (highlighted with a green box).
Revised: In the figure we illustrate both a false negative connectionerror (highlighted with a red box) and false positive connection error (highlighted with a green box).

Original: If we warp the ground truth mask towards the prediction mask (c), we observe critical pixels forminga link for the false negative connection (d), and a gap for the false positive connection (e).
Revised: If we warp the ground truth mask towards the prediction mask (c), we observe critical pixels forminga link for the false negative connection (d), and a gap for the false positive connection (e).

Original: Similarly,we can warp the prediction mask towards the ground truth, and get different sets of critical pixels forthe same topological errors (illustrations will be provided in the Supplemental Material).
Revised: Similarly,we can warp the prediction mask towards the ground truth, and get different sets of critical pixels forthe same topological errors (illustrations will be provided in the Supplemental Material).

Original: Note that for 2D images with fine structures, errors regarding 1D topological structures are the mostcrucial.
Revised: Note that for 2D images with fine structures, errors regarding 1D topological structures are the mostcrucial.

Original: They affect the connectivity of the prediction results.
Revised: They affect the connectivity of the prediction results.

Original: For 3D images, errors on 1D or 2Dtopological structures are both important, corresponding to broken connections for tubular structuresand holes in membranes.
Revised: For 3D images, errors on 1D or 2Dtopological structures are both important, corresponding to broken connections for tubular structuresand holes in membranes.

Original: We will provide more comprehensive characterization of different types oftopological structures and errors in the Supplementary Material.
Revised: We will provide more comprehensive characterization of different types oftopological structures and errors in the Supplementary Material.

Original: 3.3 Homotopy Warping Loss
Revised: 3.3 Homotopy Warping Loss

Original: Next, we formalize the proposed homotopy warping loss , which is evaluated on the critical pixels dueto homotopic warping.
Revised: Next, we formalize the proposed homotopy warping loss , which is evaluated on the critical pixels dueto homotopic warping.

Original: As illustrated in the previous section, the warping can be in both directions,from the prediction mask to the ground truth mask, and the opposite.
Revised: As illustrated in the previous section, the warping can be in both directions,from the prediction mask to the ground truth mask, and the opposite.

Original: Formally, we denote by f the predicted likelihood map of a segmentation network, and f B thecorresponding binarized prediction mask (i.e., f thresholded at 0.5).
Revised: Formally, we denote by f the predicted likelihood map of a segmentation network, and f B thecorresponding binarized prediction mask (i.e., f thresholded at 0.5).

Original: We denote by g the ground truthmask.
Revised: We denote by g the ground truthmask.

Original: First, we warp g towards f B , so that the warped mask, g ∗ has the minimal Hamming distancefrom the target f B .
Revised: First, we warp g towards f B , so that the warped mask, g ∗ has the minimal Hamming distancefrom the target f B .

Original: [Equation]
Revised: [Equation]

Original: where ◁ is the homotopic warping operation.
Revised: where ◁ is the homotopic warping operation.

Original: The pixels at which g ∗ and f B disagree are the criticalpixels and will be penalized in the loss.
Revised: The pixels at which g ∗ and f B disagree are the criticalpixels and will be penalized in the loss.

Original: We record these critical pixels due to the warping of g with amask M g , M g = f B ⊕ g ∗ , in which ⊕ is the Exclusive Or operation.
Revised: We record these critical pixels due to the warping of g with amask M g , M g = f B ⊕ g ∗ , in which ⊕ is the Exclusive Or operation.

Original: We also warp the prediction mask f B towards g .
Revised: We also warp the prediction mask f B towards g .

Original: [Equation]
Revised: [Equation]

Original: We use the mask M f to record the remaining critical pixels after warping f B , M f = g ⊕ f ∗ B .
Revised: We use the mask M f to record the remaining critical pixels after warping f B , M f = g ⊕ f ∗ B .

Original: The union of the two critical pixel masks is the complete set of critical pixels corresponding to topological errors, M = M g ∪ M f .
Revised: The union of the two critical pixel masks is the complete set of critical pixels corresponding to topological errors, M = M g ∪ M f .

Original: M contains all the locations directly related to topological structures.
Revised: M contains all the locations directly related to topological structures.

Original: Note this is different from persistent-homology-based method [25], DMT based method [26] or skeleton based method [44], which extract topological locations/structures on the predicted continuouslikelihood maps.
Revised: Note this is different from persistent-homology-based method [24], DMT based method [25] or skeleton based method [42], which extract topological locations/structures on the predicted continuouslikelihood maps.

Original: Our warping loss directly locates the topological critical pixels/structures/locationsbased on the binary mask.
Revised: Our warping loss directly locates the topological critical pixels/structures/locationsbased on the binary mask.

Original: The detected critical pixel set is sparse and less noisy.
Revised: The detected critical pixel set is sparse and less noisy.

Original: L pixel denotes the pixel-wise loss function (e.g., cross-entropy), then L warp can be defined as:
Revised: L pixel denotes the pixel-wise loss function (e.g., cross-entropy), then L warp can be defined as:

Original: [Equation]
Revised: [Equation]

Original: where ⊙ denotes Hadamard product.
Revised: where ⊙ denotes Hadamard product.

Original: L warp penalizes the topological critical locations, forcing theneural network to predict better at these locations, and thus are less prone to topological errors.
Revised: L warp penalizes the topological critical locations, forcing theneural network to predict better at these locations, and thus are less prone to topological errors.

Original: The final loss of our method, L total , is given by:
Revised: The final loss of our method, L total , is given by:

Original: [Equation]
Revised: [Equation]

Original: where L dice denotes the dice loss.
Revised: where L dice denotes the dice loss.

