GL-NeRF: Gauss-Laguerre Quadrature Enables Training-Free NeRF Acceleration

TL;DR

GL-NeRF significantly reduces the number of MLP calls needed for volume rendering, introducing no additional data structures or neural networks, showing the potential to speed up any NeRF model.

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.

Figures (6)

• 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.
• Figure 2: Verification on using the Gauss-Laguerre quadrature for volume rendering. We plot the red channel of the color function w.r.t. the ray it corresponds to. The color function remains zero in most of the interval (bottom). We use a 7th-degree polynomial to approximate the non-zero region (top). As can be seen, the color function itself is similar to a polynomial, validating the use of our approach.
• Figure 3: Point Selection strategy in GL-NeRF. We choose points along the ray that satisfy the integral from zero to the point of the volume density function should be equal to the roots of Laguerre polynomials. The points selected is then used for querying the color. In the figure above is an example of choosing $5$ points using a $5$-degree Laguerre polynomial. The number on the plot indicates the value of the integral from zero to the right boundary of the region.
• 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 2 times faster than the vanilla NeRF. In the mean time, the overall performance remains almost the same despite some minor decreases.
• Figure 5: Qualitative results on LLFF (top) and NeRF-Synthetic (bottom) datasets. We could tell from the comparisons that the drop in performances has minimal effect on the visual quality.

Theorems & Definitions (7)

• Theorem 4.1: Stone-Weierstrass theorem
• Theorem A.1
• proof
• Theorem A.2
• proof
• Theorem A.3
• proof