Gaussian Opacity Fields: Efficient Adaptive Surface Reconstruction in
Unbounded Scenes
ZEHAO YU, University of Tübingen, Tübingen AI Center, Germany
TORSTEN SATTLER, Czech Technical University in Prague, Czech Republic
ANDREAS GEIGER, University of Tübingen, Tübingen AI Center, Germany
https://niujinshuchong.github.io/gaussian-opacity-fields

Fig. 1. Applying TSDF fusion with rendered depth maps from the state-of-the-art Mip-Splatting [Yu et al. 2024a] models results in noisy and incomplete
meshes, while meshes extracted with our method are complete, smooth, and detailed. This is achieved by establishing Gaussian opacity fields from 3D
Gaussians, which enables geometry extraction by directly identifying its level-set. Moreover, we generate tetrahedral meshes from 3D Gaussians and utilize
Marching Tetrahedra to extract adaptive and compact meshes.
Recently, 3D Gaussian Splatting (3DGS) has demonstrated impressive novel
view synthesis results, while allowing the rendering of high-resolution images in real-time. However, leveraging 3D Gaussians for surface reconstruction poses significant challenges due to the explicit and disconnected natureof 3D Gaussians. In this work, we present Gaussian Opacity Fields (GOF),
a novel approach for efficient, high-quality, and adaptive surface reconstruction in unbounded scenes. Our GOF is derived from ray-tracing-based
volume rendering of 3D Gaussians, enabling direct geometry extraction
from 3D Gaussians by identifying its levelset, without resorting to Poissonreconstruction or TSDF fusion as in previous work. We approximate thesurface normal of Gaussians as the normal of the ray-Gaussian intersection
plane, enabling the application of regularization that significantly enhances
geometry. Furthermore, we develop an efficient geometry extraction method
utilizing Marching Tetrahedra, where the tetrahedral grids are induced from
3D Gaussians and thus adapt to the scene’s complexity. Our evaluations
reveal that GOF surpasses existing 3DGS-based methods in surface reconstruction and novel view synthesis. Further, it compares favorably to or evenoutperforms, neural implicit methods in both quality and speed.
CCS Concepts: • Computing methodologies → Reconstruction; Rendering; Machine learning approaches.
Additional Key Words and Phrases: Novel View Synthesis, Differentiable
Rendering, Gaussian Splatting, Surface Reconstruction, Multi-view-to-3D
Authors’ addresses: Zehao Yu, University of Tübingen, Tübingen AI Center, Germany;
Torsten Sattler, Czech Technical University in Prague, Czech Republic; Andreas Geiger,
University of Tübingen, Tübingen AI Center, Germany.
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https://doi.org/10.1145/3687937

ACM Reference Format:
Zehao Yu, Torsten Sattler, and Andreas Geiger. 2024. Gaussian Opacity
Fields: Efficient Adaptive Surface Reconstruction in Unbounded Scenes.
ACM Trans. Graph. 43, 6, Article 271 (December 2024), 13 pages. https://doi.org/10.1145/3687937

1

INTRODUCTION

3D Reconstruction from multi-view images has been a long-standing
goal in computer vision, with various applications in robotics, graphics, animation, virtual reality, and more. Since Neural Radiance Field(NeRF) [Mildenhall et al. 2020] demonstrated impressive novel view
synthesis (NVS) results with implicit representations [Mescheder
et al. 2019; Park et al. 2019] and volume rendering [Drebin et al.
1988; Kajiya and Von Herzen 1984; Levoy 1990], it has been extended
to surface reconstruction with occupancy networks [Oechsle et al.
2021] and Signed Distance Functions (SDF) [Wang et al. 2021; Yariv
et al. 2021]. While recent advancements [Li et al. 2023; Yariv et al.
2023; Yu et al. 2022a,b] have shown impressive reconstruction results, these methods are mostly limited to reconstructing foregroundobjects [Rosu and Behnke 2023] and computationally expensive to
optimize [Yariv et al. 2023]. For instance, Neuralangelo [Li et al.
2023] models the background separately with NeRFs and necessitates approximately 128 GPU hours to reconstruct a single scene.
Another research avenue focuses on directly extracting surfaces
from NeRF’s opacity field for real-time rendering [Chen et al. 2023a;
Rakotosaona et al. 2024; Reiser et al. 2024; Tang et al. 2022]. Thesemethods employ a modular workflow including opacity field training, mesh extraction, simplification, and refinement. Particularly
noteworthy is Binary Opacity Grids (BOG)[Reiser et al. 2024], which
excels at capturing intricate details in unbounded scenes throughsuper-sampling. To extract detailed surfaces, it renders depth maps
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271:2 •Zehao Yu, Torsten Sattler, and Andreas Geiger

to generate a sparse high-resolution voxel grid and applies a heuristic fusion technique to label the voxels as inside or outside. Marching cube algorithm [Lorensen and Cline 1998] is applied to extract
a high-resolution mesh with hundreds of millions of points andthousands of millions of triangles, which is then simplified using
slow post-processing techniques [Garland and Heckbert 1997]. Notably, as it focuses on NVS rather than surface reconstruction, theextracted meshes are noisy and contain fewer details in the background region, probably due to lack of regularization and contracted
space [Barron et al. 2022a] fusion.
More recently, 3D Gaussian Splatting (3DGS) [Kerbl et al. 2023]
represents complex scenes as a set of 3D Gaussians, demonstrating
photorealistic NVS results while trained efficiently and rendered
in real-time. It has been quickly extended to surface reconstruction [Chen et al. 2023b; Guédon and Lepetit 2023; Huang et al. 2024;
Yu et al. 2024b]. Notably, SuGaR [Guédon and Lepetit 2023] regularizes the 3D Gaussians to align with surfaces and employs Poisson
surface reconstruction [Kazhdan and Hoppe 2013] to extract a meshfrom rendered depth maps. 2D Gaussian Splatting (2DGS) [Huang
et al. 2024] uses 2D Gaussians instead of 3D Gaussians as a scene
representation for better surface representation and utilizes TSDF
fusion to reconstruct a mesh. While these methods have shown
improved reconstruction, they struggle with extracting fine-grained
geometry [Guédon and Lepetit 2023] and reconstructing backgroundregions [Huang et al. 2024]. A primary challenge is the inconsistency
between mesh extraction and volume rendering during training.
Specifically, Poisson reconstruction ignores the opacity and scale ofGaussian primitives and rendered depth maps are not sufficiently
reliable. Moreover, TSDF fusion struggles to accurately model thin
structures and to reconstruct unbounded scenes. Resorting to highresolution voxel grids for TSDF leads to the creation of large meshes,
similar to BOG [Reiser et al. 2024], due to the lack of adaptivity inthe grid resolution relative to the scene’s geometric complexity.
Contributions: In this paper, we propose Gaussian Opacity Fields(GOF), a novel approach to achieve efficient, high-quality, and adaptive surface reconstruction from 3D Gaussians directly. Our key
insights are threefold: First, we establish a Gaussian opacity fieldfrom a set of 3D Gaussians. Specifically, unlike projection-based
volume rendering, our method leverages an explicit ray-Gaussianintersection to determine a Gaussian’s contribution during volume
rendering. Our ray-tracing-inspired formula facilitates the evaluation of opacity values for any point along a ray. We then define theopacity of any 3D point as the minimal opacity among all training
views that observed the point. Taking the minimum opacity acrossall views achieves view independence, making the opacity field
solely a function of position. Our GOF is consistent with volume
rendering during training and enables surface extraction from 3D
Gaussians by directly identifying a level set, without resorting toPoisson reconstruction or TSDF fusion. Second, we approximate
the surface normals of 3D Gaussians as the normals of intersection planes between the ray and Gaussians. This technique allows
for the incorporation of regularizations [Huang et al. 2024] during
training, thus enhancing the fidelity of geometry reconstruction.
Third, we propose an efficient surface extraction technique based