Original: And the loss weight λ warp is used to balance the two loss terms.3.

Revised: And the loss weight λ warp is used to balance the two loss terms.3.

Original: Distance-Ordered Homotopy Warping
Revised: Distance-Ordered Homotopy Warping

Original: Even though checking whether a pixel is simple or not is easy, finding the optimal warping as in
Revised: Even though checking whether a pixel is simple or not is easy, finding the optimal warping as in

Original: Eq.
Revised: Eq.

Original: (1) and (2) is challenging.
Revised: (1) and (2) is challenging.

Original: The reason is that there are too many degrees of freedom.
Revised: The reason is that there are too many degrees of freedom.

Original: At eachiteration during the warping, we have to choose a simple point to flip.
Revised: At eachiteration during the warping, we have to choose a simple point to flip.

Original: It is not obvious which simplepoint will finally lead to a global optimum.
Revised: It is not obvious which simplepoint will finally lead to a global optimum.

Original: In this section, we provide an efficient heuristic algorithm to find a warping local optimum.
Revised: In this section, we provide an efficient heuristic algorithm to find a warping local optimum.

Original: Weexplain the algorithm for warping g towards f B .
Revised: Weexplain the algorithm for warping g towards f B .

Original: The algorithm generalizes to the opposite warpingdirection naturally.
Revised: The algorithm generalizes to the opposite warpingdirection naturally.

Original: Recall the warping algorithm iteratively flips simple points.
Revised: Recall the warping algorithm iteratively flips simple points.

Original: But there are toomany choices at each iteration.
Revised: But there are toomany choices at each iteration.

Original: It is hard to know which flipping choice will lead to the optimalsolution.
Revised: It is hard to know which flipping choice will lead to the optimalsolution.

Original: We need good heuristics for choosing a flippable pixel.
Revised: We need good heuristics for choosing a flippable pixel.

Original: Below we explain our mainintuitions for designing our algorithm.
Revised: Below we explain our mainintuitions for designing our algorithm.

Original: First, we restrict the warping so it only sweeps through the area where the two masks disagree.
Revised: First, we restrict the warping so it only sweeps through the area where the two masks disagree.

Original: Inother words, at each iteration, we restrict the candidate pixels for flipping to not only simple, but alsopixels on which g and f B disagree.
Revised: Inother words, at each iteration, we restrict the candidate pixels for flipping to not only simple, but alsopixels on which g and f B disagree.

Original: In Fig.
Revised: In Fig.

Original: 4 (c) and (f), we highlight the candidate pixels for flippingat the beginning.
Revised: 4 (c) and (f), we highlight the candidate pixels for flippingat the beginning.

Original: Notice that not all simple points are selected as candidates.
Revised: Notice that not all simple points are selected as candidates.

Original: We only choose simplepoints within the difference set Diff ( f B , g ) =
Revised: We only choose simplepoints within the difference set Diff ( f B , g ) =

Original: f B ⊕ g .
Revised: f B ⊕ g .

Original: Second, since we want to minimize the difference of the warped and target masks, we propose to flippixels within the difference region Diff ( f B , g ) .
Revised: Second, since we want to minimize the difference of the warped and target masks, we propose to flippixels within the difference region Diff ( f B , g ) .

Original: To implement this strategy efficiently, we order allpixel within Diff ( f B , g ) according to their distance from the FG/BG, and flip them according to thisorder.
Revised: To implement this strategy efficiently, we order allpixel within Diff ( f B , g ) according to their distance from the FG/BG, and flip them according to thisorder.

Original: A pixel is skipped if it is not simple.
Revised: A pixel is skipped if it is not simple.

Original: Our algorithm is based on the intuition that a far-away pixel will not become simple until nearbypixels are flipped first.
Revised: Our algorithm is based on the intuition that a far-away pixel will not become simple until nearbypixels are flipped first.

Original: To see this, we first formalize the definition of distance transform from themasks, f B and g , denoted by D f B and D g .
Revised: To see this, we first formalize the definition of distance transform from themasks, f B and g , denoted by D f B and D g .

Original: For a BG pixel of g , p , its distance value D g ( p ) is theshortest distance from p to any FG pixel of g , D g ( p )
Revised: For a BG pixel of g , p , its distance value D g ( p ) is theshortest distance from p to any FG pixel of g , D g ( p )

Original: =
Revised: =

Original: min s ∈ FG g dist ( p, s ) .
Revised: min s ∈ FG g dist ( p, s ) .

Original: Similarly, for a FGpixel of g , q , D g ( q ) = min s ∈ BG g dist ( q, s ) .
Revised: Similarly, for a FGpixel of g , q , D g ( q ) = min s ∈ BG g dist ( q, s ) .

Original: The definition generalizes to D f B .
Revised: The definition generalizes to D f B .

Original: We observe that a pixel cannot be simple unless it has distance 1 from the FG/BG of a warping mask.
Revised: We observe that a pixel cannot be simple unless it has distance 1 from the FG/BG of a warping mask.

Original: The proof is straightforward.
Revised: The proof is straightforward.

Original: Formally,
Revised: Formally,

Original: Lemma 1.
Revised: Lemma 1.

Original: Given a 2D binary mask m , a pixel p cannot be simple for m if its distance function
Revised: Given a 2D binary mask m , a pixel p cannot be simple for m if its distance function

Original: D m ( p ) > 1 .
Revised: D m ( p ) > 1 .

Original: Proof.
Revised: Proof.

