GL-NeRF: Gauss-Laguerre Quadrature Enables Training-Free NeRF Acceleration
Authors: Silong Yong
Yaqi Xie
Simon Stepputtis
Katia P. Sycara
Venue: NeurIPS 2024 poster (2024)
Keywords: Neural Radiance Fields
Real-Time NeRF
Gauss-Laguerre Quadrature
Neural Rendering
Primary area: machine_vision
TL;DR: In this work, we propose GL-NeRF, a new perspective of computing volume rendering with the Gauss-Laguerre quadrature.

Abstract: Volume rendering in neural radiance fields is inherently time-consuming due to the large number of MLP calls on the points sampled per ray. Previous works would address this issue by introducing new neural networks or data structures. In this work, we propose GL-NeRF, a new perspective of computing volume rendering with the Gauss-Laguerre quadrature. GL-NeRF significantly reduces the number of MLP calls needed for volume rendering, introducing no additional data structures or neural networks. The simple formulation makes adopting GL-NeRF in any NeRF model possible. In the paper, we first justify the use of the Gauss-Laguerre quadrature and then demonstrate this plug-and-play attribute by implementing it in two different NeRF models. We show that with a minimal drop in performance, GL-NeRF can significantly reduce the number of MLP calls, showing the potential to speed up any NeRF model. Code can be found in project page https://silongyong.github.io/GL-NeRF_project_page/.

GL-NeRF: Gauss-Laguerre Quadrature Enables Training-Free NeRF Acceleration
Silong Yong
Simon Stepputtis Carnegie Mellon University {silongy, yaqix, sstepput, sycara)@andrew. cmu edu
Yaqi Xie
Katia Sycara
Abstract
Volume rendering in neural radiance fields is inherently time-consuming due to the large number of MLP calls on the points sampled per ray. Previous works would address this issue by introducing new neural networks or data structures. perspective of computing volume
In this work, we propose GL-NeRF, rendering with the Gauss-Laguerre quadrature. GL-NeRF significantly reduces the number of MLP calls needed for volume rendering, introducing additional data structures or neural networks. The simple formulation makes adopting GL- NeRF in any NeRF model possible. In the paper, first justify the use of the Gauss-Laguerre quadrature and then demonstrate this plug-and-play attribute by implementing in two different NeRF models. We show that with a minimal drop in performance, GL-NeRF can significantly reduce the number of MLP calls, showing the potential speed up any NeRF model. Code can be found in project page httts:://sioong::: github.
_projecttppaee
Introduction
Neural Radiance Fields (NeRFs) 27 have shown promising results for synthesizing images from novel views. Plenty of works extend NeRF towards different aspects applicable in the real world (see related works for details). The core component for NeRF's success is volume rendering, which requires approximating an integral by densely sampling points along the ray and evaluating volume density and radiance using neural networks for them. In practice, dense set of points is evaluated by expensive operations like neural network inferences for a single pixel, which could be redundant. Works have been done to reduce the time needed for rendering images, aiming at providing NeRF with real-time rendering ability works, they propose different approaches for achieving real-time rendering by introducing new networks, new data structures, etc. Therefore, each individual work requires training from scratch with a specific optimization goal. In this work, we propose novel lightweight method that could be implemented in any existing NeRF-based models that require volume rendering without further training. In contrast existing works, our approach introduces additional representation neural network and is training- -free. We make minimal modifications to the computation of the volume rendering integral, making it rely much fewer samples.
24 29 8] Despite the promising results shown by these
Our approach arises from revisiting the volume rendering integral, the key discovery that with simple change of variable, we can turn the integral into a pure exponentially weighted integral of color. This specific form has a Gauss quadrature (i.e. the Gauss-Laguerre quadrature) which best approximates it mathematically. Naturally, we propose to use the Gauss-Lagueree quadrature to directly compute the volume rendering integral, which we call GL-NNeR (Gauss Laguerre-NeRRY leading to much lower computational cost for approximating the integral and therefore lower time and memory usage. Computing the points needed for the integral requires a dense evaluation of per-point density. However, the efficiency for this step can be improved using modern techniques
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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calla
@MP calls
NeRF uniform sample rendering
Gauss-Laguerre Quadraturo rendering (ours)
Figure 1: GL-NeRF method overview. The vanilla volume rendering in NeRF requires uniform sampling in space. This leads to a huge number of computationally heavy MLP calls since we have to assign each point a color value. Our approach, GL-NeRF, significantly reduces the number of points needed for volume rendering and selects points in the most informative area.
like factorized tensors [6] Benefiting from the guarantee of the highest precision Gauss quadrature provides, only a very small number of fixed points could provide comparable results to the heavy and redundant strategy NeRF adopts, leading free speedup.
To verify the use of the Gauss-Laguerre quadrature, we conduct an empirical study on the landscape of color function. We also analyze the relationship between our approach and other techniques that aim to reduce the sample points for NeRF We demonstrate the plug-and-play property of our method by directly incorporating it into vanilla NeRF and TensoRF models that are already trained on Nynteey and LLFF datasets. Furthermore, we showcase the drop in time and memory usage as direct outcome of reducing the computational cost.
GL-NeRF provides different perspective for computing volume rendering and has the potential be a direct plug-in in for existing NeRF- based products. Specifically, our contributions are three-fold. We propose GL-NeRF, a brand new perspective for computing volume rendering with the Gauss-Laguerre quadrature with no additional component introduced. We analyze the validity of using the Gauss- Laguerre quadrature for volume rendering integral and the relationship between our approach and existing sample-efficient NeRFs. We demonstrate that GL-NeRF could be incorporated into any NeRF model without further training. To the best our knowledge. GL-NeRF the first method that could be used without training in any NeRF models thanks to the simple formulation. We showcase that GL-NeRF reduces computational cost, time and memory usage while keeping the rendering quality.
2 Related work
Volume rendering. Volume rendering has been widely used in computer graphics and vision
applications (25 43 7. It maps a 3D scene onto 2D images by a weighted integral over the color of the points along the corresponding rays with function of opacity (volume density) as weight. In practice, the integral is approximated using finite over sampled points along the ray as derived in (25). Implicit scene models like NeRF [27]. Plenoxels (8 and 3D gaussians [18] and most of their follow-up all adopt this technique as the render pipeline. Since randomly sampling in space for approximating the integral may bring unnecessary information (i.e. sampling in empty space) that may cost extra computation, plenty of works aim address that by introducing different techniques for better approximation of the component needed for volume rendering integral (i.e. volume density, radiance) 40 29
23 21 36). PL-NeRF [40] proposes to use piecewise linear function for approximating the volume density throughout the space, leading to fewer points needed for the "fine" stage sampling proposed by 277 AutoInt and DIVeR 23 44 introduce a neural network for
approximating the integral of volume density instead of using Monte-Carlo sampling. DONeRF (29 reduces the sampled point needed for computing the integral by introducing depth oracle neural network that predicts the surface position of the underlying scene and samples the points near the

