Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper proposes a novel technique for segmentation, especially for data with fine-scale structure. The authors questioned the conventional segmentation technique, which evaluates the segmentation on a pixel-by-pixel basis and overlooks the perspective of topological structure, and proposed a method to solve this problem. The proposed method proposes a Deep Learning structure that introduces a method for handling topological data, and compares its effectiveness with a typical baseline method.
strength_and_weaknesses: Strength

- We raised the issue of conventional segmentation technology and created a method to solve it.
- Experiments show that the proposed method significantly outperforms the conventional method

Weakness

- Before the substantive discussion, one major concern is that this paper is a difficult text to understand in terms of the proposed methodology. One of the reasons for this is the use of uncommon terms and symbols without explanation, although the terminology may be common knowledge among the authors. It is difficult to present all of them, but the following are representative ones.
	- In P6L1, Y remains undefined. A short time later, Y is shown to be ground truth, but it is extremely difficult to read. Also, not a big deal, but $L_{bce}$ is marked as cross-entropy, but I don't know what bce stands for.
	- In equation (1), $Y\circ S$ is written, but I don't know what it means.
	- The terminology regarding TDA, persistent homology, seems to be different from the common one. A persistent diagram in persisitent homology is one that extracts changes in the features of a figure that vary with filtration by a certain parameter. The proposed PD seems to refer to a method to create a figure (a single figure) by connecting point clouds when the parameter is a certain value. The proposed method is a major obstacle to understanding the correction by the simplex construction and its filtration, which seems to play an important role.
- This method seems to clarify the segment boundaries and compare them with the ground truth while adjusting the degree of clarification. I don't understand why ground truth Y and $S_{skelton}$ are combined in (1) with the above understanding, partly because the meaning of $Y\circ S$ is not clear. Also, in fig.1, Y and $S_{skelton}$ still do not seem to be combined, which is inconsistent with equation (1). This part should clarify definitions and implications.
- I assume that by comparing skelton and ground truth with well-defined boundaries, we are trying to make the topological information, i.e., the information of segmented chunks, similar, but why use cross-entropy for this purpose? I think you are looking at pixel-by-pixel information after all. I think there is a way to get more chunky information on a Morse theory basis. A simple idea is the Weierstrass distance of a persistent diagram (in the general sense, not in the sense of this paper) using Morse theory-based sub-level filtration, for example.
- The results shown in the experiment are an evaluation metric for topological information. This is not surprising, since the proposed method takes topology information into account, whereas conventional methods do not. If these indices are more valid than conventional indices for segmentation evaluation, they should be discussed. Alternatively, a comparison should be made with the evaluation indices provided by conventional methods.
clarity,_quality,_novelty_and_reproducibility: The novelty of the concepts and methods is recognized, but there are doubts about the clarity of the description of the methods. This makes it difficult to fully assess the quality of the methods.
summary_of_the_review: It is interesting to point out the problems with conventional segmentation methods and to propose methods to solve them. On the other hand, it is difficult to judge the value of the proposed method because it is unclear why the issue is important and there are many parts of the text that are unclear, such as the proposed method. At the very least, the sentence structure should be reviewed.

**Conclusion following discussion with the authors**

The problem of sentence structure, which hinders understanding of the proposed method, has been greatly improved. There is still room for improvement of the method, but it does provide some progress on the segmentation problem. I recognize that this is a positive result, and I will raise my score.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The focus of the paper is on extraction of structures like vessels in 2D, 3D image datasets.  The central idea in the paper is a deep learning algorithm that can deal with topological representations.  Topological structure variability is modeled by a one-parameter family facilitated by discrete Morse theory and persistent homology.  Unlike other schemes that deal with pixelwise probabilities for describing segmentation variability, this paper learns a probabilistic model that performs inference directly in topological (structural) representation space.  Results are shown to illustrate that the new method outperforms other methods that rely on pixelwise probabilities. Moreover, the utility of the uncertainty estimates on the topological structures for ease of editing is illustrated.
strength_and_weaknesses: Key strengths:
The deep architectural formulation and integration of ideas from morse theory and persistence homology is interesting.

Weaknesses:
The topic of vessel segmentation or topological structure estimation from 2D/3D data is something that has been studied in the literature for several decades.  While the paper claims that the new method outperforms other baselines, it appears that the new technique's performs is very close in performance to that of DMT method.  The impact of the connectivity improvements on end application goals must be described in order to really gauge in what concrete scenarios the DMT method was outperformed.    I feel that explicit knowledge of the noise characteristics of the sensing modality are critical for adaptively rank ordering possible branch hypotheses (for a given input image instance) in low signal to noise conditions where the improvements are being sought.  While I understand the rationale for the prior network and the KL divergence loss between the prior and posterior network parameter estimates,  it will be helpful to see in the appendix details of how well the instance specific priors look for specific images.  



clarity,_quality,_novelty_and_reproducibility: The idea is novel.  
Paper is well written and clearly contrasts the strengths of the work against other baselines used. 
The description in the paper seems adequate to reproduce the experiments.  There is no statement about that the code will be made available for ease of reproducibility.
summary_of_the_review: Overall I find the paper quite interesting and feel it has value. The main concern I have is on the justification of the empirical claims and in having more details on the inner working of the devised pipeline (please see weakness section).
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: This paper presents a new method for learning topological representations of 2D and 3D images. In contrast to existing work, the paper proposes a method that leverages the intrinsic structure of the "representation space", making it possible to derive (a) topological representations in a probabilistic fashion, while (b) also enabling and facilitating the analysis of structural uncertainty. The new method is then used to to primarily address segmentation task, exhibiting high-quality performance in different data sets.

