# DehazeNeRF: Multiple Image Haze Removal and 3D Shape Reconstruction using
Neural Radiance Fields

Wei-Ting Chen1,2*
Wang Yifan1*
Sy-Yen Kuo2
Gordon Wetzstein1
1Stanford University 2National Taiwan University

###### Abstract

Neural radiance fields (NeRFs) have demonstrated state-of-the-art performance
for 3D computer vision tasks, including novel view synthesis and 3D shape
reconstruction. Yet, these methods fail in adverse weather conditions that are
crucial for applications such as autonomous driving. To address this
challenge, we introduce DehazeNeRF as a framework that robustly operates in
hazy conditions. DehazeNeRF extends the volume rendering equation by
physically realistic terms that model atmospheric scattering. These act as
inductive network biases in our pipeline and, together with several
regularization strategies, allow DehazeNeRF to demonstrate successful multi-
view haze removal, novel view synthesis, and 3D shape reconstruction where
existing approaches fail.

††footnotetext: * indicates equal contribution.

##  1 Introduction

Neural radiance fields (NeRFs) [46] have emerged as a powerful approach to
solving 3D computer vision problems, such as novel view synthesis and 3D
reconstruction [64]. Existing NeRF††We refer to neural radiance fields and
neural surfaces [71] collectively as “NeRF” in this paper. approaches are
successful in many scenarios, but they fail to accurately reconstruct the
geometry and appearance of a 3D scene in adverse weather conditions such as
haze (see Fig. 6). This poses a critical limiting factor for many real-life
applications, including autonomous driving among others.

The problem of applying NeRF to hazy input images is the ambiguity between
light-reflecting solid surfaces and light-scattering atmospheric particles in
free space – both are interchangeably modeled as a light-emitting volume by
NeRF and cannot be disambiguated. Most state-of-the-art dehazing methods adopt
feedforward neural networks [62, 1], but applying these to the input views as
a preprocessing step to standard NeRFs leads to poor reconstruction results,
in part because these techniques tend to overfit to the training data and
generalize poorly to real-world data [55, 61].

Figure 1: Given a set of posed hazy images, our method estimates the haze-free
scene using a neural radiance field that includes physically based terms for
scattering. Here, we show results of an experimentally captured scene.

To address this limitation, we develop a differentiable physically based 3D
haze model that seamlessly integrates into the volume rendering framework used
by NeRF. In the absence of scattering, our model is equivalent to NeRF. For
the inverse problem of estimating the scene parameters from multiple posed
input views, our 3D haze model is used in conjunction with several
regularization strategies that disentangle solid object surfaces from
scattering volumetric particles. Our proposed framework jointly learns the 3D
shape and haze components of a scene and allows for significantly improved and
multi-view-consistent novel view synthesis as well as more accurate 3D shape
reconstruction compared to previous work. Our approach does not require large
collections of paired hazy-clear images and it circumvents the sim2real gap of
data-driven dehazing methods by building an inductive bias based on the
physics of scattering into the network architecture.

Specifically, our contributions include:

  * •

extending the volume rendering equation of NeRF by a physically based 3D haze
image formation model to accurately model the in-scattering phenomenon
prevalent in hazy conditions;

  * •

introducing multiple physically inspired inductive biases as well as
optimization regularizers to effectively disambiguate the surface appearance
thus achieving accurate clear-view appearance and geometry reconstruction
using only hazy image as inputs.

We demonstrate state-of-the-art results using simulation and experimental data
for 3D multi-view construction and novel view synthesis under hazy condition.
Code and data will be made available
††https://www.computationalimaging.org/publications/dehazenerf for research
purposes.

##  2 Related Work

DehazeNeRF integrates knowledge from several areas of research. We briefly
review these next.

