Adaptive Graduated Non-Convexity for Point Cloud Registration

We greatly appreciate your positive feedback and constructive suggestions on our paper. We are glad that you found our approach innovative. Below, we respond to your specific comments:

W1: In the experimental results, it seems there is no time analysis. It is better to compare the proposed method to existing ones in terms of time cost;

• We appreciate your suggestions. Regarding the analysis of time cost, we also understand that this is an important consideration. As you said, in this study, we propose a novel method that can significantly improve registration accuracy without additional training procedures. Our main focus is to verify the accuracy and robustness of the proposed method in handling different outliers, so we prioritize these metrics in the experimental section. Given our research objectives, we choose to focus on the RE, TE, RR performance of the algorithm. In addition, we have modified Table 1 to provide the runtime and computational overhead information required by Hessian, which will provide a complete performance picture. Thank you for your understanding and support.

W2: The experiments on Indoor Scenes may be insufficient, as this is the main field that registration methods focus on. I would suggest adding different descriptors on both 3DMatch and 3DLoMatch, those descriptors should cover a wide range of outlier rates. (as on synthetic data, the test covering different outlier rates is included, but on real data, there is no such setting) Moreover, using FCGF descriptors on 3DMatch but Predator on 3DLoMatch is weird. It looks like that FCGF doesn't perform well on 3DLoMatch, which leads to bad results for the proposed method. The authors have to choose another descriptors for 3DLoMatch.

• We fully agree that it is important to add different descriptors in indoor scene experiments. In the preliminary experimental design, a single descriptor was selected for 3DMatch and 3DLoMatch, mainly based on the usage frequency of these descriptors. We have added experiments with two other descriptors regarding SC2-PCR and MAC. Specifically, 3DMatch uses both FCGF and FPFH. 3DLoMatch uses both FCGF and Predator (Tables 3 and 4 in revised manuscript). We are pleased to find that our method can still achieve better results among different descriptors. We believe that these supplementary experiments will make our paper more comprehensive.

W3: More visualization results on indoor and outdoor scenes are encouraged. And it is also better to directly visualize the correspondences before and after using the proposed method.

• We agree that more visualization results will help to more intuitively demonstrate the performance of our method. We have added more visualization results for indoor and outdoor scenes (Figures 3 and 4 in supplementary material). We believe that these supplementary materials will further enhance the quality and persuasiveness of the paper. Thanks again for your suggestions.

W4: Some closely related works may be missing:

[1]. Yu et al. CoFiNet: Reliable Coarse-to-fine Correspondences for Robust Point Cloud Registration, NeurIPS 2021;

[2]. Ao et al. SpinNet: Learning a General Surface Descriptor for 3D Point Cloud Registration, CVPR 2021

[3]. Qin et al. Geometric Transformer for Fast and Robust Point Cloud Registration, CVPR 2022;

[4]. Want et al. You Only Hypothesize Once':' Point Cloud Registration with Rotation-equivariant Descriptors, ACM MM, 2022;

[5]. Yu et al. RIGA: Rotation-Invariant and Globally-Aware Descriptors for Point Cloud Registration, TPAMI 2024

[6]. Yu et al. Rotation-Invariant Transformer for Point Cloud Matching, CVPR 2023;

• Thank you for your thoughtful suggestions. These related works provide valuable references for this paper and have important implications and guiding significance for future research. We have added these related works in the revised version. (Yu et al., 2021), (Ao et al, 2021), (Qin et al, 2022), (Wang et al, 2022), (Yu et al, 2024), (Yu et al, 2023), (Qin et al, 2023)

Finally, we would like to express our gratitude again for your time and effort in reviewing our paper. Please do not hesitate to let us know if you have any further concerns or comments. We would be happy to address them.

Dear Reviewer k1rY,

We sincerely appreciate your time and effort in reviewing our paper and providing valuable suggestions.

Since there will be no second stage of author-reviewer discussions, and the discussion phase is drawing to a close, we would to confirm whether our responses have effectively addressed your concerns. If you require further clarification or have additional concerns, please don’t hesitate to contact us. We are more than willing to continue our communication with you.

Best regards.

The Authors

Thank you for your thoughtful feedback and for taking the time to assess our revisions. We sincerely appreciate your recognition of the paper and your positive comments. We would add time analysis in the final manuscript to further improve our paper.