ACM Trans. Graph., Vol. 43, No. 6, Article 271. Publication date: December 2024.

on tetrahedra-grids. Recognizing that 3D Gaussians effectively indicate potential surface locations, we focus opacity evaluations onthese areas. In particular, we use the center and corners of 3D bounding boxes around the 3D Gaussian primitives as vertex sets for the
tetrahedral mesh. Upon assessing the opacity at tetrahedral points,
we utilize the Marching Tetrahedra algorithm for triangle mesh extraction. Given that our opacity fields challenge the assumption that
opacity changes linearly, we further implement a binary search algorithm to accurately identify the opacity field’s level set, substantiallyenhancing the quality of the resulting surfaces.
To demonstrate the effectiveness and efficiency of GOF, we carryout extensive experiments across three challenging datasets [Barron
et al. 2022a; Jensen et al. 2014; Knapitsch et al. 2017]. Our results
indicate that GOF not only matches but, in some cases, surpasses
the performance of existing SDF-based methods, while being much
faster. Moreover, GOF outperforms all other 3DGS-based methods
in both surface reconstruction and novel view synthesis.

2 RELATED WORK
2.1 Novel view synthesis
NeRF [Mildenhall et al. 2020] utilizes a multi-layer perception (MLP)
for scene representation, including geometry and view-dependent
appearances. The MLP is optimized via a photometric loss throughvolume rendering [Drebin et al. 1988; Kajiya and Von Herzen 1984;
Levoy 1990; Max 1995]. Subsequent enhancements have focused on
optimizing NeRF’s training using feature-grid representations [Chenet al. 2022; Fridovich-Keil et al. 2022; Kulhanek and Sattler 2023;
Müller et al. 2022; Sun et al. 2022] and improving rendering speed
via baking [Hedman et al. 2021; Reiser et al. 2021, 2023; Yariv et al.
2023]. Moreover, NeRF has been adapted to address challenges inanti-aliasing [Barron et al. 2022b, 2023] and unbounded scene modeling [Barron et al. 2022a; Zhang et al. 2020]. More recently, 3D
Gaussian splatting [Kerbl et al. 2023] represents complex scenes
with 3D Gaussians. It demonstrated impressive NVS results while
being optimized efficiently and rendering high-resolution images
in real-time. Subsequent works improved its rendering quality via
anti-aliasing [Yu et al. 2024a] or extended it to dynamic scenes
modeling [Zhou et al. 2024], and more [Chen and Wang 2024]. In
this work, we extend 3DGS for high-quality surface reconstruction
through the development of Gaussian Opacity Fields. We further
introduce an efficient tetrahedron grid-based mesh extraction algorithm to extract scene adaptive and compact meshes.

2.2

3D reconstruction

3D Reconstruction from multi-view images is a fundamental problem in computer vision. Multi-view stereo methods [Schönberger
et al. 2016; Yao et al. 2018; Yu and Gao 2020] often employ complex
multi-stage pipelines that include feature matching, depth estimation, point clouds fusion, and ultimately, surface reconstruction
from aggregated point clouds [Kazhdan and Hoppe 2013]. In contrast, neural implicit methods [Oechsle et al. 2021; Wang et al. 2021;
Yariv et al. 2021] significantly simplify the pipeline by optimizing an implicit surface representation via volume rendering. After
optimization, triangle meshes can be extracted easily with Marching Cubes [Lorensen and Cline 1998] at any resolution. Notable

Gaussian Opacity Fields: Efficient Adaptive Surface Reconstruction in Unbounded Scenes •advancements have been made through the adoption of more expressive scene representations [Li et al. 2023], advanced training
strategies [Li et al. 2023], and the integration of monocular priors [Yuet al. 2022b]. Despite these advances, these methods are mostly limited in reconstructing foreground objects [Rosu and Behnke 2023]
and are computationally expensive to optimize [Li et al. 2023; Yariv
et al. 2023]. Furthermore, the use of high-resolution grids for capturing fine details often results in excessively large meshes. By contrast,
we establish Gaussian Opacity Fields using 3DGS [Kerbl et al. 2023],
which facilitates fast training. We utilize Marching Tetrahedra [Doiand Koide 1991; Shen et al. 2021] to extract adaptive, compact, anddetailed meshes in unbounded scenes.

2.3

Surface Reconstruction with Gaussians

Inspired by the impressive NVS performance of 3D Gaussian Splatting (3DGS) [Kerbl et al. 2023], researchers have proposed to utilize
3D Gaussians for surface reconstruction. Recent efforts [Chen et al.
2023b; Yu et al. 2024b] have integrated 3D Gaussians with neural implicit surfaces, optimizing a Signed Distance Function (SDF)
network and 3D Gaussians jointly. While these approaches marksome advancements, they also inherit the shortcomings associatedwith implicit surfaces as previously discussed. Other studies havefocused on surface extraction from optimized Gaussian primitives
through post-processing techniques [Dai et al. 2024; Guédon and
Lepetit 2023; Huang et al. 2024; Turkulainen et al. 2024]. Notably,
SuGaR [Guédon and Lepetit 2023] and GaussianSurfels [Dai et al.
2024] adopt Poisson surface reconstruction [Kazhdan and Hoppe
2013] to extract meshes from rendered depth maps. Meanwhile, 2D
Gaussian Splatting (2DGS) [Huang et al. 2024] employs TSDF fusion
for this purpose. Though these methods achieve improved reconstructions, they face challenges in capturing fine-grained geometry
and adequately reconstructing background areas. In this work, wederive Gaussian Opacity Fields directly from 3D Gaussians. Our GOF
is consistent with the volume rendering process for rendering RGB
images. It enables direct surface extraction by identifying a levelset, without resorting to Poisson reconstruction and TSDF fusion.
Additionally, we propose a tetrahedron grid-based technique formesh extraction, resulting in adaptive and detailed meshes.

3

METHOD

Given multiple posed and calibrated images, our goal is to reconstruct the 3D scene efficiently while allowing detailed and compact
surface extraction and photorealistic novel view synthesis. To thisend, we first construct Gaussian Opacity Fields (GOF) from 3D Gaussians, enabling geometry extraction directly by identifying a levelset, eliminating the need for Poisson reconstruction or TSDF fusion.
Next, we extend two effective regularizers from 2DGS [Huang et al.
2024] to our 3D Gaussians, improving reconstruction quality. Finally,
we propose a novel tetrahedra-based method to extract detailed andcompact meshes from GOFs with marching tetrahedra.

by center p𝑘 ∈ R3 , scaling matrix S𝑘 ∈ R3×3 , and rotation R𝑘 ∈R3×3 parameterized by a quaternion:
1

Modeling

Similar to prior works [Kerbl et al. 2023; Zwicker et al. 2001], we
represent the scene with a set of 3D Gaussian primitives {G𝑘 |𝑘 =1, · · · , 𝐾 }. The geometry of each 3D Gaussian G𝑘 is parameterized−1