###  2.1 3D Scene Reconstruction with NeRF

Neural radiance fields (NeRFs) [46] were introduced for novel view synthesis
applications. By combining differentiable volumetric rendering from classic
computer graphics and learnable radiance field, NeRF can jointly optimize for
the geometry and appearance of a static 3D scene from posed RGB images. As the
performance of the original NeRF noticeably deteriorates under imperfect
capturing conditions, much research has investigated the robustness of NeRF
under a variety of challenging conditions, such as blur [40], noise [52],
reflection [19], low resolution [69], low dynamic range [22, 45], and
occlusion [41, 8]. In this paper, we address haze, an unexplored yet common
scenario in real-life captures. Our method leverages a scattering-aware
rendering equation to model the haze phenomenon and disambiguate the surface
and haze by using physically inspired inductive biases. Among those, we adopt
a surface-like density field parameterization [71] to encourage solid
geometry, although other surface priors may be equally applied in our method
[49, 75, 71, 50].

###  2.2 Dehazing

We provide a brief summary of existing dehazing methods and refer to [62] for
a comprehensive review. Early works aim to design strong priors to recover the
transmission map, global atmospheric light, and scene radiance [83, 5, 13, 20,
2], thus they struggle on scenes where these priors do not hold [39]. More
recently, data-driven methods [39, 7, 10, 43, 79, 18] achieve better quality
using deep learning and large datasets [32], but have difficulty generalizing
to images in the real world [61]. More related to our work are multi-image
dehazing methods, which exploit the consistency between neighboring frames.
Similar to single-image approaches, multi-image approaches can be divided into
prior-based or data-driven categories. The former adds photo-consistency
regularization between neighboring frames using jointly estimated or known
depth [80, 33, 14], while the latter achieves more temporal stability by
fusing features from a small number of neighboring frames [57, 72, 82]. Yet
both approaches are subject to the aforementioned problems exhibited in the
single-image scenario. Furthermore, none of these dehazing methods estimates a
holistic 3D structure, and therefore they are not directly applicable to novel
view synthesis.

#### Estimating Scattering Coefficients.

In addition to estimating the 3D geometry and haze-free appearance in novel
views, we also jointly optimize for the scattering coefficient. Prior work
mainly computes the scattering coefficient based on handcrafted priors such as
average saturation [17], dark channel mean [9], polarization [58], and gamma
correction [26]. In data-driven approaches, the scattering coefficient can be
estimated explicitly as an intermediary output [74, 68]. Our method directly
estimates the scattering coefficient in 3D, while at the same time utilizing
2D priors such as dark channel mean to address the ambiguity with other
physical properties such as airlight.

#### Seeing through Scattering Media.

Various methods have been developed to solve the challenging problem of
imaging through and within scattering media. These methods can be classified
as interference of light, relying on ballistic photons, and being based on
diffuse optical tomography. For methods based on interference of light, they
leverage information in the speckle pattern to reconstruct the image [54, 29,
3] or adopt wavefront shaping to focus light through or within scattering
media [21, 66]. On the other hand, ballistic photons can avoid scattered
photons since they can travel through a medium without scattering and can be
isolated by adopting time-gating [56, 70], coherence-gating [23, 11], or
coherent probing and detection of a target at different illumination angles
[28]. The last class is to adopt non-line-of-sight imaging techniques [67, 38,
51, 36, 12, 37, 77] or diffuse optical tomography [4, 15, 35] to recover
objects by modeling and inverting scattering of light explicitly. However,
these methods all require exotic hardware setups, which are usually expensive,
and are limited to some scenarios such as long propagation distance of light
or scale of resolution. Our approach removes the scattering effect using
inverse rendering, requiring only a set of RGB images of a hazy scene. Similar
principles have been explored in concurrent and independent work [31] for
underwater scene reconstruction.

##  3 Method

Our method, DehazeNeRF, extends the volume rendering equation to accurately
reconstruct the geometry and appearance robust to hazy conditions. Our key
idea is to introduce a series of important biases in the network architecture
along with regularizers in the loss function that together underpin physically
based scattering phenomena.

