# phyrecon_physically_plausible_neural_scene_reconstruction__bfd20887.pdf

PHYRECON: Physically Plausible Neural Scene Reconstruction

Junfeng Ni1,2 , Yixin Chen2 , Bohan Jing2, Nan Jiang2,3, Bin Wang2, Bo Dai2, Puhao Li1,2, Yixin Zhu3, Song-Chun Zhu1,2,3, Siyuan Huang2

Equal contribution Work done as an intern at BIGAI 1 Tsinghua University 2 State Key Laboratory of General Artificial Intelligence, BIGAI 3 Peking University https://phyrecon.github.io

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.

1 Introduction

3D scene reconstruction is 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 in 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 on neural implicit representation rely solely on rendering supervision, lacking explicit incorporation of physical constraints, such as those from physical simulators. This oversight compromises their ability to optimize 3D shapes for stability, as the models are 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 is due to overly-smoothed and averaged optimization

38th Conference on Neural Information Processing Systems (Neur IPS 2024).

Previous Method

t=0 t=50 Final state

t=0 t=50 Final state t=0 Final state

Physical Simulator

Differentiable Rendering

Monocular Cues

Differentiable Physics Simulation In Isaac Gym

In Isaac Gym

Figure 1: Illustration of PHYRECON. We leverage both differentiable physics simulation and differentiable rendering to 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 multi-modal inconsistency (e.g., depth-normal), multi-view inconsistency, and inaccurate predictions deviating from the true underlying 3D geometry. As a 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 rendering and differentiable physical simulations for learning implicit surface representations. Specifically, we propose a 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 inconsistencies, we propose a joint uncertainty model describing both rendering and physical uncertainty. The rendering uncertainty identifies and mitigates inconsistencies arising from multi-view geometric priors, while the physical uncertainty reflects the dynamic trajectory of 3D contact points in 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 a physics-guided pixel sampling based on physical uncertainty to help recover slender structures.

We conduct extensive experiments on real datasets including Scan Net [9] and Scan Net++ [76], and the synthetic dataset Replica [62]. Results demonstrate that our method significantly surpasses all state-of-the-art methods in 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, our main contributions are three-fold:

1. We 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.

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 physics-guided 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-art methods. Our results exhibit substantial stability improvements, signaling broader potential for physics-demanding applications.

2 Related Work

Neural Implicit Surface Reconstruction With the increasing popularity of neural implicit representations [48, 79, 35, 6, 49, 7, 40] in 3D reconstruction, recent studies [50, 64, 75] have bridged the volume density in Neural Radiance Fields (Ne RF) [45, 80] with the iso-surface representation, e.g., 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 reconstruction 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.

Incorporating Priors into Neural Surface Reconstruction Various priors, such as Manhattanworld assumptions [20, 18], monocular geometric priors (i.e., normal [63] and depth [77] from off-theshelf model [55, 12, 2]) 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 a 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 to 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 topics such as object generation [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, 24]. In the context of scene reconstruction, previous work primarily focuses on applying collision loss on 3D bounding boxes to adjust their translations and rotations [11, 3], 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, we 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 a refined optimization of object shapes and physical stability.

Given an input set of N posed RGB images I = {I1, . . . , IN} and corresponding instance masks S = {S1, . . . , SN}, our objective is to reconstruct each object and the background in the scene. Fig. 2 presents an 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 a given point p, with the sign indicating whether

Differentiable physics simulation

Depth, Normal prior & GT Instance mask Depth, Normal & Instance mask

Adjusted rendering

Physical uncertainty map

Detach gradient

Interpolation

Depth & Normal uncertainty map

Guided pixel sampling

Uncertainty

Figure 2: Overview of PHYRECON. We incorporate explicit physical constraints in the neural scene reconstruction through a differentiable particle-based physical simulator and a 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 in Sec. 3.3 to 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 a multi-layer perceptron (MLP) 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 {pi = o + tiv | i = 0, 1, . . . , n 1} are sampled, where ti is the distance from the sample point to the camera center. We predict the signed distance s(pi) and the color ci for each point along the ray, and the normal ni is the analytical gradient of the s(pi). The predicted color ˆ C(r), depth ˆD(r), and normal ˆ N(r) for the ray r are computed with the unbiased rendering method following Neu S [64]:

i=0 Tiαici, ˆD(r) =

i=0 Tiαiti, ˆ N(r) =

i=0 Tiαini, (1)

where Ti is the discrete accumulated transmittance and αi is the discrete opacity value, defined as:

j=0 (1 αj), αi = max Φu(s(pi)) Φu(s(pi+1))

Φu(s(pi)) , 0 , (2)

where Φu(x) = (1 + e ux) 1 and u is a learnable parameter.