Original: Assume the foreground has a pixel value of 1 and p is a background pixel with a index of ( i, j ) .
Revised: Assume the foreground has a pixel value of 1 and p is a background pixel with a index of ( i, j ) .

Original: Consider the m -adjacent ( m =4) for p .
Revised: Consider the m -adjacent ( m =4) for p .

Original: Since D m ( p ) > 1 , then we have m ( i − 1 , j )
Revised: Since D m ( p ) > 1 , then we have m ( i − 1 , j )

Original: = m
Revised: = m

Original: ( i + 1 , j ) =
Revised: ( i + 1 , j ) =

Original: m
Revised: m

Original: ( i, j − 1) = m ( i, j + 1) = 0 .
Revised: ( i, j − 1) = m ( i, j + 1) = 0 .

Original: In this case, p is not 4-adjacent to any FG connected component,violating the 1) of Definition 1 .
Revised: In this case, p is not 4-adjacent to any FG connected component,violating the 1) of Theorem 1 .

Original: Consequently, pixel ( i, j ) is not a simple point.
Revised: Consequently, pixel ( i, j ) is not a simple point.

Original: This also holds forforeground pixels.
Revised: This also holds forforeground pixels.

Original: This lemma  naturally generalizes to 3D case.
Revised: This lemma is naturally generalized to 3D case.

Original: Lemma 1 implies that only after flipping pixels with distance 1, the other misclassified locationsshould be considered.
Revised: Lemma 1 implies that only after flipping pixels with distance 1, the other misclassified locationsshould be considered.

Original: This observation gives us the intuition of our algorithm.
Revised: This observation gives us the intuition of our algorithm.

Original: To warp g towardsf B , our algorithm is as follows: (1) compute the difference set Diff ( f B , g ) as the candidate set ofpixels; (2) sort candidate pixels in a non-decreasing order of the distance transform D g ; (3) enumeratethrough all candidate pixels according to the order.
Revised: To warp g towardsf B , our algorithm is as follows: (1) compute the difference set Diff ( f B , g ) as the candidate set ofpixels; (2) sort candidate pixels in a non-decreasing order of the distance transform D g ; (3) enumeratethrough all candidate pixels according to the order.

Original: For each iteration, check if it is simple.
Revised: For each iteration, check if it is simple.

Original: If yes, flipthe pixel’s label.
Revised: If yes, flipthe pixel’s label.

Original: It is possible that this algorithm can miss some pixels.
Revised: It is possible that this algorithm can miss some pixels.

Original: They are not simple when thealgorithm checks, but they might become simple as the algorithm continues (since their neighboringpixels get flipped in previous iterations).
Revised: They are not simple when thealgorithm checks, but they might become simple as the algorithm continues (since their neighboringpixels get flipped in previous iterations).

Original: One remedy is to recalculate the distance transform after one round of warping, and go throughremaining pixels once more.
Revised: One remedy is to recalculate the distance transform after one round of warping, and go throughremaining pixels once more.

Original: But in practice we found this is not necessary as this scenario is veryrare.
Revised: But in practice we found this is not necessary as this scenario is veryrare.

Original: The effectiveness of the proposed warping strategy is illustrated in Supplementary Material.
Revised: The effectiveness of the proposed warping strategy is illustrated in Supplementary Material.

Original:  Experiments
Revised:  Experiments

Original: We conduct extensive experiments to demonstrate the effectiveness of the proposed method.
Revised: We conduct extensive experiments to demonstrate the effectiveness of the proposed method.

Original: Sec.
Revised: Sec.

Original: 11introduces the datasets used in this paper, including both 2D and 3D datasets.
Revised: 4introduces the datasets used in this paper, including both 2D and 3D datasets.

Original: The benchmarkmethods are described in Sec. 
Revised: The benchmarkmethods are described in Sec. 4.

Original: We mainly focus on topology-aware segmentation methods.
Revised: We mainly focus on topology-aware segmentation methods.

Original: Sec.
Revised: Sec.

Original: 13describes the evaluation metrics used to assess the quality of the segmentation.
Revised: 4describes the evaluation metrics used to assess the quality of the segmentation.

Original: To demonstrate theability to achieve better structural/topological performances, besides standard segmentation metric,such as DICE score, we also use several topology-aware metrics to evaluate all the methods.
Revised: To demonstrate theability to achieve better structural/topological performances, besides standard segmentation metric,such as DICE score, we also use several topology-aware metrics to evaluate all the methods.

Original: Severalablation studies are then conducted to further demonstrate the efficiency and effectiveness of thetechnical contributions (Sec. 4.1).
Revised: Severalablation studies are then conducted to further demonstrate the efficiency and effectiveness of thetechnical contributions (Sec. 4.1).

Original: [Figure]
Revised: [Figure]

Original: results compared with the standard UNet.
Revised: results compared with the standard UNet.

Original: The proposed warping loss can help to correct the topological errors red circles).
Revised: The proposed warping loss can help to correct the topological errors red circles).

Original: The sampled patches are from four different
Revised: The sampled patches are from four different

Original: Datasets.
Revised: Datasets.

Original: We conduct extensive experiments tovalidate the efficacy of our method.
Revised: We conduct extensive experiments tovalidate the efficacy of our method.

Original: Specifically,we use four natural and biomedical 2D datasets( RoadTracer [4], DeepGlobe [13], Mass. [35],
Revised: Specifically,we use four natural and biomedical 2D datasets( RoadTracer [4], DeepGlobe [12], Mass. [34],

Original: DRIVE [46]) and one more 3D biomedicaldataset ( CREMI 2 ) to validate the efficacy of thepropose method.
Revised: DRIVE [44]) and one more 3D biomedicaldataset ( CREMI 2 ) to validate the efficacy of thepropose method.

