Title: SplitNeRF: Split Sum Approximation Neural Field for Joint Geometry, Illumination, and Material Estimation
Venue: NeurIPS 2024
Accepted: yes
Keywords: inverse rendering

neural rendering

physically-based
Abstract: We present a novel approach for digitizing real-world objects by estimating their geometry, material properties, and environmental lighting from a set of posed images with fixed lighting. Our method incorporates into Neural Radiance Field (NeRF) pipelines the split sum approximation used with image-based lighting for real-time physically based rendering. We propose modeling the scene's lighting with a single scene-specific MLP representing pre-integrated image-based lighting at arbitrary resolutions. We accurately model pre-integrated lighting by exploiting a novel regularizer based on efficient Monte Carlo sampling. Additionally, we propose a new method of supervising self-occlusion predictions by exploiting a similar regularizer based on Monte Carlo sampling. Experimental results demonstrate the efficiency and effectiveness of our approach in estimating scene geometry, material properties, and lighting. Our method attains state-of-the-art relighting quality after only ${\sim}1$ hour of training in a single NVIDIA A100 GPU.

SplitNeRF: Split Sum Approximation Neural Field for Joint Geometry, Illumination, and Material Estimation

Abstract

We present a novel approach for digitizing real-world objects by estimating their geometry, material properties, and environmental lighting from a set of posed images with fixed lighting. Our method incorporates into Neural Radiance Field (NeRF) pipelines the split sum approximation used with image-based lighting for real-time physically based rendering. We propose modeling the scene's lighting with a single scene-specific MLP representing pre-integrated image-based lighting at arbitrary resolutions. We accurately model pre-integrated lighting by exploiting a novel regularizer based on efficient Monte Carlo sampling. Additionally, we propose a new method of supervising self-occlusion predictions by exploiting a similar regularizer based on Monte Carlo sampling. Experimental results demonstrate the efficiency and effectiveness of our approach in estimating scene geometry, material properties, and lighting. Our method attains state-of-the-art relighting quality after only ∼ 1 hour of training in a single NVIDIA A100 GPU.

Predictions

Relighting

Metalness

Roughness

R

R

。

Render

Render

Albedo

Normals

方

R

R

Courtyard

Interior

Sunrise

Sunset

682

1 Introduction

The idea of creating realistic and immersive digital environments has piqued the imagination of countless science fiction authors, science fiction directors, and scientists. In the past few years, the fields of computer graphics and computer vision have advanced so much that we are capable of

creating photo-realistic environments [6, 28, 21], as well as capturing real-world environments in a way that allows us to render new photo-realistic views [25, 32]. However, the creation of digital twins [9] of objects that can be integrated within photo-realistic environments still requires artists to meticulously hand-design realistic object meshes, materials, and lighting. While this is feasible for generating a few scenes, large-scale digitization requires automatic ways of reconstructing real-world objects along with their corresponding material properties.

In this work, we address the problem of object inverse rendering: extracting object geometry, material properties, and environment lighting from a set of posed images of the object. Inverse rendering enables the seamless integration of virtual objects into different environments with varying illumination conditions from simple image captures taken by commonplace camera sensors.

Neural rendering methods, such as Neural Radiance Fields (NeRF) [18, 1, 34], have revolutionized novel view synthesis, 3D reconstruction from images, and inverse rendering. By directly modeling outgoing radiance at each point in 3D space, NeRF methods excel at accurately recovering scene geometry and synthesizing novel views. However, a drawback of this approach is that the learned radiance representation entangles environment lighting with the rendered scene's properties, making it challenging to recover material properties and illumination. Due to the success of NeRFs in reconstructing scenes, several works have proposed modifications to enable inverse rendering [29, 3, 16, 10, 41]. These works build upon NeRF by decomposing radiance into a function of illumination and material properties but differ in their ways of modeling lighting and reflections. We build upon these methods with the main goal of efficiency without sacrificing reconstruction quality or the ability to recover high-frequency illumination details.