I think the authors addressed most of my concerns well. I will keep my positive score. However, I still suggest the authors adding more time analysis for the overall comparisons with existing methods, not just only showing the running time of some designed blocks with out references.

W5: I recommend visualizing the outputs from competitive methods in Figure 4 and using bright colors to enhance clarity in the visualizations (minor).

• Thank you for your suggestion regarding the visualization of the figure. We have redrawn Figure 4 with bright colors. More visualizations of the registration results with competitive methods are provided in the Figure 1 of supplementary material.

W6: In Tables 3 and 4, it is unclear if "ER" and "Et" refer to rotation error and translation error, respectively. If so, for consistency with the previous definitions, they should be labeled as "RE" and "TE" as mentioned in line 344.

• Thank you for your careful reading. We have corrected ER and Et to RE and TE in the revised manuscript to maintain consistency with the previous definitions. At the same time, we have checked and ensured the accuracy and consistency of all terms in the manuscript.

W7: I am not convinced of the accurate performance of this method, as the criteria for successful registration (line 431) in Tables 3 and 4 seem too lenient for both indoor and object-level settings. Similar concerns also arise for Table 5.

• Thank you for your attention to the performance evaluation criteria of this method. We understand that you have doubts about the strictness of the successful registration criteria in Tables 3, 4, and 5. These criteria are not set arbitrarily, but refer to the evaluation criteria in the comparative methods [1, 2, 3] (Zhang et al, 2023; Chen et al, 2022b; Huang et al, 2024). These references provide us with recognized evaluation indicators and threshold settings to ensure the scientificity and rationality of our experiments.
• Regarding the issue that the threshold setting may appear too loose, it is because the point cloud registration problem on the 3DMatch, 3DLoMatch, and KITTI datasets is very challenging. First, the point cloud overlap rate in these datasets is generally low, which means that fewer feature pairs can be used for registration, which increases the difficulty of registration. Second, the noise and occlusion contained in the point cloud data will interfere with the feature extraction and matching process, increasing false matches. For more information on the challenges of the dataset, please see Table 1 in supplementary material. We have cited the above references in the revised manuscript and further explained the reasons for our selection of these criteria.

References

[1] 3D Registration with Maximal Cliques, CVPR, 2023

[2] SC^2-PCR: A Second Order Spatial Compatibility for Efficient and Robust Point Cloud Registration, CVPR, 2022

[3] Efficient and Robust Point Cloud Registration via Heuristics-guided Parameter Search, TPAMI, 2024

W8: Please correct the spelling of "black-Rangarajan" duality. It would also be beneficial to include a brief sentence explaining what this duality is about for the reader’s clarity. In line 305, please change the reference from "(line 4)" to "(line 5)"

• We appreciate your careful review and valuable suggestions. We have corrected it to "black-Rangarajan" duality and then added a brief sentence with reference. "This duality provides a way to convert traditional line process methods and robust statistical methods into each other (Black & Rangarajan, 1996)." In addition, we have also corrected the reference in line 305 from "(line 4)" to "(line 5)".

Q1: Given that computing the Hessian is typically expensive, could the authors please discuss whether alternative routines, such as Adam or RMSprop, could serve as equivalent methods for scaling? How would these compare in terms of performance and convergence speed?

• We recognize that calculating the Hessian matrix does have the problem of high computational cost. Regarding whether optimization algorithms such as Adam or RMSprop can be used instead of Hessian for scale adjustment, we believe that although these two algorithms perform well in many tasks, their core mechanisms are different from the goals of our method. Adam and RMSprop mainly adjust the learning rate by adaptively estimating the first moment and second moment [1, 2]. However, they do not directly utilize second-order information and differ from the Hessian-based methods in terms of convergence speed and final performance. Adam and RMSprop do not explicitly consider the structural information of the Hessian, so they may not be able to effectively capture the global optimal solution when facing the point cloud registration optimization problem with strong non-convexity. Regarding your suggestion of using the Adam and RMSprop algorithms as alternatives, we believe that this is a research direction worthy of further exploration in the point cloud registration task.

  References

  [1] Adam: A method for stochastic optimization, 2014

  [2] An overview of gradient descent optimization algorithms, 2016

Thank you for taking the time to review our work. We have carefully addressed the questions and concerns raised, and have made updates to the manuscript to improve clarity and understanding. We will address each point below.