Original: More details about the datasetsand the split of the training and validation subsets are included in Supplementary Material.
Revised: More details about the datasetsand the split of the training and validation subsets are included in Supplementary Material.

Original: Baselines.
Revised: Baselines.

Original: We compare the results of ourmethod with several state-of-the-art methods.
Revised: We compare the results of ourmethod with several state-of-the-art methods.

Original: The standard/simple UNet (2D/3D) UNet [41,10 ] is used as a strong baseline and the backbone for other methods.
Revised: The standard/simple UNet (2D/3D) UNet [39,10 ] is used as a strong baseline and the backbone for other methods.

Original: The other baselinesused in this paper include RoadTracer [4], VecRoad [50], iCurb [54], DIVE [18], VGGUNet [36], TopoLoss [25], clDice [44] and
Revised: The other baselinesused in this paper include RoadTracer [4], VecRoad [48], iCurb [52], DIVE [17], VGGUNet [35], TopoLoss [24], clDice [42] and

Original: DMT [26].
Revised: DMT [25].

Original: More descriptions and implementation details of these baselines are included in
Revised: More descriptions and implementation details of these baselines are included in

Original: Supplementary Material.
Revised: Supplementary Material.

Original: Evaluation Metrics.
Revised: Evaluation Metrics.

Original: We use both pixelwise ( DICE ) and topology-aware metrics ( ARI ,
Revised: We use both pixelwise ( DICE ) and topology-aware metrics ( ARI ,

Original: Warping Error and Betti number error )to evaluate the performance of the proposedmethod.
Revised: Warping Error and Betti number error )to evaluate the performance of the proposedmethod.

Original: More details about the evaluation metrics are provided in Supplementary Material.
Revised: More details about the evaluation metrics are provided in Supplementary Material.

Original: Implementation Details.
Revised: Implementation Details.

Original: For 2D images, we use ( m, n ) = (4, 8) to check if a pixel is simple ornot; while ( m, n ) = (8, 26) for 3D images.
Revised: For 2D images, we use ( m, n ) = (4, 8) to check if a pixel is simple ornot; while ( m, n ) = (8, 26) for 3D images.

Original: We choose cross-entropy loss as L warp /L pixel for all theexperiments, except the ablation studies in Tab. 3.
Revised: We choose cross-entropy loss as L warp /L pixel for all theexperiments, except the ablation studies in Tab. 3.

Original: .org/
Revised: .org/

Original: For 2D datasets, the batch size is set as 16, and the initial learning rate is 0.01.
Revised: For 2D datasets, the batch size is set as 16, and the initial learning rate is 0.01.

Original: We randomly croppatches with the size of 512 × 512 and then feed them into the 2D UNet.
Revised: We randomly croppatches with the size of 512 × 512 and then feed them into the 2D UNet.

Original: For 3D case, the batch sizeis also 16, while the input size is 128 × 128 × 16 .
Revised: For 3D case, the batch sizeis also 16, while the input size is 128 × 128 × 16 .

Original: We perform the data normalization for the singlepatch based on its mean and standard deviation.
Revised: We perform the data normalization for the singlepatch based on its mean and standard deviation.

Original: We use PyTorch framework (Version: 1.7.1) to implement the proposed method.
Revised: We use PyTorch framework (Version: 1.7.1) to implement the proposed method.

Original: A simple/standard
Revised: A simple/standard

Original: UNet (2D or 3D) is a used as baseline and the backbone.
Revised: UNet (2D or 3D) is a used as baseline and the backbone.

Original: For the proposed method, as well as theother loss function based baselines, to make a fair comparison, we use the same UNet as backbone.
Revised: For the proposed method, as well as theother loss function based baselines, to make a fair comparison, we use the same UNet as backbone.

Original: And the training strategy is to train the UNet with dice loss first until converge, and then add theproposed losses to fine-tune the models obtained from the initial step.
Revised: And the training strategy is to train the UNet with dice loss first until converge, and then add theproposed losses to fine tune the models obtained from the initial step.

Original: All the experiments are performed on a Tesla V100-SXM2 GPU (32G Memory), and an Intel(R)
Revised: All the experiments are performed on a Tesla V100-SXM2 GPU (32G Memory), and an Intel(R)

Original: Xeon(R) Gold 6140 CPU@2.30 GHz.
Revised: Xeon(R) Gold 6140 CPU@2.30 GHz.

Revised: [Table]

Original: [Table]
Revised: [Table]

Original: Qualitative and Quantitative Results.
Revised: Qualitative and Quantitative Results.

Original: In
Revised: In

Original: Fig.
Revised: Fig.

Original: 6, we show qualitative results from different datasets.
Revised: 6, we show qualitative results from different datasets.

Original: Compared with baseline UNet, ourmethod recovers better structures, such as connections, which are highlighted by red circles.
Revised: Compared with baseline UNet, ourmethod recovers better structures, such as connections, which are highlighted by red circles.

Original: Our final loss is a weighted combination of thedice loss and warping-loss term L warp .
Revised: Our final loss is a weighted combination of thedice loss and warping-loss term L warp .

Original: Whenλ warp = 0, the proposed method degrades toa standard UNet.
Revised: Whenλ warp = 0, the proposed method degrades toa standard UNet.

Original: The recovered better structures (UNet and Warping columns in Fig.
Revised: The recovered better structures (UNet and Warping columns in Fig.