Object-compositional Scene Reconstruction Following previous work [68, 69, 32], we consider the compositional reconstruction of k objects utilizing their corresponding masks, and we treat the background as an object. More specifically, for a scene with k objects, we predict k SDFs at each point p, and the j-th (1 j k) SDF represents the geometry of the j-th object. Without loss of generality, we set j = 1 as the background object and others as 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 sj(p), j = 1, 2, . . . , k, 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, we propose a particle-based physical simulator and a highly efficient method to transition implicit SDF, which is adept at modeling visual appearances, to explicit representations conducive to physics simulation.

Surface Points Marching Cubes (SP-MC) Existing methods [13, 43] fall short of attaining a satisfactory balance between efficiency and precision in extracting surface points from the SDF-based implicit representation. Thus, we develop a novel algorithm, SP-MC, with differentiable and efficient parallel operations and surface point refinement leveraging the SDF network f( ).

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 Pcoarse (black) are derived through linear interpolation and (e) the fine-grained points Pfine (purple) are obtained by querying the SDF network f( ).

Fig. 3 illustrates the SP-MC algorithm. Formally, let S RN N N denote a signed distance grid of size N obtained using f( ), where S(p) 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 x, y, and z axis, respectively. For example, shifting along the x axis means Sx(i, j, k) = S(i + 1, j, k). Then each vertex p V can be found through the element-wise multiplication of S and the shifted grids Sd {x,y,z}, together with its sign-flipping neighbor vertex denoted as pshift. The coarse surface points Pcoarse are determined along the grid edge via linear interpolation:

Pcoarse = p + S(p) S(p) S(pshift)(pshift p) | p V . (3)

Finally, Pcoarse are refined using their surface normals and signed distances by querying f( ):

Pfine = {p f(p) f(p) | p Pcoarse} . (4)

Benefiting from all operations that are friendly for parallel computation, SP-MC is capable of extracting 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 a 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 Euler time integration. Besides gravity, the changes in the rigid body state are induced by contact and friction forces, which are resolved using an impulse-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 Pfine until reaching stability or maximum simulation steps. Utilizing the trajectories and contact states, we present our simple yet effective physical loss: Lphy = P

p Pcontact p p0 2 , where the initial position of each point p is denoted as p0 and its first contact position with the supporting plane as p . Our physical loss intuitively penalizes the object s movement and is only applied to the contact points Pcontact instead of all the surface points. If the physical loss is homogeneously applied to all the surface points, it will contradict the rendering cues when the object is unstable, leading to degenerated results. The contact points Pcontact, 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.

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 ud and normal uncertainty un of a 3D point as view-dependent representations [51, 58, 71], which we utilize the appearance network g( ) to predict along with the color c for a 3D point p:

g : (p R3, n R3, v R3, f R256) 7 (c R3, ud R, un R), (5)

where n is the normal at p, v is the viewing direction, and f is a geometry latent feature output from the SDF network f( ).

The depth uncertainty ˆUd(r) and normal uncertainty ˆUn(r) of ray r are computed using ud and un through volume rendering, similar to Eq. (1). Subsequently, we modulate the L2 depth loss LD(r) and L1 normal loss LN(r) based on the rendering uncertainty:

LDU (r) = ln(| ˆUd(r)| + 1) + LD(r)

| ˆUd(r)| , LNU (r) = ln(| ˆUn(r)| + 1) + LN(r)

| ˆUn(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, we represent the physical uncertainty field with a dense grid Gu-phy RN N N. The physical uncertainty uphy(p) R 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 Pcontact. For each contact point, we track its trajectory from p0 to p , forming the uncertain point set Pu. We design a loss function Lu-phy to update Gu-phy:

Lu-phy = ξ X

p Pu uphy(p), (7)

where ξ > 0 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 ˆUphy(r) of the pixel corresponding to ray r is computed via volume rendering.

Finally, we present the rendering losses re-weighted by both rendering and physical uncertainty:

LDU (r) ˆUphy(r) + 1 , LN = X

LNU (r) ˆUphy(r) + 1 , LS = X

LS(r) ˆUphy(r) + 1 . (8)

The segmentation loss LS(r) is weighted by physical uncertainty only since we use view-consistent instance masks following prior work [68, 69, 32]. We detach ˆUphy(r) in the above losses, ensuring the physical uncertainty field Gu-phy reflects accurate physical information from the simulator.

Physics-Guided Pixel Sampling A critical observation is that intricate object structures occupy only a minor portion of the image, leading to a 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 a physics-guided pixel sampling strategy to enhance the learning of detailed 3D structures. Using physical uncertainty, we calculate

the sampling probability of each pixel in the entire image by: p(r) = ˆUphy(r) P r R ˆUphy(r) .

3.4 Training Details

In summary, our overall loss function is: L = LRGB + LD + LN + LS + Lphy + Lu-phy + Lreg, where the loss weights are omitted for simplicity. Following prior work [16, 77, 32], we introduce

Mono SDF RICO Object SDF++ Ours

Scan Net++ Scan Net Replica

Figure 4: Qualitative results of indoor scene reconstruction. Examples from Scan Net++ [76], Scan Net [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.

