ND-SDF: Learning Normal Deflection Fields for High-Fidelity Indoor Reconstruction

## Abstract

Abstract Neural implicit reconstruction via volume rendering has demonstrated its effectiveness in recovering dense 3D surfaces. However, it is non-trivial to simultaneously recover meticulous geometry and preserve smoothness across regions with differing characteristics. To address this issue, previous methods typically employ geometric priors, which are often constrained by the performance of the prior models. In this paper, we propose ND-SDF , which learns a N ormal D eflection field to represent the angular deviation between the scene normal and the prior normal. Unlike previous methods that uniformly apply geometric priors on all samples, introducing significant bias in accuracy, our proposed normal deflection field dynamically learns and adapts the utilization of samples based on their specific characteristics, thereby improving both the accuracy and effectiveness of the model. Our method not only obtains smooth weakly textured regions such as walls and floors but also preserves the geometric details of complex structures. In addition, we introduce a novel ray sampling strategy based on the deflection angle to facilitate the unbiased rendering process, which significantly improves the quality and accuracy of intricate surfaces, especially on thin structures. Consistent improvements on various challenging datasets demonstrate the superiority of our method.

## 1 Introduction

3D surface reconstruction Chen et al. (); Ye et al. (; ); Wang et al. (); Ye et al. (; ); Liu et al. (); Li et al. () aims to recover watertight, dense 3D geometry from multi-view images Hartley & Zisserman (), representing a significant research area in computer vision and graphics. The recovered surfaces have proven invaluable for many downstream tasks, including robotic navigation, AR/VR, and smart cities.

Recently, coordinate-based networks Mildenhall et al. (); Barron et al. (); Müller et al. (); Martin-Brualla et al. (); Zhang et al. (); Ye et al. (); Huang et al. (); Ming et al. () are drawing increasing attention, due to their remarkable performance in the task of novel view synthesis. Inspired by implicit representations of the scene, many subsequent works have introduced Signed Distance Functions (SDF) Park et al. () or occupancy Li et al. () to parameterize the 3D geometry. Notably, some of these approaches Yariv et al. (); Wang et al. () outperform other methods in recovering dense surfaces by representing scene geometry with implicit distance fields and integrating them with differentiable rendering frameworks to enhance surface reconstruction.

However, recovering high-fidelity surfaces remains challenging, as relying solely on color images for supervision often results in an underconstrained problem, especially for regions like textureless walls and ceilings. Previous methods Yu et al. (); Wang et al. () have attempted to mitigate this issue by incorporating auxiliary supervision. For instance, MonoSDF employs monocular cues from a pretrained model as pseudo-ground truth to supervise the reconstruction process, which alleviates the issue in textureless regions. However, the large errors due to domain gaps in monocular cues and the inconsistencies in view-dependent prior guidance often lead to visible detail loss and erroneous surfaces Yu et al. (), as depicted in Figure 4 of the experimental section.

Noting that prior models are highly accurate in simpler regions like floors and walls but struggle with complex geometries. Based on this insight, we introduce ND-SDF for high-fidelity indoor reconstruction (Figure 1). Our core innovation is the construction of a deflection field that adaptively learns the geometric deviations between the actual scene geometry and the prior geometry derived from normal priors. The inherent deviation is defined as the angular difference between the true scene normals and the prior normals. By aligning the deflected scene normals with the normal cues, our model can adaptively learn the deviation, which encourages accurate recovery of intricate structures maintaining fine details without being misled by erroneous priors.

In addition, we propose a novel adaptive deflection angle prior loss that leverages prior normals for differential supervision of high and low-frequency areas, achieving an optimal balance between smoothness and detail. Our observations indicate that large angle deviations are primarily concentrated in thin and fine-grained structures. Building on this insight, we introduce deflection-angle guided optimization to proactively facilitate the recovery of detailed structures. Furthermore, we address a significant bias issue and propose deflection angle guided unbiased rendering to improve the reconstruction of small or thin structures, as shown in Figure 1.
In summary, we present the following contributions:

- •
We propose a novel scene attribute field, named normal deflection field, which adaptively learns the deviations between the scene and normal priors. This method aids us in distinguishing between detailed and textureless areas. Therefore, we can restore more fine structures while ensuring the smoothness and integrity of the scene.
- •
Empirically assuming that prior models incur larger errors in complex areas, we employ the deflection angle to discriminate between high and low-frequency regions. Consequently, we propose a novel adaptive deflection angle prior loss that dynamically adjusts the utilization of distinct cues, thereby achieving a balance between complex structures and smooth surfaces. Furthermore, we utilize the deflection angle to guide ray sampling and photometric optimization, facilitating the restoration of finer-grained structures.
- •
To address the inherent bias issue of neural surface rendering, we integrate the unbiasing method Zhang et al. () to implement an adaptive, unbiased rendering strategy. This approach facilitates the recovery of extremely thin structures without compromising scene fidelity. Our method outperforms previous approaches significantly in indoor reconstruction evaluations, and this superiority is validated through extensive ablation experiments.

## 2 Related Work

##### Neural Surface Representation of 3D scenes

Recently, the representation of 3D scenes using neural fields has gained popularity due to their expressiveness and simplicity. NeRF-type approaches Mildenhall et al. (); Barron et al. (); Müller et al. (); Fridovich-Keil et al. (); Sun et al. () have proposed encoding coordinate-based density and appearance of scenes by utilizing simple multilayer perceptrons (MLPs) and explicit structures such as voxel grids, resulting in photorealistic novel view synthesis. However, these approaches fail to accurately recover surfaces due to the lack of constraints on density. To address this issue, subsequent works, such as VolSDF Yariv et al. () and Neus Wang et al. (), implicitly represent signed distance functions (SDFs) and employ a SDF-density conversion function to encourage precise surface reconstruction. Other methods Li et al. (); Rosu & Behnke (); Wang et al. (); Yariv et al. (); Fu et al. (); Zhang et al. () have also been proposed, introducing different representations and optimization techniques to further enhance reconstruction quality and efficiency. Nevertheless, these aforementioned methods struggle to handle indoor scenes with a large number of low-frequency areas (e.g., walls and floors), as photometric loss becomes unreliable in such regions.

##### Neural reconstruction in Indoor Environments

Due to the complex layouts of indoor scenes, additional auxiliary data are required for reasonable reconstruction. Manhattan-SDF Guo et al. () employs the Manhattan assumption and semantic priors to jointly regularize textureless regions such as walls and floors. HelixSurf Liang et al. () achieves intertwined regularization iteratively by combining neural implicit surface learning with PatchMatch-based multi-view stereo (MVS) Barnes et al. () robustly and efficiently. Other works Yu et al. (); Wang et al. () propose utilizing monocular priors from pretrained models to achieve smooth and complete reconstruction. However, directly applying normal or depth priors Yu et al. () may lead to undesirable reconstruction results due to the unreliable nature of these priors. NeuRIS Wang et al. () filters out unreliable priors based on the assumption that areas with rich 2D visual features are more error-prone. H2O-SDF Park et al. () simply utilizes uncertainty prior to re-weight the normal prior loss. Nevertheless, both of these methods exhibit weak generalization capabilities. For example, the visual feature hypothesis Wang et al. () struggles to differentiate between rich-textured planar areas. Similarly, prior-guided re-weighting Park et al. () is further constrained by the domain gap between the prior model and the scene. On the other hand, DebSDF Xiao et al. () effectively designs an uncertainty field based on a probabilistic model to guide the loss function, leading to more robust and accurate reconstructions. In contrast to methods with weaker generalization capabilities and those relying on probabilistic models, our proposed method, ND-SDF, learns a Normal Deflection field that enables the dynamic adaptation of priors based on their characteristics. This field directly models geometrically meaningful SO(3) residuals, resulting in highly detailed surface reconstructions while maintaining smoothness and robustness.

Figure: Figure 2: Overview of Our method. We utilize multi-resolution hash grids $\gamma_{L}$ as scene representation. The core of ND-SDF is the normal deflection field, which we represent using quaternions predicted by the deflection network (denoted as $f_{d}$). We align the deflected normals with the prior normals to learn the deviation between the scene and the priors. To distinctly supervise high and low-frequency areas, we employ an adaptive deflection angle prior loss, ensuring both smoothness and detail. Furthermore, we utilize the deflection angle $\Delta\theta$ to distinguish complex structures, enabling angle guided sampling and color loss to facilitate the recovery of intricate surface details. Lastly, we combine the unbiased rendering method Zhang et al. () to ensure the generation of extremely thin structures indoors.
Refer to caption: x2.png

## 3 Method

The primary objective of our Normal Deflection Signed Distance Function (ND-SDF) is to reconstruct dense surfaces from calibrated multi-view images, with a primary focus on indoor scenes with established priors.
To achieve this, we introduce a novel component known as the normal deflection field. This field is designed to quantify and learn the deviations between actual scene geometries and their corresponding normal priors.
By integrating this field, our method can adaptively supervise both high and low-frequency regions in the scene. This adaptive supervision is crucial for preserving fine details while ensuring overall surface smoothness.