W1: While the paper presents experimental validation using a toy example to illustrate the effects of the adaptive scheme on convergence iterations, it lacks critical information on runtime and the computational overhead required for Hessian. Including runtime data in Table 1 would be beneficial, as it would provide a complete picture of performance by accounting for the overhead of Hessian computation and binary search.

• We completely agree with you that information on runtime and the computational overhead required for Hessian would provide a complete picture of performance. We have added two columns to Table 1 specifically to show the runtime data under different experimental settings.

Table 1: Comparison of different updating methods of $\zeta$.

• These results show that although there is a certain overhead in Hessian computation and binary search, the overall runtime is still within an acceptable range with better performance. We will include these changes in the revised version and ensure that our study can provide readers with deeper insights.

W2: The paper contains many acronyms, such as FGCF and LSQ, without definitions. It is important to clarify these terms early on to enhance reader understanding. Additionally, several notations, like "r" in Equation 1, are introduced without explanation. These should be defined as soon as they are presented. Equations 6 through 11 also contain unexplained notations, which can confuse readers.

• We have provided clear definitions for all acronyms in the revised manuscript and explained the meaning of each symbol immediately upon its first appearance. This enhances the readability and understanding of the paper.

W3: The assertion that “…. H in Eq. 2 obtained at µ_{k+1} is ensured to remain positive definite, then the new estimate z_{k+1} is guaranteed to be in the same convergence domain as the previous iteration” requires citation or proof. Providing a reference would strengthen the validity of this claim.

• In the {k+1} iteration, if we update the scale to µ_{k+1} ensuring that the Hessian remains positive definite, then we have likely ensured that the new z_{k+1} is in the same convergence domain as the previous iteration. This is true since we initialize optimization for the {k+1} iteration at z_{k} and converge to the local minimum by IRLS. We have cited relevant references in the paper to support this assertion (Andrew&Gao, 2007; Kohetal., 2007; Ochsetal., 2013).

W4: An insightful addition would be to include experiments assessing the impact of a multi-task knowledge-sharing mechanism with fixed scaling in Table 6, alongside the number of iterations and runtime required for convergence. Currently, the errors without knowledge sharing are worse than those of the fixed scaling scheme, raising questions about whether the adaptive scale adjustment primarily enhances convergence speed rather than accuracy. Adding columns to compare knowledge sharing in fixed scaling settings and the number of convergence stages would clarify the contributions of each component.

• We understand your concern about whether the adaptive scale adjustment primarily improves convergence speed rather than accuracy. We have added columns in Table 6 to compare knowledge sharing in fixed scaling settings and the number of convergence stages to clarify the contribution of each component. As shown in Table 6 in revised manuscript, the adaptive scale adjustment reduces the number of stages by about half, while effectively reducing both the rotation error and the translation error. As you said, the errors without knowledge sharing are worse. The reason is that tracking the local minimum sometimes leads to a solution far away from the ground truth. The adaptive scale adjustment of graduated non-convexity effectively not only effectively improves the registration speed, but also improves the registration accuracy. The knowledge sharing strategy overcomes the severe failure conditions caused by high outliers to jump out of the local minimum.

Q2: The paper states that "little attention has been paid to how the scaling factor is determined." Please provide references to existing techniques that address this issue. Including comparisons to these methods would enhance the context of this work.

• Thank you for pointing out the need for references. In the literature, some methods explore the issue of choosing a scaling factor. We have included these references (Hazan et al., 2016; Le & Zach, 2020) in the revised manuscript.

Q3: Once the Hessian is computed, how valuable is its inclusion in the cost function? I would expect this to lead to faster convergence with minimal overhead. Can authors provide insights on this aspect?

• Thank you for your comments. Indeed, the calculation of the Hessian matrix will bring a certain amount of time overhead. In our method, the introduction of the Hessian matrix is mainly to provide more accurate second-order derivative information, which helps the algorithm to more effectively adjust the step size when approaching the optimal solution, thereby achieving faster convergence. We have added two columns to Table 1 in revised manuscript specifically to show the time cost under different experimental settings. These results show that although there is some overhead in the Hessian computation, better convergence is achieved with less overhead.

• In the future, we believe it would be possible to consider optimizations (e.g., using approximation methods) to try to reduce this overhead further.