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Algorithm Gauss Laguerre Quadrature for Volume Rendering
Input: ray direction d, ray origin step size At, sample number M, Gauss- quadrature weight look-up table Lu- point look-up table L.p
1: fiin: fmax = Reylnerseccouudnngoxdd,, 0)
2: if' Emin tax then return bg__ollllclloo color
3: Initialize 1 = mmn transmittance = 1.0, already sampled point number n =
4: while nax do
5: if (n = m) then break
6: pos = o + + d
7:
a GerVolumeeensityypps
8:
(T)
9:
Inext x Al o
10:
if < Ip! and Text 2 1p [n] then
11:
Laguerre = (Lpln)
z (Tnext - x)
12:
pOs seuple + (t + Laguerre At At) d ray ray_colors color+ L [n] GetColor(pos eample)
13:
14:
-=
1= Al.
16:
T= * exp( At)
bg_weight sum :I)
18:
ray_ pay_color>> = ray_color + bg weight * bg color return _ol_r
surface, which contributes the most to the visual effect in the images. MCNeRF 13 Monte-Carlo rendering and denoising do sample efficient rendering, but it still introduces denoiser network that requires per-scene training. Different from these previous works, Our work proposes to the Gauss- itself, introduces no additional neural networks or data structures and remains in the simplest version, leading to its adaptability into any existing work that relies volume rendering integral.
proposes to use
quadrature to directly improve the precision of the volume rendering integral
NeRFs. Neural Radiance Fields (NeRFs) have proved to be a powerful tool for novel view syn- thesis [27]. It uses coordinate-based multi-layer perceptron (MLP) to represent the scene and render high-fidelity images from different views. The render is done by pixel-wise volumetric ren- dering (25) with density and color evaluated using the MLP on hundreds of sampled points along the ray. For modeling high-frequency information in the scene, NeRF uses positional encoding to map the input coordinates onto high-frequency bands. The success of NeRF has triggered an explosive emergence of follow-up works. There plenty of works focusing improving extending the ability of NeRF towards different aspects. Aliasing along xy coordinates has been tackled (3], unbounded scenes 4
47 39 [4], dynamic scenes [32 22 30 and scenes with semantic information [1 37 49 19 have been well explored and demonstrated the potential of implicit scene representation with NeRF. Nonetheless, NeRF requires plenty of time for training and rendering, blocking its way of being used for real-time rendering. The bottleneck of the computation time is the MLP used. There are two main branches of work for extending NeRF towards real-time rendering. The first branch introduces different data structure [46) | 14] 33 other branch, in which our method falls, improves the sample efficiency of the model 20 29 31 38
8 6 for scene representation. An-
to accelerate NeRF rendering process. While previous works draw their intuition from the underlying physies perspective and thus need different formulations of the sampling strategy and different neural network architecture for predicting the surface position of the underlying scenes, we propose our method based on mathematical observation while maintaining the overall pipeline. Benefiting from this, our work could be seamlessly incorporated into any existing NeRF-related works without further training. On the other hand, despite being derived from the mathematical perspective, our method still intuitively satisfies the underlying physical constraints.

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3 Preliminaries
3.1 NeRF and volume rendering
NeRF 27 is powerful implicit 3D scene model for novel view synthesis. At the core of its rendering ability is volume rendering. NeRF uses coordinate-based MLP encode the scene, assigning volume density (opacity) and radiance (color) to spatial points. When used for synthesizing new views, it casts a ray r(t) = O + td through the pixel to rendered, sample points along the ray and compute volume density and radiance for these points. These values are then aggregated together using T to give the color the pixel.
C(r) = Σ w,c(r(t)),
where
1y = T2(1 exp( ccp(-o((((,)))6))
(2)
T. erp( Σ (r((t))))) 1=0
represents the sampled position along the ray and 6; 4++1 - t; is the distance between two nearby sampled points. NeRF uses an MLP to represent volume density a and color c. The loss function for training NeRF is simply the square error between rendered pixel colors and the corresponding pixel colors over batch rays R. Variants NeRF like [en) use different representations for volume density and color, but the process of volume rendering remains the same.
L Σ
|[((r) - C(r)
PER
3.2 Gauss quadrature
An n-point Gauss quadrature 10 is method for numerical integration that guarantees yield exact results for integral of polynomials of degree 2n - or less, which is the highest possible precision integral by quadrature. Intuitively, consider approximating integral using
for approximating quadrature as in Eq. 5
plo)dss Σ w.p(x.))
(5)
where p(r) is a polynomial of degree 2n - 1, w(x) a weight function and is the interval for com- puting the integral. We first give the definition of orthogonality of two polynomials Pm) and pm (æ)
[ Pm (m)())p = 0,
(6)
where Pmr is of degree m, Pm (z) is of degree and * n. use long division for p(x) to obtain
p(x) - (()) +r()).
where L, polynomial of degree that is orthogonal any polynomials that have degree less than n (i.e. degree Legendre polynomial), g(x) and r(x) are both polynomials with degree less than n. Then
[
[ple)de p(xr)de
(.)) + [[(aa)de
(
is orthogonal to any polynomials with degree less than n, the first term on the right hand side of Eq. should equal to 0. Since it doesn't 't contribute to the computation of the integral, we may also neglect when computing the quadrature. Therefore, we should choose x; that satisfies Ln (#r;) = 0 16). With this intuition bearing in mind, carefully choosing the weights w; for computing the quadrature would help us precisely calculate Eq. 5 because we have n points to compute the second term on the right hand side of Eq. which is integral of polynomial of degree less than
Since L,,
(x:) L,,xx)

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In general, given function f(x), Gauss quadrature computes its integral on [-1, 1] using
[ f(x)do wi (a:),
where F2 ........ corresponds to root of the orthogonal polynomials on [-1.1]. This
quadrature called Gauss-Legendre quadrature since the orthogonal polynomials on [-1, with weight function g(x)) 1 are Legendre polynomials. An n-th degree Legendre polynomial takes
the form [35 15
d n" drn (x² 1)a
(r(r)
(10)
and wy is computed using Eq. 11 as shown
W.
(1 (1-23)|P((x)))
(11)
Gauss-Laguerre quadrature is an extension of Gauss quadrature for approximating integrals following the form of
1 ( Ews(( (12)
In this case, the weight function is g(x) e. the integral interval is [0, oo). x, corresponds to the root of Laguerre polynomials
1 _ 1)"x", n!! dx
L, (x)
(13)
66-
class of polynomials that are orthogonal over the interval [0, oo) with respect to the weight function g(x) e = The weight for comput- ing the quadrature is computed as
0.4
F2
Wi ()+
(14)
Ray
While the computation for x; and w; is compli- cated, in practice we can use look up table to store corresponding x, and w, for given n.
Figure 2: Verification on using the Lagus-Leguerre quadrature for volume rendering. We plot the red channel of the color function w.r.t. the ray it cor- responds to. The color function remains zero in most of the interval (bottom). We use 7th-degree polynomial to approximate the non-zero region (top). As can be seen, the color function itself is similar polynomial, validating the of our approach.
4 GL-NeRF
We developed algorithm based simple observation of the integral for volume rendering. Eq. is an approximation to the integral
C(r)= /"' (t))
(15)
where
T(t) [00)=exp[->> J (r(ss))ss).
(16)
4.1 Volume rendering and Gauss-Laguerre quadrature
5

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Let
dersty
x(1) [ a(r(s))dss
(17)
Volume density (o)
we have
dr
dt =(r(t))
(18)
Since (((()) 2 xrl) is monotonically non-decreasing function of I, therefore, x has unique correspondence with on increasing intervals. With this observation, we can do change of variables for Eq. 13 get
0.26
1.41
3.60
7.09
12.64
Ray samples (4)
C(r) = S" ((t))
Figure 3: Point Selection strategy in GL-NeRF. We choose points along the ray that satisfy the in- (19) tegral from zero to the point of the volume density function should be equal to the roots of Laguerre polynomials. The points selected then used for querying the color. In the figure above is an exam- ple of choosing points using a 5-degree Laguerre polynomial. The number on the plot indicates the value the integral from zero to the right bound- ary of the region.
I"' c=((((t), d) dr
dt
dt
calte) -faeet e =<<(((()), djdr. (((())
As can be seen from Eq. 19 the integral for volume rendering is weighted integral of c(r(z),)) with the weight function be g(x) = e- We can extend the integral interval from (ar(t)), x(ty)] [0, oo) since the integral between [0, x(tx)) and (z(tt), oo) are zero. Thus, we have
c(r)= "e(((((()) d)dr,
(20)
a pure exponentially weighted integral with respect to the color function, which is of the exact same form as required by the Gauss-Laguerre quadrature.
4.1.1 Gauss-Laguerre quadrature for volume rendering
As discussed Sec. 3] the Gauss- Laguere quadrature guarantees the highest algebraic precision when computing integral over polynomials. To perform the Gauss-Laguerre quadrature for volume rendering integral calculation, natural question arises: the color function a polynomial, or can it be approximated by polynomial with satisfactory error rate?
To answer this question, first analytically give out a fundamental theorem, and then empirically approximate the color function with polynomials.
Theorem 4.1 (Stone- Weierstrass theorem). Suppose f is continuous real-valued function defined on the real interval (a, b). For every 0, there exists a polynomial such that for all x in (a, b), have [f(x) p(z)] <
Since the pixel color is contributed by points that have a density larger than a threshold (i.e. regions near the surface), we can overlook the points in the empty space and only analyze the remaining part of the color function. In Fig. we plot representative of how the color function looks like. As can be seen from the figure, it has a major region with values greater than zero while the others remain zero. When approximating the non-zero region with a 7-th degree polynomial, we have relative eтror rate lower than 6.3%. While the relative error is not sufficiently low, we argue that we can increase the degree to better approximate since it's cintinuous by nature. On the other hand, this specific landscape is fluctuated and for most of the cases, the enror rate could be smaller than 1%. This suggests that Theorem 4. holds our case, thus Gauss-Laguerre quadrature can be used for computing the volume rendering integral.
4.1.2 Point selection in GL-NeRF
Different from NeRF's sampling strategy, the Gauss-L Laguerre quadrature enables us use deterministic point selection strategy for the color samples. Recall Eq. i7 is the integral variable