The operational framework of our approach is depicted in Figure 2. In subsequent sections, we delve into a comprehensive discussion on the implementation of the normal deflection field. Additionally, we explore various strategies that leverage this field to enhance the fidelity of surface details, thereby promoting a more accurate surface reconstruction.

### 3.1 Preliminaries

Volume Rendering NeRF assumes that a ray $\textbf{r}(t)=\textbf{o}+t\textbf{v}$ is emitted from viewpoint o in direction v, where $t$ denotes the distance from the viewpoint. $N$ points are sampled along the ray, i.e. $\textbf{x}_{i}=\textbf{o}+t_{i}\textbf{v},i\in\{1,2...N\}$. Given each point’s volume density $\sigma_{i}$ and color $\textbf{c}_{i}$, the color of the ray $\hat{\textbf{C}}(\textbf{r})$ can be synthesized using volume rendering techniques as:

$$ $\hat{\textbf{C}}(\textbf{r})=\sum_{i=1}^{N}T_{i}\alpha_{i}\textbf{c}_{i},\ \alpha_{i}=1-\exp{(-\sigma_{i}\delta_{i})},T_{i}=\prod_{j=1}^{i-1}(1-\alpha_{j }),$ (1) $$

where $\alpha_{i}$ is the opacity of $i$-th ray segment, $T_{i}$ denotes the transmittance of light reaching the $i$-th segment, $\delta_{i}=t_{i}-t_{i-1}$ denotes distance between adjacent samples.

A color loss supervised using ground-truth (GT) color is utilized to optimize the neural fields:

$$ $\mathcal{L}_{\mathrm{color}}=\sum_{{\textbf{r}}\in\mathcal{R}}\|\hat{\textbf{C }}(\textbf{r})-{\textbf{C}(\textbf{r})}\|_{1},$ (2) $$

where $\mathcal{R}$ is the sampled ray set, C(r) is the GT color.

SDF-induced Volume Rendering The rise of neural surface reconstruction is attributed to seminal works like VolSDF and Neus, which propose specific SDF to volume density conversion, allowing optimization via volume rendering. The final surface, denoted as $S$, is equivalent to the zero level-set of the implicit distance field, i.e., $S=\{\textbf{x}\in\mathbb{R}^{3}|s(\textbf{x})=0\}$, where $s(\textbf{x})$ is the SDF value.

We follow VolSDF Yariv et al. (), employing the Laplace Cumulative Distribution Function (CDF) to model the relationship between SDF and volume density:

$$ $\sigma(\textbf{x})=\frac{1}{\beta}\Psi_{\beta}\left(-s(\textbf{x})\right)= \left\{\begin{array}[]{ll}\frac{1}{2\beta}\exp\left(\frac{-s(\textbf{x})}{ \beta}\right)&\text{ if }s(\textbf{x})\leq 0\\ \frac{1}{\beta}-\frac{1}{2\beta}\exp\left(\frac{s(\textbf{x})}{\beta}\right)& \text{ if }s(\textbf{x})>0\end{array}\right.,$ (3) $$

where $\Psi_{\beta}$ denotes the Laplace CDF, $\beta$ denotes the variance.
Depth $\hat{D}(\textbf{r})$ and normal $\hat{\textbf{N}}(\textbf{r})$ at the surface intersection are synthesized using the same approach as that employed for synthesizing colors:

$$ $\hat{D}(\textbf{r})=\sum_{i=1}^{N}T_{i}\alpha_{i}t_{i},\hat{\textbf{N}}( \textbf{r})=\sum_{i=1}^{N}T_{i}\alpha_{i}\textbf{n}_{i}.$ (4) $$

Here, $\textbf{n}_{i}$ is the analytical gradient of the SDF network at point $i$.
Following MonoSDF Yu et al. (), we utilize monocular depth and normal priors generated from pretrained model Eftekhar et al. () to supervise the rendered depth and normal:

$$ $\begin{array}[]{c}\mathcal{L}_{\text{depth }}=\sum_{\textbf{r}\in\mathcal{R}} \|w\hat{D}(\textbf{r})+q-D(\textbf{r})\|^{2}\\ \mathcal{L}_{\text{normal }}=\sum_{\textbf{r}\in\mathcal{R}}\|\hat{\textbf{N}} (\textbf{r})-{\textbf{N}(\textbf{r})}\|_{1}+\left\|1-{\hat{\textbf{N}}(\textbf {r})}^{T}{\textbf{N}(\textbf{r})}\right\|_{1}\end{array},$ (5) $$

where the two coefficients $(w,q)$ obtained by least square algorithms are utilized to align the scale between monocular depth and rendered depth, i.e.,$w\hat{D}+q\approx D$.

We encode scene geometry using Instant-NGP Müller et al. () $\gamma_{L}$ and a shallow MLP $f_{g}$, that is, $(s(\textbf{x})\in\mathbb{R}^{3},\textbf{z}(\textbf{x})\in\mathbb{R}^{256})=f_{
g}(\textbf{x},PE(\textbf{x}),\gamma_{L}(\textbf{x}))$, where ${\textbf{z}(\textbf{x})}$ denotes the latent geometry feature.

Also, we utilize the eikonal term Gropp et al. (); Yariv et al. () to regularize the shape of SDF in 3D space:

$$ $\mathcal{L}_{\text{eik }}=\frac{1}{N}\sum_{i=1}^{N}\left(\left\|\nabla s\left( {\textbf{x}}_{i}\right)\right\|_{2}-1\right)^{2},$ (6) $$

Following Neuralangelo Li et al. (), we further employ numerical gradients to enhance surface geometric consistency and use curvature loss for smoothness. The detailed definitions of them are provided in the appendix.

### 3.2 Normal deflection field

A significant challenge in indoor 3D reconstruction is achieving a balance between the smoothness of overall surfaces and the intricacy of complex structures. Traditional approaches relying solely on photometric loss have proven inadequate for accurately capturing smooth areas, particularly in texture-deficient regions. Later methods often leverage auxiliary data, such as normal priors, to enhance the reconstruction quality in textureless areas.
However, the uniform application of normal priors across different scene types can impair the recovery of complex structures. This is primarily due to the variable reliability of these priors in diverse regions, where they may not accurately reflect the underlying geometry.
To address this issue, we propose the development of a Normal Deflection Field. This field is designed to dynamically represent the existing 3D angular deviation between the actual scene normals and the provided normal priors. By doing so, it effectively circumvents potential misguidance arising from inconsistent priors,
thereby enabling a more reliable and nuanced reconstruction of both smooth and complex indoor structures.

Specifically, we choose quaternion as the deflection form, which is a lightweight rotation representation. The quaternion can be parameterized by a single MLP $f_{d}$:

$$ $\textbf{q}_{i}=f_{d}(\textbf{x}_{i},\textbf{v}_{i},\textbf{n}_{i},\textbf{z}_{ i}),$ (7) $$

where $\textbf{q}_{i}=(\textbf{q}_{i}^{0},\textbf{q}_{i}^{1},\textbf{q}_{i}^{2},
\textbf{q}_{i}^{3})$ is the deflection quaternion (normalized default) for the sampled $\textbf{x}_{i}$. The quaternion at the surface intersection point is synthesized using volume rendering techniques, analogous to the way NeRF synthesizes colors:

$$ $\textbf{Q}(\textbf{r})=\sum_{i=1}^{N}T_{i}\alpha_{i}\textbf{q}_{i},$ (8) $$

We deflect the rendered normal using the synthesized deflection quaternion $\textbf{Q}(\textbf{r})$:

$$ $\hat{\textbf{N}}^{d}(\textbf{r})=\textbf{Q}(\textbf{r})\otimes\hat{\textbf{N}} (\textbf{r})\otimes\textbf{Q}^{-1}(\textbf{r}),$ (9) $$

where $\hat{\textbf{N}}^{d}$ denotes the deflected rendered normal, $\textbf{Q}^{-1}$ denotes the inverse of Q, also known as the conjugate, and $\otimes$ is a quaternion multiplication operation. Since a quaternion can be represented in trigonometric form, i.e. $\textbf{Q}=\cos{\frac{\theta}{2}}+\sin{\frac{\theta}{2}}(\textbf{u}^{1}i,
\textbf{u}^{2}j,\textbf{u}^{3}k)$, this operation is to rotate the rendered normal around the quaternion axis $\textbf{u}=(\textbf{u}^{1},\textbf{u}^{2},\textbf{u}^{3})$ by $\theta$ degrees.