Q4: How valid is it to consider different optimization stages as different tasks? Could authors elaborate on this concept and its implications for the proposed method?

• Thank you for your attention to the concept of different optimization stages. It is an innovative idea to treat different optimization stages as different tasks, which is based on the observation that the shape of the cost function or the optimal solution has certain similarities at various stages of the optimization process. This approach is inspired by multi-task learning, where different but related tasks can improve the optimization performance by sharing knowledge.

• In this paper, we achieve the collaborative optimization of non-convex cost functions at different levels through a multi-task knowledge sharing mechanism, which helps to improve the success rate of point cloud registration under the challenging condition of a high outlier rate. The ablation experiments in Table 6 further verify the effectiveness of the strategy. We find that without knowledge sharing, the registration results are very poor. The reason is that the point cloud registration method is at risk of settling in the local minimum under high outliers, which will lead the results far away from the true solution.

Q5: How is the final value of \mu_{final}determined?

• Thank you for pointing this out. The choice of \mu_{final} was based on the results of experimental analysis. We experimentally evaluated the impact of different \mu_{final} values on the point cloud registration performance and selected the value that achieved the best registration accuracy and robustness on multiple datasets. In addition, we considered computational efficiency to ensure that the selected \mu_{final} value can complete the optimization process in a reasonable time.

Q6: Could authors add extra columns in Tables 3 and 4 to indicate the range of initial total translation and rotation offsets? The translation errors of greater than 6 cm and 9 cm for even the best performances seem unacceptable for practical applications in small indoor settings.

• Thank you for your suggestion. We have listed the contents of all datasets in detail in Table 1 of supplementary material. We also provide a further introduction to the datasets. We agree that in some precise indoor environment application scenarios, such as industrial parts inspection. Large translation errors and rotation errors are unacceptable. However, the point cloud registration problem on the experimental dataset in this paper is very challenging. Point clouds have different overlap rates, noise, occlusion, and other interferences. These factors reduce the number of valid feature pairs and increase false matches. The registration performance of all methods in small indoor environments is the average result on this dataset. Not all models exceed the translation error of 6 cm and 9 cm.

Finally, we would like to express our gratitude again for your time and effort in reviewing our paper. Please do not hesitate to let us know if you have any further concerns or comments. We would be happy to address them.

Dear Reviewer dmuo,

We sincerely appreciate your time and effort in reviewing our paper and providing valuable suggestions.

We understand that you may be extremely busy at this time. But we would to confirm whether our responses have effectively addressed your concerns. We hope that you could kindly update the rating if your questions have been addressed. We are also happy to answer any additional questions before the rebuttal ends.

Best regards,

The Authors

Thank you for your continued attention and valuable feedback on our paper.

C1: Thank you for your response and for carefully addressing the comments. However, considering the presence of other (adaptive) schemes in the literature, I feel the paper would be more complete with a comparison to these methods. It would be helpful to include comparisons in terms of convergence steps, runtime, and accuracy to provide a more complete context for the work.

• Thank you for your feedback and suggestions. We have included a comparison with existing scale adaptive schemes in the supplementary material (in Table 2 in supplementary material). Our method AGNC is compared with two different scaling schemes, GradOpt (Hazan et al., 2016) and ASKER(Le & Zach, 2020). Table 2 reports the results of convergence steps, runtime, and relative accuracy. Compared with the second-ranked ASKER method, the convergence steps are reduced by 11 stages, the runtime is reduced by 3.92 and with higher accuracy. The results show that our AGNC achieves the best results.
• These additions more fully demonstrate the advantages of our approach. Thank you very much for your suggestions, and we have added these improvements to the supplementary materials.

C2: While I understand that these thresholds have been adopted from the literature, I would appreciate it if you could provide the initial translation and rotation offset ranges in the tables as this would give a deeper understanding of the scale of errors (Table 1 in the supplementary material does not contain this information). Although these criteria are taken from literature, I find them somewhat lenient. Given that these are considered successful registrations, I would expect that applying a refined registration method to these results would yield the correct registration with minimal errors. If this is not the case, it would be helpful to include a discussion in the text regarding the specific applications that can tolerate up to 30 cm of translation error indoors, and similarly for outdoor settings. This would provide a better understanding of the practical implications of these thresholds.