###  3.1 Preliminary on Neural Radiance Fields

Neural Radiance Fields (NeRFs) [46] map a 3D sample point $\mathbf{p}$ into a
color $\mathbf{c}$ and volume density $\sigma$. Considering only emission from
classic volume rendering [27, 63], the expected color ${C}(\mathbf{r})$ of a
camera ray $\mathbf{r}(t)=\mathbf{o}+t\mathbf{d}$ with the near and far
boundary $t_{n}$ and $t_{f}$ can be written as

$\displaystyle{C}(\mathbf{r},\mathbf{d})=\int_{t_{n}}^{t_{f}}T(t)\sigma(\mathbf{r}(t))c(\mathbf{r}(t),\mathbf{d})\
dt\;\textrm{with}$ (1) $\displaystyle
T(t)=\mathrm{exp}\left(-\int_{t_{n}}^{t}\sigma(\mathbf{r}(t^{\prime}))\
dt^{\prime}\right),$ (2)

where $T(t)$ is the accumulated transmittance between the ray section $t_{n}$
to $t$. The predicted pixel value is then compared to the ground truth
$\widehat{C}(\mathbf{r},\mathbf{d})$ for optimization.

###  3.2 3D Haze Formation

To address the 3D dehazing problem, we propose an alternative rendering
equation to the image formation model. We start from the radiative transfer
equation (RTE) [6, 65], which describes the behavior of light in a medium that
absorbs, scatters and emits radiation. Assuming, a ray
$\mathbf{r}\left(t\right)=\mathbf{o}+t\mathbf{d}$ hits a surface point at
$\mathbf{r}\left(t_{0}\right)$, the incident radiance at the near image plane
$t_{n}$ can be divided into three parts [53]:

$\displaystyle C(\mathbf{r},\mathbf{d})$
$\displaystyle=C_{\textrm{emission}}(\mathbf{r})+C_{\textrm{surface}}(\mathbf{r})+C_{\textrm{in-
scattering}}(\mathbf{r})$ $\displaystyle
C_{\textrm{emission}}(\mathbf{r},\mathbf{d})$
$\displaystyle=\int_{t_{n}}^{t_{0}}\epsilon\left(\mathbf{r}\left(t\right),\mathbf{d}\right)T_{\sigma_{t}}\left(t\right)dt$
$\displaystyle C_{\textrm{surface}}(\mathbf{r},\mathbf{d})$
$\displaystyle=C_{e}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)T_{\sigma_{t}}\left(t_{0}\right)$
$\displaystyle C_{\textrm{in-scattering}}(\mathbf{r},\mathbf{d})$
$\displaystyle=\int_{t_{n}}^{t_{0}}c_{\textrm{s}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\sigma_{s}\left(\mathbf{r}\left(t\right)\right)T_{\sigma_{t}}\left(t\right)dt,$

where $\epsilon$ is the emission, $C_{e}$ is the outgoing radiance at the
surface intersection,
$c_{\textrm{s}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)$ is the in-
scattered light and $\sigma_{s}$ is the scattering coefficient. In particular,
the transmittance here is computed from the attenuation coefficient
$\sigma_{t}$, _i.e_.,
$T_{\sigma_{t}}\left(t\right)=\exp\left(-\int_{t_{n}}^{t}\sigma_{t}(t^{\prime})dt^{\prime}\right)$,
where $\sigma_{t}=\sigma_{a}+\sigma_{s}$ including the absorption and out-
scattering effect. For common haze formation, the participating particles are
considered non-luminous [48], therefore we can drop the emission part, which
leads to

$\displaystyle\begin{split}C(\mathbf{r},\mathbf{d})={}&C_{e}(\mathbf{r}\left(t_{0}\right),\mathbf{d})T_{\sigma_{t}}\left(t_{0}\right)+\\\
&\int_{t_{n}}^{t_{0}}c_{\textrm{s}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\sigma_{s}\left(\mathbf{r}\left(t\right)\right)T_{\sigma_{t}}\left(t\right)dt.\end{split}$
(3)

Following NeRF [46], we represent the surface as a continuous density field
with emission
$\epsilon\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\coloneqq
c\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\sigma\left(\mathbf{r}\left(t\right)\right)$.
Meanwhile, the absorption part in the attenuation $\sigma_{t}$ can be
interpreted as the surface density $\sigma$, since the volume density $\sigma$
is equal to absorption coefficient $\sigma_{a}$ in that they both determine
the probability of a photon or a ray terminating at a given location. As a
result, we can write the rendering equation as

