PhyRecon: Physically Plausible Neural Scene Reconstruction
Authors: Junfeng Ni
Yixin Chen
Bohan Jing
Nan Jiang
Bin Wang
Bo Dai
Puhao Li
Yixin Zhu
Song-Chun Zhu
Siyuan Huang
Venue: NeurIPS 2024 poster (2024)
Keywords: Multi-view Reconstruction
Neural Implicit Surface Reconstruction
Physically Plausible Reconstruction
Primary area: machine_vision
TL;DR: We propose to leverage both differentiable rendering and differentiable physics simulation for neural scene reconstruction

Abstract: We address the issue of physical implausibility in multi-view neural reconstruction. While implicit representations have gained popularity in multi-view 3D reconstruction, previous work struggles to yield physically plausible results, limiting their utility in domains requiring rigorous physical accuracy. This lack of plausibility stems from the absence of physics modeling in existing methods and their inability to recover intricate geometrical structures. In this paper, we introduce PHYRECON, the first approach to leverage both differentiable rendering and differentiable physics simulation to learn implicit surface representations. PHYRECON features a novel differentiable particle-based physical simulator built on neural implicit representations. Central to this design is an efficient transformation between SDF-based implicit representations and explicit surface points via our proposed Surface Points Marching Cubes (SP-MC), enabling differentiable learning with both rendering and physical losses. Additionally, PHYRECON models both rendering and physical uncertainty to identify and compensate for inconsistent and inaccurate monocular geometric priors. The physical uncertainty further facilitates physics-guided pixel sampling to enhance the learning of slender structures. By integrating these techniques, our model supports differentiable joint modeling of appearance, geometry, and physics. Extensive experiments demonstrate that PHYRECON significantly improves the reconstruction quality. Our results also exhibit superior physical stability in physical simulators, with at least a 40% improvement across all datasets, paving the way for future physics-based applications.

PHYRECON: Physically Plausible Neural Scene Reconstruction
Junfeng Nil.2*t, Yixin Chen?*, Bohan Jing", Nan Jiang Bin Wang?; Bo Dai", Puhao Li! Yixin Zhu?, Song-Chun Zhu SSyuan Huang?
Equal contribution Work done as an intern at BIGAI Tsinghua University 2 State Key Laboratory of General Artificial Intelligence, BIGAI PPeking University
https://phyrecon. gi thub. io
Abstract
We address the issue of physical implausibility in multi-view neural reconstruction. While implicit representations have gained popularity in multi-view 3D reconstruc- tion, previous work struggles to yield physically plausible results, limiting their utility in domains requiring rigorous physical accuracy. This lack of plausibility stems from the absence of physics modeling in existing methods and their inability recover intricate geometrical structures. In this paper. we introduce PHYRE- CON, the first approach to leverage both differentiable rendering and differentiable physics simulation learn implicit surface representations. PHYRECON features a novel differentiable particle-base sentations. Central to this design is efficient transformation between SDF-based implicit representations and explicit surface points via our proposed Surface Points Marching Cubes (SP-MC), enabling differentiable learning with both rendering and physical losses. Additionally, PHYRECON models both rendering and physical uncertainty to identify and compensate for inconsistent and inaccurate monocular geometric priors. The physical uncertainty further facilitates physics-guided pixel sampling to enhance the learning of slender structures. By integrating these tech- niques, our model supports diffcrentiable joint modeling of appearance, geometry, and physics. Extensive experiments demonstrate that PHYRECON significantly improves the reconstruction quality. Our results also exhibit superior physical stability in physical simulators, with at least a 40% improvement across all datasets, paving the way for future physics-based applications.
physical simulator built on neural implicit repre-
Introduction
3D scene reconstruction fundamental in computer vision, with applications spanning graphics, robotics, and more. Building on neural implicit representations [45], previous methods [18, 63, 77] have utilized multi-view images and monocular cues to recover fine-grained 3D geometry via volume rendering [10]. However, these approaches have overlooked physical plausibility, limiting their applicability physics-intensive tasks such as embodied AI [17, 1, 30] and robotics [15, 26, 60].
The inability to achieve physically plausible reconstruction arises primarily from the lack of physics modeling. Existing methods based neural implicit representation rely solely on rendering supervi- sion, lacking explicit incorporation of physical constraints, such as those from physical simulators. This oversight compromises their ability optimize 3D shapes for stability, as the models not informed by the physics that would ensure realistic and stable structures in the real world.
Additionally, these methods often ignore thin structures, focusing instead on optimizing the substantial parts of objects within images. This limitation due to overly-smoothed and averaged optimization
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Diftremabble Rendering
Plysic Simulation
Previous Method
In Isane Glym
Monocular Cues
t-0
t-50
Final state
PHyRECON
In Isnac Gym
Physical Simulator
Final state
[-50
Final
Figure 1: Illustration of PHYRECON. We leverage both differentiable physics simulation and differentiable rendering learn implicit surface representation. Results from previous methods [32] fail to remain stable in physical simulators or recover intricate geometries, while PHYRECON achieves significant improvements in both reconstruction quality and physical plausibility.
results from problematic geometric priors, including molli-moddl inconsistency (e.g., depth-normal), multi-view inconsistency, and inaccurate predictions deviating from the true underlying 3D geometry. As result, they struggle to capture the slender structures crucial for object stability, leading to reconstructions that lack the fine details necessary for accurate physical simulation.
To address these limitations, we present PHYRECON, the first effort to integrate differentiable render- ing and differentiable physical simulations for learning implicit surface representations. Specifically, we propose differentiable particle-based physical simulator and an efficient algorithm, Surface Points Marching Cubes (SP-MC), for differentiable transformation between SDF-based implicit representations and explicit surface points. The simulator facilitates accurate computation of 3D rigid body dynamics subjected to forces of gravity, contact, and friction, providing intricate details about the current shapes of the objects. Our differentiable pipeline efficiently implements and optimizes the implicit surface representation by coherently integrating feedback from rendering and physical losses.
Moreover, to enhance the reconstruction of intricate structures and address geometric prior inconsis- tencies, we propose a joint uncertainty model describing both rendering and physical uncertainty. The rendering uncertainty identifies and mitigates inconsistencies arising from multi-vien geometric pri- ors, while the physical uncertainty reflects the dynamic trajectory of 3D contact points the physical simulator, offering precise and interpretable monitoring of regions lacking physical support. Utilizing these uncertainties, we adaptively adjust the per-pixel depth, normal, and instance mask losses to avoid erroneous priors in surface reconstruction. Observing that intricate geometries occupying fewer pixels are less likely to be sampled, we propose physics-guides pixel sampling based on physical uncertainty to help recover slender structures.
We conduct extensive experiments on real datasets including ScanNet [9] and ScanNet++ [76], and the synthetic dataset Replica [62]. Results demonstrate that our method significantly surpasses all oatt--of-the-ars methods both reconstruction quality and physical plausibility. Through ablative studies, we highlight the effectiveness of physical loss and uncertainty modeling. Remarkably, our approach achieves significant advancements in stability, registering at least a 40% improvement across all examined datasets, as assessed using the physical simulator Isaac Gym [39].
In summary, main contributions are three-fold:
introduce the first method that seamlessly bridges neural scene reconstruction and physics simulation through a differentiable particle-based physical simulator and the proposed SP-MC that efficiently transforms implicit representations into explicit surface points. Our method enables differentiable optimization with both rendering and physical losses.

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2. We propose a novel method that jointly models rendering and physical uncertainties for 3D reconstruction. By dynamically adjusting the per-pixel rendering loss and sryssiss-guidee pixel sampling, our model significantly improves the reconstruction of thin structures.
3. Extensive experiments demonstrate that our model significantly enhances reconstruction quality and physical plausibility, outperforming state-of-the-ar methods. Our results exhibit substantial stability improvements, signaling broader potential for physics-demanding applications.
Related Work
With the increasing popularity of neural implicit rep- resentations [48, 79, 35, 6, 49, 704 in 3D reconstruction, recent studies [50, 64, 75] have bridged the volume density in Neural Radiance Fields (NeRF) [45, 80] with the iso-surface representation, egg. occupancy [42] or signed distance function (SDF) [52] to enable reconstruction from 2D images. Furthermore, advanced methods [68, 29, 69, 32] achieve compositional scene reconstruction by decomposing the latent 3D representation into the background and foreground objects. Despite achieving plausible object disentanglement, these methods fail to yield physically plausible recon- struction results, primarily due to the absence of physics constraints in existing neural implicit reconstruction pipelines. This paper addresses this limitation by incorporating both appearance and physical cues, thereby enabling surface learning with both rendering and physical losses.
Neural Implicit Surface Reconstruction
Incorporating Priors into Neural Surface Reconstruction Various priors, such Manhattan- world assumptions [20, 18), monocular geometric priors (i.e,, normal [63] and depth [77] from off-the- shelf model [55, 12, 2D) and diffusion priors [70, 34], have been employed to improve optimization and robustness in surface reconstruction, especially in texture-less regions of indoor scenes. However, these priors primarily involve geometry and appearance, neglecting physics priors despite their critical role in assessing object shapes and achieving stability. This paper proposes to incorporate physical priors through a differentiable simulator, with physical loss to penalize object movement and joint uncertainty modeling of both rendering and physics. Rendering uncertainty filters multi-view inconsistencies in the monocular geometric priors, akin to prior work [41, 73, 51, 71]. The physical uncertainty employs the 3D contact points from the simulator offer accurate insights into areas lacking physical support, which helps modulate the rendering losses and guides pixel sampling.
Physics in Scene Understanding There has been an increasing interest in incorporating physics commonsense in the scene understanding community, spanning various topies such as object gen- eration [47, 14, 61, 44, 43], object decomposition [36, 82, 5], material simulation [23, 31, 72, 81], scene synthesis [54, 74], and human motion generation [57, 59, 22, 78, 66, 21, 4, 8, 25, 65, 24J. In the context of scene reconstruction, previous work primarily focuses applying collision loss 3D bounding boxes to adjust their translations and rotations [11, 31, and penalize the penetration between objects [79, 69]. Ma et al. [38] controls objects within a differentiable physical environment and integrates differentiable rendering to improve the generalizability of object control in unknown rendering configurations. In this paper, propose to incorporate physics information into the neural implicit surface representation through a differentiable simulator, significantly enhancing both the richness of the physical information and the granularity of the optimization. The simulator not only furnishes detailed object trajectories but also yields detailed information about the object shapes in the form of 3D contact points. With physical loss directly backpropagated to the implicit shape representation, our approach leads to refined optimization of object shapes and physical stability.
3 Method
Given an input set of N posed RGB images I (!,...,
.a....,,y |N} and corresponding instance masks = {Si.... Sy}. our objective is to reconstruct each object and the background in the scene. Fig. 2 presents overview of our proposed PHYRECON.
3.1 Background
Volume Rendering of SDF-based Implicit Surfaces We utilize neural implicit surfaces with SDF to represent 3D geometry for implicit reconstruction. The SDF provides a continuous function that yields the distance s(p) to the closest surface for given point p, with the sign indicating whether
3

