reviews: 
Comments: This submission takes monocular dynamic NeRF one steps further and tries to discover rigid objects parts. It uses D-NeRF-style monocular RGB input and aims for general articulated scenes. To this end, it uses forward and backward deformation modeling and a cycle consistency loss between them on the feature level (doing this on the features instead of offsets is novel). Then, at test time, the forward motions are merged into an adaptively determined number of rigid groups. Experiments evaluate the reconstruction and part segmentation quality. The results are almost exclusively on quasi-multi-view/D-NeRF-style synthetic scenes.
Strengths: Extending dynamic NeRF towards part discovery is interesting, as also evidenced by a few related works in this direction. 

The proposed method is quite straightforward and thus elegant. It does not appear to have overly involved components.

Using cycle consistency on motion features rathter than the final offsets is interesting and novel.

The reconstruction quality on the D-NeRF dataset is high (other datasets are unclear). The same goes for the part discovery, which gives very accurate looking qualitative results.

A few simple downstream applications like basic editing or tracking are presented.
Weaknesses: Related Work and Method:

(A) I am not convinced that "Eulerian" and "Lagrangian" are the right terms. Standard deformable NeRFs like D-NeRF or Nerfies use backward deformation modeling. The paper proposes forward modeling instead. These terms should at the very least be mentioned explicitly somewhere. Both forward and backward modeling focus on particles and are direct functional inverses of each other. They thus resemble the Lagrangian perspective. As for example mentioned in Tretschk et al. State of the Art in Dense Monocular Non-Rigid 3D Reconstruction, I think it's more accurate to think of methods like NSFF that do not have a proper canonical model and rather directly regress the radiance field at each coordinate point in time and space as Eulerian. Such methods do not have a notion of particle, while both forward and backward deformation methods do. Or, to be more precise, they only use temporal consistency losses with auxiliary scene flow between *neighboring* frames and thus only have a notion of *instantaneous* flow to *temporally neighboring* frames at coordinate points in world space, which is exactly what Eulerian flow is. All of this is of course not ultimately relevant for the point of the paper in any way, only the presentation/writing, and I'm willing to hear why the notions proposed in the paper are more appropriate.

(B) The submission argues from the intractability of tracking a particle with backward deformation modeling. But for example Cai et al. Neural Surface Reconstruction of Dynamic Scenes with Monocular RGB-D Camera use an invertible deformation model that would allow to long-term track a particle in a straightforward manner. They lack expressivity and are computationally expensive but that's a more involved argument than the one in the submission.

(C) Liu et al. also propose cycle consistency of forward and backward modeling for dynamic NeRF in their NeurIPS 2022 paper DeVRF. It does this on the motion and not the feature level though. I think this should be discussed in the paper.

(D) It might be good to discuss how the proposed method relates to articulated NeRF methods like Wei et al. Self-supervised Neural Articulated Shape and Appearance Models. 

(E) While not necessary, please consider citing Nerfies when introducing the idea of using an SE(3) parametrization, since they were the first to do that for dynamic NeRFs.

(F) Maybe mention that category-specific methods, e.g. those focusing on the human body or face, do quite easily allow for part understanding and editing/re-posing, etc. and that the proposed method focuses on general dynamic scenes.

(G) The lower half of page 5 talks about averaging all the features in a group. I assume all these features come from all the points in the batch? But what is in the batch? An entire image? I.e. how is good, reproducible coverage ensured such that averaging in this manner always gives pretty much the same final feature for a certain group across different views?

(H) What loss function provides a gradient for D_L, the Lagrangian decoder? All losses except for the cycle consistency loss only go into the canonical model and the Eulerian part. Where is the total variation loss applied exactly?

(I) I don't understand what the group merging module is used for. My only guess is this: During training, the initial, too-large number of groups is fixed/always used. Then, after training is finished, the group merging module is used to determine the final number of groups? So the APE cost is never used for backpropagation?

(J) The Lagrangian/forward module assumes articulated rather than fully non-rigid motion due to its per-group averaging operation. Please state this explicitly.

Experiments:

(K) I am concerned that video results are only shown for the synthetic D-NeRF dataset, which is effectively a multi-view dataset (as discussed in Gao et al. Monocular Dynamic View Synthesis: A Reality Check). I'd much prefer to see the video results of the real-world monocular scenes used in the paper. And the datasets from Figures 6 and 7 (although these also use the quasi-multi-view setting). Otherwise, I will assume that reconstruction only works for quasi-multi-view synthetic scenes and not for input that is truly monocular or real.

(L) Please add video results for part discovery on multiple scenes of the HyperNeRF dataset. Currently, only a handful of still images are shown. Otherwise, I will assume that these are severely cherry-picked and part discovery does not work on real-world scenes.

(M) The same applies for tracking. Using a well-textured canonical space like here https://github.com/pablopalafox/npms for example would help in visualizing the tracking/correspondence quality.

(N) Please add LPIPS scores everywhere. It's standard for NeRF papers (see e.g. the original NeRF paper) and PSNR and SSIM are quite similar, while LPIPS is much more perceptual.

(O) There appear to be no ablations. For example, applying the cycle consistency loss on features vs. positions. Or using SE(3) vs. offsets. And loss ablations, e.g.: Why is the per-point color loss useful? What happens without the total variation loss? Also, an ablation of whether the forward model helps *in reconstructing* would be good. Is this additional component useful? If it isn't, then the proposed part discovery portion of the method (i.e. the group merging module) might be added flexibly as a post-processing step to existing backward dynamic NeRFs to discover parts, which would be interesting to know.

(P) Can Watch-It-Move, the most relevant related work, not be applied to the D-NeRF dataset? I'm curious about the parts it would discover.

(Q) I assume that part segmentation is evaluated in 2D image space rather than in 3D due issues with the unobservable insides of objects?
Recommendation: 8: accept, good paper
Reviewer confidence: 5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.
Soundness/correctness: 2 fair
Clarity: 2 fair