Original: 6)demonstrates that our warping-loss helps thedeep neural networks to achieve better topological segmentations.
Revised: 6)demonstrates that our warping-loss helps thedeep neural networks to achieve better topological segmentations.

Original: More qualitative results areprovided in Supplementary Material.
Revised: More qualitative results areprovided in Supplementary Material.

Original: Tab. 1 shows quantitative results for three 2D image datasets, RoadTracer, DeepGlobe and The
Revised: Tab. 1 shows quantitative results for three 2D image datasets, RoadTracer, DeepGlobe and The

Original: Massachusetts dataset and one 3D image dataset,
Revised: Massachusetts dataset and one 3D image dataset,

Original: CREMI.
Revised: CREMI.

Original: The best performances are highlightedwith bold.
Revised: The best performances are highlightedwith bold.

Original: The proposed warping-loss usuallyachieves the best performances in both DICEscore and topological accuracy (ARI, Warping
Revised: The proposed warping-loss usuallyachieves the best performances in both DICEscore and topological accuracy (ARI, Warping

Original: Error and Betti Error) over other topology-awaresegmentation baselines.
Revised: Error and Betti Error) over other topology-awaresegmentation baselines.

Original: Note that we only report the mean performances here for space limitation.
Revised: Note that we only report the mean performances here for space limitation.

Original: We also include standard deviations anduse t-test to determine the statistical significancein the Supplementary Material.
Revised: We also include standard deviations anduse t-test to determine the statistical significancein the Supplementary Material.

Original: 4.1 Ablation studies
Revised: 4.1 Ablation studies

Original: To further explore the technical contributionsof the proposed method and provide a roughguideline of how to choose the hyperparameters,we conduct several ablation studies.
Revised: To further explore the technical contributionsof the proposed method and provide a roughguideline of how to choose the hyperparameters,we conduct several ablation studies.

Original: Note thatall the ablation studies are conducted on the
Revised: Note thatall the ablation studies are conducted on the

Original: RoadTracer dataset.
Revised: RoadTracer dataset.

Original: The impact of the loss weights.
Revised: The impact of the loss weights.

Original: As seen in
Revised: As seen in

Original: Eq. 4, our final loss function is a combination ofdice loss and the proposed warping loss L warp .
Revised: Eq. 4, our final loss function is a combination ofdice loss and the proposed warping loss L warp .

Original: The balanced term λ warp controls the influenceof the warping loss term, and it’s a dataset dependent hyper-parameter.
Revised: The balanced term λ warp controls the influenceof the warping loss term, and it’s a dataset dependent hyper-parameter.

Original: The quantitative results for different choices of λ warp are illustrated in
Revised: The quantitative results for different choices of λ warp are illustrated in

Original: Tab. 2.
Revised: Tab. 2.

Original: For the RoadTracer dataset, the optimal value is 1 × 10 − 4 .
Revised: For the RoadTracer dataset, the optimal value is 1 × 10 − 4 .

Original: From Tab. 2, we can find thatdifferent choices of λ warp do affect the performances.
Revised: From Tab. 2, we can find thatdifferent choices of λ warp do affect the performances.

Original: The reason is that, if λ warp is too small, theeffect of the warping loss term is negligible.
Revised: The reason is that, if λ warp is too small, theeffect of the warping loss term is negligible.

Original: However, if λ warp is too large, the warping loss term will compete with the L dice and decrease the performance of the other easy-classified pixels.
Revised: However, if λ warp is too large, the warping loss term will compete with the L dice and decrease the performance of the other easy-classified pixels.

Original: Notethat within a reasonable range of λ warp , all the choices contribute to better performances comparedto baseline (row ‘0’, standard UNet), demonstrating the effectiveness of the proposed loss term.
Revised: Notethat within a reasonable range of λ warp , all the choices contribute to better performances comparedto baseline (row ‘0’, standard UNet), demonstrating the effectiveness of the proposed loss term.

Original: Thebest choices of λ warp for each specific dataset are included in Supplementary Material.
Revised: Thebest choices of λ warp for each specific dataset are included in Supplementary Material.

Original: [Table]
Revised: [Table]

Original: The choice of loss functions.
Revised: The choice of loss functions.

Original: The proposed warping loss is defined on the identified topologicalcritical pixels.
Revised: The proposed warping loss is defined on the identified topologicalcritical pixels.

Original: Consequently, any pixel-wise lossfunctions can be used to define the warping loss
Revised: Consequently, any pixel-wise lossfunctions can be used to define the warping loss

Original: L warp /L
Revised: L warp /L

Original: pixel .
Revised: pixel .

Original: In this section, we investigate howthe choices of loss functions affect the performances.
Revised: In this section, we’d like the investigate how the choices of loss functions affect theperformances.

Original: The quantitative results are show in Tab. 3.
Revised: The quantitative results are showin Tab. 3.

Original: Compared with mean-square-error loss (MSE) or Dice loss, the cross-entropy loss (CE) achieves best performances in terms of topological metrics.
Revised: Compared with mean-square-error loss(MSE) or Dice loss, the cross-entropy loss (CE) achieves best performances in terms of topologicalmetrics.

Original: On the other hand, all these three choices perform better than baseline method (row ‘w/o’, standard UNet), which further demonstrates the contribution of the proposed loss term. 
Revised: On the other hand, all these three choices perform better than baseline method (row ‘w/o’,standard UNet), which further demonstrates the contribution of the proposed loss term. Note that

Original: [Table]
Revised: [Table]

Original: The efficiency of the proposed loss.
Revised: The efficiency of the proposed loss.