To achieve these goals, we rely on the split sum approximation [11], which is commonly used in efficient image-based lighting techniques and has been successfully applied for inverse rendering before [3, 20]. This approximation involves splitting the surface reflectance equation into two factors: one responsible for pre-integrating illumination and the other for integrating material properties. The separation allows us to estimate pre-integrated illumination using a Multi-Layer Perceptron (MLP). This manner of modeling the pre-integrated illumination function is inspired by the modeling of radiance fields, which model a complex integral of lighting and material properties using an MLP. Correspondingly, our illumination representation inherits beneficial properties observed with the modeling of radiance fields such as smoothness. While MLPs representing pre-integrated illumination have been previously exploited [14, 13], previous works do not supervise the MLP's learning, leading to physically inaccurate illumination representations. We propose a novel regularizer based on Monte Carlo sampling to ensure accurate learning of illumination.

However, the split sum approximation on its own does not take into account self-occlusions. This hinders material property estimation since shadows tend to be incorrectly attributed to objects' albedo. Thus, we derive an occlusion factor to alleviate the effect of self-occlusions. This factor is then approximated via Monte Carlo sampling and used to supervise an occlusion MLP.

Altogether, our method is capable of attaining state-of-the-art relighting results with under an hour of training on a single NVIDIA A100 GPU.

Contributions. We claim the following contributions:

(i) We propose a novel representation for pre-integrated illumination as a single MLP with a corresponding regularization to ensure learning a physically-meaningful representation.

(ii) We derive a method for approximating the effect of self-occlusions on pre-integrated lighting and use it to supervise an occlusion MLP improving material estimation.

(iii) We demonstrate the effectiveness of our method in extracting environmental lighting, geometry, and material properties by achieving competitive reconstruction and relighting quality on both synthetic and real data with under one hour of training on a single NVIDIA A100 GPU.

2 Related works

The problem of digitizing real-world objects and environments has long been a subject of active research in computer vision and computer graphics. We approach this problem through the lenses of neural rendering and neural inverse rendering; paradigms with lots of recent attention. We now provide a brief overview of related works in these areas.

2.1 Neural rendering and 3D reconstruction

Novel view synthesis is the task of rendering new views of a scene given a set of observations of the scene. Neural Radiance Fields (NeRF) [18] and its variants [1, 34, 19, 4, 26] have demonstrated remarkable success in the task of novel view synthesis. NeRF directly models the volumetric scene function by predicting radiance and density at each 3D point in space while supervising learning with a photometric reconstruction loss. Due to its success in implicitly learning accurate 3D reconstructions, several works have branched out to reconstruct accurate meshes through neural rendering [22, 31]. Signed Distance Function (SDF)-based methods [36, 40, 37, 12] model density as a function of the SDF to obtain well-defined surfaces. By increasing sharpness during training in the conversion from SDF to density these methods can transition from volume rendering to surface rendering as they train. While effective, these methods suffer from entangled representations of scene geometry, material properties, and lighting. Our work follows the surface rendering pipeline proposed in [36] but reformulates the radiance prediction to disentangle environment lighting and material properties.

2.2 Neural inverse rendering

The task of inverse rendering is a long-standing problem in computer graphics which consists of estimating the properties of a 3D scene such as shape, material, and lighting from a set of image observations. The success of neural rendering methods for novel view rendering and 3D reconstruction has led to a variety of works [2, 45, 43, 29, 3, 20, 46, 42, 14, 16, 30] exploiting neural rendering for inverse rendering. Due to the challenging nature of this problem, multiple simplifying assumptions have been adopted. Some works simplify the modeling of lighting by using low-frequency representations such as spherical gaussians [2, 45, 43, 29, 46, 39, 10] or low-resolution environment maps [2, 43, 46]. While this approximation generally allows for closed-form solutions of the rendering integral, it does not capture natural high-frequency illumination. Our work leverages the split sum approximation [11], proposed for real-time rendering of image-based global illumination to enable the learning of high-frequency environment lighting. The split sum approximation has been adopted by several inverse rendering methods [3, 20, 17, 13, 14]. These works represent preintegrated lighting with an autoencoder-based illumination network [3, 13], with a set of learnable images for different roughness levels [20, 17], or with an MLP with integrated spherical harmonic encoding as input [14]. Autoencoder-based methods rely on learnt illumination features incompatible with existing rendering pipelines. Learnable images are susceptible to noise and require re-integrating illumination whenever the base illumination is updated. Lastly, using integrated encodings to avoid integrating light leads to a representation that is not physically based. In contrast, we propose modeling pre-integrated lighting as the output of an MLP paired with a novel regularization, which ensures the network correctly learns to represent physically-based pre-integrated lighting. An issue arising from the split sum approximation is that pre-integrated illumination is blind to geometry and does not account for the occlusion of light sources due to geometry throughout the scene. Our work tackles this issue by supervising the prediction of ambient occlusion through Monte Carlo sampling.