G𝑘 (x) = 𝑒 − 2 (x−p𝑘 ) 𝚺𝑘 (x−p𝑘 )𝑇(1)where 𝚺𝑘 ∈ R3×3 is a covariance matrix defined as 𝚺𝑘 = R𝑘 S𝑘 S𝑇𝑘 R𝑇𝑘 .
Ray Gaussian Intersection: Instead of projecting 3D Gaussians
to 2D screen space and evaluating the Gaussian in 2D, we evaluate
the contribution of a Gaussian to a ray with explicit ray-Gaussianintersection [Gao et al. 2023; Keselman and Hebert 2022]. As we will
see in Section 3.2, this crucially enables evaluation of opacity valuesof arbitrary 3D points in contrast to the projection-based Gaussian
splatting mechanism of [Kerbl et al. 2023] where 3D information
is lost during the 3D-to-2D projection step. The ray-Gaussian intersection is defined as the point where the Gaussian reaches its
maximum value along the ray. Specifically, given a camera centerat o ∈ R3 and a ray direction r ∈ R3 , any point x ∈ R3 along theray can be written as x = o + 𝑡r, where 𝑡 is depth of the ray. We
first transform the point x to the local coordinate system of the 3D
Gaussian and normalize the point by its scale:o𝑔 = S𝑘−1 R𝑘 (o − p𝑘 )(2)

r𝑔 = S𝑘−1 R𝑘 r

(3)x𝑔 = o𝑔 + 𝑡r𝑔(4)

In this local coordinate system, the Gaussian value at any point
along the ray becomes a 1D Gaussian, which is described by:
1 𝑇

1

G𝑘1𝐷 (𝑡) = 𝑒 − 2 x𝑔 x𝑔 = 𝑒 − 2 (r𝑔 r𝑔 𝑡 +2o𝑔 r𝑔 𝑡 +o𝑔 o𝑔 )
𝑇

2

𝑇

𝑇

(5)

The maximum of Eq. 5 is achieved when the quadratic term reaches
maximum. Therefore, the closed-form solution for Eq. 5 is𝑡∗ = −

𝐵
𝐴(6)

where 𝐴 = r𝑇𝑔 r𝑔 and 𝐵 = o𝑇𝑔 r𝑔 . Note that our formula is equivalent
to the one presented in [Keselman and Hebert 2022], where the rayGaussian intersection is computed directly in world space. However,using the normalized Gaussian coordinate by transforming the rayoffers a clearer geometric interpretation and simplifies defining thenormal of the intersection plane, as we will show in Sec. 3.3.
Now, we define the contribution E of a Gaussian G𝑘 for a given
camera center o and ray direction r as:
E (G𝑘 , o, r) = G𝑘1𝐷 (𝑡 ∗ )(7)Volume Rendering: Similar to 3DGS [Kerbl et al. 2023], the colorof a camera ray is rendered via alpha blending according to theprimitive’s depth order 1, . . . , 𝐾:c(o, r) =

𝐾
∑︁
𝑘=1

3.1

271:3

c𝑘 𝛼𝑘 E (G𝑘 , o, r)𝑘Ö
−1(1 − 𝛼 𝑗 E (G𝑗 , o, r))

(8)

𝑗=1

where c𝑘 is the view-dependent color, modeled with spherical harmonics, and 𝛼𝑘 ∈ [0, 1] is an additional parameter that influences
the opacity of Gaussian 𝑘. To efficiently render an image, we utilize
the same tile-based rendering process as in 3DGS [Kerbl et al. 2023].
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271:4 •Zehao Yu, Torsten Sattler, and Andreas Geiger

Fig. 2. Illustration of ray tracing volume rendering. Evaluating a 3D
Gaussian along a ray results in a 1D Gaussian, which has a closed-form
solution for when it reaches maximal value. We define the opacity along
the ray as monotonously increasing until it reaches the maximal value andconstant afterward.

3.2

Fig. 3. Definition of Gaussian’s normal. We approximate the Gaussian’s
normal as the normal of the ray-Gaussian intersection plane. First, wetransform the ray into the Gaussian coordinate system and normalize itusing the Gaussian’s scales. In this normalized coordinate system, the rayGaussian intersection plane is perpendicular to the ray, making the normal
the inverse of the ray direction. Finally, we transform the intersection planeback to world space by reversing the normalization process.

3.3

Gaussian Opacity Fields

One significant benefit of using explicit ray-Gaussian intersection
instead of projection is that it allows evaluating the opacity value ortransmittance of any 3D point x along the ray. Let’s first consider
the case when there is only a single Gaussian G𝑘 along the ray. In
this case, we define the opacity of any 3D point along the ray as(
G𝑘1𝐷 (𝑡) if 𝑡 ≤ 𝑡 ∗O𝑘 (G𝑘 , o, r, 𝑡) =G𝑘1𝐷 (𝑡 ∗ ) if 𝑡 > 𝑡 ∗where x = o + 𝑡r. Intuitively, the opacity (reverse of transmittance)increases until it reaches its maximal value, and remains constantafterwards as illustrated in Figure 2. Therefore, the opacity at any
point along a ray given a set of Gaussians can be defined similar tothe volume rendering process in Eq. 8 as:

Optimization

Optimizing 3D Gaussians with pure photometric loss leads to noisy
results as 3D reconstruction from multi-views is an underconstrained
problem [Barron et al. 2022a; Zhang et al. 2020]. Therefore, we extend the regularization terms in 2DGS [Huang et al. 2024] to optimize our 3D Gaussians, including a depth distortion loss and anormal consistency loss.
Depth Distortion: We apply the depth distortion loss [Huang et al.
2024], which is originally proposed in Mip-NeRF 360 [Barron et al.
2022a], to the ray-Gaussian intersection to concentrate Gaussians
along the ray:∑︁L𝑑 =𝜔𝑖 𝜔 𝑗 |𝑡𝑖 − 𝑡 𝑗 |
(11)
𝑖,𝑗

(10)

where 𝑖, 𝑗 index over Gaussians contributed to the ray and 𝜔𝑖 =
Î −1
𝛼𝑘 E (G𝑘 , o, r) 𝑘𝑗=1(1−𝛼 𝑗 E (G𝑗 , o, r)) is the blending weight of the𝑖−th Gaussian and 𝑡𝑖 is the depth of the intersection point in Eq. 6.
However, the distortion loss minimizes both the distance between
Gaussians and the weights of each Gaussian whereas minimizing theweights of the Gaussians could lead to an increase in alpha values
for Gaussians that are blended first, which results in exaggerated
Gaussians, resulting in floaters. Therefore, we detach the gradientof weights 𝜔𝑖 and only minimize the distance between Gaussians.

We refer to O(x) as the Gaussian Opacity Field (GOF) since it is an
opacity field derived from 3D Gaussians. Our GOF shares similarities
to the visual hull [Laurentini 1994] or space carving [Kutulakos andSeitz 2000]. But instead of using a silhouette where the opacity valueof a ray (all points on the ray) is either 1 or 0, GOF uses volume
rendering to evaluate opacity for each point from 3D Gaussians.
Our GOF is consistent with the volume rendering process for
RGB rendering during training. With GOF, we can extract surfaces
directly by identifying their level sets, similar to UNISURF [Oechsle
et al. 2021], without resorting to Poisson reconstruction [Guédon
and Lepetit 2023] or TSDF fusion [Huang et al. 2024]. We will discuss
our method for efficient and adaptive mesh extraction in Section 3.4.
We also note that GOFs are a general formula as long as the scene
representation is a set of 3D Gaussians. For example, we can use it
to extract a mesh from a pre-trained 3DGS [Kerbl et al. 2023] or MipSplatting [Yu et al. 2024a] model, where projection-based volume
rendering is used for training, as we will show in the experiments.

