# Response to Reviewer ojV5
We thank the reviewer for the constructive comments. We will do a careful proofreading and fix all the typos. For example, for lines 65-68, we modify it as ‘the estimated 6D pose should naturally be equivariant with…’;

#### Q1: clarify main contribution of the paper
* **[Misunderstanding of the contributions]** We are well aware of the reviewer’s concerns on intra-category shape variations and poor canonical shape reconstruction, and that is exactly where our novelty and main contribution come from--by proposing leveraging SE(3) equivariance and designing an effective framework. And our method doesn’t use residual pose predictions for progressive alignment, but could get accurate pose in a single forward pass during the inference stage. We don’t claim novelty over the branch design for canonical reconstruction and pose regression.
* **[Contribution 1]** We are the first one to explore self-supervised category-level 6D pose estimation from point clouds when there are no CAD models available, allowing intra-category shape variations, and occlusions-introduced partialness, no need of iterative alignment, and handling both symmetric and non-symmetric categories;
* **[Contribution 2]** We propose the key idea of using networks preserving SE(3) equivariance, which could disentangle pose and shape effectively, and solve this problem elegantly;
* **[Contribution 3]** Our proposed framework can achieve accurate pose estimation results that are comparable or even beat supervised SOTA methods, for both complete and partial point cloud input in a single forward pass. The secret is that by using rotational-invariant features, our model could learn to predict aligned shape reconstruction in canonical space for different instances through pure end-to-end self-supervised training, and by using rotation-equivariant features on divided SO(3) groups, our pose branch can give accurate SO(3) regression and associated translation;

#### Q2: ablation studies on effects of quality of reconstructed shape on pose estimation
The reconstructed shape indeed affects pose estimation largely and this is exactly what motivates us to leverage equivariant neural networks, through which we demonstrate for the first time that self-supervised pose estimation methods are able to achieve comparable performance with supervised methods.

* **[Equivariance is the key for high quality aligned reconstruction]** In our major experiments on both complete and partial input, we show that our reconstructed canonical shapes are naturally aligned well against intra-category variations. Also for categories like symmetric bottle, the predicted pose is still quite accurate even the reconstructed shape is not complete(in poor quality), check further visualization in figure 3 and figure 4 in our supp.;
* **[Ablation studies on reconstruction qualities]** If using general non-equivariant neural networks(like KPConv) for shape prediction, we won’t be able to get category-level aligned canonical shape reconstructions, which would largely affect the accuracy of pose estimation, we list our ablation studies for shape backbone here. Figure 5 in our supplementary also gives a better visualization for drifted canonical shape reconstruction if using KPConv.
<center>Different Backbones for Canonical Shape Reconstruction</center>

| Dataset | Shape Backbone | Pose Backbone | Mean R_err | Median R_err | Mean T_err | Median T_err | 5deg. Acc. | 5deg.0.05 Acc. |
| :---------------- | :------------- | :------------ | :--------- | :----------- | :--------- | :---------------------------------- | :------------ | :---------- |
| Complete airplane | EPN | EPN | **23.09** | **1.66** | / | / | **0.87** | / |
| Complete airplane | KPConv | EPN | 133.89 | 169.96 | / | / | 0.00 | / |
| Partial airplane | EPN | EPN | **3.47** | **1.58** | **0.02** | **0.02** | **0.95** | **0.88** |
| partial airplane | KPConv | EPN | 134.34 | 174.98 | 0.07 | 0.08 | 0.05 | 0.05 |

# Response to Reviewer 8FBM
We thank the reviewer for the constructive advices, and will properly address the reviewer’s suggestions on our paper writing. We tune contribution point 3 as ‘our proposed method achieves accurate pose estimation that is comparable or surpasses existing supervised SOTA methods’.
#### Q1: ablations with different amounts of training data.

We evaluate on the same validation sets for both the airplane and the chair categories, but with different training data following the settings below:
- 25% or 50% data by using less instances, while keeping the same number of viewpoints;
- 25% or 50% data by reducing viewpoints per instance, while keeping the same number of instances;
- 200% data by increasing the viewpoints;