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Rendering
Le
Depth Nommal uncertainty map
Uncertrinty
Rendering
Image
Aljusted rendering
Image
p.
Mel dp
loss
Object SDDs
Rendering
Lig
Ex
SP-MC
Differentable plasics simulation
Interpciation
$1
SP-MC:
Depth, Nornal
Instance mask
Detoch Physical uncertainty mup
Cp
& lastanca mask
Depth. Normal price
Eupply
Gruphy
Mphy
Rendering
Guided sampling
m
Figure 2: Overview of PHY RECON. We incorporate explicit physical constraints in the neural scene reconstruction through differentiable particle-based physical simulator and differentiable transformation (i.e., SP-MC) between implicit surfaces and explicit surface points in Sec. 3.2. To learn intricate 3D structures, we introduce rendering and physical uncertainty Sec. address the inconsistencies in the geometric priors and guide the pixel sampling.
the point lies inside (negative) or outside (positive) the surface. We implement the SDF function as multi-layer perceptron (MIP) network f((), and similarly, the appearance function as g(-).
For each camera ray r (o, v) with o as the ray origin and v as the viewing direction, n points (p: -0+ 4 0, 1, ,, 1} are sampled, where 4 is the distance from the sample point to the camera center. We predict the signed distance s(p,) and the color C; for each point along the ray, and the normal n. is the analytical gradient of the sp.). The predicted color Õ(r), depth D(r), and normal (r) for the ray computed with the unbiased rendering method following NeuS [64]]
C(r)= = ETaa D(r) = - Ratt N(r)
Σ Ta
(1)
200
2
where T; is the discrete accumulated transmittance and a is the discrete opacity value, defined as:
max (a(s(p:)) P (n(s(pi+1))
T
II ==
az),
(s(())))
where (x( (1 + "=) and u is learnable parameter.
Object-compositional Scene Reconstruction Following previous work [68, 69, 32], consider the compositional reconstruction of objects utilizing their corresponding masks, and we treat the background object. More specifically, for scene with i objects, we predict SDFs each point p. and the j-th (1 < j < k) SDF represents the geometry of the j-th object. Without loss of generality, set j
the background object and others the foreground objects. In subsequent sections of the paper, we denote the j-th object SDF at point p as sj (p). The scene SDF s(p) is the minimum of the object SDFs, i.e., s(p) min 8y (p),.j 1, ...... which is used for sampling points along the ray and volume rendering in Eqs. (1) and (2). See more details in the Appx. A.
3.2 Differentiable Physics Simulation in Neural Scene Reconstruction
To jointly optimize the neural implicit representations with rendering and physical losses, propose a particle-based physical simulator and highly efficient method to transition implicit SDF, which is adept modeling visual appearances, explicit representations conducive physics simulation.
Surface Points Marching Cubes (SP-MC) Existing methods [13, 43] fall short of attaining satisfactory balance between efficiency and precision in extracting surface points from the SDF-based implicit representation. Thus, we develop novel algorithm, SP-MC, with differentiable and efficient parallel operations and surface point refinement leveraging the SDF network f-)).

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1.2
Las
x
-0.3
F2
0.5
<5
-22
-08
<0
5
08
0.6
45
0.48
soliit
Original SDF grid
Shifted SDF grid (slong x asis)
(c) crossing localization
Inserpolation
(e) Retinement
Figure 3: Illustration of SP-MC. (a-b) We first shift the SDF grids, and (c) localize the zero-crossing vertices V (blue). (d) The coarse surface points Posase (black) are derived through linear interpolation and (e) the fine-grained points Pine (purple) are obtained by querying the SDF network f(-)-
Fig. 3 illustrates the SP-MC algorithm. Formally, S E RNNNNN denote a signed distance grid of size N obtained using f()), where S(p) E R denotes the signed distance of the vertex p to its closest surface. SP-MC first locates the zero-crossing vertices V. where a sign change occurs from the neighboring vertices in signed distances, by shifting the SDF grid along the y. and a axis, respectively. For example, shifting along the æ axis means (i. k) = S(i + 1, 1,j.k) k). Then each vertex p É V can be found through the element-wis multiplication of S and the shifted grids {dE( together with its sign-flipping neighbor vertex denoted p The coarse surface points Powarse are determined along the grid edge via linear interpolation:
P Powaree
={
-(p+
S(p) S(p) - hiin
(p dhil p) p)IPEv].
(3)
Finally, Powarse are refined using their surface normals and signed distances by querying f()):
Punc 11p-fo f(p) f p Pocarse)
Benefiting from all operations that are friendly for parallel computation, SP-MC capable of extract- ing object surface points faster than the Kaolin [13] algorithm with less GPU memory consumption. In the meantime, it also achieves unbiased estimation of the surface points compared with direct thresholding the signed distance field [43]. which is crucial for learning fine structures. For detailed algorithm and quantitative computational comparisons, please refer to the Appx. B.
Particle-based Physical Simulator To incorporate physics constraints into the neural implicit reconstruction, we develop fully differentiable particle-based simulator implemented with Diff- Taichi [19]. Our simulator is designed for realistic simulations between rigid bodies, which are depicted as a collection of spherical particles of uniform size. The simulator captures the body's dynamics through translation and rotation, anchored to a reference coordinate system. The body state at any time is given by its position, orientation, linear and angular velocities. The total mass and the moment of inertia are intrinsic physical properties of the rigid body, reflecting its resistance to external forces and torques. During forward dynamics, the states are updated through time using explicit Fuler time integration. Besides gravity, the changes in the rigid body state induced by contact and friction forces, which are resolved using an impulse-bant based method. For more implementation details, please refer to the Appx. C.
The simulator is used to track the trajectory and contact state of the object's surface points Pine until reaching stability or maximum simulation steps. Utilizing the trajectories and contact states, we present our simple yet effective physical loss: Eyhn = Epe EpePssu Pauus ||p' p"112 where the initial position of each point p is denoted as pd and its first contact position with the supporting plane as p'. Our physical loss intuitively penalizes the object's 'ss movement and is only applied to the contact points P Poomact instead of all the surface points. If the physical loss is homogeneously applied to all the surface points, will contradict the rendering cues when the object is unstable, leading to degenerated results. The contact points P Perntact. on the other hand, indicate the areas of object instability. As shown in the experiment section, our design leads to stable 3D geometry under the coordination between the appearance and physics constraints.
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3.3 Joint Uncertainty Modeling
Apart from incorporating explicit physical constraints, we propose a joint uncertainty modeling approach, encompassing both rendering and physical uncertainty, to mitigate the inconsistencies and improve the reconstruction of thin structures, which is crucial for object stability.
Rendering Uncertainty The rendering uncertainty is designed to address the issue of multi-view and multi-modal inconsistencies in monocular geometry priors, including depth uncertainty and normal uncertainty. We model the depth uncertainty us and normal uncertainty un of a 3D point as view-dependent representations [51, 58, 71], which we utilize the appearance network g(-) predict along with the color e for a 3D point p:
g (p E R², E R v E R f E 2 + (c E R², E R, Un E R),
(5)
where n is the normal at p. v is the viewing direction, and f is geometry latent feature output from the SDF network f(().
The depth uncertainty Ua(r) and normal uncertainty (r) of ray r are computed using da and un through volume rendering, similar to Eq. (1). Subsequently, we modulate the L2 depth loss (D(r) and L1 normal loss (r) based on the rendering uncertainty:
Cp(r) Lo (r) = In( In((a(rr) +1)+ [(a(T))'
CNo (r) = In( + +1+ Ex(r) (r,
(6)
Intuitively, the uncertainty-aware rendering loss can assign higher uncertainty to the pixels with larger loss, thus filtering the inconsistent monocular priors when the prediction from one viewpoint differs from others. Fig. 6 depicts how higher uncertainty precisely localizes inconsistent monocular priors.
Physical Uncertainty We introduce physical uncertainty to capture object instability from physics simulations. Specifically, represent the physical uncertainty field with dense grid Gu-phy N The physical uncertainty Uphy (p) RR for any 3D point p is obtained through trilinear interpolation. Gu.phy is initialized to zero and updated after each simulation trial using the 3D contact points P Poontact. For each contact point, we track its trajectory from pe to p', forming the uncertain point set P... We design loss function Lu-phy to update Gu-phyi
Supphy - Σ Uphy(P))
PEP.
where & >> O represents the increasing rate for Gu-phy- Thus, areas with high physical uncertainty indicate that the object lacks support and requires the development of supporting structures. The physical uncertainty (r) of the pixel corresponding to ray computed via volume rendering.
Finally, we present the rendering losses re-weighted by both rendering and physical uncertainty:
(r)
LNu(r)
Ls(r)
LD Σ PER (phy(r)+11
LN Σ PER (p)g(r)+11 +1 Ls -s=- E PER Uphy(r)+1
(8)
The segmentation loss Cs(r) is weighted by physical uncertainty only since we use view-consistent instance masks following prior work [68, 69, 32]. We detach Uphy (r) in the above losses, ensuring the physical uncertainty field Gu-phy reflects accurate physical information from the simulator.
Physics-Guider Guided Pixel Sampling A critical observation that intricate object structures occupy only a minor portion of the image, leading to lower probability of being sampled despite their significance in preserving object stability. Effectively pinpointing these fine geometries is perceptually challenging, yet physical uncertainty precisely outlines the areas where the object lacks support and requires special attention for optimization. Hence, we introduce physics- guided pixel sampling strategy to enhance the learning of detailed 3D structures. Using physical uncertainty, we calculate Uphy(r)
the sampling probability of each pixel in the entire image by: p(r)
ΣPER Uphy(()
3.4 Training Details
In summary, our overall loss function is: C LNGB + LD + LN + Lss + f.ph + fow + Leres where the loss weights are omitted for simplicity. Following prior work [16, 77, 32], we introduce

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NeanNet++
ScanNet
Replica
MonoSDF
RICO
ObjectSDFF+
Figure 4: Qualitative results of indoor scene reconstruction. Examples from ScanNet++ [76].
ScanNet [9], and Replica [62] demonstrate that our model produces higher quality reconstructions compared to the baselines. Our results contain finer details for slender structures (e.g.. chair legs and objects on the table) and plausible support relations, which are highlighted in the zoom-in boxes.
Creg regularize the unobservable regions for background and foreground objects, as well as the implicit shapes through an eikonal term.
To ensure robust optimization and coordination among various components, empirically divide the training into three stages. the first stage, we exclusively leverage the rendering losses with rendering uncertainties. In the second stage, introduce physical uncertainty into rendering losses and enable physics-guided pixel sampling. Finally, we integrate the physical loss using learning curriculum that gradually increases the physical loss weight. The staged training allows the simulator to pinpoint more accurate contact points, essential for effective optimization of shape through the physical loss. For further details about the loss, training, and implementation details, please refer to Appx. A.
physical simulator to incorporate
Experiments
We assess the effectiveness of PHYRECON by evaluating scene and object reconstruction quality, as well as object stability. We present additional results in Appx. D and limitations in Appx. E.
4.1 Settings
Datasets We conduct experiments both the synthetic dataset Replica [62] and real datasets Scan- Net [9] and ScanNet++ [76]. We use scenes from Replica following the setting MonoSDF [77] and ObjectSDF++ [69]. For ScanNet, use the 7 scenes following RICO [32]. Additionally, con- duct experiments on scenes from the more recent real-world dataset ScanNet++ with higher-quality images and more accurate camera poses. More data preparation details in Appx. D.I.
Baselines and Metrics We choose MonoSDF [77], RICO [32], and ObjectSDF++ [69] as the baseline models. For reconstruction, measure the Chamfer Distance (CD), the F-score (F-score),