Normal Consistency: A key challenge of applying 2DGS’s normal
consistency regularization [Huang et al. 2024] to 3D Gaussians isthat the gradient of 3D Gaussians always points outwards from thecenters. Consider a simple case when we render a single isotropic
3D Gaussian to image space. The rendered result is a 2D Gaussian in the image plane. The gradient of this 2D Gaussian always
points outward from the projected 2D center, meaning the rendered
normals at two different pixels will differ as long as the directions
from the projected 2D Gaussian center to the pixel coordinates vary.
Moreover, the normal at the projected 2D center is not well-defined.
This ambiguity makes the optimization difficult.
To mitigate this issue, we define the normal of a 3D Gaussianas the normal of the intersection plane given a ray direction. An
illustration is shown in Figure 3. Specifically, we begin by transforming the ray into the Gaussian coordinate system and normalizing it
using the Gaussian’s scales, as described in Eq. 3. In the normalized
coordinate system, the ray-Gaussian intersection plane is perpendicular to the ray. Therefore, the normal is the inverse ray direction

O(o, r, 𝑡) =𝐾∑︁𝛼𝑘 O𝑘 (G𝑘 , o, r, 𝑡)𝑘Ö
−1(1 − 𝛼 𝑗 O 𝑗 (G𝑗 , o, r, 𝑡))

(9)

𝑗=1

𝑘=1

As a 3D point might be visible by any training view, we define theopacity of a 3D point x as the minimal opacity value among all
training views or viewing directions:O(x) = min O(o, r, 𝑡)(o,r)

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Gaussian Opacity Fields: Efficient Adaptive Surface Reconstruction in Unbounded Scenes •3DGS [Kerbl et al. 2023]

3DGS + Our Densification

Mip-Splatting [Yu et al. 2024a]

271:5

Mip-Splatting + Our Densification

Fig. 4. Comparison of densification strategy on the Mip-NeRF 360 Dataset [Barron et al. 2022a]. Applying our densification to 3DGS [Kerbl et al.
2023] and Mip-Splatting [Yu et al. 2024a] significant improves the NVS results. Please note that the glass regions are blur in both 3DGS and Mip-Splatting,while our method renders the image faithfully.

−r𝑔 . Next, we reverse the normalization to transform the intersection plane back to world space, where the plane’s normal is given
by −S𝑘−1 r𝑔 . Finally, we apply the inverse of the world-to-Gaussianrotation to transform the normal back to world space, yielding thenormal direction of the intersection plane as n𝑖 = −R𝑇𝑘 S𝑘−1 r𝑔 , whichshould then be normalized to ensure it is a valid unit vector. It
is important to note that in world space, the intersection plane is
not necessarily perpendicular to the ray direction, but the plane’s
normal is always perpendicular to the intersection plane.
After defining the normal of a Gaussian, we apply the depthnormal consistency regularisation:∑︁L𝑛 =𝜔𝑖 (1 − n𝑖⊤ N)(12)
𝑖where 𝑖 indexes over intersected Gaussians along the ray, 𝜔 denotes
the blending weight, and N is the normal estimated by the gradientof the depth map [Huang et al. 2024].
Final Loss: Finally, we optimize our model from an initial sparse
point cloud using multiple posed images with the following loss:
L = L𝑐 + 𝛼 L𝑑 + 𝛽L𝑛(13)where L𝑐 is an RGB reconstruction loss combining L1 with the
D-SSIM term from [Kerbl et al. 2023], while L𝑑 and L𝑛 are regularization terms. Note that, we utilize the decoupled appearance
modeling proposed in VastGaussian [Lin et al. 2024] to model theuneven illumination for the Tanks and Temples dataset [Knapitsch
et al. 2017], where a small convolutional neural network is used
to predict image-dependent colors such that the model will not
fake inconsistent illumination with geometry. While other variants
could also be applicable, we use VastGaussian’s solution due to its
demonstrated improvements in large-scale scenarios and its ease ofreimplementation.
Improved Densification: As the optimization starts from a sparse
point cloud, it is necessary to increase the number of 3D Gaussians tobetter reconstruct the scene. We follow the densification strategy in
3DGS [Kerbl et al. 2023]. Specifically, the densification of a Gaussian(either by cloning or splitting) is guided by the magnitude of theview-space position gradient 𝑑𝐿
𝑑x , where x is the center of projected𝑑𝐿
Gaussian. Mathematically, 𝑑x sums over pixels p𝑖 that the Gaussian
contributed to:𝑑𝐿 ∑︁ 𝑑𝐿 𝑑p𝑖
=
(14)
𝑑x
𝑑p𝑖 𝑑x𝑖

If the norm of the gradient ∥ 𝑑𝐿
𝑑x ∥ 2 is above a predefined threshold𝜏x , the Gaussian is chosen as the candidate for densification.
However, we found that this metric is not effective in identifying
overly blurred areas, due to its inability to distinguish between
well-reconstructed regions and those where the gradient signals
from different pixels negate each other, leading to minimal overall
gradient magnitudes. Therefore, we propose a simple modification
to the metric that accumulates the norms of the individual pixel
gradients instead:∑︁ 𝑑𝐿 𝑑p𝑖
𝑀=
∥
∥
(15)
𝑑p𝑖 𝑑x𝑖
Our metric 𝑀 better indicates regions with significant reconstruction errors, resulting in better reconstruction and novel view synthesis results, as shown in Figure 4.

3.4

Surface Extraction

Post-training, the conventional step towards surface or triangle
mesh extraction involves densely evaluating the opacity values
within regions of interest, a technique well-suited to simple scenarios like the DTU Dataset [Jensen et al. 2014], as done in previous
work [Oechsle et al. 2021; Wang et al. 2021; Yariv et al. 2021]. For alarge-scale unbounded scene, some have adopted dense evaluation
in a contracted space [Yariv et al. 2023]. However, dense evaluation
for grids incurs substantial computational costs due to the cubic
growth of complexity with grid resolution. Unfortunately, capturing
fine details necessitates high-resolution grids, leading to significant
overhead. Alternative sparse grids may reduce dense evaluation but
still result in huge meshes, often comprising hundreds of millionsof points and billions of faces. Simplifying such large and complex
meshes typically requires a slow post-processing step. For instance,mesh simplification in BOG [Reiser et al. 2024] requires approximately 4 hours. To circumvent these challenges, we introduce anovel method for extracting adaptive and compact meshes using
tetrahedral grids and marching tetrahedra.
Tetrahedral Grids Generation: Our primary insight is that theposition and scale of 3D Gaussian primitives serve as reliable indicators for the presence of surfaces. To capitalize on this, we define a
3D bounding box around each Gaussian primitive, where the extent
of the 3D box is 3 times the Gaussian scales. The 3D box’s center has
the highest opacity and its corners have the smallest opacity. Notethat we do not consider the opacity 𝛼 of Gaussian primitives but
it could be used to filter out Gaussians with low opacity value. We
ACM Trans. Graph., Vol. 43, No. 6, Article 271. Publication date: December 2024.

271:6 •Zehao Yu, Torsten Sattler, and Andreas Geiger(a) Marching Cubes(b) Marching Cubes with Binary Search

Fig. 5. Comparison of applying binary search to Marching cubes.
Strong step artifacts can be observed in Marching cubes results since thelinear assumption does not hold in our GOF. Applying our binary searchalgorithm eliminates this artifact.