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LLFF
LLFF
LLFF
40
0.30
Vanila
0.25
Ours
PSNR
25
0.90
0:20
20
.585
0.16
15
0.80
0.10
10
Vanilla
Vanilla
0.06
Ours
x-
Ours
0.00
6.4
6.6
6.8
7.0
7.2
7.4
7.6
5.4
6.6
68
7.0
7.2
7.4
7.6
6.4
6.5
6.8
7.0
7.2
7.4
7.6
Time(s)
Time(s)
Time
1.2 x faster
1.2 x faster
1.2 x faster
Blender
Blender
Blender
8
1.00
0.30
Varilla
0.95
0.25
PSNR
0.20
20
U1.88
0.16
15
0.80
0:10
10
Vanilla
0.75
Vanilaa
0.05
Ours
Ours
0.70
0.00
16
15
Time(s)
Time(s)
Time
2 faster
2x
2 x faster
x faster
Figure 4: Comparison between GL-NeRF and vanilla NeRF in terms of render time and quantitative metrics. Each point on the figure represents an individual scene. We showcase that with the drop of computational cost GL-NeRF provides, the average time needed for rendering one image is 1.2 to times faster than the vanilla NeRF. In the mean time, the overall performance remains almost the same despite some minor decreases.
for Eq. 20 This means if want to use the Gauss-Laguerre quadrature approximate Eq. 20
we have to choose points I; that are the root of ntt--degree Laguerre polynomials. Since every I; has corresponding following Eq. 17] we can choose t; based on given value of xú, as depicted in Fig. 3 Specifically, we want the integral Eq. 17 to be equal to the roots of an reh--degree Laguerre polynomial. Fig. gives an example of n
In the figure, the numbers in the five regions filled with different colors indicate the integral value of the volume density function from zero to the right boundary of the regions. A pseudocode for GL.-NeRF rendering shown in Algo.
4.1.3 Intuitive understanding of the points selected using the Gauss-Laguerre quadrature
Since the points near the surface contribute the most to the final color of the pixel as discussed in 200 29, 31], the optimal point selection strategy should choose points near the surface. The volume density, on the other hand, increases remarkably near the surface and remains close to zero at other areas. Therefore, the integral value of it Eq. [7 should also increases significantly near the surface and remains almost unchanged throughout the rest of the space. Therefore, most of the points chosen using GL-NeRF should lie around the surface of the underlying scene. Consider a case when 8, we want to choose points t,, 1,,...., 8 such that (t,)) in Eq. 17 should be equal to the value given in the look-up table Tab. Notice that the first few value for x; (say first three) are small so that they could be reached by the integral of volume density near the surface easily. These values have relatively larger weights assigned to them. Evaluating the color of these points using neural network and summing them up using the weights w; given Eq. following Eq. 12 would contribute mostly to the pixel color. Notice that even though the last few Jy are quite large and may not be reached by Eq. [7 along the ray, their corresponding weights are so small that they almost couldn't it affect the final result of the pixel color. Hence, the points selected
0.17
3.69 x 10
0.90
4.19 x 10
2.25
1.76 x 10-1
4.27
3.33 x 10-2
7.05
2.79 x 10-3
10.76
9.08 x 10-5
15.74
8.49 x 10-7
22.86
1.05 10-9
Table 1:
Gauss-Laguerre
quadrature when = 8.
look-up table
7

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TensoRF's PSNR: 20.76
TensoRF PSNR: 21.34
TensoRF Ours
TensoRF
Figure 5: Qualitative results on LLFF (top) and NeRF-Synthetin (bottom) datasets. We could tell from the comparisons that the drop in performances has minimal effect on the visual quality.
Dataset
Methods
Avg. MLPs|
PSNR
SSIM
LPIPS
TensoRF
118.51
26.51
0.832
0.135
LLFF
ours
25.63
0.797
0.146
TensoRF
31.08
32.39
0.957
0.032
NeRF-Synthetic
ours
30.99
0.945
0.048
Table Quantitative comparison. We demonstrate that our method has minimal performance drop while significantly reducing the number of color MLP calls.
using GL-NeRF also correspond to the points near the surface, like in previous works 220 29 31 that
design different neural networks for estimating the surface position, but only without any additional neural networks. Therefore, thanks to the nice property of the Gauss quadrature, ideally we can select the optimal points for computing volume rendering integral if the volume density estimation oracle.
Experiments
Datasets and evaluation metrics. We evaluate our method the standard datasets: NeRF- Synthetic and Real Forward Facing Datas) 26 as in [27] with two different models, i.e. conduct Vanilla NeRF (27, TensoRF [6] and InstantNGP (28). Since method is training-
render-only experiments with the vanilla volume rendering method and our method. We plot the standard render quality evaluation metrics PSNR, SSIM [42) and LPIPS (88 with respect to the average time needed for rendering one image for each scene in Vanilla NeRF. We also report the metrics with averaged color MLP calls for TensoRF and InstantNGP. For Vanilla NeRF, we use 32 points for our method while the network is trained with more than 100 points. For TensoRF and InstantNGP, the results are produced with MLP calls if not otherwise mentioned. More details can be found in Sec. A.
5.1 Comparison with baselines

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1
3
8 Ori GT
Figure 6: Effect sample number. The first five columns correspond to the number sampled points on top. The sixth column shows the result of the original sampling strategy adopted in TensoRF (Ori). The last column is the ground truth visualization the details in the scene. Our method could achieve comparable results using only 4-8 points while the original strategy requires more than 100 points. The blurriness in the first two columns is inherently the inaccuracy of piece-wise constant density estimation.
Dataset
Methods
PSNRT
SSIM
LPIPS
Vanilla
27.62
0.88
0.073
LLFF
ours
27.21
0.87
0.087
Vanilla
30.63
0.95
0.037
NeRF-Synthetic
ours
29.18
0.93
0.056
Table Quantitative comparison when training with GL-NeRF. Vanilla refers to the vanilla NeRF and its sampling strategy while ours refers to replacing the fine sample stage in vanilla NeRF with our sampling strategy, i.e. GL-NeRF. The result for Vanilla NeRF is produced by rendering using more than 100 points while GL-NeRF only uses 32 points.
We showcase that method can be used for rendering novel views based on pretrained NeRF without further training. We plotted the quantitative metrics of GL-NeRF and original NeRF for an intuitive comparison in Fig. It shows that our method achieves comparable results the original NeRF while requiring less computation, leading to 1.2 to times faster rendering. We also observed drop in memory usage due to the fewer MLP calls we have. We further implement our method with TensoRF [6]. As can be seen from Tab. 5, our method significantly reduces the number of MLP calls needed for volume rendering while the rendering quality only drops a little. We observe that the minimal drop in the performance has little effect on the quality of the image. Some qualitative comparisons can be found in Fig. 5 Other than TensoRF, we implement method on top of InstantNGP (8 showcase the plug-and-play attribute of our method. Our method performs similarly on Blender dataset to InstantNGP as shown in Tab,
Table 4: Ablation study on the number of points sampled. The more points have, the better the performance will be. With points, our method is comparable to the original sampling strategy in TensoRF.
Point number
PSNR
SSIM
LPIPS
23.49
0.752
0.166
2
24.90
0.782
0.142
3
25.38
0.791
0.145
4
25.63
0.797
0.146
8
26.10
0.812
0.142
Ori
26.51
0.832
0.135
5.2 Discussion on acceleration
The reason why the speed-up in Vanilla NeRF doesn't lead to real-time performance is that it has an- other heavy neural network estimating the volume density. While our method needs cheap density estimation, it can be easily achieved by recent efforts in NeRF like factorized tensors [ Therefore, reducing the number of color MI.P calls needed could lead to real-time performance as shown by previ- work [13]. We therefore follow MC-NeRF [3] and develop real-time renderer based on WebGL