The deviation between the scene and priors is learned by minimizing the difference between the deflected rendered normal and the prior normal. Thus, we define the deflected normal loss:

$$ $\mathcal{L}^{d}_{{normal}}=\sum_{\textbf{r}\in\mathcal{R}}\|\hat{\textbf{N}}^{ d}(\textbf{r})-\textbf{N}(\textbf{r})\|_{1}+\left\|1-\hat{\textbf{N}}^{d}( \textbf{r})^{T}\textbf{N}(\textbf{r})\right\|_{1},$ (10) $$

### 3.3 Adaptive deflection angle prior loss

We assume that the learned deviation is encompassed within the actual deflected angle when the scene normal is deflected. This deflected angle ($\Delta\theta$) is computed as follows:

$$ $\Delta\theta=\arccos{(\hat{\textbf{N}}(\textbf{r})\cdot\hat{\textbf{N}}^{d}( \textbf{r}))},$ (11) $$

where $\Delta\theta\in[0,\pi]$. Empirically, the deviation between the scene and priors increases as the structure becomes more complex. This intuitively suggests that the deflection angle is small in smooth regions and large in high-frequency zones. Based on this, the adaptive deflection angle normal prior loss is proposed to dynamically adjust the utilization of differing priors based on their characteristics:

$$ $\begin{array}[]{cc}\mathcal{L}^{ad}_{normal}=&\sum_{\textbf{r}\in\mathcal{R}}{ g^{d}(\Delta\theta)\mathcal{L}^{d}_{normal}(\hat{\textbf{N}}^{d}(\textbf{r}),{ \textbf{N}}(\textbf{r}))}+g(\Delta\theta)\mathcal{L}_{normal}(\hat{\textbf{N}} (\textbf{r}),{\textbf{N}}(\textbf{r}))\end{array},$ (12) $$

where $g^{d}$ and $g$ are modulation functions that adjust the weights of the deflected and original normal loss terms based on the deflection angle. We similarly define an adaptive depth prior loss:

$$ $\mathcal{L}^{ad}_{depth}=\sum_{\textbf{r}\in\mathcal{R}}{g(\Delta\theta) \mathcal{L}_{depth}(\hat{D}(\textbf{r}),{D}(\textbf{r}))}.$ (13) $$

As $\Delta\theta$ increases, $g^{d}(\Delta\theta)$ increases, and $g(\theta)$ decreases, since we should less apply priors in complex areas indicated by large deflection angles. The combined adaptive normal and depth prior loss is termed the adaptive deflection angle prior loss (as illustrated in Figure 2); see appendix for detailed definitions of the proposed modulation functions.

### 3.4 Deflection angle guided optimization

Through the utilization of the proposed adaptive prior loss, we dynamically adjust the utilization of various priors, significantly enhancing the quality of reconstruction. However, this alone is insufficient for generating more complex structures. Fundamentally, merely learning the deviation does not endow the capability to reconstruct additional details. Recognizing that large deflection angles indicate complex areas, we introduce three deflection angle guided optimization methods specifically designed to facilitate the recovery of more thin and fine-grained structures.

##### Deflection angle guided sampling.

Textureless areas, such as walls, constitute a significant portion of indoor scenes. These regions converge rapidly by leveraging prior knowledge, partly due to their inherent simplicity. To prevent continuous oversampling of well-learned smooth regions, we proactively sample more rays in complex areas to capture finer details. The sampling process is naturally guided by the learned deflection angles (see supplementary for details).

##### Deflection angle guided photometric loss.

To further enhance detail recovery, we impose additional photometric loss on complex areas identified by the deflection angles. The original color loss term is re-weighted based on the deflection angle, resulting in a re-weighted color loss.

$$ $\mathcal{L}^{d}_{color}=\sum_{\textbf{{r}}\in\mathcal{R}}{w_{color}({\Delta \theta}(\textbf{r}))\|\hat{\textbf{C}}(\textbf{r})-{\textbf{C}(\textbf{r})}\|_ {1}},$ (14) $$

where ${\Delta\theta}(\textbf{r})$ denotes the actual deflected angle of the sampled ray r, and $w_{color}$ is the re-weighting function guided by ${\Delta\theta}$ (see appendix for details).

##### Deflection angle guided unbiased rendering.

Accurately reconstructing thin structures, such as chair legs, remains challenging. This limitation arises from the inherent bias issues in SDF-induced volume rendering, manifesting in two ways: (1) According to TUVR Zhang et al. (), the derivative of the rendering weight $\frac{\partial W(t)}{\partial t}$ is influenced by various angle differences between ray directions and scene normals. This results in a non-maximum value of $w$ at the surface intersection point, thereby degrading the surface quality; (2) For rays passing close to the object, the Laplace CDF assigns a high density to points near the surface, which leads to incorrect rendering weights, and subsequently affects the depth and normal compositing. This issue causes thin structures to disappear during training. Therefore, it is crucial to apply an unbiased rendering method during reconstruction. We illustrate this biased scenario in Fig. 3, highlighting the significance of ensuring unbiasedness.

Inspired by TUVR Zhang et al. (), we transform SDF to density using an unbiased function:

$$ $\sigma(\textbf{r}({t_{i}}))=\frac{1}{\beta}\Psi_{\beta}\left(\frac{-f_{g}( \textbf{r}({t_{i}}))}{\left|f_{g}^{\prime}(\textbf{r}({t_{i}}))\right|}\right),$ (15) $$

where $\textbf{r}(t_{i})=\textbf{o}+t_{i}\textbf{v}$ represents a sampled point on the ray, and $f_{g}$ is the geometry network, which predicts the SDF value. This transformation effectively alleviates bias issues, as illustrated in Fig 3. However, we observe that simply applying it results in the overall surface failing to converge. Therefore, we only partially apply it to thin structures indicated by the learned deflection angles.

Figure: Figure 3: We present a scenario in which a ray passes close to a chair leg. Along the ray, point A is situated near the chair leg, while point B is located at the ray-ground intersection. In the absence of unbiasing, abnormal density and rendering weight are observed at point A, whereas density should be exclusively assigned to point B.
Refer to caption: x3.png

### 3.5 Optimization

The overall loss function is defined as:

$$ $\mathcal{L}=\mathcal{L}_{color}^{d}+\lambda_{1}\mathcal{L}_{eik}+\lambda_{2} \mathcal{L}_{curv}+\lambda_{3}\mathcal{L}^{ad}_{normal}+\lambda_{4}\mathcal{L} ^{ad}_{depth},$ (16) $$

The weights $\lambda_{1}...\lambda_{4}$ are utilized to balance the importance of these loss terms.

## 4 Experiment

**Table 1: Quantitative results on the ScanNet dataset. Our method substantially outperforms previously best-performing methods across all metrics to date.**
| Method | Acc$\downarrow$ | Comp$\downarrow$ | Pre$\uparrow$ | Recall$\uparrow$ | Chamfer$\downarrow$ | F-score$\uparrow$ |
| --- | --- | --- | --- | --- | --- | --- |
| COLMAP Schönberger et al. () | 0.047 | 0.235 | 71.1 | 44.1 | 0.141 | 53.7 |
| VolSDF Yariv et al. () | 0.414 | 0.120 | 32.1 | 39.4 | 0.267 | 34.6 |
| Neus Wang et al. () | 0.179 | 0.208 | 31.3 | 27.5 | 0.194 | 29.1 |
| NeuRIS Wang et al. () | 0.050 | 0.049 | 71.7 | 66.9 | 0.050 | 69.2 |
| MonoSDF Yu et al. () | 0.035 | 0.048 | 79.9 | 68.1 | 0.042 | 73.3 |
| HelixSurf Liang et al. () | 0.038 | 0.044 | 78.6 | 72.7 | 0.042 | 75.5 |
| DebSDF Xiao et al. () | 0.036 | 0.040 | 80.7 | 76.5 | 0.038 | 78.5 |
| H2OSDF Park et al. () | 0.032 | 0.037 | 83.4 | 76.9 | 0.035 | 79.9 |
| Ours | 0.031 | 0.036 | 84.0 | 80.3 | 0.034 | 82.0 |

Datasets We conducted experiments on four indoor datasets: ScanNet Dai et al. (), Replica Straub et al. (), TanksandTemples Knapitsch et al. (), and ScanNet++ Yeshwanth et al. (). ScanNet contains 1513 indoor scenes captured with an iPad, processed using the BundleFusion algorithm to obtain camera poses and surface reconstruction. Replica is a synthetic dataset comprising 18 indoor scenes. Each scene features dense geometry and high dynamic range textures. TanksandTemples is a large-scale 3D reconstruction dataset including high-resolution outdoor and indoor environments. We followed the splits from MonoSDF and applied the same evaluation settings. We also conducted experiments on ScanNet++, which includes 460 indoor scenes captured using laser scanners. These scenes offer high-quality dense reconstructions and images. For testing, we selected six scenes from ScanNet++.