Lreg to 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, we empirically divide the training into three stages. In the first stage, we exclusively leverage the rendering losses with rendering uncertainties. In the second stage, we introduce our physical simulator to incorporate physical uncertainty into rendering losses and enable physics-guided pixel sampling. Finally, we integrate the physical loss using a 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.

4 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 on both the synthetic dataset Replica [62] and real datasets Scan Net [9] and Scan Net++ [76]. We use 8 scenes from Replica following the setting of Mono SDF [77] and Object SDF++ [69]. For Scan Net, we use the 7 scenes following RICO [32]. Additionally, we conduct experiments on 7 scenes from the more recent real-world dataset Scan Net++ with higher-quality images and more accurate camera poses. More data preparation details are in Appx. D.1.

Baselines and Metrics We choose Mono SDF [77], RICO [32], and Object SDF++ [69] as the baseline models. For reconstruction, we measure the Chamfer Distance (CD), the F-score (F-score),

Initial state 10 steps 50 steps Final state

RICO Object SDF++ 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 on physical stability.

Dataset Method Scene Recon. Obj. Recon. Obj. Stability

CD F-Score NC CD F-score NC SR (%)

Mono SDF 3.94 78.14 89.37 - - - - RICO 3.87 78.45 89.64 4.29 85.91 85.45 26.43 Object SDF++ 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

Mono SDF 8.97 60.30 84.40 - - - - RICO 8.92 61.44 84.58 9.29 73.10 79.44 29.68 Object SDF++ 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

Mono SDF 3.87 85.01 88.59 - - - - RICO 3.86 84.66 88.68 4.16 80.38 84.30 32.89 Object SDF++ 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 Mono SDF [77]. These metrics are evaluated for both scene and object reconstruction across all datasets. As Mono SDF is designed to reconstruct the whole scene, we 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 is 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 in Appx. D.2.

4.2 Results

Tab. 1 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 in Isaac Gym [39]. Fig. 6 showcases visualizations of several critical components in our 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 in object stability, outperforming all baselines by at least 40% across all datasets. This signifies a significant

Image Random sampling Depth prior Depth uncertainty map

Rendered depth

Physical uncertainty map Guided pixel sampling

Normal prior

Normal uncertainty map Rendered normal Rendered instance mask

GT Instance mask

Reconstructed mesh

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 a better-reconstructed 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. 4 and 5 and 2) enhanced learning of intricate structures, facilitated by joint uncertainty modeling and physics-guided pixel sampling. Results from RICO [32] and Object SDF++ [69] in Fig. 5 fail to capture all the chair legs, leading to inherent instability.

Reconstruction Quality Note that the improvement of the physical plausibility does not come under the sacrifice of reconstruction quality. From the results in Tab. 1, our method surpasses all the state-of-the-art 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 on the tables, shown in the zoom-in views.

Joint Uncertainty Modeling Lastly, we discuss the intermediate results of our joint uncertainty modeling exemplified on the Scan Net 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 physicsguided pixel sampling, informed by the physical uncertainty map, prioritizes intricate structures over the random sampling strategy. Moreover, it 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 in 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 Scan Net++ dataset.

RU PU PS PL Scene Recon. Obj. Recon. Obj. Stability

CD F-Score NC CD F-score NC SR (%)

3.82 78.64 89.52 4.13 86.26 85.42 25.28 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 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 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 uncertainty (PU), and physics-guided pixel sampling (PS). Tab. 2 and Fig. 7 illustrate quantitative and qualitative comparisons, respectively. Key findings are as follows:

(a) Baseline (b) +PL (c) +RU (d) +RU+PU (e) +RU+PU+PS (f ) Ours

Figure 7: Visual comparisons for ablation study. PL denotes physical loss, RU for rendering uncertainty, PU for physical uncertainty and PS for physics-guided sampling.

Image Input View Novel View

Input View Novel View

Figure 8: Qualitative examples for failure cases.

1. Physical loss significantly improves 3D object stability. However, due to the insufficient regularization 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. 2. Rendering uncertainty, physical uncertainty, and physic-guided pixel sampling collectively contribute to enhancing the reconstruction quality, particularly on 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 to degenerated object shapes, e.g., bulges, to maintain physical stability. This is due to insufficient supervision from the rendering losses. Fig. 8 (b) illustrates that objects may be 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 be addressed by incorporating topological regularization to penalize disconnected parts of the object or further enhance the physical simulator.

5 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-based physical simulator and joint uncertainty modeling, facilitating efficient optimization with both rendering and physical losses. Extensive experiments validate the effectiveness of PHYRECON, showcasing its significant outperformance of all state-of-the-art methods in terms of both reconstruction quality and physics stability, underscoring its potential for future physics-demanding applications.