create tetrahedral grids with the center and corners of 3D bounding boxes. Inspired by Tetra-NeRF [Kulhanek and Sattler 2023], we
employ the CGAL Library [Jamin et al. 2024] for Delaunay triangulation to construct tetrahedral cells. The generated tetrahedralcell might connect points across significant distances. Therefore,
we employ a filtering step for the tetrahedral cells, removing any
cell whose edges connect non-overlapping Gaussians. Gaussians are
considered non-overlapping when the length of the edge connecting them exceeds the sum of their maximum scales, i.e., maximum
3-sigma extent dimension.
Efficient Opacity Evaluation: To efficiently evaluate the opacity ofthe vertices of the tetrahedral grid, we design a tile-based evaluation
algorithm, inspired by 3DGS [Kerbl et al. 2023]. Specifically, we firstproject the vertices to image space and identify the corresponding
tiles for these projections. These points are then organized according
to their tile ID. For each tile, we retrieve the list of points that are
projected within it, project these points again to identify the pixelthat it falls inside and identify the Gaussians that contribute tothis pixel. Finally, we enumerate all points to evaluate their opacity
based on the pre-filtered list of Gaussians. Note that this process
is iterated over all training images. Then we take the minimum
opacity over all training images as the opacity for the vertices of
the tetrahedral grid. An overview of the algorithm is provided in
the supplementary material.
Binary Search of Level Set: Upon determining the opacity values
for the tetrahedral grid, we proceed to extract triangle meshes using
the Marching Tetrahedra method [Shen et al. 2021]. Traditionalalgorithms, including Marching Cubes [Lorensen and Cline 1998]
and Marching Tetrahedra, typically rely on linear interpolation toapproximate the level set, presuming the underlying field’s linearity. This assumption, however, misaligns with the characteristics
of our Gaussian Opacity Field, leading to artifacts due to linear interpolation, as shown in Figure 5 (a). To overcome this discrepancy
and accurately identify the level set within our non-linear opacity
field, we relax the linear assumption to a monotonically increasing
assumption. This adjustment allows for the implementation of a binary search algorithm to precisely identify the level set. In practice,
we found that conducting 8 iterations of binary search—effectively
simulating 256 dense evaluations—yields consistent and reliable
outcomes. A comparison highlighting the improvements of binary
search is shown in Figure 5.
ACM Trans. Graph., Vol. 43, No. 6, Article 271. Publication date: December 2024.

Table 1. Quantitative results on the Tanks and Temples
Dataset [Knapitsch et al. 2017]. Reconstructions are evaluatedwith the official evaluation scripts and we report F1-score and average
optimization time. The results of implicit methods are taken from
Neuralangelo [Li et al. 2023] and the results of explicit methods are takenfrom 2DGS [Huang et al. 2024], where the mesh of 3DGS [Kerbl et al. 2023]
and 2DGS [Huang et al. 2024] are extracted with TSDF fusion and SuGaR
uses Poisson reconstruction. GOF outperforms all 3DGS-based surface
reconstruction methods by a large margin and performs comparably with
the SOTA neural implicit methods while optimizing significantly faster.

Barn
Caterpillar
Courthouse
Ignatius
Meetingroom
Truck
Mean
Time

4

NeuS
0.29
0.29
0.17
0.83
0.24
0.45
0.38
>24h

Implicit
Geo-Neus Neuralangelo
0.33
0.70
0.26
0.36
0.12
0.28
0.72
0.89
0.20
0.32
0.45
0.48
0.35
0.50
>24h
>24h

SuGaR
0.14
0.16
0.08
0.33
0.15
0.26
0.19
>1h

Explicit
3DGS 2DGS
0.13
0.41
0.08
0.23
0.09
0.16
0.04
0.51
0.01
0.17
0.19
0.45
0.09
0.32
14.3 m 15.5 m

Ours
0.51
0.41
0.28
0.68
0.28
0.59
0.46
24.2 m

EXPERIMENTS

We conduct a thorough evaluation of our Gaussian Opacity Fields(GOF), comparing its surface reconstruction and novel view synthesis against leading methods. We further validate the effectivenessof its key components through ablation studies.

4.1

Implementation Details

We build GOF upon the open-source 3DGS code base [Kerbl et al.
2023] and implement custom CUDA kernels for ray-tracing-based
volume rendering, regularizations, and opacity evaluation. Regularization parameters are set to 𝛼 = 1000 for bounded scenes,
𝛼 = 100 for unbounded scenes, and 𝛽 = 0.05 for all scenes, following 2DGS [Huang et al. 2024]. As our improved densification
strategy increases the number of primitives for the same hyperparameters, we use a higher opacity threshold 0.05 instead of 0.005
for Gaussian pruning, resulting in a similar number of primitives
post-training for a fair comparison. Similar to 3DGS, we stop densification at 15k iterations and optimize all of our models for 30k
iterations. For mesh extraction, we adapt the Marching Tetrahedra
algorithm [Shen et al. 2021] from the Kaolin library [Fuji Tsang et al.
2022] with our binary search algorithm and extract the mesh for
the 0.5 level-set. All experiments are conducted on an NVIDIA A100
GPU.

4.2

Geometry Evaluation

We first compare against both SOTA implicit and explicit surface
reconstruction methods on the Tanks and Temples Dataset. Reconstructions are evaluated only for the foreground objects since theground truth point clouds do not contain background regions. As
shown in Table 1, our method is competitive with the leading implicit approaches [Li et al. 2023] while being much more efficient
to be optimized. Note that most implicit approaches [Li et al. 2023;
Wang et al. 2021] only reconstruct the foreground objects, while ourmethod can reconstruct detailed meshes also for the backgroundregions, which is of great importance for mesh-based real-timerendering [Reiser et al. 2024]. Furthermore, while our method isslightly slower than 3DGS [Kerbl et al. 2023] and 2DGS [Huang et al.

Gaussian Opacity Fields: Efficient Adaptive Surface Reconstruction in Unbounded Scenes •Neuralangelo [Li et al. 2023]

SuGar [Guédon and Lepetit 2023]

2DGS [Huang et al. 2024]

Ours

271:7

GT

Fig. 6. Surface Reconstruction on the Tanks and Temples Dataset [Knapitsch et al. 2017]. We show the rendered normal maps from extract meshes
and GT images for reference. Neural implicit methods, such as Neuralangelo [Li et al. 2023], model the foreground with SDF and the background with
NeRF. Therefore, only the foreground mesh is extracted. SuGaR’s mesh is noisy [Guédon and Lepetit 2023] and 2DGS fails at reconstructing backgroundregions [Huang et al. 2024]. In contrast, our method can reconstruct detailed surfaces for both foreground objects and background regions.

explicit

implicit

Table 2. Quantitative comparison on the DTU Dataset [Jensen et al. 2014]. We report the Chamfer distance and average optimization time. The resultsof 3DGS [Kerbl et al. 2023] and SuGaR [Guédon and Lepetit 2023] are taken from 2DGS [Huang et al. 2024]. Our method achieves the highest reconstruction
accuracy among other explicit methods.
NeRF [Mildenhall et al. 2021]
VolSDF [Yariv et al. 2021]
NeuS [Wang et al. 2021]
Neuralangelo [Li et al. 2023]
3DGS [Kerbl et al. 2023]
SuGaR [Guédon and Lepetit 2023]
GaussianSurfels [Dai et al. 2024]
2DGS [Huang et al. 2024]
Ours

24
1.90
1.14
1.00
0.37
2.14
1.47
0.66
0.48
0.50

37
1.60
1.26
1.37
0.72
1.53
1.33
0.93
0.91
0.82

40
1.85
0.81
0.93
0.35
2.08
1.13
0.54
0.39
0.37

55
0.58
0.49
0.43
0.35
1.68
0.61
0.41
0.39
0.37

63
2.28
1.25
1.10
0.87
3.49
2.25
1.06
1.01
1.12

65
1.27
0.70
0.65
0.54
2.21
1.71
1.14
0.83
0.74

2024] due to the ray-Gaussian intersection computation, it significantly outperforms all SOTA 3DGS-based methods in terms of reconstruction quality. A qualitative comparison is shown in Figure 6.
GOF reconstructs fine-detailed surfaces for both foreground objects
and the background regions. By contrast, meshes extracted from
SuGaR [Guédon and Lepetit 2023] are noisy, while 2DGS [Huang
et al. 2024] fails to extract geometry for the background regions.
We further compare against SOTA surface reconstruction methods on the DTU dataset [Jensen et al. 2014]. As shown in Table 2, ourmethod outperforms all other 3DGS-based methods [Dai et al. 2024;
Guédon and Lepetit 2023; Huang et al. 2024]. Despite a performance
gap with the leading implicit reconstruction method [Li et al. 2023],
GOF’s optimization is much faster. This performance disparity is
attributed to the DTU dataset’s strong view-dependent appearance.
Utilizing a better view-dependent appearance modeling [Verbin
et al. 2022] or a coarse-to-fine training strategy [Li et al. 2023] could
potentially improve the reconstructions.