3 Methodology

Our method aims to extract a scene's geometry, material properties, and illumination from a set of posed images of the scene. We accomplish this by incorporating a decomposed formulation of radiance into a surface rendering pipeline. In the following sections, we begin with an overview of the surface rendering pipeline. We then detail the physically-based radiance formulation, which allows us to decompose radiance into illumination and material properties. Next, we describe our proposed MLP representation for illumination along with the additional loss term it requires. Afterward, we derive a method for estimating an occlusion factor to account for visibility within the split sum approximation. Finally, we describe additional regularization used to facilitate learning.

3.1 Overview of neural rendering

Neural volume rendering relies on learning two functions: σ ( x ; θ ) : R 3 ↦→ R which maps a point in space x onto a density σ , and L o ( x , ω o ; θ ) : R 3 × R 3 ↦→ R 3 that maps point x viewed from direction ω o onto outgoing radiance L o . The parameters θ that define the density and radiance functions are typically optimized to represent a single scene by using multiple posed views of the scene. To learn these functions, they are evaluated at multiple points along a ray r ( t ) = o -tω o , t ∈ [ t n , t f ] , defined

Γ)ΓLΓ[ΓHΓG

Γ/ΓHΓDΓUΓQΓDΓEΓOΓH

ΓΛ

ΓΛ

ΓΛ

ΓΛ

by the camera origin o ∈ R 3 , pixel viewing direction ω o , and camera near and far clipping planes t n and t f . A pixel color for the ray can then be obtained through volume rendering via:

In practice, a summation of discrete samples along the ray is used to approximate the integral. This volume rendering process allows us to supervise the learning of implicit functions L o and σ , in a pixel-wise fashion through the reconstruction loss:

where R is a batch of rays generated from a random subset of pixels from training images.

The learned geometry can be improved if, instead of directly predicting density σ , a signed distance field (SDF) is learned and then mapped to density. To this end, we follow the SDF formulation proposed in NeuS [36]. Learning a valid SDF requires the use of an additional Eikonal loss term L Eik. For more details, please refer to [36].

Since volume density σ depends only on a point's position in space while output radiance L o depends on both position and viewing direction, neural rendering networks are typically split into a spatial network and a radiance network. As shown in fig. 2, we maintain the spatial network to estimate density along with additional material properties but rely on a physically-based [23] radiance estimation instead of a radiance network.

3.2 Physically-based rendering

Given knowledge of a scene's geometry, material properties, and illumination, it is possible to model the outgoing radiance L o ( x , ω o ) reflected at any position x of an object's surface in direction ω o by integrating over the hemisphere Ω defined by the surface's normal n using the reflectance equation:

where L i is the incoming radiance, a is the object's diffuse albedo, and k d and f s are material properties dependent on the object's Bidirectional Reflectance Distribution Function (BRDF). For clarity, we omit from notation the dependency of incoming radiance on ω i as well as the dependency of material properties on position x . Radiance L o has diffuse and specular components L d and L s . Image-based lighting methods often employ the split sum approximation to calculate specular lighting L s by splitting the integral into two components: one containing the incoming light L i , and one depending only on material properties. We use the Disney [11] microfacet BRDF parameterized

by albedo, metalness, and roughness. The specular component is modeled with the Cook-Torrance GGX [33, 35] BRDF, leading to the following approximation:

where D ( ω i , ω r , ρ ) is the microfacet normal distribution function dependent on the direction of light reflection ω r as well as the surface roughness ρ . The term on the right can be pre-computed as in [11] since it depends only on the BRDF and not a scene's lighting. The term on the left in Equation (4) depends on the scene's lighting and the microfacet distribution function D ( ω i , ω r , ρ ) . The following sections refer to this term as g ( ω r , ρ ) and aim to estimate it with an MLP representation. As shown in fig. 2, we estimate an object's albedo ˆ a , metalness ˆ m , and roughness ˆ ρ from the spatial network.