Implementation details Our method was implemented using PyTorch Paszke et al. (). The image resolution for all scenes is 384×384. We obtained normal and depth cues using Omnidata Eftekhar et al. (). Multi-resolution hash grids were utilized for scene representation. Both the geometry network and color network consists of two layers, each with 256 nodes. Our network was optimized using AdamW Loshchilov & Hutter () optimizer with a learning rate of 1e-3. The weights for loss terms were: $\lambda_{1}=0.05$, $\lambda_{2}=0.0005$, $\lambda_{3}=0.025$, $\lambda_{4}=0.05$. Upon adequate initialization of the deflection field, we initiated deflection angle guided sampling, photometric optimization, and unbiased rendering. All experiments were conducted on an NVIDIA TESLA A100 PCIe 40GB, with each iteration sampling 4$\times$1024 rays, totaling 128,000 training steps. The training time was about 14 GPU hours. Our hash encoding resolution spanned from $2^{5}$ to $2^{11}$ across 16 levels, with the initial activation level set to 8 and activation steps set to 2000.

Metrics In accordance with prior researches, we employ six standard metrics to evaluate the reconstructed meshes: Accuracy, Completeness, Chamfer Distance, Precision, Recall, and F-score. Additionally, normal consistency is used to evaluate the Replica dataset.

Baselines We compare our approach with the following methods: (1) traditional MVS techniques COLMAP Schonberger & Frahm (); Schönberger et al. (); (2) basic neural implicit methods, including VolSDF and NeuS; (3) methods incorporating auxiliary data, including MonoSDF and NeuRIS; (4) other indoor reconstruction methods, such as HelixSurf Liang et al. ().

Figure: Figure 4: Qualitative results on ScanNet. Compared to previous state-of-the-art methods, our approach generates more thin and fine-grained structures, such as chair legs and detailed table layouts.
Refer to caption: x4.png

**Table 2: F-score on the TanksandTemples dataset, evaluated at a resolution of 768.**
| Method | Auditorium | Ballroom | Courtroom | Museum | Mean |
| --- | --- | --- | --- | --- | --- |
| NeuRIS Wang et al. () | 0.79 | 0.84 | 2.78 | 1.34 | 1.13 |
| MonoSDF Yu et al. () | 3.17 | 3.70 | 13.75 | 5.68 | 6.58 |
| Ours | 4.37 | 6.45 | 16.28 | 10.30 | 9.35 |

### 4.1 Comparisons

ScanNet: In Figure 4, we present qualitative results and in Table 1, quantitative results are displayed. These quantitative results are averaged over the selected four scenes. ND-SDF significantly surpasses previous best-performing works, achieving state-of-the-art performance. Specifically, we achieve the highest F-score, a reliable metric that reflects reconstruction accuracy by balancing both Accuracy and Completeness. When compared with HelixSurf and MonoSDF (as shown in Figure 4), our approach accurately captures thin structures, such as chair legs and lamp rings. The results highlight the effectiveness of our proposed deflection methods, which substantially enhance the surface accuracy and the recovery of thin and fine-grained structures in indoor environments.

**Table 3: Quantitative results on the ScanNet++ dataset. We select 6 scenes and calculate their average metrics. Baked-Angelo is the best-performing reconstruction method tested without priors.**
| Method | Acc$\downarrow$ | Comp$\downarrow$ | Pre$\uparrow$ | Recall$\uparrow$ | Chamfer$\downarrow$ | F-score$\uparrow$ |
| --- | --- | --- | --- | --- | --- | --- |
| VolSDF Yariv et al. () | 0.070 | 0.102 | 0.405 | 0.339 | 0.086 | 0.368 |
| Baked-Angelo Yu et al. () | 0.152 | 0.039 | 0.543 | 0.718 | 0.095 | 0.614 |
| MonoSDF Yu et al. () | 0.057 | 0.032 | 0.651 | 0.703 | 0.044 | 0.675 |
| Ours | 0.056 | 0.024 | 0.667 | 0.785 | 0.040 | 0.721 |

**Table 4: Quantitative results of Replica dataset, significantly surpassing previous methods.**
| Method | Normal C.$\uparrow$ | Chamfer$\downarrow$ | F-score$\uparrow$ |
| --- | --- | --- | --- |
| Unisurf Oechsle et al. () | 90.96 | 4.93 | 78.99 |
| MonoSDF Yu et al. () | 92.11 | 2.94 | 86.18 |
| Ours | 92.85 | 2.57 | 91.6 |

Replica: Following MonoSDF Yu et al. (), we used their pre-processed masks to filter out abnormal priors. Quantitative results are presented in Table 4, where our approach achieves the highest F-score and the lowest Chamfer Distance, significantly surpassing previous methods. Refer to the supplementary material for visualization results.

Tanks and temples: We also conducted experiments on the advanced split of T&T, which features several challenging large-scale indoor scenes. As shown in Table 2, ND-SDF surpasses previous methods that utilize priors in terms of quantitative performance. Additional qualitative results can be found in the supplementary material, where our method recovered substantial detailed structures.

**Table 5: F-score of Different Priors Models.**
| Method |  | Omnidata Eftekhar et al. () | Wizard Fu et al. () |
| --- | --- | --- | --- |
| Base |  | 0.700 | 0.661 |
| Ours |  | 0.750 | 0.730 |

ScanNet++: ScanNet++ comprises numerous complex indoor scenes, each offering high-resolution GT mesh and RGB data. Since no public results are available for ScanNet++, we trained several baselines methods, including VolSDF, MonoSDF, and Baked-Angelo Yu et al. (). Among these, only MonoSDF utilized monocular priors as we did. The quantitative results are summarized in Table 3, demonstrating the state-of-the-art performance of our method across all metrics. For additional visualization results, please refer to the appendix.

**Table 6: Quantitative Results of Ablation for Different Modules. ‘Base’ can be viewed as a simple combination of MonoSDF and Neuralangelo. ‘ND’ denotes applying the deflection field and the deflected normal loss term $L^{d}_{normal}$. ‘AP’ refers to the adaptive deflection angle prior loss terms including $L^{ad}_{normal}$ and $L^{ad}_{depth}$. ‘DO’ denotes the two deflection angle guided optimizations, namely sampling and photometric loss. ‘DU’ denotes the deflection angle guided unbiased rendering.**
| Method | Base | ND | AP | DO | DU | — | Acc$\downarrow$ | Comp$\downarrow$ | Prec$\uparrow$ | Recall$\uparrow$ | Chamfer$\downarrow$ | F-score$\uparrow$ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Base | ✓ | $\times$ | $\times$ | $\times$ | $\times$ | — | 0.046 | 0.046 | 0.576 | 0.640 | 0.046 | 0.606 |
| ModelA | ✓ | ✓ | $\times$ | $\times$ | $\times$ | — | 0.044 | 0.042 | 0.601 | 0.666 | 0.043 | 0.632 |
| ModelB | ✓ | ✓ | ✓ | $\times$ | $\times$ | — | 0.038 | 0.032 | 0.637 | 0.726 | 0.035 | 0.679 |
| ModelC | ✓ | ✓ | ✓ | ✓ | $\times$ | — | 0.040 | 0.030 | 0.631 | 0.740 | 0.035 | 0.681 |
| Ours | ✓ | ✓ | ✓ | ✓ | ✓ | — | 0.038 | 0.031 | 0.648 | 0.728 | 0.034 | 0.686 |

Figure: Figure 5: Visualization Results of the Models after Ablation. Detailed definitions of these models are provided in Table 6.
Refer to caption: x5.png

### 4.2 Ablation Study

#### 4.2.1 Ablation of different modules

The learned deviations play a crucial role in locating high-frequency regions. Consequently, we employ deflection angles for sampling, photometric optimization, and unbiased rendering to facilitate the recovery of thin and fine-grained structures. We conducted detailed ablation experiments to assess the effectiveness of these modules, resulting in five post-ablation models: Base, ModelA, ModelB, ModelC, and Ours. For quantitative results and comprehensive definitions, please refer to Table 6. Figure 5 visually illustrates how the proposed components enhance overall performance.