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Blender
Avg.
Chair
Drums
Ficus
Hotdog
Lego
Mat.
Mic
Ship
InstantNGP
32.05
34.13
25.61
31.91
36.32
34.72
29.73 26.54
29.09
34.92
PSNR
Ours
30.35
33.08
25.07
30.13
34.78
33.05
33.02 27.15
Table 5: Per-scene results on Blender dataset between InstantNGP and ours. We demonstrate that GL-NeRF is able be plugged into ANY NeRF models.
Method
PSNR1
SSIM
LPIPS
FPS
TensoRF
33.28
0.97
0.016
5.84
ours
33.09
0.97
0.016
22.34
Table 6: Comparison between our method and TensoRF Lego scene using WebGL-based renderer. The result is collected from an AMD Ryzen 9 5900HS CPU. GL-NeRF is able to provide almost real-time rendering while remaining similar quality TensoRF:
and train a small variant of TensoRF with channels for each density component and color component and 32 as hidden size for the color MLP. The result on Lego in the Blender dataset shown in Tab, 5.4 GL-NeRF is able to provide almost real-time performance in WebGL with similar quality as TensoRT running on an AMD Ryzen 5900HS CPU thanks the reduced number of color MLP calls.
5.3 Ablation studies
We further study the effect of sampled points per ray. We conduct experiments using the TensoRF model on the L.LFF dataset. We found that 8 points per ray already shows comparable results to the original sampling strategy that uses more than 100 points. Quantitative comparison can be found in Tab. 4 Qualitatively, in Fig. we found that less number of points would lead to blurrier results. Since the points selected using GL-NeRF intuitively correspond to where the surface is, we argue that the blurriness comes from the inherent inaccuracy of piece-wise constant density estimation.
5.4 Discussion on GL-NeRF usage for training
While we mainly showcase that GL-NeRF general alternative to the sampling strategy for volume rendering at test time, it is also capable of being used for training. We demonstrate this by replacing the fine sample stage in Vanilla NeRF with GL-NeRF and show the result in Tab. [ and Tab. 9 GL-NeRF is able to produce on-par results with the vanilla sampling strategy but use much smaller number of points, i.e. 32 for GL-NeRF and more than 100 for vanilla NeRF.
Conclusion
In this paper, we propose GL-NeRF, novel approach for calculating the volume rendering integral. We show that with a simple change of variable, the Gauss-Laguerre quadrature can be used for computing the volume rendering integral. Thanks the highest algebraic precision guaranteed by the Gauss-Laguerre quadrature, GL-NeRF significantly reduces the number of MLP calls needed for the volume rendering integral. We justify the of the Gauss-Laguerre quadrature theoretically and empirically and showcase the plug-ane and play attribute of GL-NeRF in two different NeRF models. Experiments show the potential of GL-NeRF being used for accelerating any existing NeRF model. We also demonstrate that GL-NeRF can be used for training vanilla NeRF, providing a potential new direction for neural rendering research.
Limitations. While GL-NeRF shows promising results in reducing the number of MLP calls needed, it still affects the rendering quality despite the theoretical guarantee the highest precision. How to improve the performance that it would meet the theoretical results would be interesting.
10

---

Acknowledgement
The authors would like to thank the author of MC-NeRF for the useful tips on developing the WebGL renderer. This work has been funded in part by the Army Research Laboratory (ARL) award W911NT-23-2-0007, DARPA award FA8750-23-2-1015, and ONR award N00014-23-1-2840.
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13

---

A
Appendix
Here we introduce the implementation details, give a brief introduction of the Gauss-Laguerre quadrature and present our quantitative results NeRF-Synthetic and LLFF datasets.
A.1 Implementation details
reproducible PyTorch implementation of the original NeRF (27). We implement our method by changing the hierarchical sampling strategy into our point selection method using the
For Vanilla NeRF, our experiments are conducted based upon NeRF-PyTorch 45
quadrature.
follow the standard setting as done in [27 to train "coarse" and a "fine" network for evaluation. We use learning rate of 5 X 10-4 that exponentially decays to 5 10-3 over the course of optimization. Each scene is trained for 200k iterations using single NVIDIA RTX 6000 GPU. We use 128 coarse samples and 32 fine samples to test our method. Since the density estimation for "coarse" and "fine" network are not aligned, we test our method using only the "fine" network by first using "coarse" samples query it for estimation of density, then use GL-NeRF to select 32 points for final rendering as discussed in Sec. 4.1.2 LLFF scenes are trained and tested with 64 coarse samples and 64 fine samples for baseline method, while NeRF-Synthetic scenes require 128 coarse samples and 64 fine samples. For TensoRF, we directly use the pretrained checkpoints in the folder VM48 provided by the authors. The qualitative results are produced by 4 neural network calls if not otherwise mentioned. For InstantNGP, we build our code on top of the public Py PyToch implementation [12] and train it with the default setting. The final results for GL-NeRF are also produced by 4 neural network calls.
A.2 Gauss-Laguerre quadrature
The Gauss- Lagsure quadrature is an approximation formula for computing integrals over the semi- infinite interval [0, +oo) with the weight function e and reads
e e
Σ (xf( k==0
(21)
Here To, 21, ..: 2n E [0, +oo) are the zeros of the Laguerre polynomial Ln+1 Ln+1(x) of degree (n + 1):
d=+1 Tnti (2) (n + 1)! dra+I (x= e-²),
for n = -1,0,1, and the coefficients
Wk
3.k (xx)
k= 0,2,,
(22)
From the Leibniz formula, it is easy to see that L (I) is a polynomial of degree and the coefficient
of xt is (-1)" ml In particular, we have
1 n²²
Lo = -1=1-2;
L2
2x + 1,...
2
The fundamental property of the Laguerre polynomials is
Theorem A.1. The Laguerre polynomials In, = (x((x) are orthogonal with respect to the weight function z that is,
| tcc
0,
+ m,
- (I)
1, m.
Proof. Assume m 5 n and set (x(x) = -= From the Leibniz formula follows that, for j < (
Ie (x) is product of = and polynomial of degree (k 1) and thereby g(6) (0)(() ( (+00)
14

---

for j < k. Thus, we deduce that
n!m! / e (n()
(n) gm (x)do
" (z)dg))
1 (x) Im (1))l
In
(23)
1e 9m [x)) [x)] dglh (x)
sht f(m 2) (x) 1e Im 1
[e (e"99h") (z) (x)] dg (x)
=(-1)" - gn(x)]e" d (x) (x)dz
By the Leibniz formula, we have
e g/mm//)) (z)
(n)
n!
(0)(n-19gim fl (x)
(n j}lj!
j=0
n!!
(m (m++)((a (x)
Im
(n - jolj!
j=0
and thereby
+oc
n!m!
e Lnn (m))dd
=(-1)"
n!
x n 9 ((++)))))))))))) (z)dz
(n - jーレー
1=
n! nl(-1)n+m+d
| to0
n
(n m j)ز 9m (r)dr
(n إز!زز
n!( al(-1)ntm+)
+00
n!
(n - m - j)m 1" -e- dx
(n j}زز!
n-m
n!
(-1)m+) nl(n= - 5)! (n - m - j]}
-(-1)"
(n - j!!ز!
-=0
(n!)²
(n - m)!
=(-1)n+m
j!(n m j)l (-13
(n m)!
1=0
(n!)²
1-1)n+m
1)n-m
(n m)!
Here the second equality is similar to that in 23 and the fourth uses
n- e dx = (n- - j.
This completes the proof.
15