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Initial state
10 sleps
sleps
Final state
RICO
ObjectSDI
Ours
Figure 5: Object trajectory during simulation. Our method enhances the physical plausibility of the reconstruction results, which can remain stable during dropping simulation in Isaac Gym.
Table 1:
Quantitative results of 3D reconstruction. Our model reaches the best reconstruction quality across all datasets and significantly improves physical stability.
Scene Recon.
Obj. Recon.
Obj. Stability
Dalaset
Method
CDL
F-Scoref
NC1
CD1
F-scoret
NC1
SR (%) 1
MonoSDF
3,94
78.14
89.37
RICO
3.87
78.45
89.64
4.29
85.91
85.45
26.43
ScanNet++
ObjectSDF++
3.79
79.12
89.57
4.08
86.32
85.32
25.28
Ours
3.34
81.53
90.10
3.28
87.21
86.16
78.16
MonoSDF
8.97
60.30
84.40
RICO
8.92
61.44
84.58
9.29
73.10
79.44
29.68
ScanNet
ObjectSDF++
8.86
61.68
85.20
9.21
74.82
81.05
26.56
Ours
8.34
63.01
86.57
7.92
75.54
82.54
70.31
MonoSDF
3.87
85.01
88.59
RICO
3.86
84.66
88.68
4.16
80.38
84,30
32.89
Replica
ObjectSDF++
3.73
85,50
88.60
3,99
80.71
84.22
30.26
Ours
3.68
85.61
89.45
3.86
81.30
84.91
77.63
and Normal Consistency (NC) following MonoSDF [77]. These metrics are evaluated for both scene and object reconstruction across all datasets. As MonoSDF is designed to reconstruct the whole scene, only evaluate its scene reconstruction quality. To evaluate the physical plausibility of the reconstructed objects, we report the Stability Ratio (SR). defined as the ratio of the number of physically stable objects against the total number of objects in the scene. The assessment conducted using dropping simulation via the Isaac Gym simulator [39] to avoid bias in favor of our method. We provide more details on our evaluation procedure Appx. D.2.
4.2 Results
Tab. and Fig. 4 present the quantitative and qualitative results of the neural scene reconstruction. In Fig. 5, we visualize the trajectory for the reconstructed object during dropping simulation Isaac Gym [39]. Fig. showcases visualizations of several critical components in methods, including the sampling strategy. volume renderings of the geometric cues, and the uncertainty maps. Our method significantly outperforms all baselines, and we summarize the key observations as follows.
Physical Stability From the results in Tab. 1, our method realizes significant improvements object
stability, outperforming all basclines by at least 40% across all datasets. This signifies significant

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Imaaee
Handom sampling
Depth uncertinty
Depih peice
Neemal prior
GI Instance mask
Gf mesh
uncertaioty mup
piucl sampling
unceraiity map
depth
Reconstrcted
Figure 6: Joint uncertainty modeling, The physical uncertainty pinpoints the regions critical for stability, efficiently guiding the pixel sampling. The rendering uncertainties can alleviate the impact of inconsistent geometry cues, leading to better-reconstrruced mesh than the GT.
stride towards achieving physically plausible scene reconstruction from multi-view images. The notable stability improvement is attributed to two essential factors: 1) the integration of physical loss and physical uncertainty ensures the reconstructed structure closely aligns with the ground floor, as evident in the examples from Figs. and 5 and 2) enhanced learning of intricate structures, facilitated by joint uncertainty modeling and physics-guided pixel sampling. Results from RICO [32] and ObjectSDF++ [69] in Fig. fail capture all the chair legs, leading to inherent instability.
Reconstruction Quality Note that the improvement under the sacrifice of reconstruction quality. From the results in Tab. 1, our method surpasses all the state-of-the-ar methods across three datasets in all the reconstruction metrics. As also shown in Fig. 4, while the baseline methods are capable of reconstructing substantial parts of objects, e.g., sofa or tabletop, they struggle with the intricate structures of objects. In contrast, our model achieves much more detailed reconstruction, e.g., the vase and lamp the tables, shown in the zoom-in views.
the physical plausibility does not come
Joint Uncertainty Modeling Lastly, we discuss the intermediate results of our joint uncertainty modeling exemplified on the ScanNet dataset. As shown in Fig. 6, the physical uncertainty adeptly pinpoints the regions critical for remaining stability, such as the chair legs and table base. The physics- guided pixel sampling. informed by the physical uncertainty map, prioritizes intricate structures over the random sampling strategy. Moreover, modulates rendering losses, particularly useful for instance mask loss where chair legs are absent in the ground-truth instance mask. Meanwhile, the depth and normal uncertainty maps identify inconsistencies the geometric prior, e.g. miss detections in the normal prior and the overly sharp depth prior. Collectively, they contribute to the physically plausible and detailed reconstructed mesh, surpassing the ground truth.
Table 2: Ablation results on ScanNet++ dataset.
Scene Recon.
Obj. Recon.
Obj. Stability
RU
PU
PS
PL
CDJ,
F-Score
NC1
CDJ
F-scoret
NC1
SR (%)
X
X
3.82
78.64
89.52
4.13
86.26
85,42
25.28
X
3,68
79.35
89.55
3.97
86.35
85.26
68.96
3.46
80.73
89.63
3.65
86.83
85.53
47.12
X
3.42
80.68
89.67
3.46
86.97
85.45
56.32
3.31
81.64
89.94
3.34
87.17
85.47
60.91
V
V
V
3.34
81.53
90.10
3.28
87.21
86.16
78.16
4.3 Ablation Study
We conduct ablative studies on the physical loss (PL), rendering uncertainty (RU), physical uncer- tainty (PU), and physies-guided pixel sampling (PS). Tab. and Fig. illustrate quantitative and qualitative comparisons, respectively. Key findings are as follows:

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Image
(a) Baseline
(b) +PL
+RU
(4) -RU-PU
(e) -RUHPU-PS
(f) Ours
Figure 7: Visual comparisons for ablation study. PL denotes physical loss, RV for rendering uncertainty, PU for physical uncertainty and PS for physics-guide sampling.
Image
View
Novel View
View
Novel View
Figure 8: Qualitative examples for failure cases.
1. Physical loss significantly improves 3D object stability. However, due to the insufficient regular- ization of thin structures and imprecise contact points in the simulation, adding physical loss alone will overshadow the rendering losses, leading to degenerated object shapes to sustain stability. Rendering uncertainty, physical uncertainty, and physic-guided pixel sampling collectively con- tribute to enhancing the reconstruction quality, partieularly thin structures. However, despite the advancements, the reconstructed results still struggle to maintain physical plausibility without direct physical supervision during the optimization.
3. When all components are introduced, the enhanced reconstruction of thin structures leads to more meaningful contact points in simulation, thus enabling effective joint optimization with physical loss. This not only improves the objects' stability but also preserves their reasonable shapes, leading to robust reconstruction with both physical plausibility and fine details simultaneously.
4.4 Failure Cases
We present and diagnose representative failure examples. Fig. 8 (a) demonstrates that in regions scarcely observed in the input images, optimizing with the physical loss may lead degenerated object shapes, e.g., bulges, to maintain physical stability. This is due to insufficient supervision from the rendering losses. Fig. (b) illustrates that objects may divided into several disconnected parts, which also appear stable in the simulation. This is a common limitation of the current neural reconstruction pipeline and may addressed by incorporating topological regularization to penalize disconnected parts of the object further enhance the physical simulator.
Conclusion
In conclusion, this paper introduces PHYRECON as the first approach to leverage both differentiable rendering and differentiable physics simulation for learning implicit surface representations. Our framework features a novel differentiable particle-basee physical simulator and joint uncertainty modeling, facilltatiigg efficient optimization with both rendering and physical losses. Extensive experiments validate the effectiveness of PHYRECON, showcasing its significant outperformance of all state-of-the-an methods terms of both reconstruction quality and physics stability, underscoring its potential for future physics-demandee applications.
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Acknowledgement
The authors thank the anonymous reviewers for their constructive feedback. Y. Zhu is supported in part by the National Science and Technology Major Project (2022ZD01 14900), the National Natural Science Foundation of China (62376031), the Beijing Nova Program, the State Key Lab of General Al at Peking University, and the PKU-BingJi Joint Laboratory for Artificial Intelligence.
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Appendix
In Appx. A, first delineate the comprehensive PhyRecon algorithm, followed by providing details about the loss functions and implementation. Subsequently, we delve into the SP-MC Algorithm in Appx. B, offering detailed comparison with other methods for extracting surface points. Next, we describe our differentiable particle-based rigid body simulator in Appx. C, covering both its theoretical derivation and concrete implementation. We also provide additional experimental details, including data preparation and evaluation metrics in Appx. D. Finally, we discuss the limitations in Appx. E, along with the potential negative impacts of our research in Appx. F. For comprehensive understanding of the qualitative results and stability comparisons, we recommend viewing the supplementary video with detailed visualizations and animations.
A Model Details
A.1 PHYRECON Overview
In Alg. I, we illustrate the process of one training iteration of PHYRECON. Below, we provide further explanation.
First, we sample pixels from an image guided by physical uncertainty to enhance the sampling probability of slender structures of objects. Then, we sample points along a ray according to NeuS [64]. For each point, we predict individual object SDFs (8.,,,...,, 8k), color c, depth uncertainty ta- and normal uncertainty un using f()) and g()), and obtain physical uncertainty from the physical uncertainty grid Gu-phy through trilinear interpolation. Subsequently, we compute color Ò, depth D, normal N, instance logits H, depth uncertainty Ua. normal uncertainty Ün. and physical uncertainty Uphy for each ray via volume rendering [10]. In particular, the physical uncertainty of the pixel corresponding to ray r computed via:
(p) = Σ Tpgg.p
(A1)
Note that the loop for each ray and each point here is parallelized during actual computation.
For each foreground object, we extract its surface points pobj and background surface points pog using SP-MC. Utilizing our proposed physical simulator, we track the trajectories and contact states of the object points. For each point p E pob,, we identify its initial position as pd and its first contact position with the supporting plane p'. The object points that have made contact with the supporting plane are collectively denoted as Poomaec- If a surface point p does not contact the supporting plane, p' remains equal to p". The initial and contact positions of the contact points are used to compute the physical loss Lphy, which penalizes object movement in the simulator.
To update the physical uncertainty grid Gu-phy, we construct the uncertain point set P, by interpo- lating from the initial position p" to the first contact position p' for all the points in Peomaer- The physical uncertainty grid is optimized through Lu-phy- Note we only record the essential information and do not output the full trajectory of all object points during simulations to reduce CUDA memory usage.
Finally, we compute the depth, normal, and instance semantic loss, re-weighted by both rendering and physical uncertainty using Eq. (15) from the main paper. Additionally, calculate the color loss and regularization loss terms. Finally, we optimize the network f()), g(-), and the physical uncertainty grid Ga-pky using the aforementioned losses.
A.2 Loss Function Details
RGB Reconstruction Loss To learn the surface from images input, we need minimize the difference between ground-truth pixel color and the rendered color. We follow the previous work [77] here for the RGB reconstruction loss:
CROB Σ [(((r) -
(A2)
PER
16