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 (2022ZD0114900), the National Natural Science Foundation of China (62376031), the Beijing Nova Program, the State Key Lab of General AI at Peking University, and the PKU-Bing Ji Joint Laboratory for Artificial Intelligence.

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In Appx. A, we first delineate the comprehensive Phy Recon algorithm, followed by providing details about the loss functions and implementation. Subsequently, we delve into the SP-MC Algorithm in Appx. B, offering a 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 a 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. 1, 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 Neu S [64]. For each point, we predict individual object SDFs {s1, s2, ..., sk}, color c, depth uncertainty ud, 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 ˆ C, depth ˆD, normal ˆ N, instance logits ˆ H, depth uncertainty ˆUd, normal uncertainty ˆUn, and physical uncertainty ˆUphy for each ray via volume rendering [10]. In particular, the physical uncertainty of the pixel corresponding to ray r is computed via:

i=0 Tiαiuphy,i. (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 P obj and background surface points P bg using SP-MC. Utilizing our proposed physical simulator, we track the trajectories and contact states of the object points. For each point p P obj, we identify its initial position as p0 and its first contact position with the supporting plane as p . The object points that have made contact with the supporting plane are collectively denoted as Pcontact. If a surface point p does not contact the supporting plane, p remains equal to p0. 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 Pu by interpolating from the initial position p0 to the first contact position p for all the points in Pcontact. 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, we calculate the color loss and regularization loss terms. Finally, we optimize the network f( ), g( ), and the physical uncertainty grid Gu-phy using the aforementioned losses.

A.2 Loss Function Details

RGB Reconstruction Loss To learn the surface from images input, we need to minimize the difference between ground-truth pixel color and the rendered color. We follow the previous work [77] here for the RGB reconstruction loss:

r R || ˆC(r) C(r)||1, (A2)

Algorithm 1 Phy Recon per Training Iteration

1: Physics-Guided Pixel Sampling {r} # Guided by physical uncertainty 2: For each r do 3: Neu S ray sampling {p} 4: For each p do 5: // SDF prediction 6: s1, s2, ..., sk, z = f(p) 7: s = min(s1, s2, ..., sk) 8: σ = Φ(s) # Φ: transform SDF to density 9: // Appearance and rendering uncertainty prediction 10: c, ud, un = g(p, n, v, z) 11: // Physical uncertainty via trilinear interpolation 12: uphy = Interp(Gu-phy, p) 13: end for 14: ˆ C, ˆD, ˆ N, ˆ H, ˆUd, ˆUn, ˆUphy = Volume Rendering( ) 15: end for 16: For j=2,...,k do 17: // Background surface points extraction 18: P bg = SP-MC(s1) 19: // Object surface points extraction 20: P obj = SP-MC(sj) 21: // Differentiable Physics Simulation 22: {p0, p }, Pcontact = Simulator(P obj, P bg) # p0: initial position, p : first contact position 23: // Physical uncertainty points extraction 24: Pu = P

p Pcontact Interp(p0, p ) 25: // Physics-related losses 26: Lu-phy = ξ P

p Pu uphy(p)

27: Lphy = P

p Pcontact L1(p0, p ) 28: end for 29: // Rendering losses modulated by ˆUd, ˆUn, ˆUphy 30: Compute Lc, Ld, Ln, Ls, Lreg 31: Optimize network f( ), g( ) and physical uncertainty grid Gu-phy

where ˆC(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 ˆD(r) and the depth estimation D(r) from the Marigold model [27]:

r R ||(w ˆD(r) + q) D(r)||2, (A3)

where w and q are the scale and shift values to match the different scales. We solve w and q with a least-squares criterion, which has the closed-form solution. Please refer to the supplementary material of [77] for a detailed computation process.

Normal Consistency Loss We utilize the normal cues N(r) from Omnidata model [12] to supervise the rendered normal through the normal consistency loss, which comprises L1 and angular losses: LN = X

r R || ˆ N(r) N(r)||1 + ||1 ˆ N(r)T N(r)||1. (A4)

The volume-rendered normal and normal estimation will be transformed into the same coordinate system by the camera pose.

Semantic Loss Building on previous work [68, 32], we transform each point s SDFs into instance logits h(p) = [h1(p), h2(p), . . . , hk(p)], where

hj(p) = γ/(1 + exp(γ sj(p))). (A5)

Here, γ is a fixed parameter. Subsequently, we can obtain the instance logits ˆ H(r) Rk of pixel corresponds to the ray r using volume rendering as:

i=0 Tiαihi. (A6)

We minimize the semantic loss between volume-rendered semantic logits of each pixel and the ground-truth pixel semantic class. The semantic objective is implemented as a cross-entropy loss:

j=1 hj(r) log hj(r). (A7)

The hj(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:

i (|| min 1 j k sj(pi)||2 1). (A8)

The Eikonal loss is applied to the gradient of the scene SDF, which is the minimum of all the SDFs.