---

The orthogonality of the Laguerre polynomials ensures that the (z)x)'s 's are linearly independent and I 1 (x) has (n + 1) distinct zeros To. #1,-. Fn in [0, +00) With the zeros, the coefficients
are chosen so that the following (n + 1) equalities
c er de
(24)
hold for j j 0, 1.... This leads to system of (n+11 linear algebraic equations for the unknowns wk and the corresponding coefficient matrix is the Vandermonde matrix (x+]( (x)+)))+++))) The latter invertible since the zeros are distinct and therefore the wy are uniquely determined. The specific expressions of the wg 's are given in
remarkable that all the coefficients wy are non non-negative. This important property ensures the stability and convergence of the Gauss-Laguers quadrature [II. Moreover, we have
Theorem A.2. The algebraic precision of the Gauss-Laguers quadrature 27 (2n + 1) exactly. Namely, "Rs" in 22 is "=" if f(x) polynomial of degree (2n + 1) and is not "=" f(x) is polynomial with degree higher than (2n + 1).
Proof. Notice that
(n + 1)!Lm+1(2)) = (-1)*+1 To 2k)
polynomial of degree (n + 1). Since
e " 122) +
Σ Wk 12
the precision is less than +1n+1). On the other hand, for any polynomial p = p(x) of degree (2n 1) there are two polynomials of degree such that p(x) = g(x) La+ ( + r(x). Notice that g(x) can be written linear combination of Lo(x), La (z), Ln (1). Compute
En wap(or)) = En_o wx[g(<n) i + r(ax)]
En wgr(zx)
So *r((ddd
Jo e (-((g(x) [[(xx) Lm+i + r(=)ddr
Here the third equality is due 24 (the choice wx) and the fourth due the orthogonality of Ln+1 (x) and q(x) with respect to the weight function. Hence the proof is complete.
For further details on the Gauss- quadrature and for other Gauss quadratures, the interested reader is referred to the book
A.3 Proof for Theorem 4.1
Here we present proof of the well-known Stone- Weierstrass theorem, basically from rephrase the theorem here for readability.
5
We
Theorem A.3. Suppose f =(x) : [0, + (00,,00) continuous. Then for any e > 0 there is
polynomial p(x) satisfying
sup
[s(z) p(x)) <e.
=E(0.1]
Namely, polynomials are dense in the Banach space C' 1].
16

---

NeRF-Synthetic
Avg.
Chair
Drums
Ficus
Hotdog
Lego
Mat.
Mic
Ship
TensoRF
32.39
34,68
25.58
33.37
36.81
35.51
29.54
33.59
30.12
PSNR1
Ours
30.99
33.98
25.15
30.41
35.75
33.80
27.32
32.52
30.12
TensoRF
0.96
0.98
0.93
0.98
0.98
0.98
0.94
0.98
0.88
SSIM Ours
0.94
0.98
0.92
0.96
0.97
0.97
0.91
0.98
0.87
TensoRF
0.032
0.014
0.059
0.015
0.017
0.009
0.036
0.012
0.098
LPIPS
Ours
0.048
0.019
0.068
0.043
0.031
0.015
0.088
0.029
0.095
LLFF
Avg.
Fern
Flower
Fortress
Horns
Leaves
Orchid
Room
Trex
TensoRF
26.51
25.31
28.22
31.14
27.64
21.34
20.02
31.80
26.61
PSNR1
Ours
25.63
24.11
27.24
30.41
26.86
20.76
18.91
30.82
25.94
TensoRF
0.83
0.82
0.86
0.89
0.86
0.75
0.66
0.95
0.89
SSIM
Ours
0.80
0.76
0.82
0.87
0.83
0.72
0.59
0.92
0.87
TensoRF
0.135
0.161
0.121
0.084
0.146
0.167
0.204
0.093
0.108
LPIPS
Ours
0.146
0.181
0.115
0.089
0.146
0.146
0.255
0.122
0.118
Table 7: Per-scene quantitative comparison between TensoRF and ours.
Proof. Fix f = f(x) C(O, 1] and e > Since f fa) is continuous on the bounded closed interval [0, 1]. it is bounded and uniformly continuous, meaning that there are positive numbers M > 0 and 6 5(c) > such that
[f(x)) M,
[f()) f(v) <
(25)
for any v,y [0, 1] satisfying | y/ < 8.
M. be positive integer. Consider the nith-order Bernstein polynomial 5 With 5 fixed, let 2 8Cc
p(z) = Σ f( (
Ch (1-x)= 1)
km0
with C a! kJ]
Notice that, for x E [0,
Chaa (1-1))10000 =z)a=k 2
che (1 -)n (x x)"
(26)
Σ *Ckha((1 2z)nk
x (1 z)n
cln kyl
n. Σ
n
(k- ( - k1 2)--1- (27)
(n-1))
k=1
= nx(c + x-m-1 = nx,
Σ 12C x n=== Σ -1- + Σ k(k 1)0h x) k=0
0=
nz + Σ ke - 1)
n!! x (1 z
(28)
nzr + n(n 1)x²
Σ
(n 2)1
2 2 n
2]! byp
næ + n2nn - 1)x²
17

---

NeRF-Synthetic
Avg.
Chair
Drums
Ficus
Hotdog
Lego
Mat.
Mic
Ship
Vanilla
30.63
34.32
25.80
29.54
35,49
29.53
29.04
31.78
29.52
PSNRT
Ours
28.56
30.82
24.08
26.62
32.70
28.78
27.19
31.34
27.03
Vanilla
0.95
0.98
0.93
0.97
0.97
0.95
0.95
0.97
0.87
SSIMI
Ours
0.93
0.96
0.90
0.94
0.96
0.94
0.92
0.97
0.84
Vanilla
0.042
0.014
0.052
0.021
0.034
0.042
0.035
0.044
0.092
LPIPS
Ours
0.070
0.050
0.098
0.055
0.059
0.047
0.070
0.044
0.135
LLFF
Avg.
Fern
Flower
Fortress
Horns
Leaves
Orchid
Room
Trex
Vanilla
27.62
26.82
28.37
32.59
28.83
22.38
21.20
32.87
27.93
PSNR1
Ours
26.53
26.27
28.19
31.12
26.81
22.27
20.99
30.38
26.24
SSIM Vanilla
0.88
0.86
0.89
0.93
0.90
0.82
0.74
0.96
0.92
Ours
0.85
0.84
0.88
0.89
0.86
0.81
0.73
0.93
0.89
Vanilla
0.074
0.097
0.064
0.030
0.070
0.113
0.122
0.041
0.052
LPIPS
Ours
0.090
0.106
0.066
0.064
0.096
0.115
0.125
0.075
0.075
Table 8: Per-scene quantitative comparison between Vanilla NeRF and ours.
Then, for any x E [0, 1]. we deduce from 25 that
[f(x) p(x)==
iff(x - f( | 1 (1 x)n
k=
[f(x) f( b
k=
f( k
S(
Σ fl(())
-( x)
n
folu-k/n/cb
k:ll--k/n|26
Σ
+2M
Σ
1Chaa (1 =z)
2
k:/2-k/n|<6
2//l2//n22
(nx k)2
S2
6 + 2M
Σ
Ck (1 (1-x)aa
x))
n²8²
k:/2-k/n|26
6 +
2M
(nx kk) (1-x)n-A
2
22
k=0
2M
2M nd² x(1 x)
+
(n²x²
2nxnx + no + n(n 1)x
2
n282
8
+
2
2
This completes the proof.
A.4 Quantitative results on NeRF-Synthetic and LLFF datasets
In this part, we present our per-scene guantitative results on NeRF-Synthetic and LLFF datasets in Tab. [ for Vanilla NeRF and in Tab. 7 for TensoRF. The full result for training using GL-NeRF is presented in Tab, 9
18

---

Blender
Avg.
Chair
Drums
Ficus
Hotdog
Lego
Mat.
Mic
Ship
Vanilla
30.63
34.32
25.80
29.54
35.49
29.53
29.04
31.78
29.52
PSNR
Ours
29.18
32.43
24.38
26.92
33,91
29.49
27.27
31.55
27.47
Vanilla
0.95
0.98
0.93
0.97
0.97
0.95
0.95
0.97
0.87
SSIMT
Ours
0.93
0.97
0.91
0.94
0.96
0.95
0.92
0.97
0.84
Vanilla
0.037
0.014
0.052
0.021
0.034
0.042
0.035
0.044
0.092
LPIPS
Ours
0.056
0.029
0.087
0.050
0.052
0.038
0.065
0.046
0.122
LLFF
Avg.
Fern
Flower
Fortress
Horns
Leaves
Orchid
Room
Trex
Vanilla
27.62
26.82
28.37
32.59
28.83
22.38
21.20
32.87
27.93
PSNRT
Ours
27.21
26.63
28.05
31.93
28.05
22.35
21.12
32.51
27.01
Vanilla
0.88
0.86
0.89
0.93
0.90
0.82
0.74
0.96
0.92
SSIM
Ours
0.87
0.85
0.88
0.91
0.88
0.81
0.74
0.95
0.90
Vanilla
0.073
0.097
0.064
0.030
0.070
0.113
0.122
0.041
0.052
LPIPS
Ours
0.087
0.121
0.075
0.043
0.089
0.117
0.131
0.053
0.069
Table
Quantitative results when training on Blender and LLFF Datasets.
19