Ground Truth

Roughness = 0.0

Roughness = 0.2

Roughness = 0.5

Roughness = 1.0

Ours

3.3 Pre-integrated illumination MLP representation

We propose to estimate the pre-integrated lighting g ( ω r , ρ ) at different roughness levels through a pre-integrated illumination MLP ˆ g ( ω r , ρ ) . Please refer to Appendix A.2 for details on how ˆ g ( ω r , ρ ) is used to calculate ˆ L d and ˆ L s . The predictions ˆ g should accurately represent the environment lighting at different levels of roughness. We achieve this through a loss term based on Monte Carlo estimates ¯ g of the original integral for varying roughness and reflected directions using the predicted environment map ˆ L i ( ω ) which can be extracted from ˆ g querying perfect specular reflections ˆ g ( ω, 0) .

where the set S consists of paired samples of directions ω s and roughness ρ s . Directions are taken uniformly on a sphere, and half the roughness samples are taken uniformly in the range [0 , 1] with the other half fixed to 1 to ensure correct learning of diffuse lighting. Please refer to Appendix A.1 for a detailed derivation of ¯ g ( s ) . The set Ω of light direction samples is also taken uniformly on a sphere. While a different sampling could lead to reduced variance, we utilize uniform spherical sampling for ω i to be more computationally efficient. Uniform spherical sampling allows us to share light samples across the batch of predictions, thus reducing the number of evaluation calls to the light function ˆ g ( ω, 0) . We visualize both g and ˆ g in fig. 3 for a specific scene.

3.4 Occlusion factors

The split sum approximation does not consider the occlusion of light sources due to geometry. Occlusions can be incorporated by multiplying incoming light L i by a binary visibility function V i :

Albedo

Ground Truth

Albedo

Ours

Occlusion

Ours

Albedo

w/o Occlusion Loss

Occlusion

w/o Occlusion Loss

with V i taking a value of 1 when there are no occlusions and 0 when incoming light is occluded by geometry. Both diffuse and specular integrals can be rewritten to incorporate visibility via occlusion factors o d ( x ) and o s ( x ) which multiply the split sum diffuse and specular lighting terms respectively. We propose learning the occlusion factors o ( x ) with an MLP by supervising them with Monte Carlo estimates ¯ o ( x ) using the predicted geometry. Please refer to Appendix A.3 for the derivation of Monte Carlo estimates ¯ o ( x ) . Given the Monte Carlo estimates, we supervise the predicted occlusion terms ˆ o ( x ) as follows:

where the sample set X is a random subset of the points sampled for volume rendering, and the weights w are the corresponding normalized volume rendering weights. Weighting the loss function by the volume rendering weights is required so that the occlusion prediction focuses only on learning surface points. The output radiance at each point in space is then calculated as follows:

where γ maps the predicted output radiance ˆ L o from linear to sRGB space.

3.5 Material regularization

To better learn material properties, we introduce a soft regularizer to reduce the prediction of metallic materials. This encourages the model to prefer explaining outgoing radiance through albedo and roughness whilst still allowing the prediction of metallic materials. We implement this regularization as a weighted L 2 loss with the same weighting as for the occlusion loss in Equation (7). That is,

4 Experiments

4.1 Baselines

We compare against Nerfactor [45], NVDiffRec [20], NVDiffRecMC [8], NeRO [14], NMF [16], and TensoIR [10]. Due to the differing evaluation methodologies among these works, we train all baseline methods following publicly released code and report metrics as detailed in the following.

4.2 Experimental setup

Datasets. We report results using the NeRFactor [45] dataset along with extended versions of the NeRF Blender [18] (Blender) and the RefNeRF Shiny Blender [34] (Shiny Blender) datasets. The NeRFactor dataset consists of four synthetic scenes, where test images are rendered under eight different low-frequency lighting conditions. The Blender dataset consists of eight synthetic scenes representing a mix of glossy, specular, and Lambertian objects, while the Shiny Blender dataset consists of six highly reflective synthetic scenes. To showcase the ability of our model to estimate high-frequency environment lighting, we extend the Blender and Shiny Blender datasets by rendering all objects under seven novel high-frequency lighting conditions. All models are trained using 100 posed images, and evaluated on 20 test images consisting of novel views for each lighting condition. Finally, we report qualitative results on real-world objects using the CO3D [24] dataset. Each object in the dataset consists of a set of images captured along a circular path along with automatically extracted foreground segmentation masks. We estimate each image's camera pose with Colmap [27].