Analysis of Normal Deflection Field The Base model strictly adheres to monocular cues, leading to significant detail loss due to misguided priors in complex areas. In ModelA, we introduced the deflection field and directly applied the deflected normal loss to learn deviations. The improvement in the F-score to 0.632 (as shown in Table 6) confirms the superiority of the deviation learning approach. However, we observed wrinkles (as depicted in Figure 5) in textureless regions like walls. This issue arises from the lack of constraints on the deflection, resulting in inefficient utilization of priors in smooth regions. In Model B, the introduction of the adaptive deflection angle prior loss significantly enhances reconstruction quality, striking a balance between smoothness and detail.

Analysis of deflection angle guided optimization While ModelB substantially improves reconstruction quality compared to the Base, it still struggles with finer structures (as depicted in Figure 5). In ModelC, we introduced deflection angle guided sampling and color loss to emphasize surface details. Finally, we incorporated an adaptive unbiased rendering method to facilitate the recovery of thin structures. Figure 5 illustrates that Ours recovers a wide range of complex and fine structures, such as skull models and iron swabs, surpassing the Base model. As indicated in Table 6, Ours achieved the highest F-score, quantitatively outperforming other models. This ablation study strongly validates the effectiveness of the proposed modules in restoring high-fidelity surfaces.

#### 4.2.2 Ablation of different prior models

Our method learns deviations between the scene and normal priors predicted by a specific pretrained model. It is essential to evaluate its effectiveness across cues generated by different pretrained models. To achieve this, we employed the state-of-the-art method Geowizard Fu et al. () to generate both depth and normal cues. The quantitative results, summarized in Table 5, demonstrate that our method significantly enhances surface quality irrespective of the cues utilized.
This experiment illustrates the superiority of our approach. It invariably improves reconstruction accuracy in scenarios where prior knowledge is used.

## 5 Conclusion

We have presented ND-SDF, a novel approach that learns deviations between the scene and normal priors for high-fidelity indoor surface reconstruction. We introduce an adaptive deflection angle prior loss to dynamically supervise areas with varying characteristics. By identifying high-frequency regions based on deflection angles, we employ angle guided optimization to generate thin and fine-grained structures. Our method recovers a substantial number of complex structures, as demonstrated by extensive qualitative and quantitative results that confirm its superiority. ND-SDF efficiently adapts to unreliable priors in challenging areas, significantly addressing the detail loss issue caused by large prior errors in regions with fine structures.

Limitation ND-SDF is confined to reconstruction scenarios involving priors. Our approach exhibits limited capability in handling areas with less coverage. Due to inherent ambiguities in observations, the deflection field may learn incorrect structures and converge to local optima. We encourage the reader to refer the supplementary material for more analysis and results. In the future, we may explore multi-view consistency constraints to enhance reconstruction quality.

## Appendix / supplementary material

Figure: Figure 6: Architecture of ND-SDF. We utlize multi-resolution hash grids $\gamma_{L}$ for scene representation. The geometry network $f_{g}$, color network $f_{c}$, and deflection network $f_{d}$ are all constructed using simple multilayer perceptrons (MLPs).
Refer to caption: x6.png

## Appendix A Implementation details

### A.1 Architecture

When considering the inputs to the deflection network, two scenarios are contemplated: (1) No deflection is required when the prior is completely accurate. In this case, the quaternion axis aligns with the scene normals, indicating that the deflection axis is closely related to the scene geometry; (2) Since the prior model is view-dependent, the quaternion for alignment with the prior must also be view-dependent. Consequently, we define:

$$ $\textbf{q}=f_{d}(\textbf{x},\textbf{v},\textbf{n},\textbf{z}),$ (17) $$

where $\textbf{q}=(\textbf{q}_{0},\textbf{q}_{1},\textbf{q}_{2},\textbf{q}_{3})$ represents the deflection quaternion. As detailed in Section 3.2, the quaternion can be expressed in trigonometric form as $\textbf{q}=\cos(\theta/2)+\sin(\theta/2)\left(\mathbf{u}^{1}i+\mathbf{u}^{2}j+
\mathbf{u}^{3}k\right)$. Within the deflection network, the rotation axis is initialized along the x-axis, i.e., $\mathbf{u}=(1,0,0)$, and the rotation angle $\theta/2$ is set to $\pi/2$. By default, the network normalizes the output quaternions.

We observe that the structure of the deflection network mirrors that of the color network Yariv et al. (), denoted as $\textbf{c}=f_{c}(\textbf{x},\textbf{v},\textbf{n},\textbf{z})$. However, considering that the color network exclusively models scene-related radiance, while the deflection network encapsulates a broader range of prior deviations, we separate these two networks.

### A.2 Progressive warm-up

In the early stages of training, deflection fields may introduce negative effects, such as noise. Before the rough formation of scene geometry, these deflection quaternions are random and erroneous. If unconstrained, deflecting the true normals during this initial phase can lead to reconstruction errors, such as surface protrusions.

To address this issue, we propose a gradual activation strategy for deflection fields, allowing them to become effective as the surface takes shape. Specifically, we design a ”quat anneal” policy to enable the deflection effect to progress gradually. Given the surface normals synthesized by volume rendering, denoted as $\hat{\textbf{N}}(\textbf{r})$, and the synthesized quaternion $\textbf{Q}(\textbf{r})=(\textbf{Q}_{0},\textbf{Q}_{1},\textbf{Q}_{2},\textbf{Q
}_{3})$, we first decompose $\textbf{Q}(\textbf{r})$ into its trigonometric representation:

$$ $\left\{\begin{aligned} \frac{\theta}{2}&=\arccos{\textbf{Q}_{0}}\\ \textbf{u}&=\frac{[\textbf{Q}_{1},\textbf{Q}_{2},\textbf{Q}_{3}]}{\sin{\frac{ \theta}{2}}}\end{aligned}.\right.$ (18) $$

Here, u denotes the rotation axis corresponding to the quaternion $\textbf{Q}(\textbf{r})$, and $\theta$ represents the rotation angle. We then utilize the scene normals to warm up the deflection axis, starting with zero rotation:

$$ $\left\{\begin{aligned} \frac{\theta_{iter}}{2}&=prog_{q}\times\frac{\theta}{2} \\ \textbf{u}_{iter}&=prog_{q}\times\textbf{u}+(1-prog_{q})\times\hat{\textbf{N}} (\textbf{r})\end{aligned}.\right.$ (19) $$

Here, the parameter $prog_{q}$ linearly increases from 0 to 1 during the training process within the process interval $[0,anneal_quat_end]$. Once the training progress reaches $anneal_quat_end$, the warm-up phase concludes, and $prog_{q}$ becomes equal to 1.

The warm-up deflection quaternion is computed as:

$$ $\textbf{Q}_{iter}(\textbf{r})=cos{\frac{\theta_{iter}}{2}}+sin{\frac{\theta_{ iter}}{2}}\times\textbf{u}_{iter}.$ (20) $$

During the initial training stage, the deflection angles start at zero and gradually increase, allowing the model to effectively leverage prior knowledge to optimize smooth regions and approximate the scene geometry. We consider it to be adequately initialized upon the warm-up phase concludes. Throughout this process, the deflection field gradually adapts to more complex structures and learns the deviations between the true geometry and the normal priors. Quantitative results with and without progressive warm-up are presented in Table 7.

**Table 7: Ablation of Warm-Up Strategy on ScanNet++. The warm-up strategy can better facilitate the initialization of the deflection field, resulting in improved surface reconstruction quality.**
| Method | w/o warm-up | w warm-up |
| --- | --- | --- |
| F-score | 0.667 | 0.686 |

### A.3 Detailed Implementation Aspects

#### A.3.1 Adaptive deflection angle prior loss

Figure: Figure 7: Visualization of the Deflection Angle. We visualize the angle map and the angle mask using thresholds of 5°, 15°, and 25°.
Refer to caption: x7.png

In Section 3.3, we introduce two modulation functions, namely $g^{d}$ and $g$ (Eq. 12), which adjust the weights of the deflected and original normal loss terms based on the deflection angle.

Here provide the specific forms for both functions. We employ a shifted logistic function to define them:

$$ $\left\{\begin{aligned} g^{d}(\theta)&=\frac{1}{1+e^{-s_{0}}(\theta-\theta_{0}) }\\ g(\theta)&=1-g^{d}(\theta)\end{aligned}.\right.$ (21) $$

Here, the parameter $s_{0}$ controls the steepness of $g^{d}$, and $\theta_{0}$ denotes the offset term. In our experiments, we set $s_{0}=12.5,\theta_{0}=\frac{\pi}{12}$ for all scenes.

As shown in Figure 7, a deviation angle of less than 5° consistently indicates a simple flat region, whereas an angle greater than 15° typically denotes a complex structural region.