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: We prove the claim of reducing computation in Sec. 5 the analysis and
justification using the LaguerL quadrature are in Sec. 4.1.1 and Sec. 4.1.2
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
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The code can be found in https://silongyorgg http://illoogyooggggthh.....bb.. ott.a//.o..o.tt.....oooooo-----. io/iio//LL-- NeRF project_page/
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//nips. //nipscc//pie wwcppbblc/guuiicuccccccccoocoopp /eegguddsse/Ceebbbinninn
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Reviews:
[{"review_id": "6EiqqhcabG", "reviewer": "Reviewer_GERG", "summary": "This paper presents a method for reducing the number of color samples needed for volume rendering in Neural Radiance Fields. The method works by applying Gauss-Laguerre quadrature as a replacement for the importance sampling used by some NeRF methods to reduce the number of calls needed for their fine color MLPs. This approach is shown to yield some efficiency improvements on plain NeRF as well as TensoRF models.", "questions": "It is strange that timings are reported for vanilla NeRF but not TensoRF, which would presumably see a larger gain. Is there a reason for this?\n\nI would suggest that the authors try to show quantifiable improvement on a more recent baseline. If there were a substantial run-time speed up for TensoRF, that would be good, but something lime Mip-NeRF 360 or Zip-NeRF would be even better as that would show how GL compares to another method which tries to draw samples near the surface.", "limitations": "It is mentioned, but I think the writing could make it a lot more clear that only the color samples are being reduced, as this significantly affects where the method would actually be expected to provide a speedup. As it is, this is quite easy to miss and could lead to misunderstanding if one does not read the method carefully.\n\nI don't think there are any notable concerns with the paper regarding societal impact.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The mathematical aspects of the paper are detailed and well supported. It is very clear what the method is trying to achieve and how the Gauss-Laguerre formulation is being applied. Generally, the quality of explanation is good and not too hard to follow, even though the math is fairly dense.\n\nOverall, I think this is an original idea with some likely applications in volume rendering.", "weaknesses": "The main issue with this paper is that the experiments do little to give an idea of how the proposed method would compare to the current state of the art, which has progressed quite significantly beyond the baselines shown here. While it is quite believable that the GL method outperforms the naive hierarchical sampling of the original NeRF, I have significant doubts about that holding when applied to something more recent like the proposal networks from Mip-NeRF 360 and Zip-NeRF.\n\nGiven the weak evaluation, I would lean towards rejecting.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for your time and insightful comments. Your valuable suggestions have significantly elevated our work. In light of your comments, we have proposed a brand new mathematical perspective for volume rendering and proved its effectiveness on speedup using two baselines. We will address your concerns below.\n\n> The main issue with this paper is that the experiments do little to give an idea of how the proposed method would compare to the current state of the art.\n\nWe validated it with 3 baseline approaches (Vanilla NeRF,  TensoRF and InstantNGP in the rebuttal) and show that our method has a plug-and-play attribute. In the experiments, we primarily aim to demonstrate GL-NeRF’s plug-and-play attribute, and can provide comparable performances as well as reducing the sample points, which results in speedup as a free lunch. Moreover, we’d like to point out that our paper’s main contribution is on the mathematical perspective itself.\n\n\n> proposal networks from Mip-NeRF 360 and Zip-NeRF.\n\nWhile Mip-NeRF 360 focuses on anti-aliasing and Zip-NeRF is its combination with Instant NGP, we instead conduct experiments by combining GL-NeRF with InstantNGP since our focus is on the volume rendering integral itself. The results can be found in the general response and we paste it here for better reference. The result along with the results on Vanilla NeRF and TensoRF indicates that our method can be incorporated into any NeRF pipeline that relies on volume rendering regardless of the underlying representation.\n\n| Method             | PSNR  | Avg. color MLPs calls |\n| ------------------ | ----- | --------------------- |\n| Instant NGP        | 32.05 | 30.90                 |\n| Ours + Instant NGP | 30.35 | 4.00                  |\n\n\nOn the other hand, our method needs no training, therefore can be plugged into any existing pipeline, which is the main advantage of our work compared to e.g. the proposal networks from Mip-NeRF 360 and Zip-NeRF.\n\n> It is strange that timings are reported for vanilla NeRF but not TensoRF, which would presumably see a larger gain. Is there a reason for this?\n\nAs mentioned in MCNeRF, NVIDIA GPUs have specialized components to accelerate the neural network inferences, therefore evaluating on these devices with more sample points may not cost that much. Therefore we instead implement a WebGL-based renderer as WebGL is a more accessible platform that is agnostic to the underlying hardware[1, 2, 3, 4]. The result is in the General Response section and we paste it here for better reference. As can be seen from the table, our method achieves comparable quality as TensoRF and achieves almost real-time on WebGL running on a AMD Ryzen 9 5900HS CPU ***by only reducing the sampling points***.\n\n| Method         | PSNR  | SSIM | LPIPS | FPS   |\n| -------------- | ----- | ---- | ----- | ----- |\n| TensoRF        | 33.28 | 0.97 | 0.016 | 5.84  |\n| Ours + TensoRF | 33.09 | 0.97 | 0.016 | 22.34 |\n\n>It is mentioned, but I think the writing could make it a lot more clear that only the color samples are being reduced, as this significantly affects where the method would actually be expected to provide a speedup. As it is, this is quite easy to miss and could lead to misunderstanding if one does not read the method carefully.\n\nThank you for this construction feedback. We'll modify the manuscript to make the point clearer.\n\n>[1] Gupta, Kunal, et al. \"MCNeRF: Monte Carlo rendering and denoising for real-time NeRFs.\" SIGGRAPH Asia 2023 Conference Papers. 2023.\n\n>[2] Chen, Zhiqin, et al. \"Mobilenerf: Exploiting the polygon rasterization pipeline for efficient neural field rendering on mobile architectures.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.\n\n>[3] Reiser, Christian, et al. \"Merf: Memory-efficient radiance fields for real-time view synthesis in unbounded scenes.\" ACM Transactions on Graphics (TOG) 42.4 (2023): 1-12.\n\n>[4] Yariv, Lior, et al. \"Bakedsdf: Meshing neural sdfs for real-time view synthesis.\" ACM SIGGRAPH 2023 Conference Proceedings. 2023.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tuT22Ozcod", "reviewer": "Reviewer_a3Mk", "summary": "This paper focuses on accelerating novel view synthesis using neural radiance fields (NeRF). Unlike previous works that concentrate on designing lightweight networks, this study is motivated by the specific volume rendering formula, which includes a negative exponential term in the integration function. By employing Gauss-Laguerre quadrature, the authors approximate this complex integral operation, thus improving the rendering speed of existing NeRFs. This approach is validated on two backbones: the original NeRF and TensoRF, demonstrating speed improvements ranging from 1.2X to 2X.", "questions": "In the experiment, the authors validated the proposed method on two backbones (the original NeRF and the TensoRF). However, neither of these represents the current fastest method. It is suggested to compare the proposed method with Instant NGP, DVGO, or other faster alternatives. Such comparisons could not only better verify the plug-and-play capability but also significantly enhance the impact of the paper.", "limitations": "n/a.", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The idea is intriguing and represents a promising exploration originating from the specific volume rendering formula.\n2. This paper is highly theoretical.", "weaknesses": "see Questions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the time and thoughtful feedback you've given. Your suggestions have greatly improved our work. Considering your insights, we have provided a highly theoretical framework for calculating volume rendering and validated it using 2 baselines. We will address your concerns below.\n\n>In the experiment, the authors validated the proposed method on two backbones (the original NeRF and the TensoRF). However, neither of these represents the current fastest method. It is suggested to compare the proposed method with Instant NGP, DVGO, or other faster alternatives. Such comparisons could not only better verify the plug-and-play capability but also significantly enhance the impact of the paper.\n\n\nSince Instant NGP is a representative of grid-based representation for volume density, we combined GL-NeRF with Instant NGP as suggested and compared between the versions. We believe doing so would provide a more comprehensive perspective for the plug-and-play attribute of GL-NeRF. As can be seen from the table, our method achieves comparable performance with instant NGP using only $\\frac{1}{8}$ number of color MLP calls. Since the experiment using TensoRF on WebGL has shown that reducing the color MLP calls can lead to significant speedup, here we simply showcase the color MLP calls needed in the table. The experiment indicates that as long as the underlying radiance fields pipeline relies on volume rendering, GL-NeRF can be an ***off-the-shelf replacement*** for dense sampling / hierarchical sampling, agnostic to the underlying representation.\n\n| Method             | PSNR  | Avg. color MLP calls |\n| ------------------ | ----- | --------------------- |\n| Instant NGP        | 32.05 | 30.90                 |\n| Ours + Instant NGP | 30.35 | 4.00                  |", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SXoiSZYNUY", "reviewer": "Reviewer_cdYt", "summary": "This paper proposes a computational method for volume rendering using Gauss-Laguerre quadrature. In the context of NeRF, volume rendering is performed by evaluating MLPs (or other data structures) at a sequence of query points on a ray and integrating the weighted results. The proposed method reduces the number of evaluations without much performance degradation by computing the integrations using Gauss-Laguerre quadrature. The proposed method has been embedded in vanilla NeRF and TensoRF for validation, and the reviewer reports reductions in computation time and memory usage. The reviewer acknowledges and appreciates the effectiveness of the proposed method and looks forward to future discussions on the generality of the proposed method (other backbones and learning time applications).", "questions": "- Can the proposed method be used for learning? If so, the proposed method could be a very powerful tool. (identical to the description in the Weaknesses section).\n- Why are the results for TensoRF not shown in Figure 4?\n- The sample points up to 8 are very small compared to the use of more than 100 sample points in vanilla NeRF. Could the performance of vanilla NeRF be exceeded with more sample points? In other words, vanilla NeRF also approximates integration by discretization, and this sampling density is constant. We believe that a comparison of vanilla NeRF and the proposed method in terms of sample points vs. image quality will more convincingly demonstrate the effectiveness of the proposed method.", "limitations": "- I agree with the limitations described in the conclusion. This reviewer believes that the proposed method contributes to real-time rendering, but it better to be validated to make this claim in the future.", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- As emphasized in the paper, the approach proposed in this paper is a replacement of integral computations, not a learning of sampling or a change in data structure. Therefore, it is applicable and highly available for various NeRF variants that use volume rendering.\n- The position of the proposed method in NeRF research is clearly stated, especially the related work is very clearly described as an introduction to the proposed method.\n- In the introduction of the method, the reviewer's first question was l.177, and the paper answers the question. This helps the understanding of the paper and increases the credibility of the proposed method.", "weaknesses": "- The proposed method results in dense sampling near the surface, but can it handle translucent objects? Intuitively, there seems to be a correlation between density solidity and rendering quality. It would be desirable to discuss for which scenes the proposed method is effective and for which scenes it is not.