• Thank you for your careful reading and valuable comments. We agree that counting the initial translation and rotation offset ranges helps readers understand the scale of the errors. We have added columns to the table to explicitly show the initial translation and rotation offset ranges (in Table 1 in supplementary material). It can be observed that in addition to the interference of noise, outliers, and limited overlap, the initial rotation and translation errors are also very large.
• We understand your concern about the evaluation criteria. The point cloud registration problem in the datasets used in the experiments is very challenging. These datasets contain noise, outliers, and sensor differences, which make the registration task very difficult. Therefore, although the evaluation criteria seem loose, they reflect the actual difficulty of performing registration under these conditions. All compared methods include the refined method in their papers to minimize the error. Despite these optimizations, it is still very difficult to further reduce the registration error. Under idealized conditions, without interference such as outliers or noise, point cloud registration can achieve very accurate results. However, in the real world, registration is still a complex task, especially when considering the imperfections of the data.
• Based on your suggestion, we have also added and discussed two practical application examples. (in Section 4 in supplementary material) One is a virtual reality scenario, and the other is an autonomous driving scenario. Translation errors within the tolerable range are acceptable for both.
• We agree with the reviewer that further reducing the error is essential to improve the applicability of the registration. This will be the core goal of our future research. We are very grateful for your suggestions, which will help us better demonstrate the practical significance and application value of our work. We have fully considered and implemented these improvements in the revised manuscript.

Thank you for addressing the review. While I am not certain if a 30 cm offset is still considered acceptable in virtual reality, I do believe that a 60 cm threshold (for Table 5) is likely too large to be considered within a tolerable range. In the context of environmental modeling for autonomous driving, I imagine that such a significant error could lead to critical localization issues that may lead to other problems—such as vehicles drift off the road, failing to follow lanes (without land detection in place), or even colliding with pre-modeled objects in the map, like traffic lights. Please feel free to correct me if my understanding here is incorrect.

Comment: Thank you for addressing the review. While I am not certain if a 30 cm offset is still considered acceptable in virtual reality, I do believe that a 60 cm threshold (for Table 5) is likely too large to be considered within a tolerable range. In the context of environmental modeling for autonomous driving, I imagine that such a significant error could lead to critical localization issues that may lead to other problems—such as vehicles drift off the road, failing to follow lanes (without land detection in place), or even colliding with pre-modeled objects in the map, like traffic lights. Please feel free to correct me if my understanding here is incorrect.

• Thank you for your in-depth review and valuable comments of our paper. We completely agree with your perspective that in high-precision application scenarios within autonomous driving, such as precise lane keeping, even minor errors can have serious consequences. Your concern regarding the 60 cm threshold is entirely justified, as it could indeed impact the positioning accuracy and safety of autonomous vehicles.
• In practice, the autonomous driving system does not solely rely on point cloud registration results but continuously corrects and optimizes vehicle positioning through real-time data updates and advanced multi-sensor registration and fusion technologies. These technologies work in concert to provide continuous, high-precision positional information. Therefore, even if there are certain errors in point cloud registration, these errors may be corrected by subsequent sensor data fusion processes to ensure the safe operation of the vehicle. Correct positioning and path planning can also be maintained in the face of lane changes. Of course, we also recognize that lower registration errors will further enhance the safety of autonomous driving.
• To further validate the performance of our AGNC method under more stringent error requirements, we have adjusted the threshold and tested the performance of AGNC under a translation error threshold of 30 cm. Compared to the original threshold of 60cm, where the AGNC method achieved an average registration success rate of 99.12%, under the updated threshold of 30cm, the AGNC method achieved an average registration success rate of 87.26%. The results demonstrate that our method still exhibits a high registration success rate under more stringent conditions.
• We believe that these supplementary results can better respond to your concerns about error tolerance. Thank you again for your feedback. We have improved the discussion section of the supplementary materials.

Dear reviewer dmuo,

Thank you once again for your valuable insights. As the discussion period is coming to an end, we would like to confirm whether we have adequately addressed your concerns. If you have any additional comments or concerns, we would be happy to address them during the remainder of the discussion period. We have noticed that you have improved your score. We sincerely appreciate your recognition and the score adjustment.

Sincerely,

Authors

Q1: For the experiments on 3DMatch/3DLoMatch dataset to be registered with only one feature, I think it is not enough, please refer to the experimental setup of SC2-PCR and MAC. In addition, 3DMatch uses FCGF and 3DLoMatch uses Predator for feature extraction, which is kind of intentional to show the reader better results.