Additionally, we visualize the modulation function $g^{d}(\Delta\theta)$ in Figure 8, with $(s_{0},\theta_{0})$ set to $(12.5,\frac{\pi}{12})$. For regions with a deviation angle less than 5°, weights greater than 0.9 are assigned to the original normal loss term, ensuring the effective incorporation of normal priors in smooth regions.

To further demonstrate the effectiveness of the proposed adaptive deflection angle prior loss, we visualize three normal loss heatmaps: the naive normal loss, the deflected normal loss, and the adaptive deflection angle normal loss (as shown in Figure 9). The black box highlights thin structures and ambiguous regions where monocular priors are largely unreliable. In these areas, the naive normal loss heatmap shows significant errors, while both the deflected and adaptive losses exhibit significantly lower errors by aligning the deflected scene normals with the normal priors (GT). In summary, the adaptive prior loss can dynamically adjust the weighting between the naive and deflected loss terms, promoting globally correct convergence while fully leveraging the normal prior.

Figure: Figure 8: Modulation Functions defined in Eq. 21.
Refer to caption: x8.png

Figure: Figure 9: Heatmap comparison of naive, deflected, and adaptive deflection angle normal loss.
Refer to caption: x9.png

#### A.3.2 Deflection angle guided optimization

In Section 3.4, we introduce three deflection angle guided optimization methods, including ray sampling, photometric optimization, and unbiased rendering.

To implement ray sampling guided by the deflection angle, we dynamically maintain a deflection angle map ($\Delta\bar{\theta}$) for each image during the training process. We update the angle map using the following equation:

$$ $\Delta\bar{\theta}(\textbf{r})=\max{(\Delta\bar{\theta}^{old}(\textbf{r}) \times\eta,\Delta\theta(r))},$ (22) $$

where r is the sampled ray and $\eta$ is the decay coefficient. We initialize the angle map to zero at the beginning.

In the sampling phase, we first calculate the per-pixel sampling probability based on the angle map, then use this probability map for inverse sampling, such that areas with greater deflection magnitude tend to have more rays sampled.

We employ a scaled and shifted logistic function to calculate the per-pixel sampling probability:

$$ $p(\textbf{r}_{i})=1+\frac{t_{1}}{1+e^{-s_{1}}(\Delta\bar{\theta}(\textbf{r}_{i })-\theta_{1})},$ (23) $$

where $\Delta\bar{\theta}$ represents the maintained angle map, and $p(\textbf{r}_{i})\in[1,t_{1}+1]$, with the steepness controlled by $s_{1}$. We then normalize the probability map for inverse sampling using the following equation:

$$ $p(\textbf{r}_{i})=\frac{p(\textbf{r}_{i})}{\sum_{i}p(\textbf{r}_{i})}.$ (24) $$

For the angle guided color loss, we similarly employ a scaled and shifted logistic function to re-weight the original color loss term based on the deflection angle (Eq. 14):

$$ $w_{color}(\Delta\theta(\textbf{r}))=1+\frac{t_{2}}{1+e^{-s_{2}}(\Delta{\theta} (\textbf{r})-\theta_{2})}.$ (25) $$

Regarding the unbiased rendering guided by the deflection angle, we first define a bias confidence using another variant of the logistic function to compute it:

$$ $cfd({\textbf{r}})=\frac{1}{1+e^{-s_{3}}(\Delta\bar{\theta}(\textbf{r})-\theta_ {3})}.$ (26) $$

Note that we utilize the maintained angle map to calculate the confidence instead of the deflection angle $\Delta\theta(\textbf{r})$ computed from current iteration, as unbiased densities of the sampled points along the ray are required prior to the volume compositing process. This then leads to the deflection angle guided unbiasing:

$$ $\sigma(\textbf{r}({t_{i}}))=\frac{1}{\beta}\Psi_{\beta}\left(\frac{-f_{g}( \textbf{r}({t_{i}}))}{cfd(\textbf{r})\left|f_{g}^{\prime}(\textbf{r}({t_{i}})) \right|+1-cfd(\textbf{r})}\right).$ (27) $$

In practice, we set the parameters as follows: $(s_{1}=25,\theta_{1}=\frac{\pi}{12},t_{1}=4),(s_{2}=25,\theta_{2}=\frac{\pi}{1
2},t_{2}=2),(s_{3}=25,\theta_{3}=\frac{\pi}{18})$.

#### A.3.3 Numerical Gradient

Following Neuralangelo Li et al. (), we employ numerical gradients w.r.t multi-resolution hash grids to promote surface consistency, as proved effective in their work. The numerical gradient is computed as follows (considering only the SDF value output of geometry network $f_{g}$):

$$ $\nabla_{x}f_{g}\left(\mathbf{x}_{i}\right)=\frac{f_{g}\left(\gamma_{L}\left( \mathbf{x}_{i}+\boldsymbol{\epsilon}_{x}\right)\right)-f_{g}\left(\gamma_{L} \left(\mathbf{x}_{i}-\boldsymbol{\epsilon}_{x}\right)\right)}{2\epsilon}.$ (28) $$

Here, $\epsilon_{x}=[\epsilon,0,0]$ signifies the incremental step size along the x-axis. The method for calculating the numerical gradient along the y or z-axis remains the same.

We also utilize the curvature loss Li et al. () to further encourage the smoothness of the reconstructed surfaces. This loss term is defined as:

$$ $\mathcal{L}_{\text{curv }}=\frac{1}{N}\sum_{i=1}^{N}\left|\nabla^{2}f_{g}\left (\mathbf{x}_{i}\right)\right|.$ (29) $$

This term decreases exponentially during training, similar to Neuralangelo. In fact, our tests indicate that it has minimal impact on surface smoothness, resulting in only a slight improvement compared to its absence. We attribute this to the use of monocular cues, as the cues are particularly accurate in smooth regions.

## Appendix B Extensive ablation studies

### B.1 Ablation of partial unbiased rendering

In Section 3.5, we introduced an unbiased function. We partially apply this technique to thin structures indicated by the learned deflection angles, without doing so may result in an inability to converge and a decrease in reconstruction quality.

We attribute this issue to the inconsistency in perspective inherent in the transform function. We compute the derivative of $f_{g}$ (considering only its SDF value output) w.r.t the sampling distance $t_{i}$ as follows:

$$ $f^{\prime}_{g}(\textbf{r}(t_{i}))=\nabla f_{g}(\textbf{o}+t_{i}\textbf{v}) \cdot\textbf{v}=\textbf{n}_{i}\cdot\textbf{v},$ (30) $$

where v is the ray direction and $\textbf{n}_{i}$ represents the geometry normal at point $\textbf{x}_{i}$. It means that $\frac{\partial f_{g}(\textbf{r}(t_{i}))}{\partial t_{i}}$ is equivalent to the cosine of the ray direction and the normal.

Consider another ray in space, denoted as $\textbf{r}^{\prime}(t)=\textbf{o}^{\prime}+t\textbf{v}^{\prime}$. Assuming that this ray also passes through $\textbf{x}_{i}$, sampling the same point such that $\textbf{x}_{j}=\textbf{o}^{\prime}+t_{j}\textbf{v}^{\prime}=\textbf{x}_{i}$.
According to Eq. 15, the volumetric densities of $\textbf{x}_{i}$ and $\textbf{x}_{j}$ under the transform function are given by:

$$ $\sigma(\textbf{r}({t_{i}}))=\frac{1}{\beta}\Psi_{\beta}\left(\frac{-f_{g}( \textbf{r}({t_{i}}))}{\left|f_{g}^{\prime}(\textbf{r}({t_{i}}))\right|}\right) ,\sigma(\textbf{r}^{\prime}({t_{j}}))=\frac{1}{\beta}\Psi_{\beta}\left(\frac{- f_{g}(\textbf{r}^{\prime}({t_{j}}))}{\left|f_{g}^{\prime}(\textbf{r}^{\prime}( {t_{j}}))\right|}\right).$ (31) $$

We observe that $f_{g}(\textbf{r}(t_{i}))=f_{g}(\textbf{r}^{\prime}(t_{j}))$ because $\textbf{x}_{i}=\textbf{x}_{j}$, but ${\left|f^{\prime}_{g}(\textbf{r}(t_{i}))\right|}\neq{\left|f^{\prime}_{g}(
\textbf{r}^{\prime}(t_{j}))\right|}$ because $\textbf{n}_{i}\cdot\textbf{v}\neq\textbf{n}_{j}\cdot\textbf{v}^{\prime}$ even though $\textbf{n}_{i}=\textbf{n}_{j}$. The inconsistency in perspective leads to ambiguity in the volume density at the same sampling point, causing complex indoor surfaces to fail to converge. Therefore, to reduce this ambiguity as much as possible, we choose to apply the transform function only to fine regions indicated by the deflection angle.