\n- The effectiveness of the proposed method as an integration method for oracle or trained models is certain. In general, however, volume rendering is used for both training and inference. It would be better to discuss whether the proposed method is specialized for inference or whether it could be used for learning as well.\n- We expect the availability of the proposed method to be very broad. If possible, the application of the proposed method to NeRF backbones other than vanilla NeRF and TensoRF could be considered to further demonstrate the generality of the proposed method.\n- The correspondence between the plots in Figure 4 is not clear. It would be easier to compare the results if the corresponding scenes of vanilla and the proposed method were connected by a line.\n- Small Comment: Related works -> Related work", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for dedicating your time and providing such perceptive feedback. Your recommendations have considerably enhanced our work. Based on your comments, we have provide a brand new general framework for computing volume rendering and validate its accessibility on 2 baselines. We will address your concerns below.\n\n> The proposed method results in dense sampling near the surface, but can it handle translucent objects? Intuitively, there seems to be a correlation between density solidity and rendering quality. It would be desirable to discuss for which scenes the proposed method is effective and for which scenes it is not.\n\nSince volume rendering is originally derived for pure volumetric data, it is inherently not suitable for modeling translucent objects. To verify this argument, we conduct experiments using TensoRF on the DexNeRF dataset[5] and report the result here. \n\n| Method         | PSNR  | SSIM | LPIPS |\n| -------------- | ----- | ---- | ----- |\n| TensoRF        | 24.02 | 0.86 | 0.288 |\n| Ours + TensoRF | 23.99 | 0.86 | 0.298 |\n\nThe result shows that our approach performs similarly to traditional volume rendering methods. However, it is ideal that the scenes are pure volumetric since TensoRF itself suffers at modeling translucent objects.\n\n> The effectiveness of the proposed method ... is certain. In general, however, volume rendering is used for both training and inference. It would be better to discuss whether the proposed method ... could be used for learning as well.\n\nSince our focus is the training-free aspect of the proposed method, the results of network training with it are not reported in the paper. However, GL-NeRF can be used for learning as well. We conduct experiments combining GL-NeRF with Vanilla NeRF and show the results here. The training time is also reduced by $1.2\\times$ to $2\\times$.\n\n---\nBlender\n\n| Method                          | PSNR  | SSIM | LPIPS |\n| ------------------------------- | ----- | ---- | ----- |\n| Vanilla NeRF                    | 30.63 | 0.95 | 0.037 |\n| Ours + Vanilla NeRF (training)  | 29.18 | 0.93 | 0.056 |\n| Ours + Vanilla NeRF (test-only) | 28.56 | 0.93 | 0.070 |\n\nLLFF\n\n| Method                          | PSNR  | SSIM | LPIPS |\n| ------------------------------- | ----- | ---- | ----- |\n| Vanilla NeRF                    | 27.62 | 0.88 | 0.073 |\n| Ours + Vanilla NeRF (training)  | 27.21 | 0.87 | 0.087 |\n| Ours + Vanilla NeRF (test-only) | 26.53 | 0.85 | 0.090 |\n---\n\n> We expect the availability of the proposed method to be very broad. If possible, the application of the proposed method to NeRF backbones other than vanilla NeRF and TensoRF could be considered to further demonstrate the generality of the proposed method.\n\nPlease refer to the general response for the combination of GL-NeRF with Instant NGP.\n\n> The correspondence between the plots in Figure 4 is not clear. It would be easier to compare the results if the corresponding scenes of vanilla and the proposed method were connected by a line.\n\n> Small Comment: Related works -> Related work\n\nThank you for the construction comments! We’ll modify the figure and writing as suggested.\n\n> Why are the results for TensoRF not shown in Figure 4?\n\nAs mentioned in MCNeRF, NVIDIA GPUs have specialized components to accelerate the neural network inferences, therefore evaluating on these devices with more sample points may not cost that much. Therefore we instead implement a WebGL-based renderer as WebGL is a more accessible platform that is agnostic to the underlying hardware[1, 2, 3, 4]. Please find the result in the General Response section.\n\n> The sample points up to 8 are very small compared to the use of more than 100 sample points in vanilla NeRF. Could the performance of vanilla NeRF be exceeded with more sample points? In other words, vanilla NeRF also approximates integration by discretization, and this sampling density is constant. We believe that a comparison of vanilla NeRF and the proposed method in terms of sample points vs. image quality will more convincingly demonstrate the effectiveness of the proposed method.\n\nThanks for this very interesting point. When increasing the number of points used for GL-NeRF, since the predefined Laguerre weights approach to zero and are smaller than machine precision, the performance reaches a bottleneck. Therefore, we focus on the fact that GL-NeRF, with higher precision, can reduce the sample points by a large margin. An example of the Gauss Laguerre Quadrature look-up table with n=64 is presented here. Notice that when n gets bigger, the weights get extremely small and would become smaller than machine precision (for 32-bit float, it’s $1.175494 × 10^{-38}$). We do have a comparison in terms of sample points vs. image quality with a smaller number of points shown here.\n\n| n    | weight          | x              |\n| ---- | --------------- | -------------- |\n| 1    | $0.0563$ | $0.0224$ |\n| 2    | $0.119$ | $0.118$ |\n| 3    | $0.157$ | $0.290$ |\n| ...  | ...             | ...            |\n| 62   | $1.592\\times 10^{-88}$ | $204.672$ |\n| 63   | $2.989\\times 10^{-94}$ | $218.032$ |\n| 64   | $2.089\\times 10^{-101}$ | $234.810$ |\n\n\n>[1] Gupta, Kunal, et al. \"MCNeRF: Monte Carlo rendering and denoising for real-time NeRFs.\" SIGGRAPH Asia 2023 Conference Papers. 2023.\n\n>[2] Chen, Zhiqin, et al. \"Mobilenerf: Exploiting the polygon rasterization pipeline for efficient neural field rendering on mobile architectures.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.\n\n>[3] Reiser, Christian, et al. \"Merf: Memory-efficient radiance fields for real-time view synthesis in unbounded scenes.\" ACM Transactions on Graphics (TOG) 42.4 (2023): 1-12.\n\n>[4] Yariv, Lior, et al. \"Bakedsdf: Meshing neural sdfs for real-time view synthesis.\" ACM SIGGRAPH 2023 Conference Proceedings. 2023.\n\n>[5] Ichnowski, Jeffrey, et al. \"Dex-NeRF: Using a neural radiance field to grasp transparent objects.\" arXiv preprint arXiv:2110.14217 (2021).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PRuaf4etin", "reviewer": "Reviewer_PEjS", "summary": "Volume rendering requires numerical integration for estimating output colors. This work proposes using the Gauss-Laguerre quadrature to reduce the number of samples and improve integration accuracy. The paper demonstrates that this method can be a plug-and-play module for any NeRF model. Experimental results show that with a limited drop in performance, the GL-NeRF can significantly reduce the number of ray samples and MLP calls.", "questions": "1. Why use a look-up table? Will this lead to worse performance?\n2. Why did the performance not match TensoRF? Would increasing the number of point samples help?", "limitations": "The paper mentions that it has a theoretical guarantee of the highest precision. However, there is no evidence that the current precision matches previous works.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "1. The perspective of improving quadrature for NeRF is new and interesting.\n2. The formulation of using the Gauss-Laguerre quadrature looks good, and the mathematical formulation appears rigorous.", "weaknesses": "1. The results show a performance drop of about ~2 PSNR, while the speed improvement is not significant.\n2.  The motivation is not strong. Most state-of-the-art NeRF approaches use shallow MLPs or even no MLPs, making the evaluation less expensive. Reducing ray samples does not seem to address a core issue in radiance field research.\n3. How the points are selected is unclear. The method requires approximating polynomial coefficients and resolving the roots for $x$ However, since $x$ is a highly non-linear function of $t$, finding the samples  $t$ unavoidably requires root finding along the ray, which does not seem to actually improve accuracy or efficiency.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We’re thankful for your time and the valuable insights you’ve shared. Your input has significantly advanced our project. In response to your feedback, we proposed a brand new perspective for volume rendering with a strong math foundation and validated it with experiments. We’ll address your concerns below.\n\n\n> The results show a performance drop of about ~2 PSNR, while the speed improvement is not significant.\nSince vanilla NeRF’s speed is bottlenecked by the coarse network, we showcase the significant improvement in speed using TensoRF in the general response and paste it below. Our method achieves comparable quality as the baseline while running almost real-time on a AMD Ryzen 9 5900HS CPU. \n\n| Method         | PSNR  | SSIM | LPIPS | FPS   |\n| -------------- | ----- | ---- | ----- | ----- |\n| TensoRF        | 33.28 | 0.97 | 0.016 | 5.84  |\n| Ours + TensoRF | 33.09 | 0.97 | 0.016 | 22.34 |\n\n> The motivation is not strong. Most state-of-the-art NeRF approaches use shallow MLPs or even no MLPs, making the evaluation less expensive. Reducing ray samples does not seem to address a core issue in radiance field research.\n\nWe’d like to highlight that the position of our paper is a ***brand new perspective for volume rendering***, and its ***applicability towards the general NeRF pipeline*** that relies on volume rendering. Therefore, the experiments validate that it can be combined with any existing NeRF pipeline that relies on volume rendering regardless of the underlying representation. GL-NeRF is orthogonal to existing state-of-the-art work in the sense that all other works introduce additional representation such as neural networks and grids while GL-NeRF focuses only on the volume rendering integral itself. We believe it's a promising exploration of volume rendering and would bring insights into the radiance field research. \n\n\n> How the points are selected is unclear. The method requires approximating polynomial coefficients and resolving the roots for $x$. However, since $x$ is a highly non-linear function of $t$, finding the samples $t$ unavoidably requires root finding along the ray, which does not seem to actually improve accuracy or efficiency.\n\nWe’d like to point out that we do not need to compute the polynomial coefficients and resolving the roots. All the roots and coefficients needed are well defined and can be looked up online / in numpy. Please refer to the document for the function numpy.polynomial.laguerre.laggause for detail.\n\nAs for finding $t$ on the ray, traditional volume rendering uses a piecewise constant PDF for approximating the volume density, and so do we. Therefore, there is no need to find roots along the ray for a non-linear function, we use the same approximation as done in the hierarchical sampling strategy.\n\n> Why use a look-up table? Will this lead to worse performance?\n\nThe Gauss-Laguerre Quadrature is a well-known formula in numerical analysis, therefore there is no need to solve for the roots and coefficients on the run. Since using a look-up table is $O(1)$ in terms of time complexity, therefore it will not lead to worse performance.\n\n> Why did the performance not match TensoRF? Would increasing the number of point samples help?\n\nWe’d like to first point out that GL-NeRF uses fewer points than TensoRF (4 v.s. ~32), leading to a lower performance with no visual quality drop (our qualitative results are randomly chosen from the images, not cherry-picked). On the other hand, when increasing the number of points used for GL-NeRF, since the predefined Laguerre weights approach to zero and are smaller than machine precision(for 32-bit float, it’s $1.175494 × 10^{-38}$), the performance reaches a bottleneck. Therefore, we focus on the fact that GL-NeRF, with higher precision, can reduce the sample points by a large margin.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
# General Response 
We’d like to thank all the reviewers for their valuable feedback, especially for acknowledging our contributions regarding the theoretical foundation and the effort we made to validate and show the plug-and-play attribute of GL-NeRF. Regarding common concerns among reviewers, we have added some experiments and presented the results to address these concerns.