4.3

69
1.47
0.72
0.57
0.53
1.43
1.15
0.85
0.81
0.73

83
1.67
1.29
1.48
1.29
2.07
1.63
1.29
1.36
1.18

97
2.05
1.18
1.09
0.97
2.22
1.62
1.53
1.27
1.29

105
1.07
0.70
0.83
0.73
1.75
1.07
0.79
0.76
0.68

106
0.88
0.66
0.52
0.47
1.79
0.79
0.82
0.70
0.77

110
2.53
1.08
1.20
0.74
2.55
2.45
1.58
1.40
0.90

114
1.06
0.42
0.35
0.32
1.53
0.98
0.45
0.40
0.42

118
1.15
0.61
0.49
0.41
1.52
0.88
0.66
0.76
0.66

122
0.96
0.55
0.54
0.43
1.50
0.79
0.53
0.52
0.49

Mean
1.49
0.86
0.84
0.61
1.96
1.33
0.88
0.80
0.74

Time
> 12h
>12h
>12h
> 12h
11.2 m∼ 1h
6.7 m
10.9 m18.4 mTable 3. Quantitative results on Mip-NeRF 360 [Barron et al. 2022b]
dataset. All results of the baseline methods are taken from their papers
whenever available and we rerun SuGaR [Guédon and Lepetit 2023] withthe same setting as ours. Our method achieved SOTA NVS results, especiallyin the outdoor scenes in terms of LPIPS.

NeRF
Deep Blending
Instant NGP
MERF
MipNeRF360

Outdoor Scene
PSNR ↑ SSIM ↑ LPIPS ↓
21.46
0.458
0.515
21.54
0.524
0.364
22.90
0.566
0.371
23.19
0.616
0.343
24.47
0.691
0.283

Indoor scene
PSNR ↑ SSIM ↑ LPIPS ↓
26.84
0.790
0.370
26.40
0.844
0.261
29.15
0.880
0.216
27.80
0.855
0.271
31.72
0.917
0.180

Mobile-NeRF
BakedSDF
SuGaR
BOG

21.95
22.47
22.93
23.94

0.470
0.585
0.629
0.680

0.470
0.349
0.356
0.263

27.06
29.43
27.71

0.836
0.906
0873

0.258
0.225
0.227

3DGS
Mip-Splatting
2DGS
Ours

24.64
24.65
24.34
24.82

0.731
0.729
0.717
0.750

0.234
0.245
0.246
0.202

30.41
30.90
30.40
30.79

0.920
0.921
0.916
0.924

0.189
0.194
0.195
0.184

Novel view Synthesis

To evaluate the NVS results of GOF, we further compare against
SOTA NVS methods on the Mip-NeRF 360 dataset [Barron et al.

2022a]. The quantitative results are shown in Table 3. GOF not
ACM Trans. Graph., Vol. 43, No. 6, Article 271. Publication date: December 2024.

271:8 •Zehao Yu, Torsten Sattler, and Andreas Geiger

SuGaR [Guédon and Lepetit 2023]

2DGS [Huang et al. 2024]

Ours

GT

Fig. 7. Reconstructions on the Mip-NeRF 360 Dataset [Barron et al. 2022a]. We show the rendered normal maps from extract meshes together with GT
images for reference. Our method can reconstruct detailed surfaces for both foreground objects and background regions while meshes from previous work arenoisy [Guédon and Lepetit 2023] or fail to reconstruct background regions and thin structures, such as the spokes in the bicycle scene [Huang et al. 2024].

Mip-Splatting + TSDF Fusion

Mip-Splatting + GOF

GT Image

Fig. 8. Comparison of mesh extraction methods. Applying GOF to SOTA Mip-Splatting [Yu et al. 2024a] yields significant improvements over TSDF,
enabling complete mesh extraction for background regions.

only performs slightly better than all other 3DGS-based methods interms of PSNR, but also outperforms all other methods significantly
in terms of LPIPS [Zhang et al. 2018] in the outdoor scenes. The
main improvements come from our improved densification strategy.
In the ablation, we show that adding our densification strategy to
3DGS [Kerbl et al. 2023] and Mip-Splatting [Yu et al. 2024a] improves the NVS results by a large margin. For the indoor scenes,
our results are similar to Mip-Splatting [Yu et al. 2024a], with lessthan 0.1 PSNR difference, which we attribute to our regularizations
that trade-off NVS and surface reconstruction. Our method also
outperforms Sugar [Guédon and Lepetit 2023] and 2DGS [Huang
et al. 2024] in all metrics. We show a qualitative comparison of
extracted meshes in Figure 7. Similar to our observations on theTanks and Temples dataset [Knapitsch et al. 2017], GOF can reconstruct detailed surfaces for both foreground objects and backgroundregions, while SuGaR’s [Guédon and Lepetit 2023] meshes are noisy
and have fewer details and 2DGS fails to extract meshes for the
background regions.

ACM Trans. Graph., Vol. 43, No. 6, Article 271. Publication date: December 2024.

Table 4. Ablation on the Tanks and Temples Dataset [Knapitsch et al.
2017]. The metrics are computed with the official script from the dataset.
GOF significantly improves surface reconstruction quality. Our regularization, decoupled appearance, and densification strategy modeling further
improve the results.

A. Mip-Splatting w/ TSDF
B. Mip-Splatting w/ GOF
C. Ours w/o GOF
D. Ours w/o normal consistency
E. Ours w/o decoupled appearance
F. Ours w/ minimal axis’s normal
G. Ours w/o improved densification
H. Ours

4.4

Precision ↑ Recall ↑ F-score ↑
0.15
0.25
0.16
0.40
0.33
0.36
0.37
0.45
0.39
0.41
0.35
0.37
0.49
0.39
0.43
0.46
0.36
0.40
0.52
0.39
0.44
0.54
0.42
0.46

Ablation Study

Mesh Extraction: Our GOF enables mesh extraction from 3D Gaussians directly by identifying a level set without resorting to Poissonreconstruction or TSDF fusion. Compared to using TSDF fusion

Gaussian Opacity Fields: Efficient Adaptive Surface Reconstruction in Unbounded Scenes •Step 0

Step 1

Step 3

Step 7271:9

GT Image

Fig. 9. Different number of binary search steps with marching tetrahedra on the DTU dataset [Jensen et al. 2014]. Applying our binary search to
Marching Tetrahedra [Shen et al. 2021] significantly improves the quality of extracted meshes in just few steps.

0.1

0.3

0.5

0.7

0.9

Fig. 10. Meshes extracted with different level sets on the Mip-NeRF 360 dataset [Barron et al. 2022a]. Our method supports multi-layer meshes
extraction by using different level sets for marching tetrahedra.

for mesh extraction (Table 4, C), our tetrahedra-based mesh extraction (Table 4, H) significantly improves the quality of the extracted
meshes.
To further demonstrate the effectiveness and generalizability of
our mesh extraction strategy, we apply it to a SOTA 3DGS-basedmodel, Mip-Splatting [Yu et al. 2024a], and compare it with TSDF
fusion. A comparison of the extracted meshes is shown in Figure 8.
The mesh extracted from TSDF fusion is noisy and has a lot of holeson the ground, due to the inconsistency of depth. By contrast, ourextracted mesh is more complete. Our method also extracts detailed
surfaces for the background regions. Quantitative results in Table 4(A vs. B) further indicate the effectiveness of our method.
Regularization and Appearance Modeling: As shown in Table 4(D vs. H), using the normal regularization during training significantly improves reconstruction quality, which is consistent with theobservation in 2DGS [Huang et al. 2024]. Including the decoupledappearance modeling [Lin et al. 2024] further improves the reconstruction results as shown in Table 4 (E vs. H), since the model isless likely to model view-dependent appearance with geometry.
Normal Definition: We approximate the normal of a Gaussianas the normal of the ray-Gaussian intersection plane. By contrast,
SuGaR [Guédon and Lepetit 2023] approximates the normal as one
of Ganssian’s axis directions with minimal scale. We experimented
by replacing our normal definition with the axis direction of theminimum scale and found that the F1-score decreases from 0.46 to
0.40 on the TNT dataset, as shown in Table 4 (F vs. H), validating
the effectiveness of our normal definition.
Densification on Geometry: As shown in Table 4 (G vs. H), disabling our improved densification strategy leads to a decrease in
the F1-score from 0.46 to 0.44 in the TNT dataset [Knapitsch et al.
2017], showing that our densification metric contributes positively
to geometry reconstruction.