Background Smoothness Loss Building upon RICO [32], we use background smoothness loss to regularize the geometry of the occluded background to be smooth. Specifically, we randomly sample a P P size patch every TP iterations within the given image and compute semantic map ˆ H(r) and a patch mask ˆ M(r): ˆ M(r) = 1[arg max( ˆ H(r)) = 1], (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:

m,n=0 ˆ M(rm,n) (| ˆD(rm,n) ˆD(rm,n+2d)| + | ˆD(rm,n) ˆD(rm+2d,n)|),

m,n=0 ˆ M(rm,n) (| ˆ N(rm,n) ˆ N(rm,n+2d)| + | ˆ N(rm,n) ˆ N(rm+2d,n)|),

Lbs = L( ˆD) + L( ˆ N) (A11)

Object Point-SDF Loss and Reversed Depth Loss To regularize SDFs of the object and the background, we further employ an object point-SDF loss to regulate objects within the room Lop and a reversed depth loss Lrd, following previous work [32].

Specifically, for the sampled points along the rays, we initially apply a root-finding algorithm among the background SDF of these points to determine the zero-SDF ray depth t . Then, the object point-SDF loss can be expressed as:

Lop = 1 k 1

j=2 max (0, ϵ sj(p(ti))) 1[ti > t ], (A12)

which pushes the objects SDFs at points behind the surface to be greater than a positive threshold ϵ.

Moreover, Lrd optimizes the entire ray s SDF distribution rather than focusing solely on discrete points. Specifically, the ray depths {ti|i = 0, 1, . . . , n 1} are transformed into the reversed ray depths, denoted as {ˆti|i = 0, 1, . . . , n 1}, where

ˆti = (t0 + tn 1) tn 1 i. (A13)

The reversed depth do of the hitting object is determined by the pixel s rendered semantic and db of the background. The reversed depth loss is computed as: Lrd = max(0, db do), (A14)

Finally, the regularization loss in our main paper Lreg is computed as follows: Lreg = Lbs + Lop + Lrd + LEikonal (A15)

A.3 Implementation Details

We implement our model in Py Torch [53] and utilize the Adam optimizer [28] with an initial learning rate of 5e 4. We sample 1024 rays per iteration. When incorporating physics-guided 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 Scan Net [9] and Scan Net++ [76] datasets, and 2000 epochs on Replica [62] dataset. As introduced in Sec. 3.4, training is divided into three stages. For the Scan Net [9] and Scan Net++ [76] datasets, the second and final stages begin at the 360th and 430th epochs, respectively, while for the Replica [62] dataset, these stages start at 1700th and 1980th epochs. All experiments are conducted on a single NVIDIA-A100 GPU.

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, LEikonal, Lbs, Lop, Lrd, respectively. Additionally, we set ξ = 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 to obtain a discretized signed distance field S RN N N with grid vertices denoted as P . Second, we shift the SDF grid S along the x, y, and z axis, respectively, to locate zero-crossing vertices. For example, shifting along the x axis results in Sx(i, j, k) = S(i + 1, j, k). We then obtain coarse surface points Pcoarse through linear interpolation. We refer to this step as the Shift-Interpolation operation in Alg. 2. Third, Pcoarse is refined to yield Pfine using their surface normals and signed distances. For a detailed algorithm, please refer to Alg. 2.

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 δ = 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 Pcoarse, but also ensures reliable backpropagation of the gradient in the physical loss from an explicit physical simulator to the implicit SDF-based surface representation. Directly converting the SDF into a 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 et 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 assessment of time and memory involves the transformation from SDF to surface points, which are prepared

Algorithm 2 Surface Points Marching Cubes (SP-MC) Algorithm

Input: SDF network f( ), Resolution N, Object Boundary B, Object Index j Output: Surface Points Pfine function SHIFT-INTERPOLATE(S, P , shift axis v)

// Grid shifting Sshift S shift along v M (S Sshift) < 0 # : Hadamard product, M: index mask V P [M] // Linear Interpolation Pcoarse = n p + S(p) S(p) S(pshift)(pshift p) | p V o # pshift: sign-flipping neighbor of p return Pcoarse end function 1: Get S RN N N and P by Voxelization: 2: P voxelize object boundary B in resolution N 3: S {fj(p) | p P } 4: Get Pcoarse by Shift-Interpolate Operation: 5: Pcoarse {SHIFT-INTERPOLATE(S, P , v) | v {x, y, z}} 6: Get Pfine by Refinement Step: 7: Pfine = {p f(p) f(p) | p Pcoarse} 8: return Pfine

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 Scan Net++ [76]. From the results presented in Tab. A1, SP-MC consumes considerably less running time and GPU memory compared to 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) GPU Memory (MB)

Kaolin [13] 0.482 21.327 SP-MC 0.264 13.673

0.218 (45.2%) 7.654 (35.9%)

Furthermore, we also note that Mezghanni et al. [43] propose a method for extracting surface points through direct thresholding of the SDF in the discretized signed distance field S, defined as:

Pcoarse = {p | |s(p)| < δ, p S} . (A16)

Although this method is conceptually simpler, it introduces a surface bias δ, leading to an inevitable discrepancy between the extracted points and the actual surface points. It additionally requires a 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 < δ. 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 on the cabinet and the monitor on the table

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 to enable convenient and efficient simulation of rigid bodies depicted as a collection of spherical particles of uniform size.