We can draw several interesting conclusions from the table below.
* **[1]** While we do expect the model performance to drop when we have smaller amount of data, the model's performance doesn't drop much when we only use 25% of the training instances for both airplane and chair categories from the ModelNet40 dataset.
* **[2]** We also found that when only using 25% of the original number of viewpoints(15 views per instance), the model is still able to estimate the 6D poses reasonably well on the evaluation set.
* **[3]** As expected, more viewpoints would help the model get a more accurate rotation estimation for partial inputs.
<center>Ablation study over different amounts of training data</center>

| Dataset | Training Data | Mean R_err | Median R_err | Mean T_err | Median T_err | 5deg. Acc. | 5deg.0.05 Acc. |
| :---------------- | :-------------- | :--------- | :----------- | :--------- | :---------------------------------- | :------------ | :---------- |
| Complete airplane | 100% instances | **14.24** | **1.53** | / | / | **0.91** | / |
| Complete airplane | 50% instances | 20.22 | 2.38 | / | / | 0.86 | / |
| Complete airplane | 25% instances | 18.38 | 2.41 | / | / | 0.88 | / |
| Partial airplane | 100% instances | **3.47** | 1.58 | **0.02** | **0.02** | **0.95** | **0.88** |
| Partial airplane | 50% instances | 4.53 | **1.34** | **0.02** | **0.02** | 0.94 | 0.87 |
| Partial airplane | 25% instances | 4.84 | 2.02 | **0.02** | **0.02** | 0.94 | 0.87 |
| Partial airplane | 200% viewpoints | **2.44** | 1.53 | **0.02** | **0.02** | **0.96** | **0.88** |
| Partial airplane | 50% viewpoints | 3.14 | **1.37** | **0.02** | **0.02** | 0.95 | **0.88** |
| Partial airplane | 25% viewpoints | 3.00 | 1.49 | **0.02** | **0.02** | **0.96** | **0.88** |
| Partial chair | 100% instances | **10.40** | **3.84** | **0.05** | **0.04** | **0.62** | **0.44** |
| Partial chair | 50% instances | 12.55 | 4.08 | 0.06 | 0.05 | 0.59 | 0.39 |
| Partial chair | 25% instances | 15.52 | 4.26 | 0.06 | 0.05 | 0.57 | 0.39 |
| Partial chair | 200% viewpoints | **11.41** | **3.85** | **0.05** | **0.04** | **0.61** | **0.41** |
| Partial chair | 50% viewpoints | 14.41 | 4.28 | 0.05 | 0.04 | 0.57 | 0.41 |
| Partial chair | 25% viewpoints | 13.99 | 4.35 | 0.06 | 0.05 | 0.56 | 0.38 |

#### Q2: why does the proposed method outperform EPN only for chair, and why does KPConv outperforms EPN only for bottles, but by a very large margin?

* **[Specialness of chair category]** On table 1 for complete shapes over different categories, EPN is supposed to achieve highly accurate results due to the effective neural network design and direct pose supervision, but our results are actually comparable to EPN regarding the median error. As pointed out, our model even outforms EPN on the complete chair category regarding the mean rotation error. We conject this is relevant to the unique structure of the chair category. Exploring the category-specific synergy between the reconstruction and the pose estimation could be an interesting future work.
* **[EPN doesn't handle well symmetric categories]** EPN is the SOTA method for 3D point clouds processing and has unique advantage on pose estimation tasks, however the weakness of EPN neural network is handling objects with symmetry due to its equivariant design. This weakness has been well explained in the original paper. Due to its own limitation, EPN doesn’t perform well on symmetric objects like bottles compared to KPConv.

#### Q3: quantitative ablations or examples on limitations

Here we add quantitative ablation showing how well the model will perform after being trained under biased viewpoints distribution. Specifically, we re-generate partial airplane dataset with viwpoints randomly sampled from a 1/4 sphere surface, which indeed hurts the quality of the estimated pose but superisingly not that much. Our method still predicts reasonable poses for those unseen viewpoints and novel instances, as shown below. We do find that the reconstructed shape is not complete for some cases, like the bottom part of the airplane is missing. We will add visualizations showing these limitations into our paper.

<center>Quantitative ablations for model limitation under bias viewpoints distribution</center>

| Dataset | view points | Mean R_err | Median R_err | Mean T_err | Median T_err | 5deg. Acc. | 5deg.0.05 Acc. |
| :---------------- | :------------ | :--------- | :----------- | :--------- | :----------- | :------------ | :---------- |
| Partial airplane | Unbiased | **3.47** | **1.58** | **0.02** | **0.02** | **0.95** | **0.88** |
| Partial airplane | Biased | 5.63 | 2.14 | 0.03 | **0.02** | 0.91 | 0.83 |

#### Q4: line 117: "only one network" or "the only network"?

EPN is the only equivariant neural network we found that could handle well input point clouds with different sampling patterns, shape variance, and different levels of partialness. While other equivariant networks like SE(3)-transformer also preserves equivariance, it doesn't learn well with variant inputs. We will update the text to make this more precise.

# Response to Reviewer VRcx
We thank reviewer VRcx for listing more related works and contributing critical thinkings to the experiments. We will add reference to the mentioned related works to give a better background.