Original: In this section, we’d like to investigate the efficiency of theproposed method.
Revised: In this section, we’d like to investigate the efficiency of theproposed method.

Original: Our warping algorithm containstwo parts, the distance transform and the sorting ofdistance matrix.
Revised: Our warping algorithm containstwo parts, the distance transform and the sorting ofdistance matrix.

Original: The complexity for distance transform is O ( n ) for a 2D image where n is the sizeof the 2D image, and n = H × W .
Revised: The complexity for distance transform is O ( n ) for a 2D image where n is the sizeof the 2D image, and n = H × W .

Original: H, W are theheight and width of the 2D image, respectively.
Revised: H, W are theheight and width of the 2D image, respectively.

Original: And the complexity for sorting is mlog ( m ) , wherem is the number of misclassified pixels.
Revised: And the complexity for sorting is mlog ( m ) , wherem is the number of misclassified pixels.

Original: Usually m ≪ n , so the overall complexity for the warpingalgorithm is O ( n ) .
Revised: Usually m ≪ n , so the overall complexity for the warpingalgorithm is O ( n ) .

Original: As a comparison, [25] needs O ( n 3 ) complexity to compute the persistencediagram.
Revised: As a comparison, [24] needs O ( n 3 ) complexity to compute the persistencediagram.

Original: And the computational complexity for [26], [44] are O ( nlog ( n )) and O ( n ) , respectively.
Revised: And the computational complexity for [25], [42] are O ( nlog ( n )) and O ( n ) , respectively.

Original: [Table]
Revised: [Table]

Original: TopoNet [25]
Revised: TopoNet [24]

Original: O ( n 3 ) ≈ 12 h clDice [44]
Revised: O ( n 3 ) ≈ 12 h clDice [42]

Original: O ( n ) ≈ 3h
Revised: O ( n ) ≈ 3h

Original: DMT [26]
Revised: DMT [25]

Original: O ( nlogn )
Revised: O ( nlogn )

Original: ≈ 7 h
Revised: ≈ 7 h

Original: Warping
Revised: Warping

Original: O ( n ) ≈ 4 h
Revised: O ( n ) ≈ 4 h

Original: The comparison in terms of complexity and trainingtime are illustrated in Tab.4.
Revised: The comparison in terms of complexity and trainingtime are illustrated in Tab.4.

Original: Note that for the proposedmethod and all the other baselines, we first train a simple/standard UNet, and then add the additional loss termsto fine-tune the models obtained from the initial step.
Revised: Note that for the proposedmethod and all the other baselines, we first train a simple/standard UNet, and then add the additional loss termsto fine-tune the models obtained from the initial step.

Original: Here, the training time is only for the fine-tune step.
Revised: Here, the training time is only for the fine-tune step.

Original: Theproposed method takes slightly longer training time thanclDice, while achieves the best performance over theothers.
Revised: Theproposed method takes slightly longer training time thanclDice, while achieves the best performance over theothers.

Original: As all these methods use the same backbone, the inference times are the same.
Revised: As all these methods use the same backbone, the inference times are the same.

Original: [Table]
Revised: [Table]

Original: Comparison with morphology 
Revised: Comparison with morphology postprocessing.

Original: post- processing

Original: To verify the necessity ofthe proposed method, we also compare theproposed method with traditional morphologypost-processing, i.e., dilation, erosion.
Revised: To verify the necessity of the proposed method, we also compare theproposed method with traditional morphologypost-processing, i.e., dilation, erosion.

Original: Dilationand erosion are global operations.
Revised: Dilationand erosion are global operations.

Original: Thoughclosing operation (dilates image and then erodesdilated image) could bridge some specificgaps/broken connections, it will damage the global structures.
Revised: Thoughclosing operation (dilates image and then erodesdilated image) could bridge some specificgaps/broken connections, it will damage the global structures.

Original: In practice, the gaps/broken willusually be more than a few pixels.
Revised: In practice, the gaps/broken willusually be more than a few pixels.

Original: If the kernel size is too small, the closing operation (dilate thenerode) will hardly affect the final performance; while too big kernel sizes will join the separatedregions.
Revised: If the kernel size is too small, the closing operation (dilate thenerode) will hardly affect the final performance; while too big kernel sizes will join the separatedregions.

Original: Tab.
Revised: Tab.

Original: 5 lists post-processing results on baseline U-Net.
Revised: 5 lists post-processing results on baseline U-Net.

Original:  Conclusion
Revised:  Conclusion

Original: In this paper, we propose a novel homotopy warping loss to learn to segment with better structural/topological accuracy.
Revised: In this paper, we propose a novel homotopy warping loss to learn to segment with better structural/topological accuracy.

Original: Under the homotopy warping strategy, we can identify the topologicalcritical pixels/locations, and the new loss is defined on these identified pixels/locations.
Revised: Under the homotopy warping strategy, we can identify the topologicalcritical pixels/locations, and the new loss is defined on these identified pixels/locations.

Original: Furthermore,we propose a novel strategy called Distance-Ordered Homotopy Warping to efficiently identify thetopological error locations based on distance transform.
Revised: Furthermore,we propose a novel strategy called Distance-Ordered Homotopy Warping to efficiently identity thetopological error locations based on distance transform.

Original: Extensive experiments on multiple datasetsand ablation studies have been conducted to demonstrate the efficacy of the proposed method.
Revised: Extensive experiments on multiple datasetsand ablation studies have been conducted to demonstrate the efficacy of the proposed method.