Relighting evaluation. We extract geometry from our model in the form of a triangular mesh by using marching cubes [15]. At each predicted mesh vertex, we estimate material properties in the form of an albedo, metalness, and roughness. We then render the predicted geometry using Blender's [5] physically-based shader. Material properties across faces are obtained by interpolating the predicted vertex material properties. We utilize the same Blender rendering pipeline to compute relighting metrics for baselines where explicit meshes and material properties are extracted. Otherwise, predictions are rendered using the provided relighting methodology. Before evaluating metrics, a per-channel scaling factor is computed for each scene to compensate for the albedo-lighting ambiguity. We evaluate the predicted scenes for the NeRFactor, Blender, and Shiny Blender datasets and report the average Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM) [38], and Learned Perceptual Image Patch Similarity (LPIPS) [44] in table 1 and table 2. Metrics are reported as an average across 20 test images and across all illumination maps for each dataset. This metric gives an aggregated performance measure for geometry and material property estimation. Our method attains competitive relighting performance while maintaining a low runtime.

Ground Truth

Pred. Render

Albedo

Metalness

Roughness

Normals

Pred. Env. Map

Albedo evaluation. In addition to overall relighting quality, we evaluate the ability of our method to recover albedo. We report reconstruction metrics (PSNR, SSIM, and LPIPS) on the predicted albedo in table 1. As with the relighting metrics, we apply a per-scaling factor to the albedo predictions before computing reconstruction metrics. Metrics are reported as an average across all 20 test images for each scene in the NeRFactor dataset. We do not report albedo metrics for other datasets due to the lack of ground truth. We exclude results from NMF [16] since the albedo in their lighting formulation is not comparable to the other methods. Thanks to our proposed occlusion factor and material regularization, our method is on average better able to reconstruct albedo.

Normals evaluation. We measure the pixel-wise Mean Absolute Error (MAE) between ground truth and predicted normal images to evaluate geometric reconstruction quality. The MAE is weighted by ground truth alpha values to lower the effect of prediction errors at object borders. Our method recovers a good estimate of geometry for most scenes as evidenced in tables 1 and 2.

Real-world qualitative results. In real-world scenes, the far-field illumination assumption is violated and objects don't follow any specific BRDF model as opposed to synthetic scenes. Both of these differences make inverse rendering from real-world data a much more challenging task than with synthetic data. Therefore, we provide qualitative results in fig. 5 to validate our method on real-world captures from the CO3D dataset. It can be observed that even in this challenging scenario our method is capable of recovering accurate object geometry, as well as providing a reasonable estimation of material properties and environment maps.

5 Discussions

5.1 Ablations

Occlusion loss. We visualize the effects of the proposed occlusion loss in fig. 4. Learning an occlusion factor without supervision leads to errors in the albedo predictions due to the inability to disentangle shadows from object color. By explicitly supervising an occlusion factor we observe better albedo color predictions such as visualized in the blue box in the hotdog example, and all boxes in the drums example. Additionally, shadows are better disentangled from albedo as observed in the red and green boxes for the hotdog example. Quantitatively, we measure the importance of adding the occlusion loss to our model in table 3, where it improves both relighting and albedo reconstruction.

Occlusion averaging. The occlusion factor we derive is a per-channel factor that depends on estimated lighting. However, since both are being learned jointly, we observe that training can be noisy. We find in table 3 that relighting and albedo reconstruction both improve when we supervise the occlusion factors ˆ o d and ˆ o s with their per-channel averages instead. Assuming that all channels of the occlusion factor are equal is equivalent to assuming only white light with varying intensities, which reduces noise during training and uses fewer parameters.

Drums

Ground Truth

Ours

NVDiffRec

NVDiffRecMC

NMF

TensoIR

Materials

Helmet

Teapot

Material regularization. We measure the effect of the material regularization in table 3. Penalizing metalness prediction discourages our model from explaining radiance through environment lighting with overpredicted metallic surfaces. This leads to improved albedo predictions as shown in table 3. However, as visualized in fig. 1, the loss coefficient is small enough such that our model is still capable of correctly predicting metallic surfaces.