Table 5. Ablation on the Mip-NeRF 360 [Barron et al. 2022b] dataset.
Our densification strategy improves the novel view synthesis results for
3DGS [Kerbl et al. 2023] and Mip-Splatting [Yu et al. 2024a] significantly.

3DGS
w/ our densification
Mip-Splatting
w/ our densification
Ours

Outdoor Scene
PSNR ↑ SSIM ↑ LPIPS ↓
24.64
0.731
0.234
24.62
0.743
0.199
24.65
0.729
0.245
24.77
0.745
0.205
24.82
0.750
0.202

Indoor scene
PSNR ↑ SSIM ↑ LPIPS ↓
30.41
0.920
0.189
31.10
0.928
0.174
30.90
0.921
0.194
31.18
0.926
0.180
30.79
0.924
0.184

Overall Improvements: When using the same mesh extraction
method, such as TSDF fusion (Table 4, A vs. C) or our tetrahedrabased mesh extraction (Table 4, B vs. H), the improvements of our
method over Mip-Splatting [Yu et al. 2024a] are due to additional
regularization, appearance modeling, and our improved densification strategy. Without these elements, the reconstruction quality
between our method and rasterization-based methods is comparable. For instance, disabling our normal regularization results insimilar reconstruction quality compared to Mip-Splatting with GOF(Table 4, D vs. B).
Densification on View Synthesis: To further evaluate the effectiveness of our improved densification metric on novel view synthesis, we apply it to 3DGS [Kerbl et al. 2023] and Mip-Splatting [Yu et al.
2024a]. The quantitative results, as shown in Table 5, demonstratethat our strategy improves NVS results significantly, especially interms of LPIPS [Zhang et al. 2018]. Qualitative comparisons are also
provided in Figure 4,where the glass regions are rendered with highfidelity using our densification metric.
It is important to note that our ray-Gaussian intersection formula
has similar rendering quality to rasterization-based methods [Kerbl
et al. 2023; Yu et al. 2024a]. For instance, our NVS results are similar
to those of Mip-Splatting when augmented with our densification
ACM Trans. Graph., Vol. 43, No. 6, Article 271. Publication date: December 2024.

271:10 •Zehao Yu, Torsten Sattler, and Andreas Geiger

strategy, as shown in Table 5. However, the ray-Gaussian intersection formula enables the establishment of opacity fields, improving
mesh extraction compared to TSDF fusion. Additionally, our regularization and appearance modeling further improve the reconstruction
quality, as shown previously.
Binary Search for Marching Tetrahedra: To demonstrate the effectiveness of applying binary search to Marching Tetrahedra [Shenet al. 2021], we apply binary search with different numbers of steps.
As shown in Figure 9, using binary search significantly improves
the quality of reconstructed meshes in just a few iterations.
Multi-layer Meshes: While we extract meshes for the 0.5 level set
in the main paper, our method also supports extracting meshes withdifferent level sets. As a proof of concept, we extract meshes withdifferent level sets 0.1, 0.3, 0.5, 0.7, and 0.9 and show the renderedmeshes in Figure 10. Finer structures can be extracted with a smaller
level set, but it might result in expanded meshes.

5

LIMITATIONS

We now discuss some limitations and potential future extensions ofour method.
Delaunay Triangulation Efficiency: We employ the CGAL Library [Jamin et al. 2024] for Delaunay triangulation to constructtetrahedral cells, which has 𝑂 (𝑁 log 𝑁 ) complexity. It becomes abottleneck particularly when the number of points increases. For
example, it takes around 8 minutes to construct the tetrahedral cells
for the bicycle scene in the Mip-NeRF dataset [Barron et al. 2022a].
This process could potentially be optimized by considering the spatial locality since the points are generated from 3D Gaussians, or by
employing parallel processing techniques on GPUs. Additionally,
optimizing the selection and number of Gaussian primitives used
for constructing tetrahedral grids could further improve efficiency.
Opacity Evaluation Optimization: During the binary search of
marching tetrahedra, our method evaluates the opacity of points
using all training views, which may lead to redundant computations.
Recognizing that a single view can determine a point’s minimal
opacity value suggests a more efficient approach could be developedby associating points with their respective influential training views.
View Dependent Appearance Modeling: Using spherical harmonics for view-dependent appearance modeling has limitations,such as potentially inaccurately representing reflections as geometric features. Incorporating a better view-dependent appearance
model [Verbin et al. 2022] could potentially enhance the quality ofreconstructions.
Mesh-based Rendering: While the current focus of GOF is on
surface reconstruction and novel view synthesis, leveraging theextracted meshes for real-time rendering is an interesting future
direction [Reiser et al. 2024]. This could potentially improve thequality of mesh-based rendering given the detailed and adaptive
meshes extracted with GOF.

6

CONCLUSION

We have presented Gaussian Opacity Fields (GOF), a novel method
for efficient, high-quality, and adaptive surface reconstruction in
ACM Trans. Graph., Vol. 43, No. 6, Article 271. Publication date: December 2024.

unbounded scenes. Our GOF is derived from ray-tracing-based volume rendering of 3D Gaussians, maintaining consistency with RGB
rendering. Our GOF enables geometry extraction directly from 3D
Gaussians by identifying its level set, without Poisson reconstruction or TSDF. We approximate the surface normal of Gaussians
as the normal of the ray-Gaussian intersection plane and apply
depth-normal consistency regularization to enhance geometry reconstructions. Furthermore, we propose an efficient and adaptive
mesh extraction method utilizing Marching Tetrahedra, where thetetrahedral grids are induced from 3D Gaussians. Our evaluations
reveal that GOF surpasses existing explicit methods in both surface
reconstruction and novel view synthesis. Further, GOF achieves comparable surface reconstruction results with leading implicit methods
while being able to reconstruct detailed meshes for the backgroundregions in unbounded scenes.
Acknowledgement: We thank Christian Reiser and Binbin Huangfor insightful discussions and valuable feedback throughout the
project and proofreading. ZY and AG are supported by the ERC
Starting Grant LEGO-3D (850533) and DFG EXC number 2064/1- project number 390727645. TS is supported by a Czech Science
Foundation (GACR) EXPRO grant (UNI-3D, grant no. 23-07973X).