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Algorithm PhyRecon per Training Iteration
Physics-Guided Pixel Sampling - {r}
Guided by physical uncertainty
2: For each r do
3: NeuS ray sampling + (p)
4: For each p do
5:
// SDF prediction
6:
81; $2;...; 8k, / (p)
7:
8 min(81, 82; ..... 85)
8:
0 (s()
# Φ: transform SDF to density
9:
I/ Appearance and rendering uncertainty prediction
10:
Ud, Un g(p, n, x)
11:
// Physical uncertainty via trilinear interpolation
12:
Uphy Interp( (eeterpGuuhgg-.. Gu-phy. p)
13:
end for
14:
C.D,N. H. Ua Un, Uphy VolumeRendering(?
15:
end for
16:
For
j=2,...,k do
17:
Background surface points extraction
18:
pbg = SP-MC(s1)
19:
// Object surface points extraction
20:
pobj SP-MC(s;)
21:
I/ Differentiable Physics Simulation
{pº: P'',,Pooutaatt Poontact Simulator pphii Pha)
# p": initial position, p': first contact position
23:
11 Physical uncertainty points extraction
Pu E EpePrnnc Prennc Interp(p .p')
Physics-related losses
26:
La-phy = -6> PEP,, Uphy (p)
27:
Lphy = EppePom Ln(p", .p)
28: end for
29: // Rendering losses modulated by Üd, Ün, Uphy
30: Compute Le, La. Lns Lgy Lreg
31: Optimize network f()).(+) and physical uncertainty grid Gu-phy
where (r) is the rendered color from volume rendering and C(r) denotes the ground truth.
Depth Consistency Loss Monocular depth and normal cues [77] can greatly benefit indoor scene reconstruction. For the depth consistency, we minimize the difference between rendered depth Drr) and the depth estimation D(r) from the Marigold model [27]]
LD
Σ (ww + + 13 PER
(A3)
where and are the scale and shift values match the different scales. We solve and q with least-squares criterion, which has the closed-form solution. Please refer to the supplementary material of [77] for detailed computation process.
Normal Consistency Loss We utilize the normal cues (r)) from Omnidata model [12] to su- pervise the rendered normal through the normal consistency loss, which comprises LI and angular losses:
LN
Σ (1 - N +||1 - -r) (())ll
(A4)
PER
The volume-rendered normal and normal estimation will be transformed into the same coordinate system by the camera pose.
Semantie Loss
Building on previous work [68, 32], we transform each point's SDFs into instance logits h(p) = [hi (p), h2 (p),.... le (p)], where
hy(p) = 7/(1 + +exp(r xy())))
(A5)
17

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Here, is a fixed parameter. Subsequently, we can obtain the instance logits H(r) IR of pixel corresponds to the ray using volume rendering as:
H(r)
Σ Nash
(A6)
We minimize the semantic loss between volume-rendered semantic logits of each pixel and the ground-truth pixel semantic class, The semantic objective implemented as a cross-entropy loss:
Ls = Σ (rg) (r)) PER f=1
(A7)
The (r) is the ground-truth semantic probability for j-th object, which is 1 or 0.
Eikonal Loss Following common practice, we also add an Eikonal term on the sampled points to regularize the SDF learning by:
Likonal ΣOI
min 1SiSk
sy(pi)l2 1).
(A8)
The Eikonal loss is applied to the gradient of the scene SDE, which is the minimum of all the SDFs.
Background Smoothness Loss Building upon RICO [32], we use background smoothness loss regularize the geometry of the occluded background to be smooth. Specifically, we randomly sample a P x P size patch every Tp iterations within the given image and compute semantic map Hr)) and patch mask M (r):
M(r [arg max H (r)) 11,
(A9)
wherein the mask value is 1 if the rendered class is not the background, thereby ensuring only the occluded background is regulated. Subsequently, we calculate the background depth map D(r) and background normal map N (r) using the background SDF exclusively. The patch-based background smoothness loss is then computed as:
P-1-2 1-2"d 2d
C(D)
Σ
((m,,)) O ((D(rma))) - (m))) (D(mm.)) Dr
d=u m,n=0
P-1-2"
((Ñ) Σ
Σ
(rm,,)) (IN(mm,,))) Ñ (r + (r Ñ (F
d=0 m,n=0
(A10)
Lis = L(D)+
(A11)
Object Point-SDF Loss and Reversed Depth Loss To regularize SDFs of the object and the background, we further employ object point-SDF loss regulate objects within the room Cop and a reversed depth loss Lra. following previous work [32].
Specifically, for the sampled points along the rays, we initially apply root-finding algorithm among the background SDF of these points to determine the ZDF ray depth t'. Then, the object point-SDF loss can be expressed as:
Lop
Σ max (0, (0,c-s((p(t.)) 1[t, t',
(A12)
k
which pushes the objects' SDFs points behind the surface be greater than positive threshold e.
Moreover, Lyd optimizes the entire ray's SDF distribution rather than focusing solely on discrete points. Specifically. the ray depths {tali 0, 0,1...., 1} are transformed into the reversed ray
depths, denoted as {E = 0, I,..., 1}. where
f (to + tn-1) tn-1-b-
(A13)
18

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The reversed depth do of the hitting object is determined by the pixel's rendered semantic and de of the background. The reversed depth loss computed as:
Lnd max (0, dg - do).
(A14)
Finally, the regularization loss in our main paper Creg is computed follows:
Creg Lbs + Lop + Lrd + L Eikonal
(A15)
A.3 Implementation Details
We implement our model in PyTorch [53] and utilize the Adam optimizer [28] with an initial learning rate of 5e 4. We sample 1024 rays per iteration. When incorporating shysics-guuied pixel sampling. we allocate 768 rays for physics-guided pixel sampling and the remaining 256 rays for random sampling. Our model is trained for 450 epochs on ScanNet [9] and ScanNet++ [76] datasets, and 2000 epochs on Replica [62] dataset. As introduced in Sec. 3.4, training is divided into three stages. For the ScanNet [9] and ScanNet++ [76] datasets, the second and final stages begin at the 360 and 430th epochs, respectively, while for the Replica [62] dataset, these stages start at 1700 and 1980 GPU.
epochs. All experiments are conducted on a single NVIDIA
Following previous work [77, 32], we set 1, 0.1, 0.05, 0.04, 0.05, 0.1, 0.1 and 0.1 as loss weights for LRGB, LD. LN. Ls. LEskonal, Lbs. Cop. Cra. respectively. Additionally, we set ç 100 for updating Gu-phy, initialize the loss weight for Lphy as 60, and increase it by 30 per epoch.
B Surface Points Marching Cubes (SP-MC)
Integrating a physical simulator into the learning of the SDF-based implicit representation demands highly efficient and accurate extraction of surface points. The SP-MC algorithm is inspired by the marching cube algorithm [37] which estimates the topology (i.e., the vertices and connectivity of triangles) in each cell of the volumetric grid. Since surface points are only required for simulation, we improve the operation efficiency and combine the implicit SDF network f(()) to create fine-grained surface points.
B.1 SP-MC Algorithm Details
The Surface Points Marching Cubes (SP-MC) is divided into three steps for extracting surface points from SDF-based implicit surface representation. First, the object is voxelized obtain a discretized signed distance field S E RN*N X with grid vertices denoted as P. Second, we shift the SDF grid S along the z, y. and axis, respectively, the x axis results in ( j, k) Sii 1, J. k). We then obtain coarse surface points P Perarse through linear interpolation. We refer this step as the Ihill-interpolation operation Alg. 2. Third, Penarse is refined to yield Pine using their surface normals and signed distances. For detailed algorithm, please refer to Alg. 2.
locate zero-crossing vertices. For example, shifting along
In this algorithm, the background's boundary remains the boundary of the entire scene. For the foreground object, its boundary starts as the boundary of the entire scene and is expanded by 0.1 around its current surface points' boundary after each SP-MC iteration. The updated boundary is more accurate than the entire scene, leading to improved precision in SP-MC.
Finally, we emphasize the importance of the refinement step, which not only enhances the accuracy of Powuse: but also ensures reliable backpropagation of the gradient in the physical loss from explicit physical simulator to the implicit SDF-based surface representation. Directly converting the SDF into triangle mesh, as done in Marching Cubes [37], does not support stable and effective backpropagation in case of topological change, as observed in the works by Liao et al. [33] and Remelli al. [56]. Thus, we detach the gradient in the steps involving the extraction of coarse surface points and rely on the refinement step for backpropagation.
B.2 Comparison with Other Methods for Extracting Surface Points
To quantitatively compare with existing methods in efficiency, we assess SP-MC alongside Kaolin [13], which converts the SDF field into a triangle mesh and surface points. The assess- ment of time and memory involves the transformation from SDF to surface points, which are prepared
19

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Algorithm 2 Surface Points Marching Cubes (SP-MC) Algorithm
Input: SDF network f(-), Resolution N, Object Boundary B, Object Index j
Output: Surface Points Pine
function SNTEF--NTTROLLAECC.. P. shift axis v) // Grid shifting
S <S S shift along
M <- (-( (S S <
# o: Hadamard product, M: index mask
V <- -P
|/ Linear Interpolation
P Powuree
p S(p)-Sip (p - -p) (oleev]
S(p)
# p #pain: sign-flipping neighhor of
return P Peoarse
end function
1: Get S E NxNNN and P by Voxelization:
2:
P +- voxelize object boundary B in resolution
3:
S (D)
P}
4:
Get Powarse by Shift-Interpolate Operation:
5:
Peoarse [SHIET-INTERRPLLATEE P,v) 1 U {x, v22
6:
Get Pine by Refinement Step:
7: Pine = (p- f(p) Vf(p) P Ponars))
8:
return Piie
and ready for use in both simulation and the computational graph for gradient backpropagation in both methods. We compared Kaolin and SP-MC at a grid resolution of 96 on a single A100 80GB machine, testing the average running time and GPU memory for all objects in all scenes of ScanNet++ [76]. From the results presented Tab. A1, SP-MC consumes considerably less running time and GPU memory compared Kaolin. The performance improvement of SP-MC primarily stems from its direct pursuit of surface points through simplified operations like grid-shifting and optimized parallel computation. This eliminates the need for face search in Kaolin's marching cubes, which consumes significant time and GPU memory.
Table A1: Quantitative comparison between SP-MC and Kaolin. SP-MC consumes less running time and GPU memory compared to Kaolin.
Running Time (s) 4
GPU Memory (MB) |
Kaolin [13] SP-MC
0.482
21.327
0.264
13.673
A
0.218 (45.2%)
7.654 (35.9%)
Furthermore, we also that Mezghanni al. [43] propose method for extracting surface points through direct thresholding of the SDF in the discretized signed distance field S, defined as:
Poearse {pl |s(p) 8. S}.
(A16)
Although this method is conceptually simpler, it introduces surface bias 6, leading to inevitable discrepancy between the extracted points and the actual surface points. It additionally requires higher resolution of the SDF grid to capture fine structures, since the formulation puts higher requirements on the surface points, i.e., from SDF sign flipping to SDF 8. Higher resolutions will decrease computational speed and consume more GPU memory.
B.3 Supporting Plane
We use the supporting plane to provide contact and friction for object surface points in the physics simulation. More specifically, we use SP-MC to obtain the surface points of both the object to be simulated and the supporting plane (e.g., background for objects on the floor) before each simulation. We only include the background surface points within proximity under the object surface points for the simulation to reduce the computational load. Note that our physical simulator and physical loss both support more general contacts, e.g., the box the cabinet and the monitor the table
20

---

Pilt + Ar)
((O)
q(t)
Pf(
a(t At)
r(t + Dt)
(())
(())
P,O0)
Time
Figure A1: Rigid body dynamics. The world and rigid body coordinate system are represented by solid and dashed lines respectively.
demonstrated in Fig. 1, The only difference lies in the calculation of the supporting plane during simulation. For example, for the monitor on the table, we use the table surface points instead of background surface points to calculate the supporting plane.
C Particle-based Rigid Body Simulator
Our particle-based rigid body simulator is designed enable convenient and efficient simulation of rigid bodies depicted as collection of spherical particles of uniform size.
Naming convention In this section, represent vectors and second-order tensors with bold lowercase and uppercase letters, respectively; and denote scalars using italic letters. Superscripts are utilized to index rigid bodies, whereas subscripts are employed index particles.
rigid body simulator captures the body's dynamics through translation and rotation. The body's state is anchored to a reference coordinate system, which is initially aligned with the world coordinate system and located at the body's 's center of mass. The state at any given moment is encapsulated by the tuple:
(A17)
where r(t) R3 and (t)) R4 specify the position and orientation of the reference frame in relation to the world frame at time t, respectively. q(t) = [qu. ạ1 92. 93] [ is a quaternion, where go is a scalar value indicating the rotation angle, and (4), Q2. g3] is the normalized rotation axis. v()) E IR3 and (l)) E R3 represent the linear and angular velocities, respectively.
In our simulator, rigid body is represented by collection of spherical particles of uniform size, as depicted in Fig. Al. When body consists of N such particles, all with identical mass m and initially position at (0), the total mass of the system is:
M = N m.
(A18)
Furthermore, the system's 's center mass, also serving as the origin the reference frame r(0), is determined by:
r(0) Σ m (0)
(A19)
M
Thus, the relative position of each particle within the local rigid body frame is u, = [14os 142]" p; r(0). Consequently, the moment of inertia within the reference frame can derived as:
(un 2 + uz22) 3)2 (4:
(u )m
- (uou42)m )m E [ua 442 ) )
I
(u
+ u42 )m
(A20)
(4
(u, U4z )m
{ + u11 2) 2)m
The total mass M and the moment of inertia Iref are intrinsic physical properties of the rigid body, reflecting its resistance external forces and torques. It important to note that I not constant as it varies with the body's orientation.
21