strength_and_weaknesses: This paper presents a strong high-quality contribution for obtaining topological representations of 2D and 3D images, which in turn can be used to perform downstream tasks, such as segmentation. The main strengths of the paper lie in its **firm theoretical foundation** and its **thorough experimental setup**. There are only minor issues to be rectified in the present write-up, mostly concerning the clarity of the approach: to make the paper fully accessible to an audience that may not be entirely familiar with all the concepts, some additional modifications of the text (see below) are required. Moreover, certain claims in the paper can be slightly rephrased. That being said, the paper at hand has no substantial weaknesses that would prevent publication. On aspect that slightly marred the review for me was that no code is being provided; this does not detract from reproducibility, though. 

clarity,_quality,_novelty_and_reproducibility: ## Clarity

- I stumbled over some of the claims in the abstract: specifically, since the paper already cites several related methods that employ topological ideas (Hu et al., 2019, for instance), the sentence "first deep learning base method [...]" in the abstract should be rephrased. Similar constructions happen at other places. I understand the delineation to previous work, and I definitely see the merits of this paper; hence, in my opinion, such statements are not necessary (and in fact, the related work section provides a more objective delineation that is sufficient for me).

- If "structural" is to be used as a synonym for "topological" throughout the paper, I would already remark on this in the abstract. As it stands right now, I was a little bit confused about the meaning of this.

- Would it be possible to highlight the improvements in Figure 1 better? I liked the idea of combining the segmentation map with the original data, as showcased in the appendix. I think Figure 1 would benefit from a similar treatment.

- Figure 2 (and its explanation) might be used as segue into explaining the impact of topological defects.

- When mentioning the quality of the method at the beginning of Section 3, please extend and formalise the aspect on being "close enough to the true structures".

- Please elaborate on the meaning of a "reasonably clean input". This reads vague at the first time and requires careful reading to fully understand.

- The point about extending this from 2D to 3D is repeated multiple times. I think it would be sufficient to discuss this in *one* place. Personally, I am also fine with the paper focusing only on the 2D case for now; that case is already sufficiently complex and interesting on its own!

- When explaining some of the "moving parts" of the method in Section 3.1, I would appreciate a brief discussion of how the "filling" of *partially observed* structures works in practice. The example in Figure 3 is really captivating, and it would present a great opportunity to better comment on this aspect.

- Consider adding some explanations about cubical complexes or citing related work in this area; it might not be familiar to all readers (in general, I would say that the paper already does an excellent job at presenting the necessary background information).

- The section on uncertainty quantification is missing a justification as to where the Bernoulli distribution is arising from. Is this an approximation type argument?

- When showing Figure 8, can some of the differences between images be highlighted better? Why not pick several structures that are differing the most between the samples?

## Quality

While the paper is already of high quality, making a strong, substantial contribution, there are several aspects that could be commented on in addition:

- Having described the method, what are its limitations? Can the method capture all relevant structures? Are there thing it is not particularly adept at (for instance, segmentations whose ground truth has topological defects)? Are there additional theoretical guarantees that could be highlighted?

- Please refer to the pruning procedure on page 5 when discussing Figure 2. Some of the high-level details from Appendix A.3 should also be added to to this section in order to make the main text self-contained.

## Novelty

The method is novel in that it provides a way to traverse the space of potential segmentations, making use of discrete Morse theory and persistent homology to generate appropriate structures. Delineation to existing work is done to a sufficient extent, and it is clear to me that the proposed method constitutes a substantial contribution to the field.

## Reproducibility

Concerning this aspect, I am on the fence: on the one hand, the paper describes the method in sufficient detail; on the other hand, additional code as part of the supplementary materials would have been appreciated. Nevertheless, I think that a skilled graduate student should be able to reproduce the bulk of the method.

## Minor issues

The paper is already written very well, and it was a joy to read and comment on! There are some minor issues with the text that require another pass, including some missing articles. Here are some other minor issues I found:

- The bibliography is suffering from some inconsistencies; please check capitalisation and author names again. For instance, it should be "Morse theory", not "morse theory".

- "existing [...] map is" --> "existing maps [...] are"

- "When it comes down to" --> "When it comes to"

- "an 1D" --> "a 1D"

- "an 1-stable" --> "a 1-stable"

- "all possible combination" --> "all possible combinations"

- "can't" --> "cannot" (admittedly, a personal preference)

- "different $\epsilon$'s" --> "different $\epsilon$ values"

- "changing its topology" --> "changing their topology" (I think the reference here is to *skeletons*)

- "especially in structure wise": I have trouble parsing this sentence.

- "struggling": maybe "strenuous" would be more appropriate?


summary_of_the_review: This paper was a joy to read; it provides a substantial contribution and, for the most part, presents all results and concepts in a very clear fashion. The clarity issues that I listed can be easily addressed by some restructuring and rewriting within the revision cycle. I am thus happy to provisionally endorse this paper for publication.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: There is a consensus among reviewers to accept this work. All reviewers appreciated the incorporation of topological tools into segmentation learning setup. Reviewers commented about some exposition issues that were addressed in the rebuttal in a satisfactory manner. Other concerns raised are extent of improvement over existing vessel/topological segmentation methods,  but that does not seem as a roadblock to acceptance. We encourage the authors to publish code for reproducibility and facilitate future work on this topic. 
justification_for_why_not_higher_score: The enthusiasm from this work exists but somehow limited. Can be bumped to spotlight.
justification_for_why_not_lower_score: Solid and novel framework for learning the threshold of persistence homology in 2D images. Reviewers unanimous for accepting this work.
Decision: Accept: notable-top-25%