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...(

Original: (a) Do the main claims made in the abstractand introduction accurately reflect the paper’s
Revised: a) Do the main claims made in the abstract

Original: contributions and scope?
Revised: and introduction accurately reflect the paper’scontributions and scope

Original: [Yes]
Revised: Yes]

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A]
Revised: [N/A]

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Original: them?
Revised: them?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [N/A]
Revised: [N/A]

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main ex-
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main ex-

Original: perimental results (either in the supplemental material or as a URL)?
Revised: perimental results (either in the supplemental material or as a URL)?

Original: [Yes] See the
Revised: [Yes] See the

Original: Supplemental Material.
Revised: Supplemental Material.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters,
Revised: (b) Did you specify all the training details (e g., data splits, hyperparameters,

Original: how they
Revised: how they

Original: were chosen)?
Revised: were chosen)?

Original: [Yes] See Sec. 4 and the Supplemental Material.
Revised: [Yes] See Sec. 4 and the Supplemental Material.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experi-
Revised: (c) Did you report error bars (e g., with respect to the random seed after running experi-

Original: ments multiple times)?
Revised: ments multiple times)?

Original: [Yes] See the Supplemental Material.
Revised: [Yes] See the Supplemental Material.

Original: (d) Did you include the total amount of compute and the type of resources used (e g., type
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type

Original: of GPUs, internal cluster, or cloud provider)?
Revised: of GPUs, internal cluster, or cloud provider)?

Original: [Yes] See Sec. 4.
Revised: [Yes] See Sec. 4.

Original: If you are using existing assets (e
Revised: If you are using existing assets (e

Original: g., code, data, models) or curating/releasing new assets...
Revised: g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [Yes]
Revised: [Yes]

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: using/curating?
Revised: using/curating?

Original: [N/A]
Revised: [N/A]

Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifiable

Original: information or offensive content?
Revised: information or offensive content?

Original: [N/A]
Revised: [N/A]

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of instructions given to participants and screenshots, if
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you describe any potential participant risks, with links to Institutional Review
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A]
Revised: [N/A]

Original: Structure-Aware Image Segmentation with Homotopy Warping

Original: — Supplementary Material—

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Appendix 6 shows the comparison of critical points identified by TopoNet [25] and the proposedhomotopy warping.

Original: Appendix 7 illustrates the simple points in 3D case.

Original: Appendix 8 shows the results of warping the prediction binary mask towards GT mask.

Original: Appendix 9 illustrates the topological errors in 3D case.

Original: Appendix 10 illustrates the effectiveness of the proposed warping strategy.

Original: Appendix 11 provides the details of the datasets used in this paper.

Original: Appendix 12 describes the details of the baselines used in this paper.

Original: Appendix 13 illustrates the details of the metrics used in this paper.

Original: Appendix 14 shows more qualitative results from different datasets.

Original: Appendix 15 provides stddev besides mean, and we use t-test to determine the statistical significance.

Original: Appendix 16 illustrates the loss weight (Cross-Entropy loss) for each dataset.

Original: Comparison of critical points between TopoNet [25] and HomotopyWarping

Original: Compared to the proposed method (Fig. 7(f)), the critical points identified from [25] are very noisyand often are not relevant to the topological errors.

Original: Simple points in 3D

Original: For a 3D binary image, a voxel is called a simple point if it could be flipped from foreground (FG) tobackground (BG), or from BG to FG, without changing the topology of the image [31].

Original: The followingdefinition is a natural extension of the Definition 1 in the main paper.

Original: It can be used to determinewhether a voxel is simple:

Original: Definition 2 (Simple Point Condition [31]) Let p be a point in a 3D binary image.

Original: Denote by F theset of FG pixels.

Original: Assume 6-adjacency for FG and 26-adjacent for BG.

Original: • p is 6-adjacent to just one FG connected component in N 26 ( p ) ; and

Original: • p is 26-adjacent to just one BG connected component in N 26 ( p ) .

Original: Submitted to 36th Conference on Neural

Original: [Figure]

Original: Illustration of warping prediction mask to GT

Original: In Fig.

Original: 5 of the main paper, we provide the illustration of warping the GT mask towards predictionmask.

Original: Similarly, we can warp the prediction mask towards the GT mask, and get different sets ofcritical pixels for the same topological errors, which are shown in Fig. 9.

Original: Topological errors in 3D

Original: In the main text, we provide the analysis of 1D topological structures (connections).

Original: Here we providethe illustration of 2D topological structures (holes/voids) for 3D case, which is illustrated in Fig.

Original: Note that for 3D vessel data, it’s also the 1D topological structures (connections) that matter.

Original: Theprocess of identifying the topological critical locations is the same as Fig.

Original: 4 and Fig.

Original: 5 in the maintext.

Original: And the only difference is to use 6-adjacency for FG and 26-adjacency for BG.

Original: Effectiveness of the proposed warping strategy

Original: By using Distance-Ordered Homotopy Warping, we are able to only consider all the inconsistentpixels once and flip them if they are simple.

Original: Otherwise, we need to iteratively warp all the inconsistentpixels (one non-simple pixel might become simple if its neighbors are flipped in the previous iteration).

Original: It takes 1.452s to warp a 512 × 512 image without the warping strategy, while only 0.317s with thestrategy thereby allowing the network to converge much faster.

Original: Datasets

Original: The details of the datasets used in the paper are listed as follows:

Original: Z

Original: RoadTracer :

Original: Roadtracer contains 300 high resolution satellite images, covering urban areas

Original: of forty cities from six different countries [4].

Original: Similar to setting in [4], twenty five cities(180 images) are used as training set and the rest fifteen cities (120 images) are used as the

Original: validation set.