#### Q1: can the authors analyze the impact of equivariant networks on the shape and pose separately? More ablation studies to verify the design choices.
To further analyze the impact of equivariant networks on the shape and pose separately, we add ablation studies on: 1. KPConv for pose + EPN for shape; 2. EPN for pose + KPConv for shape, as shown below. We found that equivariant network is the key to achieve good performance, especially on canonical shape estimation.

<center>Different backbones for pose and shape separately</center>

| Dataset | Shape Backbone | Pose Backbone | Mean R_err | Median R_err | Mean T_err | Median T_err | 5deg. Acc. | 5deg.0.05 Acc.|
| :---------------- | :------------- | :------------ | :--------- | :----------- | :--------- | :---------------------------------- | :------------ | :---------- |
| Complete airplane | EPN | EPN | **14.24** | **1.53** | / | / | **0.91** | / |
| Complete airplane | EPN | KPConv | 21.24 | 1.70 | / | / | 0.88 | / |
| Complete airplane | KPConv | EPN | 133.89 | 169.96 | / | / | 0.00 | / |
| Partial airplane | EPN | EPN | **3.47** | **1.58** | **0.02** | **0.02** | **0.95** | **0.88** |
| Partial airplane | EPN | KPConv | 4.24 | 1.96 | 0.03 | **0.02** | 0.93 | 0.85 |
| partial airplane | KPConv | EPN | 134.34 | 174.98 | 0.07 | 0.08 | 0.05 | 0.05 |

#### Q2: Questions on multiple hypotheses for 6D poses. Better study on the impact of the number of hypotheses (predefined groups) or directly regress poses.
* This ‘multiple pose hypothesis’ is determined by our EPN backbone, since each pose prediction is assigned to a subspace of the SO(3) space, only by using all of them we could cover the whole icosahedron rotation group and achieve rotational equivariance. The same MLP layer is shared to generate pose predictions per feature instead of using multiple heads.
* In our ablation study with SE(3)-transformer backbone(which is also SE(3) equivariant), we directly regress poses instead of using multiple hypotheses, but the results are not as good as our proposed method;
* Here we provide experiments where we set the predefined rotation group to be with 20 elements, instead of using 60. We show the results as below for both complete and partial airplane data, in which EPN60 works much better than EPN20.

| Dataset | Backbone | Mean R_err | Median R_err | Mean T_err | Median T_err | 5deg. Acc. | 5deg.0.05 Acc.|
| :---------------- | :------- | :--------- | :----------- | :--------- | :---------------------------------- | :------------ | :---------- |
| Complete airplane | EPN20 | 64.12 | 2.40 | / | / | 0.63 |/ |
| Complete airplane | EPN60 | **14.24** | **1.53** | / | / | **0.91** | / |
| Partial airplane | EPN20 | 85.53 | 5.15 | 0.03 | 0.03 | 0.50 | 0.45 |
| Partial airplane | EPN60 | **3.47** | **1.58** | **0.02** | **0.02** | **0.95** | **0.88** |

#### Q3: How stable is the training process? Result of multiple trials of experiments to show the variance of the method.

Here we show multiple runs of the same experiments on car, airplane categories for complete point clouds and partial point clouds.
* Our training process is usually stable for different categories with both complete and partial inputs, like for complete car and airplanes, also partial airplane data;
* We do observe for trainings on car category, the performance variance is much bigger than airplane category;
* Multiple runs on partial cars also show that our method has difficuly toward partial car data.

| Dataset | run_id | Mean R_err | Median R_err | Mean T_err | Median T_err | 5deg. Acc. | 5deg.0.05 Acc.|
| :---------------- | :----- | :--------------- | :------------- | :--------- | :----------------------------------------------------------- | :------------- | :---------- |
| Complete airplane | 1/2/3 | 14.24/17.47/16.23 | 1.53/1.26/1.82 | / | / | 0.91/0.89/0.91 | / |
| Complete car | 1/2/3 | 15.46/14.16/9.95 | 2.19/1.74/1.94 | / | / | 0.89/0.91/0.95 | / |
| Partial airplane | 1/2 | 3.91/3.47 | 1.75/1.58 | 0.02/0.02 | 0.02/0.02 | 0.94/0.95 | 0.86/0.88 |
| Partial car | 1/2/3 | 138.19/136.51/124.86 | 176.06/176.09/177.29 | 0.11/0.05/0.07 | 0.11/0.04/0.06 | 0.18/0.17/0.12 | 0.00/0.13/0.06 |

#### Q4: Strange visualization on partial car in figure 2.
Thanks for pointing out this, we believe is due to the poor downsampling when overlaying the points for visualization. This has been fixed in our updated visualizations.