---

Table A2: Three different contact status and their corresponding criteria. Colliding Contact, Resting Contact and Separation
Colliding Contact
Resting Contact
Separation
lipt V.
pill
< 2r
Ip pil Nel <
2r
1p
p < 2r Ve No >C
No < -E
||0.
C.1 Forward Dynamies
According Newton's 's second law, when subjected external forces and moments, rigid body's linear and angular velocities will change, altering its state. Using explicit Euler time integration, the evolution of state variable between successive time steps can be summarized follows:
v(t + At) v(t) + ALM f(1)
(A2la)
r(t+ + At) = r(t) Atv(()
(A21b)
w(t At) -(()) + AtI -:I (t)
(A21c)
q(t + At) = g(t) + [0, At
x (t)).
(A21d)
2
Here, At is the time step size; E R3 and T R are the total external forces and torques, respectively. The inertia tensor the rotated body I is given by:
I(l) Ret R² (1)
(A22)
where the rotation matrix R(t) is derived from the corresponding quaternion ((t) = [go, Mh, 22, qs)"
using the formula:
2( + 4 -
2 2 9043) 2( + o3
2(41 + 2042)
R(I)
2(1122 + 2003)
2 gom))
(A23)
2(1143 (092)
2g2233 + (0%1)
2 + 23
During scene reconstruction, changes in the state of rigid body are induced solely by gravity and contact. The gravity force is accounted for by setting f = Mg. with represents the acceleration of gravity. On the other hand, contact and friction forces resolved using impulse-based method, which is elaborated in the subsequent subsection.
C.2 Collision Detection
In order to more realistically simulate the behavior of rigid bodies, we have to detect whether and where two rigid bodies come into contact with each other during dynamic motion. Since we represent rigid bodies set of particles, collision detection between complex-shaper rigid bodies can be simplified to relatively simple inter- particle collisions.
Specifically, for two particles that are sufficiently close (particle in rigid body and particle j in rigid body b, both with radius r), we can accurately approximate the contact position using the centers of the particles and define the contact normal as:
P P
N.
(A24)
D Ill
and the relative velocity the contact points is given by:
v. = (v" + R u (w" R² Rlul))
(A25)
with the normal and tangential component are defined as:
V. (v. N.).
(A26a)
Yell v2 Vel.
(A26bb
Their relative contact status can be categorized into three types based the criteria illustrate in Tab. A2, Among these, only colliding and resting contact require further processing the simulation pipeline.
22

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C.3
Colliding Contact
The criterion v. -N < -c indicates that the two particles are approaching each other and further interpenetration will occur, therefore the simulator must separate them. The fundamental principle of impulse-based rigid body contact simulation [67] lies instantaneously adjusting the velocity to prevent subsequent interpenetration. This method eliminates the need for directly applying force over an extended period.
Adhering to the principle of conservation of momentum and Coulomb's 's friction model, we know the physically plausible relative velocity v.2 = 2 + e after contact should be:
T --
(A27a)
Dell = avell
(A27b)
max(1 n71 + 1 0).
(A27c)
where j is the coefficient of restitution and r is the friction coefficient. To achieve the desired velocity V. the required impulse J at the contact point is calculated as:
K K-'(0=
We)
(A28a)
T
K
R
R
[R ( (R'))
(A28b)
M
M
where I represents the 3 x identity matrix and the operator [ 1x transform vector into skew- symmetric matrix. Under the influence of J, the linear and angular velocity of the involved rigid bodies updated follows:
aaa 0d + J
(A29a)
Ma
wa w + ( J)
(A29b)
v² 0d w
(A29c)
M
R J).
(A29d)
These updates ensure that the bodies respond correctly to the collision, separating bouncing off each other in a manner that conserves momentum and energy as dictated by the specified restitution coefficient.
When multiple particles on a rigid body collide simultaneously, we use the average linear and angular impulses to update the rigid body's ' linear and angular velocities. If one of the rigid body during in contact a fixed boundary, such as the floor, simply setting its corresponding 1/M and I-1 to zero will adequately handle the situation.
C.4 Resting Contact
The criterion ||0.- Nell < e indicates that an object maintains continuous contact with another without significant changes position or orientation, such chair resting on the floor. Achieving stable resting contact without objects slowly penetrating each other or jittering due to numerical errors can be challenging with physics-based method.
To simplify this, our simulator adopt a strategy commonly employed in game development and robotics to enhance performance and realism. If dynamic rigid body remains stationary moves extremely slowly for a few seconds, our simulator will mark it as sleeping. Once classified as sleeping, the rigid body will be temporarily excluded from all steps in the simulation pipeline, except for collision detection. When the sleeping body come into colliding contact with another non-sleeping rigid body, it will automatically "wake up" and get back into the simulation again.
In practical implementation, we employ the following method to quantitatively assess the motion of rigid body over short historical period [46]:
rwa y + (1 7) 2
(A30)
23

---

where the weight lies within the range [0, 1]; cpp represents the upper bound of the current speed of the rigid body [46], which can be effectively approximated by:
v (v Rummax w) ( (v + Rumma X w)
(A31)
~ 2 (v' + (w" w) (umxxxmax))
(A32)
Here Umax denotes the maximum distance from any particle on the surface the body's 's center of mass, calculated once during the initialization phase. Given that our simulator is solely affected by gravity, body set to sleep rwa meets the following condition:
Twa < 1lg|| At.
(A33)
C.5 Implementation Details
For the sake of reproducible, present the pipeline of particle-based rigid body simulation during each time step as in Alg. 3. Our simulator and its gradient support are developed using DiffTaichi [19], which high-performance differentiable physical programming language designed for physical simulations, and computational science.
For all of our examples, we set time step At 0.01 s; particle radius 0.005 m; particle mass m = 0.01 kg: the coefficient of restitution = 0.0, the friction coefficient = 0.4, the relative velocity criterion e le 5 and 0.1.
Algorithm 3 The pipeline of particle-based rigid body simulator
1: Input: Initial particle positions p (0) for each rigid body i in the scene.
2: Output: Final particle positions p' (t) when their belonging rigid body reaches stable equilibrium,
with flags for all particles that have collided.
3:
4: I/ physical properties
5: For each rigid body do
6: compute mass and center of mass: M + Eq. (A18), r0)) + Eq. (A19)
7: For each particle do compute particle position in reference frame: u. < p,( r()) 9: end for
8:
10: compute inertia matrix in reference frame: Iref + Eq. (A20)
compute the maximum distance: Umax < max(u;)
12:
end for
13:
14:
For time step do
forward dynamics
16:
For each awake rigid body do
17:
apply gravity force: f(t + Mg compute linear and angular velocities v(t) +- Eq. (A21a), w(t) <- Eq. (A21b)
18:
19:
compute position and orientation: r(t) < Eq. (A21c), ((t) +- Eq. (A21d)
20:
compute rotation matrix: R(t) +- Eq. (A23)
21:
end for
22:
23:
update particles
24:
For each particle do
25:
update particle position in world space: pt(() < R(I)a + r()) end for
27:
28:
I/ collision detection and colliding contact resolve
29:
While no colliding contact do
For each pair of particles do
31:
if !!pp p < 2r then
32:
compute contact normal: N. +- Eg. (A24)
33:
compute contact velocity: v. <- Eq. (A25), Vall Eg. (A26a) v.. N₂ < 6 then compute desired contact velocity: v.a <- Eq. (A27a), Eg.. (A27a)
35:
24

---

36:
compute required impulse: J + Eq. (A28a)
if v.e > r/
38:
reactive the rigid body
39:
end if
40:
end if
41:
end if
42:
end for
43:
end while
44:
45:
update rigid body state For each rigid body do update linear and angular velocities: v(() t- Eq. (A29a), w(() t- Eq. (A29a)
46:
47:
48:
end for
49:
50:
II resting contact
51:
For each rigid body do
52:
compute historical motion information Uup + Eq. (A32), rwa Eq. (A30)
53:
if rwa < ||l|| At then
set the body to sleep
55:
+ [0, 0, 0), ~ + 00,0,0
end if
57:
end for
58:
end for
D More Experiment Details
D.1 Data Preparation
Monocular Cues We utilize pre-trained Marigold model [27] to generate depth map D for each imput RGB image. It's important to note that estimating the absolute scale in general scenes relative depth cue. Furthermore, pre-trained Omnidata model [12] to obtain normal maps N for each RGB image. While depth cues offer semi-local relative information, normal cues are local and capture geometric intricacies. Consequently, we anticipate that surface normals and depth complement each other effectively.
is challenging, thus D should be regarded
employ another
GT Instance Mask For the ScanNet++ [76] dataset, we utilize the provided GT mesh and per-vertex 3D instance annotations, along with their rendering engine to generate instance masks for each image. For the ScanNet [9] and Relica [62] datasets, we observed discrepancies in the masks provided by RICO [32] and ObjectSDF++ [69]. To ensure fair comparison with the baselines, we merged their instance masks into consistent ones. In our experiments, focused solely object- srjjcttgrrund support for simplicity and training efficiency, leaving the determination more general support relationships for future work.
D.2 Evaluation Metrics
Stability Ratio To evaluate the physical stability of a reconstructed object mesh shape, we employ the Isaac Gym [39] simulator. This involves conducting dropping simulation to determine if the shape remains stable within 5 in rotation and 5cm translation after the simulation, under the influence of gravity, contact forces, and friction provided by the ground. Next, we define the stability ratio of the scene as the proportion of the number of stable objects to the total object number in the scene. More specifically, using URDFs that include parameters such as the center of mass, mass, and inertia matrix, where the relative positions of the object shape and background are preserved in the scene. Subsequently, simulate for T = 200 steps with time step of At = 0.016s (i.e. 60H
import the object shape and the reconstructed background into the simulator
Reconstruction Metrics To evaluate 3D scene and object reconstruction, we use the CD in em, F-score with a threshold of 5em and NC following prior research [77, 32, 69]. In detail, CD comes from Accuracy and Completeness, F-score is derived from Precision and Recall, and NC is computed using both Normal-Accon Accuracy and Normal-Compledt to evaluate reconstruction only the visible areas. These metrics are defined in Tab. A3,
We follow previous work [18, 77, 32, 69]
25