The quantitative ablation results on ScanNet for partial unbiased rendering are presented in Table 8, where the improved F-score highlights the effectiveness of this strategy.

Figure 10 provides qualitative comparisons. As indicated by the blue box, the surfaces reconstructed without the proposed partial unbiased rendering exhibit wrinkles and irregularities, and the fine structural details of the chair legs (highlighted in red) are not well captured. With the application of this strategy, the surfaces become significantly smoother and more precise. These visual results underscore the advantages of the partial unbiased rendering approach.

Figure: Figure 10: Visual comparisons of partial unbiased rendering on ScanNet.
Refer to caption: x10.png

**Table 8: Ablation of Partial Unbiased Rendering on ScanNet. We observed a decrease in reconstruction quality after omitting partial unbiased rendering.**
| Method | w/o partial | w partial |
| --- | --- | --- |
| F-score | 0.707 | 0.746 |

### B.2 Ablation of adaptive monocular depth loss

Notably, we apply the same method to adjust depth supervision (Eq. 13) based on the deflection angle. This simplified treatment rests on a strong foundation for two reasons. First, both depth and normal cues share a common inductive bias, as they stem from the same pretrained model Eftekhar et al. (). Second, due to the inherent lack of scale in monocular depth, it cannot provide precise supervision. In implicit surface reconstruction with priors, depth cues are primarily utilized to facilitate convergence and accelerate surface formation. Therefore, automatically masking high-frequency areas based on the deflection angle has no adverse impact on the recovery of details.

We present the quantitative ablation results in Table 9. The improved performance underscores the importance of utilizing the adaptive deflection angle depth prior loss.

**Table 9: Ablation of adaptive depth prior loss on ScanNet++. We observed a decrease in reconstruction accuracy after omitting the adaptive depth loss.**
| Method | w/o $\mathcal{L}^{d}_{depth}$ | w $\mathcal{L}^{d}_{depth}$ |
| --- | --- | --- |
| F-score | 0.664 | 0.686 |

Figure: Figure 11: Visualization results on Replica dataset.
Refer to caption: x11.png

**Table 10: Quantitative results of the 6 selected scenes from ScanNet++. Our method achieves state-of-the-art performance.**
| Method | 036bce3393 | 0e75f3c4d9 |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc$\downarrow$ | Comp$\downarrow$ | Pre$\uparrow$ | Recall$\uparrow$ | Chamfer$\downarrow$ | F-score$\uparrow$ | Acc$\downarrow$ | Comp$\downarrow$ | Pre$\uparrow$ | Recall$\uparrow$ | Chamfer$\downarrow$ | F-score$\uparrow$ |  |
| VolSDF | 0.053 | 0.098 | 0.475 | 0.372 | 0.0755 | 0.417 | 0.053 | 0.078 | 0.395 | 0.363 | 0.065 | 0.378 |
| Baked-Angelo | 0.168 | 0.023 | 0.52 | 0.78 | 0.0955 | 0.624 | 0.121 | 0.053 | 0.525 | 0.719 | 0.087 | 0.607 |
| MonoSDF | 0.037 | 0.037 | 0.606 | 0.656 | 0.037 | 0.630 | 0.064 | 0.027 | 0.574 | 0.705 | 0.046 | 0.633 |
| ND-SDF | 0.037 | 0.031 | 0.648 | 0.728 | 0.034 | 0.686 | 0.054 | 0.018 | 0.662 | 0.864 | 0.036 | 0.750 |

## Appendix C More results

### C.1 Replica

We present additional visualization results on Replica in Figure 11.

### C.2 ScanNet++

We present detailed quantitative results of the selected six scenes from ScanNet++ in Table 10 and provide more visualization results in Figure 12. Compared to the baselines, our method yields significantly improved reconstructions both visually and quantitatively, notably excelling at capturing thin, fine-grained structures while maintaining smoothness in flat regions.

### C.3 Tanks and temples

We present additional qualitative comparisons with previous state-of-the-art baselines, including MonoSDF Yu et al. () and NeuRIS Yu et al. (), as depicted in Figure 13. It is evident that ND-SDF achieves accurate dense surfaces replete with extensive details.

As illustrated in Figure 14 and Figure 15, we further extracted the reconstructed meshes at a resolution of 2048. At this high resolution, the meshes demonstrate that our method significantly enhances the generation of small and thin structures, even in challenging large-scale indoor scenes. For all experiments, the hash grid settings were kept at their defaults, with resolutions ranging from $2^{5}$ to $2^{11}$ across 16 levels, and a hashmap size of $2^{19}$.

### C.4 More information about dataset

The qualitative results on ScanNet presented in the main content (as shown in Figure 4) demonstrate that our method captures fine details, such as the chair legs, which are even overlooked by the ground truth (GT). This highlights a known issue regarding the reliability of ScanNet GT meshes. Constructed in 2017, the ScanNet dataset is based on RGB-D data captured by an iPad with an additional structure sensor. The images suffer from relatively low resolution, motion blur, and lighting variations. Furthermore, the limitations of the bundle-fusion algorithm result in GT surfaces lacking fine structures, which can adversely affect evaluation performance in these areas. This underscores our preference for conducting quantitative ablation studies on ScanNet++, which provides more reliable evaluation results.

Figure: Figure 12: Qualitative Comparison on ScanNet++. We compare ND-SDF with several baseline methods, including VolSDF, Baked-Angelo, and MonoSDF. Notably, only MonoSDF and Ours leverage monocular priors. Our method effectively balances smoothness and detail, whereas MonoSDF loses significant fine structures, and Baked-Angelo struggles to handle textureless smooth regions, such as floors.
Refer to caption: x12.png

Figure: Figure 13: Qualitative Comparison on Tanks & Temples. We compare ND-SDF with previous state-of-the-art indoor implicit reconstruction methods.
Refer to caption: x13.png

Figure: Figure 14: Qualitative Comparison of Different extraction resolutions on Tanks & Temples. We additionally extract the recovered mesh from the implicit SDF at a resolution of 2048.
Refer to caption: x14.png

Figure: Figure 15: Qualitative Results of the extracted mesh at a resolution of 2048 on Tanks & Temples.
Refer to caption: x15.png

## Appendix D Societal impact

Our method contributes to high-fidelity indoor surface reconstruction. On the positive side, precise indoor surface reconstruction can revolutionize fields such as architecture, virtual reality, and robotics. It enables architects to visualize and optimize designs, enhances immersive experiences in virtual environments, and aids in navigation for autonomous robots. However, there are also challenges. Misuse of this technology could invade privacy, as detailed indoor reconstructions might reveal sensitive information. Striking a balance between innovation and responsible deployment will be crucial for maximizing the positive impact while minimizing potential harm.

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: - In Line 280, is this a typo on $g(\theta)$? Looks like this is a missing $\Delta$.
soundness: 3
strengths: - The proposed normal deflection fields, which correct inaccurately estimated normals from normal maps, seem reasonable to me. 
- Table 6 and Figure 5 effectively showcase the contribution of the method's design through comprehensive ablation studies.
- The experimental results demonstrate that this proposed pipeline achieves state-of-the-art reconstruction quality in indoor scenarios.
summary: This paper presents a pipeline for multi-view 3D reconstruction of indoor scenes. The pipeline is based on recently popular differentiable volume rendering methods using Signed Distance Functions (SDF), such as VolSDF and NeuS. To enhance reconstruction quality in indoor scenes, particularly in textureless regions, the paper introduces monocular priors for additional supervision, inspired by recent works like MonoSDF and NeuRIS.

A core contribution of this paper is the introduction of a normal deflection field, which helps the model identify and correct inaccuracies in reference normal images estimated from monocular models, such as the Omnidata model. Experimental results show that this deflection field aids in reconstructing fine geometric details, such as thin structures.

Additionally, the authors introduce techniques to further improve geometry reconstruction quality, including (1) an adaptive normal loss weight and (2) unbiased volume rendering on thin objects.

The experimental results are promising and include comprehensive ablation studies. Overall, the paper is well-argued and substantiated.
weaknesses: - Using unbiased volume rendering for thin structures seems reasonable. However, why not apply this unbiased volume rendering weight uniformly across all regions? The authors mention potential convergence issues with this approach, but it would be helpful if they provided a more in-depth analysis explaining the nature of these convergence issues and offered insights into why they occur.