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A

ADDITIONAL IMPLEMENTATION DETAILSIn this section, we describe our implementation details, including aminor modification to 3DGS’s densification strategy to overcome
the clustered issues resulting from clone operation and details forsurface extraction.
Clone with Sampling: We found that the position of Gaussian isrelatively stable, which is also observed in Mip-Splatting [Yu et al.
2024a]. Therefore, Gaussians cloned from the same parents remain
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271:12 •Zehao Yu, Torsten Sattler, and Andreas Geiger

Algorithm 2 Gaussian Opacity Field evaluation for a single view𝑤, ℎ: width and height of training image
𝑀, 𝑆, 𝐴: Gaussian means, covariances, and opacity
𝑃: position of points
𝑉 : view configuration of current camera
function EvaluateSingleView(𝑤, ℎ, 𝑀, 𝑆, 𝐴, 𝑉 , 𝑃)
CullGaussian(𝑀, 𝑉 )⊲ Frustum Culling
𝑀 ′, 𝑆 ′ ← ScreenspaceGaussians(𝑀, 𝑆, 𝑉 )⊲ Transform𝑇𝑔 ← CreateTiles(𝑤, ℎ)
𝐿𝑔 , 𝐾𝑔 ← DuplicateWithKeys(𝑀 ′ , 𝑇𝑔 )⊲ Indices and Keys
SortByKeys(𝐾𝑔 , 𝐿𝑔 )⊲ Globally Sort
𝑅𝑔 ← IdentifyTileRanges(𝑇𝑔 , 𝐾𝑔 )CullPoints(𝑃, 𝑉 )⊲ Frustum Culling𝑃 ′ ← ScreenspacePoints(𝑃, 𝑉 )⊲ Transform𝑇𝑝 ← CreateTiles(𝑤, ℎ)
𝐿𝑝 , 𝐾𝑝 ← CreateWithKeys(𝑃 ′ , 𝑇𝑝 )⊲ Indices and Keys
SortByKeys(𝐾𝑝 , 𝐿𝑝 )⊲ Globally Sort
𝑅𝑝 ← IdentifyTileRanges(𝑇𝑝 , 𝐾𝑝 )𝑂←1
⊲ Init Opacity
for all Tiles 𝑡 in 𝐼 do
⊲ 𝐼 is the Canvas
for all Pixels 𝑖 in 𝑡 do𝑟𝑔 ← GetTileRange(𝑅𝑔 , 𝑡)
𝐿𝑔′ ← FilterGaussians(𝑖, 𝐿𝑝 , 𝑟𝑔 , 𝐾𝑔 , 𝑀, 𝑆, 𝐴) ⊲ SelectGaussians that contributes to pixel 𝑖𝑟 𝑝 ← GetTileRange(𝑅𝑝 , 𝑡)
𝐿𝑝′ ← FilterPoints(𝑖, 𝐿𝑝 , 𝑟 𝑝 , 𝐾𝑝 , 𝑃) ⊲ Select Points thatprojected to pixel 𝑖for all Points 𝑝 in 𝐿𝑝′ do
𝑂 [𝑝] ← EvaluateOpacity(𝑖, 𝐿𝑔′ , 𝐾𝑔 , 𝑀, 𝑆, 𝐴)
end forend forend forreturn 𝑂
end function
Algorithm 3 Marching Tetrahedra with Binary Search
𝑀, 𝑆, 𝐴: Gaussian means, covariances, and opacity
𝑃, 𝐶: Tetrahedra Points and Cells
𝑉 : view configurations
𝐿: Level set value
𝑁 : Step of binary searchfunction MarchingTetrahedraBinarySearch(𝑀, 𝑆, 𝐴, 𝑃, 𝐶,𝑉)𝑂 ← EvaluateOpacityField(𝑀, 𝑆, 𝐴, 𝑉 , 𝑃)
𝐹 , 𝑃1 , 𝑃2 , 𝑂 1 , 𝑂 2 ← MarchingTetrahedra(𝑃, 𝐶, 𝑂, 𝑆)
for all i 𝑣 in {1, 2, ..., 𝑁 } do𝑃𝑚 ← MidPoint(𝑃1 , 𝑃2 )𝑂𝑚 ← EvaluateOpacityField(𝑀, 𝑆, 𝐴, 𝑉 , 𝑃𝑚 )𝑃1 , 𝑃2 , 𝑂 1 , 𝑂 2 ← ChangeEndPointsAndValues(𝑃1 , 𝑃2 , 𝑂 1 ,
𝑂 2 , 𝐿,𝑃𝑚 , 𝑂𝑚 )end for𝑉 ← LinearInterpolate(𝑃1 , 𝑃2 , 𝑂 1 , 𝑂 2 , 𝐿) return 𝐹 , 𝑉⊲
return faces and vertices for triangle meshend function

ACM Trans. Graph., Vol. 43, No. 6, Article 271. Publication date: December 2024.

clustered. To address this issue, instead of using the same position
for the cloned Gaussian, we sample a new Gaussian according to
the Gaussian’s parameter similar to the procedures for Gaussian
split [Kerbl et al. 2023]. We found this simple strategy leads to more
uniformly distributed Gaussians, as shown in Figure 11. However,
we do not observe a significant impact in terms of NVS results.
Details for Surface Extraction: We provide a pseudo-code of ourtile-based Gaussian Opacity Fields evaluation in Algorithm 1 andAlgorithm 2. Note that the evaluation takes 3D Gaussians, training
views, and 3D points as input and it does not rely on the tetrahedra
cells. Therefore, the same algorithm also applies to the Marching
Cubes Algorithm. The pseudo-code of our Marching Tetrahedra
augmented with binary search is shown in Algorithm 3, the sameidea could be applied to the Marching Cubes algorithm.
Algorithm 1 Gaussian Opacity Field evaluation
𝑀, 𝑆, 𝐴: Gaussian means, covariances, and opacity
𝑃: position of points
𝑉 : view configurations
function EvaluateOpacityField(𝑀, 𝑆, 𝐴, 𝑉 , 𝑃)𝑂←1
⊲ Init Opacity
for all Views 𝑣 in 𝑉 do𝑤, ℎ ← GetImageSize(𝑉 )𝑂 𝑣 ← EvaluateSingleView(𝑤, ℎ, 𝑀, 𝑆, 𝐴, 𝑉 , 𝑃)𝑂 ← Min(𝑂, 𝑂 𝑣 )⊲ Take Minimal Opacity
end forreturn 𝑂
end function

B

ADDITIONAL RESULTS

TSDF Fusion in Contraction Space: In the main paper, we compare GOF with 2DGS [Huang et al. 2024] with its default (bounded)settings. Specifically, 2DGS extracts meshes with TSDF fusion focusing on the foreground objects, resulting in missing geometry inthe background regions. However, 2DGS also supports extracting
meshes for the background regions by applying TSDF fusion in thecontraction space [Barron et al. 2022a], which is referred as theunbounded setting. We extract unbounded meshes for 2DGS with agrid resolution of 2048 and the resulting meshes have similar number of vertices and faces with GOF’s meshes. Then we compare GOF
with both 2DGS’s bounded and unbounded meshes. As shown in
Figure 12, while 2DGS’s unbounded setting can reconstruct meshes
for the background regions, the meshes on the background regions
are incomplete and lack details. By contrast, our method can reconstruct detailed surfaces for both foreground objects and backgroundregions.

Gaussian Opacity Fields: Efficient Adaptive Surface Reconstruction in Unbounded Scenes •3DGS [Kerbl et al. 2023]

Ours

3DGS

271:13Ours [Kerbl et al. 2023]

Fig. 11. Comparison of clone strategy on the Mip-NeRF 360 Dataset [Barron et al. 2022a]. Our clone strategy leads to more uniformly distributed
Gaussian primitives.

2DGS [Huang et al. 2024]

2DGS [Huang et al. 2024] Unbounded

Ours

GT

Fig. 12. Reconstructions on the Mip-NeRF 360 Dataset [Barron et al. 2022a]. We compare GOF with both 2DGS’s bounded and unbounded meshextraction. While 2DGS’s unbounded setting can reconstruct mesh for the background region, its meshes are incomplete and lack of details. By contrast, GOF
can reconstruct detailed meshes both for the foreground objects and background regions.

ACM Trans. Graph., Vol. 43, No. 6, Article 271. Publication date: December 2024.