$\displaystyle\begin{split}C(\mathbf{r},\mathbf{d})=&\underbrace{\int_{t_{n}}^{t_{0}}c(\mathbf{r}(t),\mathbf{d})\sigma(t)T_{\sigma+\sigma_{s}}\left(t\right)dt}_{C_{\textrm{Surface}}}+\\\
&\underbrace{\int_{t_{n}}^{t_{0}}c_{s}(\mathbf{r}(t))\sigma_{s}(t)T_{\sigma+\sigma_{s}}\left(t\right)dt}_{C_{\textrm{Haze}}}.\end{split}$
(4)

Eq. 4 formally disentangles the surface and haze, represented by
$\left\\{c,\sigma\right\\}$ and $\left\\{c_{s},\sigma_{s}\right\\}$
respectively, in a principled manner. Once successfully optimized (see the
next Section), the clear-view surfaces can be recovered using
$\left\\{c,\sigma\right\\}$:

$C(\mathbf{r},\mathbf{d})=\int_{t_{n}}^{t_{0}}c(\mathbf{r}(t),\mathbf{d})\sigma(t)T_{\sigma}\left(t\right)dt.$
(5)

Figure 2: DehazeNeRF architecture. Given a set of hazy images, our method
augments the existing NeRF pipeline (gray) with a haze module (yellow), which
explicitly models the scattering phenomenon using atmospheric light and
scattering coefficient. During training, we render the hazy reconstruction as
a composition of surface and haze, which is compared to the input hazy images
to optimize the learnable parameters (in green) jointly. During inference, we
use the surface module (gray) to render clear views.

###  3.3 Haze-aware Neural Radiance Field

Given multiple images of a hazy scene, we aim to jointly optimize for the
surface appearance and geometry, $\left\\{c,\sigma\right\\}$ as well as the
haze’s scattering coefficient and in-scattered light (atmospheric light),
$\left\\{c_{s},\sigma_{s}\right\\}$ based on the enhanced scattering-aware
rendering equation Eq. 4. However, the effects of these variables are
interdependent. In order to correctly disentangle them, our model adopts
suitable architecture designs and training regularizers to capture the
distinct physical properties of haze and surface. An overview of DehazeNeRF is
illustrated in Fig. 2.

#### Architecture.

Now we introduce inductive biases to match the physical properties of haze and
surface. For clarity, we highlight the quantities directly modeled by neural
networks in green.

_Modeling a Surface._ Recall our goal is to learn the surface appearance and
geometry, $\left\\{c,\sigma\right\\}$. Similar to previous works [46], we
model the appearance
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}c}\left(\mathbf{p},\mathbf{d}\right)$
with an MLP, which takes the sample location $\mathbf{p}$ and viewing
direction $\mathbf{d}$ as inputs. However, in order to encourage volume
density $\sigma$ to form a well-defined solid surface, instead of directly
learning the volume density, we adopt the reparameterization of the volume
density using signed distance function (SDF),
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f}\left(\mathbf{r}\left(t\right)\right)\in\mathbb{R}$,
as proposed in NeuS [71, 73]. The modified surface volume density
$\sigma\left(\mathbf{r}\left(t\right)\right)$, referred to as opaque density,
can be parameterized as
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f}\left(\mathbf{r}\left(t\right)\right)$:

$\sigma\left(\mathbf{r}\left(t\right)\right)=s\left(\Phi_{s}\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f}\left(\mathbf{r}\left(t\right)\right)\right)-1\right)\nabla{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f}\left(\mathbf{r}\left(t\right)\right)\mathbf{d},$
(6)

where $\Phi_{s}(x)$ is the sigmoid function $\Phi_{s}(x)=(1+e^{-sx})^{-1}$,
whose derivative is a bell-shaped density function centered at 0 and has a
learnable standard deviation of $\nicefrac{{1}}{{s}}$. We derive the discrete
approximate following [46, 63]. It samples $n$ points
$\left\\{\mathbf{p}_{i}=\mathbf{o}+t_{n}\mathbf{d}|n=1,...,N,t_{n}<t_{n+1}\right\\}$
along the ray. The approximate pixel color of the ray is computed based on
quadrature rule [42], yielding

