Paper: Instance Segmentation with Supervoxel Based Topological Loss Function (ICLR-2024)

Reviewer comment (Novelty):
In principle, the novelty of the contribution seems limited as an existing concept of simple points has been extended to superpixels (collection of pixels) instead.

Author response:
While the proposed extension is indeed simple to state (i.e., extension of the concept of simplicity from voxels to supervoxels), it took us a while to conceptualize this. We would like to note that the concepts on which our work is built (e.g., simple pixel, supervoxels) has been around for decades and there is an ever-growing need for topology-preserving methods. Therefore, we believe the proposed extension appears simple (and potentially limited) only after the fact. Finally, this extension would not be valuable unless it is supported by an accompanying mathematical and algorithmic framework. We are not aware of shorter, simpler ways of developing a rigorous theory without going through the multiple formal statements in our manuscript. Please also refer to our general response on this.

Reviewer comment (Experiments):
The authors should consider comparing against clDice [1] as a baseline since clDice has shown better performances among the topology-aware methods.

Author response:
We are actively working on training a model with the clDice. However, the original code base has some issues, as documented on the official github repository, which are delaying us from obtaining results. If we can resolve these, we aim to report the results of training this model to perform instance segmentation on the EXASPIM23 dataset.

Reviewer comment (Evaluation):
Please mention if the numbers in bold are just numerically better, or, if t-test [4] has been conducted to check if the performance improvement is statistically significant or not.

Author response:
The bold numbers highlight the best value across all methods. The paper has been updated to indicate this.

Reviewer comment (Presentation):
Lack of visual comparison with baseline in Figure 5.

Author response:
We have now added the visual comparison to Figure 5, as suggested by this referee.

Reviewer comment (Evaluation):
Performance metrics should be described using formulas.

Author response:
We have now added equations describing the performance metrics to the Supplementary material and referred to them in the main text when the metrics are introduced.

Reviewer comment (Experiments):
Lack of comparison with more loss functions that can supervise topology changes.

Author response:
We have now added an additional metric, the Betti error, to better quantify the results reported for the DRIVE dataset, as suggested. Briefly, the results support our previous findings where the proposed method is a close second-best for this metric and the top performer in other previously reported metrics, despite focusing on a narrower set of topological changes and the ensuing computational simplicity.

Reviewer comment (Evaluation):
Critical components should not include areas that do not affect the topology structure. In the visualization effect of Figure 4, why is the structure more slender compared to the baseline, and can the proposed loss function optimize the segmentation edge?

Author response:
We attribute this qualitative difference to the fact that a less slender structure is more likely to result in merge mistakes. The depiction of positively critical components in Figure 3 is a didactic cartoon and might not accurately portray a "typical" critical component. In actuality, positively critical components encompass incorrect voxel predictions that act as a bridge between two distinct components. (Plus, any voxels in the false positive mask connected to this bridge which may extend into a large region connected to the boundary of the objects.) It's plausible that the neural network learns to avoid less slender predictions due to these considerations.

We don't make any claims that this loss function leads to more slender predictions. For neuron segmentation, this is a less important qualitative feature and not easy to quantify.

Reviewer comment (Writing):
The paper could improve in terms of clarity in several cases. Most importantly, Algorithm 2--a critical part of the paper--is very vague and the only explanation provided about it under Corollary 3 does little to make it clearer.

Author response:
A1: Thank you for this very helpful and explicit feedback. In retrospect, we agree that the initial presentation of this section may be difficult to parse. We have completely rewritten this section to improve clarity and focus the discussion on the high level ideas. The pseudo code has as been significantly revised so that it is more intuitive and better aligned to the descriptions in the text.

Reviewer comment (Evaluation):
The method is only sensitive to the number of components, while topology is much broader e.g. bifurcations, loops. This limitation needs to be communicated.

Author response:
A2+A3: While the main definitions and initial formal statements (e.g., Thm 1) capture topological changes in structures broader than tree-like objects, as this referee pointed out, the statements and algorithms surrounding the fast implementation are restricted to tree-structured objects. Indeed, a similar algorithm based on the main definitions and deductions can be implemented in a straightforward way, except that this algorithm will be super-linear in complexity. To address the referee's concern, we have now added explanation to the main text (see discussion following Theorem 3) emphasizing that the proposed fast algorithm will not be sensitive to broader topological changes in more general graph structures, such as creation of a cavity or a loop.

Reviewer comment (Presentation):
The loss function is introduced in Definition 1, and then expanded again in Section 3.3. A better sense of direction would have been conveyed if the two were mixed and mentioned early on in Section 3 and stated that the rest of the Section focuses on finding N(y^) and P(y^) in Definition 4. Removing G(.) and H(.) and having double sums might improve readablility.

Author response:
A1: This is a great suggestion and we have updated the paper accordingly. It not only improved readability, but also shortened Section 3.3 so that we can present more visual results in Section 4.

Reviewer comment (Writing):
The definition originally provided for S(.) does not have a subscript and is clear, but the explanation provided for the case with a subscript is hard to grasp.

Author response:
A2: This extra notation is necessary to ensure that each connected component $C\in\mathcal S(\hat y_-)$ intersects with exactly one connected component in the set $\mathcal S(y)$. To address the referee's concern, we have now added an explanatory footnote after the definition with the subscript, relating to the definition without the subscript.

Reviewer comment (Writing):
In Method Section before Definition 1, it is stated that y_i \in {0,1,...,m} and y^_i \in {0,1,...,l}. Shouldn't they both sets be the same (no need for l)? Also m, l(?), and n are given without definitions.

Author response:
A3: Thank you for pointing this out, there is a typo in the manuscript. Our original submission contains the lines ``A ground truth segmentation $y = (y_1, . . . , y_n)$ is a labeling of the vertices such that $y_i\in\{0, 1,..., m\}$ denotes the label of node $i \in V$. Each segment has a label in 1,..., k and the background is marked with 0.'' Instead the last sentence should be "...Each segment has a label in 1,..., *m and the background is marked with 0", which defines m to be the number of objects in the image. 

The value l is the number of objects in the predicted segmentation. We assume that the true number of objects m is unknown at the time of inference. It is possible and usually the case that $l \neq m$, especially in neuron segmentation. We added a footnote explaining why m is not necessarily equal to l.

We also updated the first sentence in the method section to include the definition of n, it know reads as ``Let $G=(V,E)$ be an undirected graph with the vertex set $V=\{1,\ldots, n\}$. We assume that $G$ is a graphical representation of an image where the vertices represent voxels and edges are defined with respect to a $k$-connectivity''.

Reviewer comment (Theory):
Positively critical components rely on computing S(y^⊖y^+), while Algorithm 2 only requries knowing S(y⊖y^-). Why?

Author response:
A4: Thanks for pointing it out. This is actually a typo in Algorithm 2, it should read as $S(y\ominus \hat y_x)$ instead of $S(y\ominus \hat y_-)$, where $\hat y_x$ is a placeholder for $\hat y_-$ and $ \hat y_+$. We have now updated the pseudocode of the algorithm.