---

confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: 1. would it make sense to consider depth maps together to decide if the normal prior is accurate or not? For example, in most complex geometry areas such as thin structure areas, the depth is not continuous, and one may not be able to hope the normal is accurate there. 
2. what is the reason for the artifacts (which I mentioned in weakness 4), is that because without supervision the model struggles to recover the details?
3. to deal with 2, maybe add some smooth loss / geometric preserving loss on these areas that the normal prior is categorized as bad? 
4. I don't see a reason why this method only works indoor, has the author also tried it on outdoor datasets?
soundness: 3
strengths: 1. The paper is well-written and easy to follow. 
2. The normal deflection detection design is under reasonable assumptions and it offers a solution to deal with non-accurate normal prior.  
3. The angle deviation loss design fits the situation. 
4. The author provides complete experiments to show their method's performance. Including comparison with previous work quantitatively and qualitatively and ablation studies to show each module improves the results.
summary: The paper proposed a method to improve prior normal-guided neural SDF indoor scene reconstruction. Instead of supervising the normal uniformly across the whole domain, the author proposes to use a network to learn the deflect normal and desired adaptive angle prior to loss according to the deviation of the deflect normal to the prior normal. The intuition is that large deviation happens in complex structure areas where the normal prior in general is bad quality.  The author also proposes to use the re-weighted loss on normal, depth, and color rendering.  The experiments show the proposed method performs well qualitatively and quantitatively.
weaknesses: 1. The bad quality normal prior problem seems only solved half. The author reduced the loss weight on the part that the normal prior is not accurate, but not the other half, what to do to improve the method itself performance in these areas.
2. Like the other prior-based methods, the quality of the prior matters. The author assumes smooth area the prior is relatively accurate. 
3. Like the other SDF-based methods, the reconstruction seems to still struggle with the thin structures.  
4. I noticed that in most cases the method outperforms previous methods, but sometimes it creates noise and artifacts.

---

confidence: 5
contribution: 3
initial_rating: 6
presentation: 3
questions: See weakness.
soundness: 3
strengths: 1. The inaccuracy and bias are widely existed in monocular depth and normal priors, therefore, the idea of exploring the uncertainty of monocular clues to improve the reconstruction performance is reasonable and valuable.
 2. The normal deflection field is effectively incorporated throughout the training process, enabling rendered deflected normal images to be supervised by estimated normals and adaptively applied in ray sampling, photometric loss, and unbiased rendering.
3. The numerical and visualization results are attractive and impressive, which highly demonstrate the effectiveness of the proposed method.
4. The ablation studies are thorough, clearly showcasing the effectiveness of each proposed module.
summary: This paper proposes ND-SDF, which learns a normal deflection field to reduce the significant bias in monocular normal and depth priors. The normal deflection field can be rendered through volume rendering to further guide adaptive prior loss, ray sampling and unbiased rendering. The numerical and visualization results are compelling.
weaknesses: 1. Lack of citation. A similar approach of learning rotated normals to mitigate the bias between normal priors and ground truth normals was first proposed in NC-SDF[1]. While I know that NC-SDF has not released the source code so it’s difficult to directly compare with it, an appropriate clarification is necessary to distinguish the proposed idea from NC-SDF.
2. Lack of quantitative comparisons with DebSDF. DebSDF also predicts the uncertainty of monocular priors and is able to reconstruct details such as chair legs. The numerical result of DebSDF is included in the table but visual comparison with DebSDF is absent, which is insufficient.

The methodology and experiments in this paper are solid. I like this paper, however, the two major concerns above prevent me from giving a higher score. I am willing to raise my score if the concerns are well addressed.

[1]. Chen Z, Wu X, Zhang Y. NC-SDF: Enhancing Indoor Scene Reconstruction Using Neural SDFs with View-Dependent Normal Compensation. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024: 5155-5165.

---

confidence: 5
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. For Figure 3, what about the comparison with unbiased density function in NeuS, TUVR, DebSDF, HF-NeuS[3], and NeuRodin(optional, because NeuRodin is the concurrent work)[4]? These works also propose an unbiased density strategy. (visual comparison is better for good understanding)

2. In Figure 7, The deflection angle map appears to have a higher value at complex structural regions. What's the visual comparison between edge mask(EaNeuS), NCC mask(Equation 6 in NeuRIS), and uncertainty-aware mask(DDPNeRF[5], DebSDF, H2OSDF)?

3. For Equation 12 of this paper, it seems that when $\hat{\mathbf{N}}^d(\mathbf{r})$ and $\hat{\mathbf{N}}(\mathbf{r})$ perfectly fits the prior $\mathbf{N}(\mathbf{r})$, then this loss will become a perfect zero. The reviewer is concerned that this situation that causes zero loss doesn't seem to be in a good state. Is there a more comprehensive explanation about this training objective and why this training objective will work to help solve the concerns above?

4. Does the proposed normal deflection field have some advantages over the normal uncertainty field, like in DebSDF and H2OSDF?

Reference:

[3] HF-NeuS: Improved Surface Reconstruction Using High-Frequency Details. Wang et al. (Neurips 2022)

[4] NeuRodin: A Two-stage Framework for High-Fidelity Neural Surface Reconstruction. Wang et al. (Neurips 2024)

[5] Dense Depth Priors for Neural Radiance Fields from Sparse Input Views. Roessle et al. (CVPR 2022)
soundness: 2
strengths: 1. The paper writing is easy to follow.

2. The experiment covers lots of different dataset for a comprehensive comparison, like Scannet, TanksandTemples, Scannet++, Replica.

3. The ablation study indicates the effectiveness of the deflection field and adaptive angle prior when compared to its base model.
summary: This paper introduces ND-SDF and proposes to learn a Normal Deflection field to represent the angular deviation between the scene normal and the normal prior.

The paper also additionally introduces a ray sampling strategy based on the deflection angle to improve the surface quality further. The paper also proposes to use an unbiased function inspired by TUVR.
weaknesses: Major weakness:

1. Although the dataset included is various, the compared methods are not so much comprehensive(except the Scannet). The quantitative comparison on TanksandTemples, Scannet++, Replica only includes three works in 2022(NeuRIS, MonoSDF, VolSDF) and one work in 2021(Unisurf). This leads to an insufficient experiment. For example, what about the comparison with HelixSurf, TUVR, DebSDF, H2OSDF on TanksandTemples, Scannet++, Replica? The reviewer thinks the comparison without newer baselines is insufficient on TanksandTemples, Scannet++, Replica. Although Table 1 includes most newer baselines, usually only 4 scenes in Scannet are used for indoor scene reconstruction experiments. This is why the reviewer thinks that more quantitative experiments are required here.

Details: The reviewer understands that it's not necessary to run all previous baselines in all datasets. But just place here as an example. Since the unbiased weight technique is used, the reviewer thinks **at least** TUVR(CVPR2023) is a newer baseline that should be included in both the quantitative and qualitative comparisons on TanksandTemples, Scannet++, Replica datasets. (But more comparison with other newer methods will enhance the paper's experiment part.)

2. The qualitative comparison is also insufficient (only compared with older baselines). It's better to include the qualitative comparison with newer baselines in Figure 4, Figure 11, Figure 12, Figure 13.

3. Some works have both pure MLP version and hash grid version (like MonoSDF). The performance and efficiency of different versions have some differences. Since ND-SDF includes instant-NGP in its implementation, it's better to clearly indicate the version to compare within the table. (For example, "MonoSDF(mlp)", "MonoSDF(grid)".)

Minor weakness:

4. For Section 3.4, similar ideas of better ray sampling strategy have been investigated in some previous work. Like edge-aware sampling in EaNeuS[1], GaussianRoom[2], and uncertainty-aware sampling in DebSDF. Please consider adding a brief discussion about these previous methods.

5. In Figure 2, "GT depth" and "GT Normal" should be changed to "depth prior" and "normal prior".

6. The number of selected scenes in each dataset for the experiment has not been reported.

7. The dataset conducted in the ablation study has not been reported.

Reference:

[1] Edge-aware Neural Implicit Surface Reconstruction. Li et al. (ICME 2023)

[2] GaussianRoom: Improving 3D Gaussian Splatting with SDF Guidance and Monocular Cues for Indoor Scene Reconstruction. Xiang et al. (arxiv 2024)

Meta-review: The paper presents an approach to multi-view 3D reconstruction of indoor scenes via neural implicit functions and priors from monocular depth and surface estimation. To adaptively balance the influence of priors for reconstructing finer details, the paper proposes to learn a normal deflection field, which models the difference between normals from the prior and those from a pure image-based reconstruction. To make use of the learned field during optimization the paper further proposes losses and a sampling strategy. The efficacy of the proposed techniques is validated in an ablation study and comparisons against the state of the art. The reviewers unanimously recommended to accept the paper and the AC agrees with this assessment.

Decision: Accept (Spotlight)