6 Conclusion and limitations

In conclusion, we present a novel and efficient method for inverse rendering based on neural surface rendering and the split sum approximation for image-based lighting. Owing to our proposed integrated illumination MLP, we can jointly estimate geometry, lighting, and material properties in under one hour using a single NVIDIA A100 GPU. Physical accuracy of our pre-integrated MLP representation is ensured thanks to the proposed illumination regularization. Additionally, we define occlusion factors for diffuse and specular lighting so that self-occlusions are accounted for with the split sum approximation. Finall, we propose a way of supervising occlusion MLPs to learn the proposed occlusion estimators. Altogether, our method produces high-quality estimates of geometry, lighting, and material properties as measured by rendering objects under unseen views and lighting conditions.

However, due to the highly complex problem that inverse rendering presents, our method comes with some limitations. The major assumptions we rely on come from using image-based lighting, the split sum approximation, and Monte Carlo sampling. Image-based lighting assumes that light sources are located infinitely far away from the scene, leading to errors when this assumption is violated. While we have tackled the problem of missing self-occlusions within the split sum approximation, we disregard the effect of indirect illumination. This has a noticeable impact on albedo for reflective surfaces such as the 'toaster' scene. Additionally, we only consider the reflection of light and don't model transmission and subsurface scattering effects. Finally, we use a low number of uniform Monte Carlo samples for the occlusion loss leading to errors due to strong and small light sources. This is mostly noticeable in albedo predictions for objects in the synthetic Blender dataset, where shadows can still be noticed in the albedo predictions. We hope future works will tackle these limitations.

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Wu, H., Hu, Z., Li, L., Zhang, Y., Fan, C., Yu, X.: Nefii: Inverse rendering for reflectance decomposition with near-field indirect illumination. In: 2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 4295-4304 (2023). https://doi.org/10.1109/CVPR52729.2023.00418

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A Appendix

A.1 Derivation of illumination loss

In this section, we go through the derivation for the Monte Carlo approximation of pre-integrated illumination ¯ g used in eq. (5). We first split the specular light integral into two terms:

As mentioned in section 3.2, the term on the right can be precomputed so we focus on calculating an approximation for the term on the left, which we denote g ( ω r , ρ ) .

This term requires us to make two approximations to the Cook-Torrance GGX BRDF f s

to be able to approximate g as blurred environment maps as per the split sum approximation. The first approximation on the BRDF consists of assuming the multiplication between fresnel and geometric shadowing terms is approximately equal to the dot product between the normal and viewing directions: FG ≈ ⟨ ω i , n ⟩ . Thus, we have that

A.2 BRDF details

The only remaining terms to compute for obtaining diffuse and specular components are the precomputed BRDF integral and the diffuse coefficient k d which we compute following [11]. Given our material property estimates we can compute k d as follows:

Finally, we store two coefficients F 1 and F 2 in a two-dimensional lookup table as per [11] and compute the BRDF integral as follows:

The final expressions for the diffuse and specular coefficients are thus

as shown in Equation (4). The GGX (Trowbridge-Reitz) microfacet distribution function D is defined as:

where h is the half vector between ω i and ω o . The second approximation assumes the normal and viewing directions to be equal to the reflection direction. That is, n ≈ ω r and ω o ≈ ω r . This leaves us with the following simplified D :

which now does not depend on either the normal or viewing directions. Approximating both integrals with Monte Carlo sampling and taking the same number of samples, we arrive at the following expression:

We finally obtain the expression for ¯ g in eq. (5) by replacing L i with the estimates from ˆ g querying perfect specular reflections ˆ g ( ω, 0) . A side effect of the approximations used is that for ρ = 1 in eq. (15), we have that D ( ω i , ω r , 1) ≈ 1 π and together with eq. (13) we obtain that

This allows us to reuse the same network ˆ g used to approximate L s also approximate L d as follows:

A.3 Derivation of occlusion factor approximation

We now go over the derivation of the occlusion factor Monte Carlo approximation. We aim to approximate occlusion factors o d ( x ) and o s ( x ) such that the diffuse/specular components with visibility, L V d and L V s , can be computed as a multiplication between the diffuse/specular components without visibility, L d and L s , and their respective occlusion factors. For the diffuse component,

To incorporate this information, we learn this factor for the diffuse radiance. We propose learning the occlusion factor o d ( x ) with an MLP, supervising it with Monte Carlo estimates ¯ o d ( x ) using the predicted geometry. We approximate o d ( x ) with Monte Carlo sampling by taking the same number of ω i samples for both integrals:

with ω i taken from a cos-weighted sampling of the hemisphere around the normal n at location x . The probability density function sampled is given by:

This cos-weighted sampling aids in reducing variance by eliminating the dot product factor from the estimation.