Original: DeepGlobe :

Original: DeepGlobe contains aerial images of rural areas in Thailand, Indonesia and

Original: India [13].

Original: Similar to setting in [5], we use 4696 images as training set and the rest 1530images as validation set.

Original: Massachusetts :

Original: The Massachusetts dataset [35] contains images from both urban and ruralareas.

Original: Similar to the setting in [25], we conduct a three cross validation.

Original: DRIVE : DRIVE [46] is a retinal vessel segmentation dataset with 20 images (resolution584x565).

Original: CREMI :

Original: The CREMI dataset is a 3D neuron dataset 3 , whose resolution of 4 × 4 × 40 nm.

Original: We also conduct a three cross validation.

Original: Baselines

Original: The baseline methods used in this paper are listed as follows:

Original: RoadTracer [4]: RoadTracer is an iterative graph construction based method where node

Original: locations are selected by a CNN.

Original: VecRoad [50]: VecRoad is a point-based iterative graph exploration scheme withsegmentation-cues guidance and flexible steps.

Original: iCurb [54]: iCurb is an imitation learning-based solution for off-line road-curb detection

Original: method.

Original: UNet [41, 10]: The standard UNet trained with dice loss.

Original: Though lots of other segmentation

Original: methods/backbones have been proposed for image segmentation, UNet is still one of the

Original: most powerful methods for image segmentation with fine-structures

Original: [Figure]

Original: [Figure]

Original: DIVE [18]: DIVE is a popular EM neuron segmentation method.

Original: VGG-UNet

Original: [36]: VGG-UNet uses the response of selected filters from a pretrained CNN to

Original: construct a new loss function.

Original: This is one of the earliest works trying to deal with correct

Original: delineation.

Original: TopoNet

Original: [25]: TopoNet is a recent work which tries to learn to segment with correct topology

Original: based on a novel persistent homology based loss function.

Original: 8. clDice [44]: Another topology aware method for tubular structure segmentation.

Original: The basic

Original: idea is to use thinning techniques to

Original: extract

Original: the skeletons

Original: centerlines

Original: of the likelihood

Original: maps

Original: and ground truth mask.

Original: A new cldice loss is proposed based on the extracted skeletons

Original: besides traditional

Original: pixel-wise loss.

Original: DMT [26]: DMT is a topology-aware deep image

Original: segmentation

Original: method via

Original: discrete morsetheory.

Original: Instead of identifying topological critical pixels/locations, the DMT loss tries to identify the whole morse structures, and the new loss is defined on the identified morsestructures.

Original: Note that RoadTracer , VecRoad and iCurb are graph-based methods for road tracing.

Original: Graph-basedapproaches learn to explicitly detect keypoints and connect them.

Original: Since the graph is built iteratively,some detection errors in the early stage can propagate and lead to even more errors.

Original: Segmentationbased methods avoid this issue as they make predictions in a global manner.

Original: The challenge withsegmentation methods in road network reconstruction is they may fail in thin structures especiallywhen the signal is weak.

Original: This is exactly what we are addressing in this paper – using critical locationsto improve segmentation-based methods.

Original: VGG-UNet , TopoNet , clDice and DMT are topology-aware segmentation methods.

Original: Evaluation metrics

Original: The details of the metrics used in this paper are listed as follows:

Original: DICE : DICE score (also known as DICE coefficient, DICE similarity index) is one of themost popular evaluation metrics for image segmentation, which measures the overlapping

Original: between the predicted and ground truth masks.

Original: Adapted Rand Index (ARI) : ARI is the maximal F-score of the foreground-restricted Randindex [40], a measure of similarity between two clusters.

Original: The intuition is that the boundariespartition the whole binary mask into several separate regions, and the predicted and groundtruth binary masks can be regarded as two different partitions.

Original: ARI is used to measure thesimilarity between these two different partitions.3.

Original: Warping Error [27]: Warping Error is metric that measures topological disagreementsinstead of simple pixel disagreements.

Original: After warping all the simple points of ground truthto the predicted mask, the disagreements left are topological errors.

Original: The warping error isdefined as the percentage of these topological errors over the image size.

Original: Betti Error :

Original: Betti Error directly calculates the topology difference between the predictedsegmentation and the ground truth.

Original: We randomly sample patches over the predicted segmentation and compute the average absolute error between their Betti numbers and thecorresponding ground truth

Original: patches.

Original: More qualitative results

Original: We provide more qualitative results in Fig.

Original: Compared with baseline UNet, our method recoversbetter structures, such as connections, which are highlighted by red circles.

Original: The recovered betterstructures (UNet and Warping columns in Fig. 11) demonstrates that our warping-loss helps the deepneural networks to achieve better topological segmentations.

Original: More quantitative results

Original: We provide stddev besides mean, and use t-test (95% confidence interval) to determine the statisticalsignificance (highlighted with bold in Tab. 6) for RoadTracer dataset.

Original: The quantitative results showthat the proposed method performs significantly better than baselines.

Original: Loss weight parameter for each dataset

Original: As illustrated in Sec. 4.1, we finally choose cross-entropy loss (CE) as L pixel /L warp for all thedatasets.

Original: In Tab. 7, we list the weights achieving the best performances on each dataset.

Original: Failure cases

Original: In this section, we add a few failure cases from different datasets.

Original: Note that inferring topology givenan image is a very difficult task, especially near challenging spots, e.g., blurred membrane locations

Original: [Figure]

Original: [Table]

Original: or weak vessel connections.

Original: Current methods can help to improve the topology-wise accuracy

Original: [Table]

Original: whilethey are far from perfect.

Original: [Figure]