---

Table A3: Evaluation metrics. We show the evaluation metrics with their definitions that we use to measure reconstruction quality. and pa are the point clouds sampled from the predicted and the ground truth mesh. np is the normal vector at point p.
Metric
Definition
Chamfer Distance (CD)
Accunacy Completeeess
Accuracy
mean per
Ilp pp' P
p'111 min
Completeness
mean P
lpp p=112 PEP
min
F-score
2 Procision Recall Precision Recall
Precision
mean pEr
p - p-pp1 < 0.05 P EP-
min
Recall
mean E
min P - p'111 < 0.05 pEp
Normal Consistency
Normal Accuracy Compleieness
Normal Accurucy
mean
(p' s.l. p arg min Ip p'11 P'EP-
Normal Completeness
mean p' PEP*
11pp s.t. p = arg min p p'1h
D.3 Intermediate Results
In Fig. A2, we present the reconstruction results of the intermediate states across different stages. Epochs 360 and 430 mark the beginning of the 2nd and 3rd training stages. In the 1st stage, the reconstruction quality of the object's visible structure is continuously improved. In the 2nd stage, the object areas crucial for stability are optimized with the physical uncertainty. In the 3 stage, the reconstructed object reaches stability when the physical loss is introduced.
Image
50 epoch
150 epoch
250 epoch
360 epoch
430 epoch
440 epoch
450 epoch
Figure A2: Visualization of intermediate results during training. Epochs 360 and 430 mark the beginning of the ndd and 3rd training stages.
26

---

D.4 Time Consumption
We conduct additional experiments on example scenes from the ScanNet++ dataset to assess the time consumption of the rendering and physical simulation. average, takes approximately 0.23 seconds (5.13% in total time consumption) for rendering and 4.25 seconds (94.87% in total time consumption) for physical simulation (including automatic gradient calculation) during one single forward pass. Note that the time required for physical simulation is related to the number of particle points involved, will vary across different scenes and objects. The above experiments involve around 8000 object surface points and 6000 background surface points (i.e. the supporting plane) on average physical simulation.
Additionally, we compare the total training time between our method, RICO, and ObjectSDF++ as shown in Tab. A4 on ScanNet++. Integrating physical simulation has significantly enhanced the stability of the reconstruction results, although it has also increased time consumption.
Table A4: Training time comparison.
RICO
ObjectSDF++
Ours
Stage (0-360 epoch)
6.86h
7.14h
6.91h
Stage 2 (360-430 epoch)
1.45h
1.43h
2.46h
Stage 3 (430-450 epoch)
0.39h
0.41h
3.19h
Total
8.70h
8.98h
12.56h
D.5 Performance Sensitivity Analysis
Performance vs. Max Simulation Steps in a Forward Simulation our framework, the physics forward simulation continues until the object reaches stable state or the maximum simulation steps are reached. Therefore, the choice of maximum simulation steps affects both simulation time and performance. We conducted a sensitivity analysis of performance versus maximum simulation steps in a forward simulation. The results are evaluated on example scenes in ScanNet++, shown in Tab. A5. The results illustrate that increasing the maximum simulation steps generally improves the final stability of the objects. This trend becomes negligible once the maximum simulation steps exceed 100, most objects achieve stable state within 100 simulation steps. This also explains why increasing the maximum simulation steps leads to longer simulation time, though this increase not linear the number of steps. Consequently, in the experiments for main paper, we chose 100 as the maximum simulation steps.
Table A5: Performance of varying maximum simulation steps in one forward simulation.
F-Score t
NC1
Max Sim. Steps
SR (%)
Total time
Stage-3 time
scenc/obj
scene/obj
scene/obj
25
2.91/3.24
88. 88.17/86.61
91.56/85. (4
69.23
10.64h
1.27h
2.83/3.26
88.41/86.66
91. 01.82/85.00 (4
69.23
11.45h
2.08h
75
2.78/3.16
88.84/87.12
91.84/85.68
76.92
12.08h
2.71h
100
2.86/3.24
88.34/86.58
91.72/85.1 12
84.62
12.56h
3.19h
125
2.88/3.28
88.19/86.48
91.54/85.08
84.62
12,92h
3.55h
150
2.91/3.25
88.24/86.52
91.53/85.06
84.62
13.26h
3.89h
Performance vs. Total Simulation Epochs. We conducted sensitivity analysis of performance versus total simulation epochs to discuss the relationship between training time and performance after adding the physical loss in stage 3. The results are shown in Tab. A6, tested on ScanNet++. The table shows that as the number of training epochs with physical loss increases, the stability of the objects improves. However, there is no significant improvement after 445 epochs, as most objects had already achieved stability. Notably, the longest training time was observed between 430 and 435 epochs, after which the time required for simulation progressively decreased, again due to the improved object stability.
27

---

Table A6: Performance of increasing the number of training epochs with physical simulations.
CD +
F-Score 1
NC 1
Training epoch
SR (%)
Total time
A time
scene/obj
scene/obj
scene/obj
430 (no sim.)
2.95/3.31
87.84/86.32
90.42/84.83
61.54
9.37h
435
2.88/3.31
88.06/86.35
90.93/84.36
69.23
10.49h
1.12h
440
2.81/3.17
88.75/87.03
91.93/85.77
76.92
11.39h
0.90h
445
2.82/3.22
88.54/86.96
91.83/85.41
84.62
12.10h
0.71h
450
2.86/3.24
88.34/86.58
91.72/85.12
84.62
12.56h
0.46h
E Limitation
Representative failure examples in Fig. 8 present that optimizing with the physical loss may lead to degenerated object shapes, e.g., bulges, to maintain physical stability. This is due to insufficient supervision from rendering losses. The disconnected parts in the reconstruction results are another common limitation of the current neural implicit surface representation and reconstruction pipeline. This may be addressed by incorporating topological regularization to penalize disconnected parts of the object further enhancing the simulator. Plus, particle-based simulator treats all surface points from an object as rigid body, thus cannot handle soft bodics (which deform after collision) dynamic scenes.
In addition, our current framework requires additional ulation. While determining ground-object support is straightforward, identifying more complex relationships remains challenging. The implicit representation is optimized through per-object phys- ical simulation with the background, for the sake of efficient computation in the current neural scene understanding settings. However, our SP-MC, physical simulator and loss are designed to be compatible with multi-object scenarios, enabling seamless extension to joint optimization for the whole scene, ensuring versatility without loss of generality.
information for the sim-
Potential Negative Impact
3D reconstruction in general, while offering various benefits in fields like AR/VR, robotic and Embodied AI, also raises concerns about potential negative social impacts. Some these impacts include potential privacy concerns public areas, surveillance and security risks. Addressing these concerns requires careful consideration of ethical guidelines, regulatory frameworks, and responsible development practices ensure that 3D scene reconstruction deployed in manner that respects privacy, security, and societal well well-being.
28