$\displaystyle\begin{gathered}C_{\textrm{surface}}(\mathbf{r},\mathbf{d})=\sum_{n=1}^{N}\frac{\sigma^{n}}{\sigma_{t}^{n}}T_{t}^{n}\alpha_{t}^{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}c}^{n}\textrm{
with }T_{t}^{n}=\prod_{m=1}^{n-1}\left(1-\alpha_{t}^{m}\right),\end{gathered}$
(8)

where $\alpha_{t}$ denotes the discrete $\alpha$-compositional weight defined
as [71, 73]

$\alpha_{t}^{n}=\textsc{clamp}\left(1-\exp\left(-\sigma_{t}^{n}\delta^{n}\right),0,1\right)\textrm{
with }\delta^{n}=t^{n+1}-t^{n},$

where
$\sigma_{t}^{n}=\sigma^{n}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sigma_{s}}^{n}$
denotes the total attenuation at sample $n$, including the attenuation due to
surface occlusion and the out-scattering.

_Modeling Haze._ We use a low-frequency prior to compute the scattering
coefficient and atmospheric light, $\left\\{c_{s},\sigma_{s}\right\\}$, since
these components usually vary slowly in a common hazy scenes [33]. In
practice, we use a small band-limited MLP [34] for the scattering coefficient
$\sigma_{s}$ to capture inhomogenous haze. Analogous to Eq. 8, the haze color
can be approximated as

$\begin{gathered}C_{\textrm{haze}}(\mathbf{r})=\sum_{n=1}^{N}\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sigma_{s}}^{n}}{\sigma_{t}^{n}}T_{t}^{n}\alpha_{t}^{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}c_{s}}^{n}.\end{gathered}$
(9)

During optimization, the color for an arbitrary input hazy image can be
written as $C=C_{\textrm{surface}}+C_{\textrm{haze}}$. At test time, we can
reconstruct the clear-view color by discretizing Eq. 5, namely:

$\displaystyle
C_{\textrm{clear}}\left(\mathbf{r},\mathbf{d}\right)=\sum_{n=1}^{N}T_{\sigma}^{n}\alpha^{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}c}^{n},$
(10)

$T_{\sigma}^{n}=\prod_{j=1}^{n-1}\left(1-\alpha^{j}\right)\,\textrm{and
}\,\alpha^{n}=\textsc{clamp}\left(1-\exp\left(-\sigma^{n}\delta^{n}\right),0,1\right).$

#### Optimization.

While the inductive biases separate the high-frequency surface appearance and
geometry from the low-frequency color and density of the scattering medium, we
introduce further regularizers to guide the optimization process to converge
to more plausible clear-view geometry and color.

_Koschmieder Consistency._ Given an accurate depth map $D$, assuming globally
constant scattering coefficient $\bar{\sigma}_{s}$ and airlight $\bar{c}_{s}$,
the relation between a clear-view image $C_{\textrm{clear}}$ and the hazy
image $C$ can be described by the Koschmieder law [24] as

$C(\mathbf{r})=C_{\textrm{clear}}(\mathbf{r})\exp(-\bar{\sigma}_{s}D(\mathbf{r}))+\bar{c}_{s}(1-\exp(-\bar{\sigma}_{s}D(\mathbf{r})))$.

(11)

This model is widely adopted as the basis for image-based single and multiview
dehazing. The Koschmider model is an approximation of our rendering equation
Eq. 4 under the assumption of spatially-invariant (i.e., homogeneous)
scattering coefficient and an ideal surface

$\displaystyle C_{\textrm{surface}}\left(\mathbf{r}\right)$
$\displaystyle\approx
C_{\textrm{clear}}(\mathbf{r})\exp(-\bar{\sigma}_{s}D(\mathbf{r}))=\tilde{C}_{\textrm{surface}}\left(\mathbf{r}\right)$
(12) $\displaystyle C_{\textrm{haze}}\left(\mathbf{r}\right)$
$\displaystyle\approx\bar{c}_{s}(1-\exp(-\bar{\sigma}_{s}D(\mathbf{r}))=\tilde{C}_{\textrm{haze}}\left(\mathbf{r}\right),$
(13)