Naming convention In this section, we 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 to index particles.

A 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 center of mass. The state at any given moment t is encapsulated by the tuple:

r(t) q(t) v(t) ω(t)

where r(t) R3 and q(t) R4 specify the position and orientation of the reference frame in relation to the world frame at time t, respectively. q(t) = [q0, q1, q2, q3]T is a quaternion, where q0 is a scalar value indicating the rotation angle, and [q1, q2, q3] is the normalized rotation axis. v(t) R3 and ω(t) R3 represent the linear and angular velocities, respectively.

In our simulator, a rigid body is represented by a collection of spherical particles of uniform size, as depicted in Fig. A1. When a body consists of N such particles, all with identical mass m and initially position at pi(0), the total mass of the system is:

M = N m. (A18)

Furthermore, the system s center of mass, also serving as the origin of the reference frame r(0), is determined by:

r(0) = P m pi(0)

Thus, the relative position of each particle within the local rigid body frame is ui = [ui0, ui1, ui2]T = pi r(0). Consequently, the moment of inertia within the reference frame can be derived as:

P(ui12 + ui22)m P(ui0ui1)m P(ui0ui2)m P(ui0ui1)m P(ui02 + ui22)m P(ui1ui2)m P(ui0ui2)m P(ui1ui2)m P(ui02 + ui12)m

The total mass M and the moment of inertia Iref are intrinsic physical properties of the rigid body, reflecting its resistance to external forces and torques. It is important to note that I is not constant as it varies with the body s orientation.

Table A2: Three different contact status and their corresponding criteria. Colliding Contact, Resting Contact and Separation

Colliding Contact Resting Contact Separation

pa i pb j < 2r pa i pb j < 2r pa i pb j < 2r vc Nc < ϵ vc Nc < ϵ vc Nc > ϵ

C.1 Forward Dynamics

According to Newton s second law, when subjected to external forces and moments, a rigid body s linear and angular velocities will change, altering its state. Using explicit Euler time integration, the evolution of a state variable between successive time steps can be summarized as follows:

v(t + t) = v(t) + t M 1f(t) (A21a) r(t + t) = r(t) + tv(t) (A21b)

ω(t + t) = ω(t) + t I 1(t)τ(t) (A21c)

q(t + t) = q(t) + [0, t

2 ] q(t). (A21d)

Here, t is the time step size; f R3 and τ R3 are the total external forces and torques, respectively. The inertia tensor of the rotated body I is given by:

I(t) = R(t)Iref RT (t) (A22)

where the rotation matrix R(t) is derived from the corresponding quaternion q(t) = [q0, q1, q2, q3]T using the formula:

2(q2 0 + q2 1) 1 2(q1q2 q0q3) 2(q1q3 + q0q2) 2(q1q2 + q0q3) 2(q2 0 + q2 2) 1 2(q2q3 q0q1) 2(q1q3 q0q2) 2(q2q3 + q0q1) 2(q2 0 + q2 3) 1

During scene reconstruction, changes in the state of a rigid body are induced solely by gravity and contact. The gravity force is accounted for by setting f = Mg, with g represents the acceleration of gravity. On the other hand, contact and friction forces are resolved using an 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 as a set of particles, collision detection between complex-shaped rigid bodies can be simplified to relatively simple inter-particle collisions.

Specifically, for two particles that are sufficiently close (particle i in rigid body a and particle j in rigid body b, both with a radius r), we can accurately approximate the contact position using the centers of the particles and define the contact normal as:

Nc = pa i pb j pa i pb j , (A24)

and the relative velocity at the contact points is given by:

vc = (va + Raua i ) (vb + Rbub j), (A25)

with the normal and tangential component are defined as: vc = (vc Nc)Nc (A26a) vc = vc vc . (A26b)

Their relative contact status can be categorized into three types based on the criteria illustrate in Tab. A2. Among these, only colliding and resting contact require further processing in the simulation pipeline.