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Reviews:
[{"review_id": "I9XPZS7wK7", "reviewer": "Reviewer_5HaF", "summary": "This paper proposes a framework for scene reconstruction using neural implicit function with an emphasis on physical plausibility. The key idea is to integrate differentiable physical simulation into the neural scene reconstruction such that the reconstructed shape is stable under gravity. A surface points marching cube method is proposed to convert the neural surface into particles, which is differentiable, enabling the back propagation of gradients from simulation. The authors further propose rendering and physical uncertainty to identify the regions that are not consistent across multiview images and unstable under physical simulation. These uncertainties are beneficial to capture the thin structures of the reconstructed object. Experimental results on various datasets including ScanNet, ScanNet++,  and Replica are conducted to demonstrate the effectiveness of the proposed method.", "questions": "See Weaknesses.", "limitations": "The authors have provided sufficient limitations and failure cases of the proposed method.", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is well-written. The specific techniques are thoroughly discussed with enough details provided. The proposed method is more like a system consisting of multiple components. The authors have clearly put a lot of efforts to elaborate on each component in detail, as can be seen in the appendix.\n\n- The ablation study is nicely designed and well-controlled. I personally find Table 2 very informative, which clearly demonstrates the effectiveness of each proposed component.", "weaknesses": "- My major concern of the paper is its novelty. Though the authors position their paper as scene reconstruction, the examples shown in the paper are actually object-centric. In this sense, there are actually a series of works that have explored the physical stability of the generated objects represented as neural functions. Specifically, [37] handles the robustness of slender structure in shape generation, and [38] deals with stability of the generated shape. Especially the second work, it has a very similar problem setting to this paper, as they also first converted the SDF into surface points followed by a differentiable simulator. Somehow I feel the method in [38] is neater since the gradient is propagated to the shape latent code rather than a dense voxel grid as proposed in this paper. Neither [37] or [38] is compared in the experiments. This undermines the motivation and technical novelty of the proposed approach, since compared to shape generation, addressing scene reconstruction exclusively does not pose additional challenges for physical plausibility.\n\n- With the above being said, I think the technical novelty lies in the uncertainty model. However, what confuses me is whether this modeling on the whole voxel grid is necessary. From the qualitative results in Fig. 7, the rendering uncertainty models are mainly helping to grow the shape regions that are \"sparsely\" covered by the multi-view images. I guess simple post-processing methods such as region growing would be sufficient for dealing with these cases. The physical uncertainty may also be unnecessary: from Fig. 7 and the failure cases in the appendix, physical uncertainty and physical losses are nothing more than changing the geometric shape so that its center of mass lies on the supporting plane of the shape's points connected to the ground. The proposed approach, especially the part of \"physical uncertainty\",  to me is an overkill to problem. I am also not a big fan of using the term \"uncertainty\" here since there is really no probability estimation/inference. A better wording could be \"heuristics\".", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the constructive comments on our work. However, we believe the reviewer may have some misunderstandings regarding our method. We will clarify the novelty of our approach as the first effort to incorporate physical constraints into neural scene reconstruction, and we will further explain our rendering and physical uncertainty.\n\n**1. Misunderstandings about the method in our paper.**\n\nThe reviewer frequently mentioned that we use a dense voxel grid, as noted in comments like \"the gradient is propagated to a dense voxel grid as proposed in this paper\" and \"I think the technical novelty lies in the uncertainty model. However, what confuses me is whether this modeling on the whole voxel grid is necessary.\" In fact, our method uses MLPs to model SDF fields and rendering uncertainty simultaneously, while only physical uncertainty is modeled by a dense voxel grid as it reflects accurate 3D physical information from the simulator. \n\nAnother misunderstanding comes from the rendering uncertainty, which addresses the **multi-view and multi-modal inconsistencies** in monocular geometry priors rather than \"grow the shape regions that are 'sparsely' covered by the multi-view images\". For more details, please refer to Section 3.3 in our paper.\n\n**2. The novelty of incorporating physical constraints into neural scene reconstruction based on the NeRF pipeline isn't limited.**\n\nWe note that the reviewer considers our approach to be \"object-centric\" and confuses the difference between our work and [37, 38], which attempt to add physical constraints to object generation. This confusion extends to the novelty of our work, which integrates physical constraints into neural scene reconstruction. We will address these points one by one.\n\nFirst of all, our method is designed for decompositional scene reconstruction, which add object-centric concepts to neural scene reconstruction by simultaneous recovery of each object and the background. By disentangling objects in the scene, it offers object-centric information for holistic scene understanding and downstream applications like 3D reasoning, navigation or manipulation. In the meantime, it enables physical simulations for each object, which would not be possible for any simulation if we treated the entire scene as a whole.\n\nSecond, we **argue there is a significant gap** between scene-level reconstruction (our work) and object-level generation ([37, 38]). \n* **Reconstruction v.s. Generation**: [37, 38] aim to generate physically plausible objects but do not replicate input scene images. Based on the DeepSDF[1] pipeline, [37, 38] do not use any image supervision or condition. In contrast, our work aims to recover a physically plausible 3D scene from given 2D images, ensuring the results match the original images.\n* **Scene-level v.s. Object-level**: Our approach leverages scene images, which are captured while walking freely within the scene, rather than taken around a specific object. The partial observations and abundant occlusions from the scene images make scene reconstruction much more challenging than object reconstruction.\n\nTherefore, it's impossible to directly adapt methods for object generation to scene reconstruction.\n\nOur work stands as the first to integrate differentiable rendering with differentiable physical simulations for learning implicit surface representations, thereby enhancing reconstruction quality and physical plausibility. This is achieved by our design of the SP-MC algorithm, an efficient differentiable particle-based simulator, an effective physical loss and joint uncertainty modeling, all of which contribute to the improved reconstruction quality and physical constraints stability. The technical novelty is acknowledged by other reviewers:\n* \"Applying physical stability loss is a rather novel idea.\" - Reviewer mFxQ.\n* \"There are good technical contributions and the method is sound.\" - Reviewer fqR5.\n* \"The overall framework is mind-blowing and insightful.\" - Reviewer qV3n.\n\n**3. Rendering uncertainty isn't \"mainly helping to grow the shape regions that are 'sparsely' covered by the multi-view images\", and can't be substituted by \"simple post-processing methods such as region growing\".**\n\nRendering uncertainty addresses inconsistencies in **monocular geometry priors** (depth and normal prior obtained from off-the-shelf geometry estimators) caused by **multi-view** (same region yielding different results in different views) and **multi-modal** (individual prediction of depth and normal cues leading to conflicts) issues. These inconsistencies arise from the limitations of geometry estimators and the prediction mechanism of monocular images.\n\nTo the best of our knowledge, there are no post-processing methods that can efficiently identify these inconsistent regions. Plus, heuristic region growing may lead to degenerated results deviating from the image observations. On the contrary, our proposed rendering uncertainty, indirectly optimized by rendering loss during training, can locate the inconsistencies and mitigate them effectively.\n\n**4. Physical uncertainty isn't an overkill to problem.**\n\nWe argue that our modeling of physical uncertainty is essential to reconstruct the slender structures and improve object stability. While physical loss can improve the stability of reconstruction, relying solely on it may achieve stability at the expense of the object's shape. The rendering uncertainty can mitigate monocular inconsistencies but they cannot identify regions crucial for physical stability. On the contrary, we design physical uncertainty to identify these regions by leveraging simulation and adjust rendering losses accordingly. Additionally, we use physical uncertainty-guided pixel sampling to enhance learning on these regions. These combined designs significantly improve the reconstruction results, especially in areas critical for object stability, as demonstrated in the ablations from Tab.2 main paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3pHuhamX63", "reviewer": "Reviewer_mFxQ", "summary": "This paper proposes to improve implicit surface representations by considering object dynamics (physics) in their resting state. It made a clever observation that the objects should be stable with respect to their supporting planes if the surface representation is correct. This observation leads to utilizing object surface points in the physical simulator to decide if the objects are stable, by penalizing any instability, the solution encourages more plausible surface reconstruction.\n\nPhysical loss and physical uncertainty were introduced, assisted by the proposed SP-MC (Surface Points Marching Cubes) to improve surface point sampling efficiency for deriving the losses in simulations.\n\nTests on ScanNet and ScanNet++ demonstrated superior results compared to existing works.", "questions": "1. What is the ground plane, and how is it calculated/retrieved from the dataset?\n2. In the losses the author used (as shown in the appendix), a combination of losses from many existing works is used, what would be the performance if the baseline is a version that combined all of the existing losses?\n3. Section 3.3 mentioned the physical uncertainty $U_{phy}(r)$, which the author didn’t give a clear definition.", "limitations": "The author discussed certain limitations in the Appendix. See the above comments for additional concerns.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. Applying physical stability loss is a rather novel idea.\n2. The implementation of using particle-based physical simulation, although complicated, seems to be a plausible approach for understanding and observing any physical instability.\n3. The paper is well-written with many implementation details.", "weaknesses": "1. Applying this idea would require an object mask, and also knowledge of the ground. This is not a constraint in the existing work MonoSDF.\n2. The construction of Physical uncertainty (equation 7), based on the trajectory of the point before it makes contact, is not actually the same as the supporting structure of the object. To be more specific, such a trajectory is often a curve unless the object is falling without any rotation. Such a loss would lead to bulkiness in the supporting structure during reconstruction (as seen in Figure 7e).\n3. The author mentioned a couple of times about the supporting plane or ground, where is that information from?\n4. In general, this physical loss is only applicable to contact the the supporting plane, and not applicable to other contacts (such as a cup on a table).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the constructive comments on our work. We will thoroughly address each of the concerns raised.\n\n**1. Our method requires object masks, whereas MonoSDF does not.**\n\nFrom neural scene reconstruction which treats the entire scene as a whole (e.g., MonoSDF[1]), adding object decompositionality is a popular trend as they can provide more valuable object-centric information for downstream applications like 3D reasoning, navigation or manipulation. Following prior work [2, 3, 4], we use object mask supervision to disentangle each object from the background. Separating objects in the scene further makes it possible to add physical simulations to each object. With the rapid development of segmentation and tracking models, e.g., SAM [5] and SAM2 [6], we can efficiently obtain object masks from images with off-the-shelf models, which could inspire more efforts in decompositional neural scene reconstruction.\n\n**2. The trajectory of contact point is not actually the same as the supporting structure of the object.**\n\nUpdating physical uncertainty using contact trajectory may indeed deviate from the supporting structure. However, our proposed physical uncertainty is used to identify regions **potentially important for improving physical stability**, which is not used directly as a form of supervision. Our direct supervision comes from rendering supervision and the physical loss on the contact points only, with physical uncertainty playing only an auxiliary role in adjusting the loss weights and guiding pixel sampling towards these regions. As shown in our experiments and ablations, a coarse physical uncertainty is sufficient for improving slender-structure reconstruction and physical.\n\n**3. What is the supporting plane and how is it calculated?**\n\nWe use the supporting plane to provide contact and friction for object surface points in the physics simulation. More specifically, we use SP-MC to obtain the surface points of both the object to be simulated and the supporting plane (e.g, background for objects on the floor) before each simulation. Additionally, we only include the background surface points located within proximity under the object surface points for the simulation to reduce computational load. We will add this in the revision.\n\n**4. Can physical loss be applied to more general contact scenarios, such as a cup on a table?**\n\nIn fact, our physical simulator and physical loss both support more general contacts, as demonstrated in Fig. 1 and the supplementary video (e.g., the box on the cabinet and the monitor on the table). The only difference lies in the calculation method of the supporting plane during simulation. For example, for the monitor on the table, we use the table surface points instead of background surface points to calculate the supporting plane.\n\n**5. Add a baseline that combines all of the existing losses.**\n\nOur baseline (the first row, i.e., w/o RU, w/o PU, w/o PS, w/o PL) in Tab.2 combines all existing losses, which aligns with your case. Please refer to Tab.2 in the main paper for details.\n\n**6. Definition for physical uncertainty $U_{phy}(r)$.**\n\nDue to the length limitation of the paper, we briefly introduced $U_{phy}(r)$ in L194 as: \n\n\"The physical uncertainty $U_{phy}(r)$ of the pixel corresponding to ray r is computed via volume rendering\".\n\nHere we provide the following formula for calculating $U_{phy}(r)$:\n\n$$\nU_{phy}(r)=\\sum_{i=0}^{n-1} T_{i}\\alpha_{i}u_{phy,i}\n$$\n\n\nThank you for your suggestion, we will add these definitions in the revision of the paper.\n\n\n[1] Monosdf: Exploring monocular geometric cues for neural implicit surface reconstruction\n\n[2] Objectsdf++: Improved object-compositional neural implicit surfaces\n\n[3] Rico: Regularizing the unobservable for indoor compositional reconstruction\n\n[4] Object-compositional neural implicit surfaces\n\n[5] Segment and Track Anything\n\n[6] SAM 2: Segment Anything in Images and Videos\n\n[7] Marching cubes: A high resolution 3d surface construction algorithm\n\n[8] A volumetric method for building complex models from range images", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "K40Rwjyuy3", "reviewer": "Reviewer_fqR5", "summary": "The paper introduces a novel method for physically plausible 3D scene reconstruction. It enforces the reconstructions to be in a static equilibrium state, which is achieved by minimizing the displacement of 3D points over time in a differentiable rigid body simulator. The experiments are conducted on scannet/replica/scannet++, showing quite significant improvement over prior methods.", "questions": "- How expensive is the simulation compared to rendering? e.g., in a forward pass, how much time is consumed by simulation? Does the use of simulation introduce significant compute overhead? Relatedly, I'd like to see a sensitivity analysis of performance vs number of simulation steps.\n- How is the ground-plane computed for physical simulation? Does the method handle complex terrian (e.g., with a carpet on the floor, or outdoor scenes)?\n- Since the optimization is multi-staged, it will be good to show result after each stage.\n- Showing more intermediate results along the axis of optimization would help. For instance, how does the shape change from iteration 0 to final iteration?", "limitations": "The paper discussed the limitations and social impact.", "rating": 7, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The paper is easy to follow, and the visual comparisons are adequately made.