We promote this relation with

$\displaystyle\mathcal{L}_{\textrm{2D}}=\left\|C_{\textrm{surface}}\left(\mathbf{r}\right)-\tilde{C}_{\textrm{surface}}\left(\mathbf{r}\right)\right\|_{1}$
(14) $\displaystyle+$
$\displaystyle\left\|C_{\textrm{haze}}\left(\mathbf{r}\right)\\!-\\!\tilde{C}_{\textrm{haze}}\left(\mathbf{r}\right)\right\|_{1}\\!\\!+\\!\left\|C\\!-\\!\tilde{C}_{\textrm{surface}}\left(\mathbf{r}\right)\\!-\\!\tilde{C}_{\textrm{haze}}\left(\mathbf{r}\right)\right\|_{1}\\!,$

where $\bar{\sigma}_{s}$ and $\bar{c}_{s}$ are the average over the samples on
the ray, while the depth value $D\left(\mathbf{r}\right)$ is computed via the
learned surface geometry [46, 78] by accumulating over ray-length over all the
samples on a ray:

$D\left(\mathbf{r}\right)=\sum_{n=1}^{N}T_{\sigma}^{n}\alpha^{n}t^{n}.$ (15)

_Color Prior._ Without knowing the original image, the heavily attenuated
color in the hazy image can be explained by the haze but also by a dull
surface color. In order to reconstruct plausible clear-view colors, we adopt
the popular 2D prior widely used in image-based dehazing methods, Dark Channel
Prior (DCP) [20], which arises from the observation, that for most pixels in a
natural haze-free image, the minimum of three color channels is close to zero.
We apply this prior to the estimated clear image $C_{\textrm{clear}}$

$\displaystyle DC(C_{\textrm{clear}})\left(\mathbf{x}\right)$
$\displaystyle=\underset{\mathbf{y}\in\Omega\left(\mathbf{x}\right)}{\min}\left(\underset{c\in\left\\{r,g,b\right\\}}{\min}C_{\textrm{clear}}^{c}\left(\mathbf{y}\right)\right),$
(16) $\displaystyle\mathcal{L}_{\textrm{dcp}}$
$\displaystyle=\frac{1}{K}\sum\limits_{k=1}^{K}\|DC\left(C_{\textrm{clear}}\right)\|_{1}.$
(17)

###  3.4 Implementation Details

We adopt the same setting as that in HF-NeuS [71] wherever possible. This
includes the MLPs for the surface SDF,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f}$ and the
view-dependent surface color,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}c}$, as
well as the sampling strategy, the background composition, and learning rate
schedule.

#### Loss.

Our loss is composed of several terms:

$\mathcal{L}=\mathcal{L}_{\textrm{color}}+\lambda\mathcal{L}_{\textrm{eikonal}}+\alpha\mathcal{L}_{\textrm{dcp}}+\beta\mathcal{L}_{\textrm{2D}},$
(18)

where $\mathcal{L}_{\textrm{dcp}}$ and $\mathcal{L}_{\textrm{2D}}$ are the
regularizations introduced in Sec. 3.3, while the photo-consistency loss,
$\mathcal{L}_{\textrm{color}}$, is the standard NeRF loss, and the eikonal
loss, $\mathcal{L}_{\textrm{eikonal}}$, is commonly used to regularize SDF
[16],

$\displaystyle\mathcal{L}_{\textrm{color}}$
$\displaystyle=\frac{1}{K}\sum_{k=1}^{K}\left\|\widehat{C}_{k}(\mathbf{r},\mathbf{d})-C_{k}(\mathbf{r},\mathbf{d})\right\|_{1},$
(19) $\displaystyle\mathcal{L}_{\textrm{eikonal}}$
$\displaystyle=\frac{1}{KN}\sum_{k}^{K}\sum_{n}^{N}(\|\nabla
f({\mathbf{r}}_{k}(t_{n}))\|_{2}-1)^{2},$ (20)

where $\widehat{C}_{k}(\mathbf{r},\mathbf{d})$ is the pixel color. $N$ and $K$
denote the total sampling points on a ray and the total number of rays sampled
per training batch.