Similarly, for the specular component we have that

We then use the approximation for f s in eq. (13), leading to the following Monte Carlo estimate for ¯ o s ( x ) :

where ω i is now obtained by sampling the GGX distribution to reduce variance by eliminating the factor D from both integrals. The probability density function sampled in this case is the following:

which relies on the second approximation used in the previous section, and where ω h is the half-vector angle between ω i and ω o .

A.4 Implementation details

We embed our proposed lighting decomposition within an efficient implementation of NeuS [7]. We train our models for 20 , 000 steps using a warmup learning rate scheduler for the first 500 steps followed by an exponential decay scheduler. After every 2000 steps, we estimate the current geometry by using marching cubes [15] to extract the isosurface at SDF level-set 0 . The estimated geometry is used with 64 samples for the Monte Carlo estimation of occlusion factors. We use a random subset of 10% of the points from volume rendering to supervise the occlusion network to reduce time and memory requirements. 8129 light samples are used for computing illumination loss Monte Carlo estimates. The final loss is calculated as a linear combination of the proposed losses, with the following coefficients: λ rec = 10 . 0 , λ D = 10 . 0 , λ o = 0 . 01 , λ Eik = 0 . 1 , and λ m = 0 . 001 . We run all experiments on a single A100 GPU with 60 GB RAM and 6 CPU workers using an AMD EPYC 7713 64-Core processor for a total training time of ∼ 1 hour.

Network implementation details. We implement the spatial network using the progressive hash grid encoding from [12]. The hash grid consists of 16 levels with 2 features per level and a hashmap size of 2 19 entries. The base grid spatial resolution is 32 voxels, increasing by ∼ 1 . 32 each level. An MLP with a single 64 -channel hidden layer is used to produce spatial features with 13 channels along with the SDF predictions. Spatial features are then input to an MLP with two hidden layers of 256 channels each and ReLU activations to produce material property (metalness, roughness, and albedo) predictions. A separate but identical MLP is used to produce occlusion factor predictions. A sigmoid is used to map the MLP outputs to the occlusion factor and material properties' ranges of [0 , 1] . The illumination network consists of an MLP with five hidden layers with 256 channels each and ReLU activations. Both the direction and roughness vectors used as input to the illumination network are first positionally encoded as proposed in [18], using 10 frequencies for the directional input and 5 for the roughness input. A softplus function is used to map the illumination network's output to the range (0 , inf) .

A.5 Per-scene quantitative results

We report per-scene metrics for geometry, albedo, and relighting reconstruction using the NeRFactor dataset in tables 4 to 6. We report per-scene metrics for geometry and relighting using the Blender dataset in tables 7 to 10, and using the Shiny Blender dataset in tables 11 to 14. Additionally, we present qualitative results of our method visualizing the learnt illumination, material properties (metalness, roughness, and albedo), geometry, and relit renderings from our method's predictions for the Blender dataset in figs. 7 to 14 and for the Shiny Blender dataset in figs. 15 to 19.

A.6 CO3D qualitative relighting results

We present qualitative results of our method visualizing the learnt illumination, material properties (metalness, roughness, and albedo), geometry, and relit renderings from our method's predictions for the CO3D dataset in figs. 20 to 23. Even in this challenging real-world setting where images are taken from a user-captured video, our method provides good-quality results.

A.7 Additional qualitative videos

Please refer to the included video files for additional qualitative videos showing predicted renders, material properties, and relighting for the Blender, Shiny Blender, and CO3D datasets. All videos follow the same camera trajectory at a fixed distance from the center of the scene. Please note that due to objects in CO3D lacking training images viewing the upper or lower surfaces of objects, the reconstruction quality at these locations suffers. This issue could be alleviated with better video-capturing trajectories.

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Peer reviews:
comments: SUMMARY:
This paper tackles the problem of inverse rendering, which reconstructs geometry, material, and environmental lighting from a set of posed images with fixed lighting. It proposes two contributions: 1) representing pre-integrated illumination as a single MLP, 2) approximating self-occlusion on pre-integrated lighting and use it to supervise an occlusion MLP to disentangle shadows and materials.