C.3 Colliding Contact

The criterion vc Nc < ϵ 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 in 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 friction model, we know the physically plausible relative velocity v c = v c + v c after contact should be:

v c = µvc (A27a) v c = αvc (A27b)

α = max(1 η(1 + µ) vc

vc , 0), (A27c)

where µ is the coefficient of restitution and η is the friction coefficient. To achieve the desired velocity v c, the required impulse J at the contact point is calculated as:

J = K 1(v c vc) (A28a)

K = I M a + I M b [Raua i ] (Ia) 1[Raua i ] [Rbub j] (Ib) 1[Rbub j] , (A28b)

where I represents the 3 3 identity matrix and the operator [ ] transform a vector into a skewsymmetric matrix. Under the influence of J, the linear and angular velocity of the involved rigid bodies are updated as follows:

va = va + 1 M a J (A29a)

ωa = ωa + (Ia) 1(Raua i J) (A29b)

vb = vb 1 M b J (A29c)

ωb = ωb (Ib) 1(Rbub j J). (A29d)

These updates ensure that the bodies respond correctly to the collision, separating or 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 is 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 vc Nc < ϵ indicates that an object maintains continuous contact with another without significant changes in position or orientation, such as a 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 a dynamic rigid body remains stationary or 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 a rigid body over a short historical period [46]:

rwa = γ rwa + (1 γ) v2 up, (A30)

where the weight γ lies within the range [0, 1]; vup represents the upper bound of the current speed of the rigid body [46], which can be effectively approximated by:

v2 up = (v + Rumax ω)T (v + Rumax ω) (A31)

2 (v T v + (ωT ω) (u T maxumax)). (A32)

Here umax denotes the maximum distance from any particle on the surface to the body s center of mass, calculated once during the initialization phase. Given that our simulator is solely affected by gravity, a body is set to sleep if rwa meets the following condition:

rwa < g t. (A33)

C.5 Implementation Details

For the sake of reproducible, we present the pipeline of our particle-based rigid body simulation during each time step as in Alg. 3. Our simulator and its gradient support are developed using Diff Taichi [19], which is a high-performance differentiable physical programming language designed for physical simulations, and computational science.

For all of our examples, we set time step t = 0.01 s; particle radius r = 0.005 m; particle mass m = 0.01 kg; the coefficient of restitution µ = 0.0, the friction coefficient η = 0.4, the relative velocity criterion ϵ = 1e 5 and γ = 0.1.

Algorithm 3 The pipeline of our particle-based rigid body simulator

1: Input: Initial particle positions pi(0) for each rigid body i in the scene. 2: Output: Final particle positions pi(t) when their belonging rigid body reaches stable equilibrium, with flags for all particles that have collided. 3: 4: // physical properties 5: For each rigid body do 6: compute mass and center of mass: M Eq. (A18), r(0) Eq. (A19) 7: For each particle do 8: compute particle position in reference frame: ui pi(0) r(0) 9: end for 10: compute inertia matrix in reference frame: Iref Eq. (A20) 11: compute the maximum distance: umax max(ui) 12: end for 13: 14: For time step t do 15: // forward dynamics 16: For each awake rigid body do 17: apply gravity force: f(t) Mg 18: compute linear and angular velocities v(t) Eq. (A21a), ω(t) Eq. (A21b) 19: compute position and orientation: r(t) Eq. (A21c), q(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: pi(t) R(t)ui + r(t) 26: end for 27: 28: // collision detection and colliding contact resolve 29: While no colliding contact do 30: For each pair of particles do 31: if pa i pb j < 2r then 32: compute contact normal: Nc Eq. (A24) 33: compute contact velocity: vc Eq. (A25), vc Eq. (A26a) 34: if vc Nc < ϵ then 35: compute desired contact velocity: v c Eq. (A27a), Eq. (A27a)

36: compute required impulse: J Eq. (A28a) 37: if vc > r/ t 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 46: For each rigid body do 47: update linear and angular velocities: v(t) Eq. (A29a), ω(t) Eq. (A29a) 48: end for 49: 50: // resting contact 51: For each rigid body do 52: compute historical motion information vup Eq. (A32), rwa Eq. (A30) 53: if rwa < g t then 54: set the body to sleep 55: v [0, 0, 0], ω [0, 0, 0] 56: end if 57: end for 58: end for

D More Experiment Details

D.1 Data Preparation

Monocular Cues We utilize a pre-trained Marigold model [27] to generate a depth map D for each input RGB image. It s important to note that estimating the absolute scale in general scenes is challenging, thus D should be regarded as a relative depth cue. Furthermore, we employ another 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.

GT Instance Mask For the Scan Net++ [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 Scan Net [9] and Relica [62] datasets, we observed discrepancies in the masks provided by RICO [32] and Object SDF++ [69]. To ensure a fair comparison with the baselines, we merged their instance masks into consistent ones. In our experiments, we focused solely on object-ground support for simplicity and training efficiency, leaving the determination of 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 a 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, we import the object shape and the reconstructed background into the simulator 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, we simulate for T = 200 steps with a time step of t = 0.016s (i.e. 60Hz).

Reconstruction Metrics To evaluate 3D scene and object reconstruction, we use the CD in cm, F-score with a threshold of 5cm 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-Accuracy and Normal-Completeness. We follow previous work [18, 77, 32, 69] to evaluate reconstruction only on the visible areas. These metrics are defined in Tab. A3.