## TensoRF Speedup Report 
As mentioned in MCNeRF[1], NVIDIA GPUs have specialized components to accelerate the neural network inferences. Therefore, on those hardwares evaluating extra MLPs at more sample points may not incur as significant a cost as that on WebGL. Therefore we implemented a WebGL-based renderer and loaded TensoRF into it to test out the speedup GL-NeRF gives us. We follow MCNeRF to train a small TensoRF on LEGO scene in Blender Dataset that could fit in WebGL-based renderer and report its performance here. Our method achieves almost real-time performance in WebGL with similar quality as TensoRF running on a AMD Ryzen 9 5900HS CPU by reducing the sampling points for color MLP.

| Method         | PSNR  | SSIM | LPIPS | FPS   |
| -------------- | ----- | ---- | ----- | ----- |
| TensoRF        | 33.28 | 0.97 | 0.016 | 5.84  |
| Ours + TensoRF | 33.09 | 0.97 | 0.016 | 22.34 |





## Application of GL-NeRF to Other Approaches

To demonstrate the plug-and-play attribute of GL-NeRF to other approaches, we combine GL-NeRF and Instant NGP[2] for evaluation. The results are shown in the table below. ***The reason we choose Instant NGP is that it is another representative of NeRF pipeline that relies on volume rendering and we want to highlight that GL-NeRF can be incorporated into any NeRF pipeline that relies on volume rendering,  agnostic to the underlying representation.*** Since the result on WebGL using TensoRF proved that reducing the number of sample points can lead to significant speedup, here we simply give the MLP calls needed for reference.


| Method             | PSNR  | Avg. color MLPs calls |
| ------------------ | ----- | --------------------- |
| Instant NGP        | 32.05 | 30.90                 |
| Ours + Instant NGP | 30.35 | 4.00                  |


>[1] Gupta, Kunal, et al. "MCNeRF: Monte Carlo rendering and denoising for real-time NeRFs." SIGGRAPH Asia 2023 Conference Papers. 2023.

>[2] Müller, Thomas, et al. "Instant neural graphics primitives with a multiresolution hash encoding." ACM transactions on graphics (TOG) 41.4 (2022): 1-15.

Meta-review: This paper was reviewed by four experts in the field.  Based on the reviewers' feedback, the decision is to recommend the paper for acceptance.  The reviewers did raise some valuable concerns that should be addressed in the final camera-ready version of the paper. The authors are encouraged to make the necessary changes to the best of their ability.    We congratulate the authors on the acceptance of their paper!
Decision: Accept (poster)