The proposed method achieves state-of-the-art results while being very fast to train (less than one hour).

STRENGTHS:
1. The idea of representing pre-integrated illumination as MLP is interesting. The regularization is particularly novel and clever. I really like the way how pure specular MLP (roughness=0) is used to regularize the training of the illumination MLP (Eq. 5)

2. The proposed occlusion factor estimation method is also very interesting and novel. Instead of ambient occlusion, the paper uses simple trick to factor out occlusion as an independent scalar for light integral. This scalar can be computed with MC integration and is distilled into a neural network for efficient inference.

3. The quantitative result shows that the improvement over SOTA is significant, especially on relatively diffuse scene (NeRFactor dataset).

WEAKNESSES:
1. More datasets could be tested, e.g. the TensorIR dataset.

QUESTIONS:
1. The paper mentions that the relighting results were obtained via Blender's PBR shader. Does the shader use global illumination? If so it would be unfair for some of the baselines (TensorIR, NMF) as they can only render direct illumination. A more fair way would be run another experiment where all methods do not use global illumination.

LIMITATIONS:
The limitations and potential negative societal impact are discussed properly.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12

comments: SUMMARY:
This paper introduces a method for digitizing real-world objects by estimating their geometry, material properties, and environmental lighting from posed images under fixed lighting. Integrating image-based lighting techniques into Neural Radiance Field (NeRF) pipelines, the method uses a scene-specific MLP for pre-integrated lighting at arbitrary resolutions. A Monte Carlo sampling-based regularizer ensures accurate lighting representation and self-occlusion predictions.

STRENGTHS:
The method proposes a method to approximate the effect of self-occlusion to improve the material estimation

WEAKNESSES:
- The proposed method, particularly in terms of material and lighting representation, is trivial.
- The correctness of the decomposed material is flawed in the qualitative results. The albedo clearly has specular and shadow bake-in, and the metalness and roughness do not match the material in the ground truth.
- The occlusion introduced in the paper to improve material estimation is not robust enough to validate its contribution.

QUESTIONS:
In the discussion of occlusion loss, why is only the albedo prediction addressed? The paper claims that the occlusion is designed to improve the overall material estimation.

LIMITATIONS:
The proposed method decomposes lighting and material using surface reflection. However, real-world objects often have multiple material layers, such as the car showcased in the qualitative results. The clear coat reflects stronger specular light, while the lower paint layer includes both specular and diffuse reflections. The proposed model does not account for these complexities.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2024-07-13

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces SplitNeRF, a method that integrates the split sum approximation into Neural Radiance Field (NeRF) pipelines. This approach optimizes object geometry, material properties, and environmental lighting. The method employs a Multi-Layer Perceptron (MLP) to model scene-specific pre-integrated image-based lighting at arbitrary resolutions. The model is regularized using efficient Monte Carlo sampling to accurately capture pre-integrated lighting and self-occlusion effects. Experimental results indicate that SplitNeRF achieves state-of-the-art relighting quality with only about an hour of training on a single NVIDIA A100 GPU.\n\nSTRENGTHS:\n1. The method is efficient and achieves high-quality relighting results after only about an hour of training on a single NVIDIA A100 GPU, which is highly efficient compared to existing methods.\n2. By using Monte Carlo sampling for regularization, the method accurately models pre-integrated lighting and self-occlusion, leading to high-fidelity reconstruction of scene geometry and material properties. The integration of split sum approximation into NeRF pipelines is novel and effective, allowing the method to disentangle environmental lighting from material properties.\n3. The method demonstrates competitive performance on both synthetic and real datasets, showing its state-of-the-art performance in material decomposition and relighting.\n\nWEAKNESSES:\nI do not have a major concern about this paper- the technical claims are sound and the paper is overall well presented. Several minor points:\n\n1. Adding material editing visualizations in the paper could further strengthen the results and inspire downstream applications.\n2. Normal could contain a significant error in e.g. lego and coffee scenes. Can you provide more insight and analysis into this?\n\nQUESTIONS:\nSee weaknesses 1, 2.\n\nLIMITATIONS:\nLimitations discussed in the conclusions section.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-17"}