• Referring to SC2-PCR and MAC, we have introduced two other feature descriptors to the 3DMatch and 3DLoMatch experiments. Specifically, 3DMatch uses both FCGF and FPFH. 3DLoMatch uses both FCGF and Predator (Table 3 and Table 4 in revised manuscript). In this way, all experimental settings are the same as KITTI, using two different feature descriptors to more comprehensively evaluate the effectiveness of the methods. We are pleased to find that even under different conditions, our method can still achieve the best results. We have detailed these experimental settings and show the performance of our method under different feature descriptors in the revised manuscript.

Q2: It is suggested that ablation experiments should also be done on the 3DMatch data set. Only the contribution to accuracy can be seen for the simulation data set, but not for the robustness, such as the index RR.

• Thank you for your suggestion. We conducted ablation experiments on the Stanford dataset, and these experimental results effectively demonstrate the contribution of the components of our method in improving the registration accuracy. As for why we did not conduct ablation experiments on more datasets, the following are our considerations: Stanford is a simulated dataset, and we hope that all simulation experiments are completed on the same dataset. There is no need to consider the impact of different dataset characteristics on the experimental results, ensuring that the differences between experiments are only caused by the ablated components. In addition, the ablation experiments on Stanford have been able to verify the effectiveness of each component in our method. Even if changes are made to the dataset, we believe that our method can still maintain high performance.

• It is worth mentioning that we have further improved the ablation experiments. We have added columns to Table 6 to compare knowledge sharing and convergence stages in the fixed scaling setting. As shown in Table 6, adaptive scale adjustment reduces the number of stages by about half, while effectively reducing the rotation error and translation error. Nevertheless, we recognize the value of ablation experiments on more datasets. We will consider incorporating more experiments in future work to fully evaluate the performance of our method.

• In response to your question about the loss of the robustness indicator RR on the simulated dataset, we have added the content and analysis of the RR indicator (Table 2 in revised manuscript). These additional results further demonstrate the robustness of our method. Thank you for your thoughtful feedback.

Finally, we would like to express our gratitude again for your time and effort in reviewing our paper. Please do not hesitate to let us know if you have any further concerns or comments. We would be happy to address them.

Thank you for your feedback and recognition of our work. Below are our responses to your questions and comments, and we hope they will also address your concerns:

W1: The experimental setup needs to be improved. I have my doubts about the experimental setup. A major contribution of this paper is that high outlier rates can also perform well. As for the evaluation of high outlier rate, only on the simulation data set, however, as shown in FIG. 4, the current methods can achieve good registration. This does not speak to the strengths of your approach. On the contrary, on 3DMatch dataset, with high outlier rate, it is difficult for existing methods to achieve robust registration. It is suggested to add the experimental results of Outlier rate=99% in 3DMatch dataset, which can make this paper more convincing. "Accurate registration results are obtained even at the extreme 99% outlier rate." "Is a bit of an exaggeration.

• Thank you for your valuable comments on the experimental setup. We have added the experimental results of different outlier rates on the 3DMatch dataset in Figure 2 of the supplementary material. By conducting additional experiments on the 3DMatch, we can better demonstrate the advantages of our method in dealing with high outlier rates. We believe that these supplementary results will make the paper more convincing.
• We also realize that the statement may be too absolute. The performance of our method under high outlier conditions is verified based on a series of experiments, which show that our method does show better robustness than the existing technology when facing challenging data. We have revised this sentence to maintain an objective description of our method. "Experimental results have shown that this method outperforms compared methods in terms of robustness and accuracy, can obtain promising registration results even in 99% outlier rates."

W2: Some typos. What do we mean by ER and Et in Tables 3 and 4 and 5? Should it be RE and TE?

• We have corrected ER and Et to RE and TE in the revised manuscript to maintain consistency with the previous definitions. At the same time, we have checked and ensured the accuracy and consistency of all terms in the manuscript.

Dear reviewer aCKU,

We sincerely appreciate your time and effort in reviewing our paper and providing valuable suggestions. Special thanks for your recognition of the simple and feasible idea and good performance of our paper.

We hope to have addressed your concerns adequately. We are more than happy to answer any additional questions during the rebuttal period. Thank you again for your time and effort.

Best Regards,

The Authors