Table A3: Evaluation metrics. We show the evaluation metrics with their definitions that we use to measure reconstruction quality. P and P 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) Accuracy+Completeness

Accuracy mean p P

min p P ||p p ||1

Completeness mean p P

min p P||p p ||1

F-score 2 Precision Recall

Precision+Recall

Precision mean p P

min p P ||p p ||1 < 0.05

Recall mean p P

min p P||p p ||1 < 0.05

Normal Consistency Normal Accuracy+Normal Completeness

Normal Accuracy mean p P n T p np s.t. p = arg min p P ||p p ||1

Normal Completeness mean p P n T p np s.t. p = arg min p P ||p p ||1

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 3rd stage, the reconstructed object reaches stability when the physical loss is introduced.

150 epoch 250 epoch

Figure A2: Visualization of intermediate results during training. Epochs 360 and 430 mark the beginning of the 2nd and 3rd training stages.

D.4 Time Consumption

We conduct additional experiments on example scenes from the Scan Net++ 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 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 Object SDF++ as shown in Tab. A4 on Scan Net++. 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 Object SDF++ Ours

Stage 1 (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 In 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 on example scenes in Scan Net++, 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, as most objects achieve a stable state within 100 simulation steps. This also explains why increasing the maximum simulation steps leads to longer simulation time, though this increase is not linear to the number of steps. Consequently, in the experiments for our main paper, we chose 100 as the maximum simulation steps.

Table A5: Performance of varying maximum simulation steps in one forward simulation.

Max Sim. Steps CD F-Score NC SR (%) Total time Stage-3 time scene/obj scene/obj scene/obj

25 2.91/3.24 88.17/86.61 91.56/85.04 69.23 10.64h 1.27h 50 2.83/3.26 88.41/86.66 91.82/85.04 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 91.72/85.12 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 a 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 Scan Net++. 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.

Table A6: Performance of increasing the number of training epochs with physical simulations.

Training epoch CD F-Score NC SR (%) Total time 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 0 435 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 the 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 or further enhancing the simulator. Plus, our particle-based simulator treats all surface points from an object as a rigid body, thus it cannot handle soft bodies (which deform after collision) or dynamic scenes.

In addition, our current framework requires additional object-supporting information for the simulation. While determining ground-object support is straightforward, identifying more complex relationships remains challenging. The implicit representation is optimized through per-object physical 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.

F Potential Negative Impact

3D scene 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 of these impacts include potential privacy concerns in public areas, surveillance and security risks. Addressing these concerns requires careful consideration of ethical guidelines, regulatory frameworks, and responsible development practices to ensure that 3D scene reconstruction is deployed in a manner that respects privacy, security, and societal well-being.

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Justification: We provide detailed descriptions of the experimental details in the main text Sec. 3.4 and supplementary materials Appx. A and Appx. D.

Guidelines:

The answer NA means that the paper does not include experiments. The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [No]

Justification: Our experiments fixed the random seed following the previous works.

Guidelines:

The answer NA means that the paper does not include experiments. The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) The assumptions made should be given (e.g., Normally distributed errors). It should be clear whether the error bar is the standard deviation or the standard error of the mean. It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g., negative error rates). If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We provide detailed descriptions of the experimental details in the main text Sec. 3.4 and supplementary materials Appx. A and Appx. D.

Guidelines:

The answer NA means that the paper does not include experiments. The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn t make it into the paper).

9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the Neur IPS Code of Ethics https://neurips.cc/public/Ethics Guidelines?

Answer: [Yes]

Justification: We comply with the Code of Ethics in every respect.

Guidelines:

The answer NA means that the authors have not reviewed the Neur IPS Code of Ethics. If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We discuss the potential societal impacts of our work in the introduction, conclusion, and supplementary materials Appx. F.

Guidelines:

The answer NA means that there is no societal impact of the work performed. If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [NA] Justification: The paper poses no such risks. Guidelines:

The answer NA means that the paper poses no such risks. Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. 12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] Justification: The paper uses open-source datasets, and all sources have been properly cited. Guidelines:

The answer NA means that the paper does not use existing assets. The authors should cite the original paper that produced the code package or dataset. The authors should state which version of the asset is used and, if possible, include a URL. The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. If this information is not available online, the authors are encouraged to reach out to the asset s creators. 13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [NA] Justification: The paper does not release new assets. Guidelines:

The answer NA means that the paper does not release new assets. Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. The paper should discuss whether and how consent was obtained from people whose asset is used. At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. 14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? Answer: [NA] Justification: The paper does not involve crowdsourcing nor research with human subjects. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. According to the Neur IPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? Answer: [NA] Justification: The paper does not involve crowdsourcing nor research with human subjects. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the Neur IPS Code of Ethics and the guidelines for their institution. For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.