\n- There are good technical contributions and the method is sound. \n  - They present a differentiable pipeline that can propagate gradients from after-simulation states to the reconstruction, which is achieved by a differentiable mapping from neural SDF to particles (SP-MC), and explicit Euler integration for simulation (which is already differentiable). \n  - They also develop a re-weighting strategy that down-weights the loss terms with large uncertainty. The choice of uncertainty is different from term to term, measurement uncertainty is used fo depth and normal; physical uncertainty (as defined by the displacement of point over simulation) is used for segmentation as a proxy. The use of uncertainty has two major benefits: (1) the noise in the depth/normal/segmentation priors are filtered out and (2) the pixel sampling can focus on regions with high uncertainty to enhance the reconstruction quality. \n- The results are strong. The comparisons clearly shows the improvement over prior methods. They also compared to prior works quantitatively, showing better performance in terms of reconstruction quality, normal consistency, and object stability.\n- The authors ablated the design choices of physical loss, uncertainty and sampling strategy, showing the benefit of adding each component.", "weaknesses": "- It only deals with static scenes with rigid objects. The physical loss presented is not applicable to complex contact interactions, dynamic scenes or soft materials. But I don't think this is a major issue as the method works well for the indoor scene reconstruction task being considered.\n- More explanation on SP-MC is needed. Since the gradient only comes from the refinement step, I would imagine the gradient to be fairly local and noisy. How does it work when the initial shape is far from being physically plausible?\n- I'd like to suggest moving appendix C to the main text, since the physical simulator plays an important part in the overall story.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the constructive comments on our work. We will thoroughly address each of the concerns raised. Notably, we have added visualizations of intermediate results during training in the global rebuttal PDF, along with sensitivity experiments regarding the max simulation steps in a forward simulation and the total simulation epochs. We hope these additions help clarify your questions.\n\n**1. More intermediate results during optimization, particularly after each stage.**\n\nIn the global rebuttal Fig.1, we present the reconstruction results of the intermediate states across different stages:\n\nStage 1: We use rendering uncertainty to adjust the rendering loss, as shown at 50, 150, 250, and 360 epochs.\n\nStage 2: We add physical uncertainty to adjust the rendering loss and guide pixel sampling, as shown at 360 and 430 epochs.\n\nStage 3: We incorporate the physical loss, as shown at 430, 440, and 450 epochs.\n\nIn the Stage 1, the quality of the object's thin structure reconstruction is continuously improved. In the Stage 2, critical parts contributing to the object's stability are significantly optimized. In the Stage 3, the stability of the object reconstruction is greatly enhanced.\n\n**2. Comparison of time consumption between physical simulation and rendering.**\n\nWe conduct additional experiments on example scenes from ScanNet++ dataset to assess the time consumption of the rendering and physical simulation. On average, it takes approximately 0.23 seconds (5.13% in total time consumption) for rendering and 4.25 seconds (94.87% in total time consumption) for physical simulation (including automatic gradient calculation) during one single forward pass. Note that the time required for physical simulation is related to the number of particle points involved, it will vary across different scenes and objects.  The above experiments involves around 8000 object surface points and 6000 background surface points (i.e. the supporting plane) on average physical simulation.\n\nAdditionally, we compare the total training time between our method, RICO, and ObjectSDF++ as shown in the following table on ScanNet++:\n\n|                         | RICO  | ObjectSDF++ | Ours  |\n| ---------------------   | ----- | ----------- | ----- |\n| Stage 1 (0-360 epoch)   | 6.86h |    7.14h    | 6.91h |\n| Stage 2 (360-430 epoch) | 1.45h |    1.43h    | 2.46h |\n| Stage 3 (430-450 epoch) | 0.39h |    0.41h    | 3.19h |\n| total                   | 8.70h |    8.98h    | 12.56h |\n\nIntegrating physical simulation has significantly enhanced the stability of the reconstruction results, although it has also increased time consumption. Further accelerating simulation computations will be an interesting research direction. We will add these additional results and discussions in the revision.\n\n**3. Sensitivity analysis of performance vs max simulation steps in a forward simulation.**\n\nIn our framework, the physics forward simulation continues until the object reaches a stable state or the maximum simulation steps are reached. Therefore, the choice of maximum simulation steps affects both simulation time and performance. We conducted a sensitivity analysis of performance versus maximum simulation steps in a forward simulation. The results are evaluated example scenes on ScanNet++, shown in the global rebuttal Tab.1.\n\nThe table illustrates that increasing the maximum simulation steps generally improves the final stability of the objects. This trend becomes negligible once the maximum simulation steps exceed 100, as most objects achieve a stable state within 100 simulation steps. This also explains why increasing the maximum simulation steps leads to longer simulation times, though this increase is not linear with respect to the number of steps. Consequently, in the experiments for our main paper, we chose 100 as the maximum simulation steps.\n\n**4. Sensitivity analysis of performance vs total simulation epochs.**\n\nTo discuss the relationship between training time and performance after adding the physical loss in stage 3, we conducted a sensitivity analysis of performance versus total simulation epochs. The results are shown in Tab.2 in the global rebuttal, tested on ScanNet++.\n\nThe table shows that as the number of training epochs with physical loss increases, the stability of the objects improves. However, there is no significant improvement after 445 epochs, as most objects had already achieved stability. Notably, the longest training time was observed between 430 and 435 epochs, after which the time required for simulation progressively decreased, again due to the improved object stability.\n\n**5. The gradient backpropagated from SP-MC is fairly local and noisy. What if the initial shape is far from being stable?**\n\nWhile optimizing the implicit representation of the object shape, e.g., SDF, the surface points of the object are the most important. Therefore, we specifically sample surface points from the object  during optimization, following the common practice in related work (e.g., DeepSDF [1] Sec. 5 Data Preparation). In our method, the refinement stage of SP-MC accurately extract the surface points of the object, whose gradients are sufficient for optimizing the object's shape. Using all points from the coarse stage for backpropagation would result in excessive computations without significantly improving object shape optimization; the larger space outside the object surface are optimized with rendering loss in volume rendering.\n\nWhen the initial shape is far from being physically plausible, the shape will be improved locally but incrementally (shown in Fig.1 of global rebuttal), as the SP-MC sampling are conducted before every simulation. The physical loss ensures stability of the final result, together with our multi-stage training and reweighted rendering loss.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YK7ODBXegT", "reviewer": "Reviewer_qV3n", "summary": "This paper addresses scene surface reconstruction in an object-compositional format. Unlike previous work that only considered rendering constraints, the proposed PhyRecon framework incorporates physical constraints into the model training. To identify interaction points, the paper introduces a differentiable transformation between the SDF field and explicit surface points. Using these points, the authors integrate physical constraints into previous approaches to ensure reasonable interactions between objects. To further enhance model performance, they introduce a joint uncertainty modeling design for depth, normal, and physical constraints, which increases the stability of the final reconstruction results. The experimental results demonstrate the effectiveness of the framework and its components, proving beneficial for various downstream tasks.", "questions": "1. The method relies on ground-truth instance labels. It would be interesting to explore whether the proposed physical solution can handle noise in the ground-truth 2D instance mask, such as erosion and dilation. While uncertainty training might help mitigate this issue, it is crucial to understand how much the physical constraint can aid in overcoming label noise.\n2. In the particle simulator, how does the system perform when the points are sparse and non-uniformly distributed? Understanding the simulator's effectiveness in such cases would be beneficial.", "limitations": "The author mentioned their limitation and potential societal impact in the appendix.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. The overall framework is mind-blowing and insightful. Extending the neural implicit representation, which is mainly supervised by rendering loss, to include physical constraints is non-trivial. This work proposes an innovative solution in this direction with well-reasoned designs, making it more applicable to real-world applications like robotics and virtual reality.\n2. The physical constraint is novel and enhances the fidelity of object reconstruction within a compositional design. The proposed framework offers a comprehensive solution with several interesting components, such as SP-MC and physical-guided pixel sampling.\n3. The experimental results are extensive, showing significant improvements in object reconstruction quality and stability compared to previous baselines.\n4. The physical pixel-guided sampling is an interesting and important trick that enhances the effectiveness of the physical constraint.", "weaknesses": "1. The overall solution is based on ObjectSDF++ and RICO, which requires hours of optimization for training. The introduction of additional constraints in PhyRecon further increases the total training time. Improving efficiency could be a focus for future work.\n2. The training process involves three stages: initially applying rendering loss for initialization, then incorporating a physical simulator, and finally introducing a physical loss to stabilize object contact. This multi-stage optimization and the complexity of the loss functions make the entire system intricate.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the constructive comments on our work. We will thoroughly address each of the concerns raised. Notably, we have added visualizations of intermediate results during training in the global rebuttal PDF. We hope these visualizations will help address your questions about the multi-stage optimization.\n\n**1. Comparison of total training time between our method, RICO, and ObjectSDF++.**\n\nWe compare the total training time between our method, RICO, and ObjectSDF++ as shown in the following table (test on example scenes from ScanNet++):\n\n|                         | RICO  | ObjectSDF++ | Ours  |\n| ---------------------   | ----- | ----------- | ----- |\n| Stage 1 (0-360 epoch)   | 6.86h |    7.14h    | 6.91h |\n| Stage 2 (360-430 epoch) | 1.45h |    1.43h    | 2.46h |\n| Stage 3 (430-450 epoch) | 0.39h |    0.41h    | 3.19h |\n| total                   | 8.70h |    8.98h    | 12.56h |\n\nStage 1: We use rendering uncertainty to adjust the rendering loss.\n\nStage 2: We add physical uncertainty to adjust the rendering loss and guide pixel sampling.\n\nStage 3: We incorporate the physical loss.\n\nIntegrating physical simulation has significantly enhanced the stability of the reconstruction results, although it has also increased time consumption. Exploring 3DGS-based methods [1, 2, 3] to improve scene reconstruction speed and further accelerating simulation computations will be interesting research directions.\n\n**2. This multi-stage optimization and the complexity of the loss functions make the entire system intricate.**\n\nIn our paper, we use a three-stage training method for two main reasons:\n* Simulation requires a reasonably initialized shape to ensure correct contact points.\n* Compared to rendering calculations, adding simulation increases the computational cost, so we minimize the use of physical simulation as much as possible.\n\nTherefore, we initially use only rendering loss to optimize and achieve a reasonable initial shape in stage 1. We then introduce physical simulation to incorporate physical uncertainty in stage 2. In this stage, each object undergoes simulation once at the beginning of each epoch, further optimizing the object's shape, particularly in regions crucial for physical stability. In stage 3, we include physical loss, and physical simulation is applied at each iteration of an epoch. This stage has the highest computational load due to the added physical simulation at each iteration, but it is crucial for enhancing object stability. Conversely, introducing physical simulation and physical loss from the start would not only excessively increase the computational load but also hinder shape optimization due to incorrect collision information.\n\nAdditionally, in the global rebuttal Fig.1, we present the reconstruction results of the intermediate states across different stages:\n\nStage 1: as shown at 50, 150, 250, 360 epochs.\n\nStage 2: as shown at 360, 430 epochs.\n\nStage 3: as shown at 430, 440, 450 epochs.\n\nAdditionally, all loss functions in our method plays an important role in optimization. Furthermore, in practical implementations, thanks to our adjustment flexibility of rendering uncertainty and physical uncertainty, the various loss weights do not require meticulous tuning for each scene to achieve good results. \n\nIn summary, although the multi-stage training and loss designs may appear complex, they exhibit flexibility in deployment and robustness across various scenes. Future research could focus on reducing system complexity while preserving current performance levels.\n\n**3. Whether the proposed physical solution can handle noise in the ground-truth 2D instance mask?**\n\nYes, the physical solution we proposed, including physical uncertainty and physical loss, can alleviate noise in the ground-truth 2D instance mask. For example, in Fig.6 of the main paper, the final reconstruction result of our method includes legs which the GT instance mask of the chair lacks. Moreover, with the rapid development of segmentation and tracking models, e.g., SAM [4] and SAM2 [5], future efforts can be devoted to decompositional neural scene reconstruction with object masks from images obtained with off-the-shelf models.\n\n**4. How does the system perform when the points are sparse and non-uniformly distributed?**\n\nBased on our current implementations, sparsely and non-uniformly distributed particles may lead to the following two problems in contact resolution:\n* Missed collision detections: objects may pass through the ground.\n* Numerical instability: too sparse points will affect the calculation of impulse J (eq.A27a and eq.A27b) in supp Sec.C.3 Colliding Contact.\n\nThus, in our paper, we propose the SP-MC with an adaptive object bounding box (as discussed in lines 912-915) to effectively address these issues by ensuring a uniform and dense surface particle distribution.\nAdditionally, extending our particle-based rigid body simulator to accommodate particles of varying sizes and anisotropy would be an interesting future direction, thereby better supporting low-quality particle distributions.\n\n[1] 3D Gaussian Splatting for Real-Time Radiance Field Rendering\n\n[2] 2D Gaussian Splatting for Geometrically Accurate Radiance Fields\n\n[3] High-quality Surface Reconstruction using Gaussian Surfels\n\n[4] Segment and Track Anything\n\n[5] SAM 2: Segment Anything in Images and Videos", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We sincerely thank all the reviewers for their efforts in reviewing this manuscript and for their constructive comments. We particularly appreciate their recognition of our proposed framework as "mind-blowing, insightful" (qV3n) and "novel" (fqR5, mFxQ, qV3N), with "extensive experiments and ablations" (5HaF, qV3n, fqR5) demonsrating "strong results" (fqR5) and "significant improvements" (qV3n). Additionally, the reviewers recognize our paper as "well written and easy to follow with enough implementation details" (5HaF, mFxQ, fqR5).

Below, we address the questions from the reviewers respectively. Per the reviewers' requests, we also provide a additional PDF page in the global rebuttal that includes:
* Quantitative comparisons of varying maximum simulation steps in a forward simulation.
* Quantitative results from increasing the number of training epochs with physical simulations.
* Visualization of intermediate results during training.

We hope these clarifications enhance the reviewers' confidence in evaluating our work. We will integrate all the feedbacks to complement the original draft. Together, we are confident in delivering a high-quality paper beneficial to the research community.

Meta-review: The paper presents PhyRecon, a method combining differentiable rendering and physics simulation to improve 3D reconstruction using neural implicit representations. It introduces Surface Points Marching Cubes (SP-MC) to enhance surface detail and accounts for uncertainties to improve realism and stability. This approach integrates physics simulation with rendering for better 3D reconstruction and significantly improves reconstruction quality and stability. The reviews have concerns on the approach might be too complex, and focuses only on static, rigid objects. After carefully reading the paper, reviews, and rebuttals, the AC recommends to accept the paper.
Decision: Accept (poster)