Finally, because of the surface representation using SDF, we can optionally
adopt the object masks for supervision [75, 71, 73]. Specifically, given the
object mask, $M$, the mask loss $\mathcal{L}_{\textrm{mask}}$ for a sampled
ray $k$ is defined as

$\mathcal{L}_{\textrm{mask}}=\text{BCE}(M_{k},\hat{O}_{k}),$ (21)

where $\hat{O}_{k}=\sum_{i=1}^{N}T_{\sigma}^{i}\alpha^{i}$ is the total weight
for the clear-view surface color along the camera ray, and BCE is the binary
cross entropy loss.

NeuS |  COLMAP + NeuS |  ImDehaze+NeuS |  VidDehaze + NeuS |  DehazeNeRF |  Ground Truth
---|---|---|---|---|---
|  |  |  |  |
|  |  |  |  |
|  |  |  |  |
|  |  |  |  |
Figure 3: Qualitative comparison on synthetic data. Our method successfully
removes heterogeneous haze in the synthesized views, showing the best
appearance fidelity compared with baseline methods. The reconstructed geometry
is more accurate, less noisy, and contains more details.

##  4 Experiments with Synthetic Scenes

In this section, we detail our experiments using synthetic data. Our goal is
to quantifiably evaluate the contribution of each proposed component in a
controlled setting. We report our main findings here and refer the readers to
the supplement for more detailed evaluations.

###  4.1 Data Preparation

We synthesize haze using 10 scenes from the DTU dataset [25]. The scattering
coefficient is modeled using the sum of 4 scaled Gaussian blobs located inside
the spatial bounding box with a standard deviation uniformly sampled from 1.0
to 3.0; the 3-D atmospheric light is sampled from a uniform distribution in
the range $[0.7,0.9]$. 10% of the images in each synthetic scene are held out
as the test set.

###  4.2 Comparisons

#### Baselines.

We compare DehazeNeRF with the following baselines:

  1. 1.

NeuS: train HF-NeuS [73] on hazy images,

  2. 2.

ImDehaze+NeuS: train HF-NeuS on dehazed images obtained using the state-of-
the-art single-image dehazing method [18],

  3. 3.

VidDehaze+NeuS: train HF-NeuS on dehazed images obtained using the state-of-
the-art video dehazing method [82],

  4. 4.

COLMAP+NeuS: train HF-NeuS on dehazed images obtained by estimating the
transmission maps using the dense depth map from COLMAP [59, 60].

With HF-NeuS as the backbone surface model [73], all approaches observe the
same surface prior. While the first baseline neglects haze entirely, baselines
2 to 4 increasingly incorporate more multiview information for haze modeling,
with ours being the most 3D-aware and physically accurate, as it models the
spatial-variant scattering coefficient in 3D space and optimizes the 3D
geometry, surface appearance, and the haze parameters jointly. For the last
baseline, we use the method proposed by [20] to estimate the global airlight
from the object regions (masked by [76]) in all images. Then we use 300 pairs
of feature correspondences to estimate the global scattering coefficient,
where each pair computes a candidate scattering coefficient as follows
$\frac{1}{D_{b}\left(\mathbf{x}_{b}\right)-D_{a}\left(\mathbf{x}_{a}\right)}\ln\left(\frac{I_{a}\left(\mathbf{x}_{a}\right)-\bar{c}_{s}}{I_{b}\left(\mathbf{x}_{b}\right)-\bar{c}_{s}}\right)$.
$\left(I_{a},D_{a}\right)$ and $\left(I_{b},D_{b}\right)$ are RGB images and
depth maps in two views, and $\mathbf{x}_{a}$ and $\mathbf{x}_{b}$ are the
image coordinates of a pair of matched SIFT features. The final result is
obtained after filtering out negative or invalid estimations, which may occur
due to specularities and noisy depth estimation.

Qualitative Evaluation. We demonstrate some examples of the dehazed results
and reconstructed geometry in Fig. 3. Despite having a surface prior, naïvely
training HF-NeuS directly from hazy images is equivalent to averaging the
scattering-induced geometry-dependent irradiance variance observed across
different views and attributing it the surface color. Consequently, the view
synthesis is hazy and blurred. For two-stage strategies, the rendered results
have color distortion of various degrees, as indicated by the PSNR evaluation
in Tab. S.1, since it is difficult to accurately estimate the airlight and
coefficient when the presented data does not comply with the specific
assumptions or fall in the distribution of the training data. Moreover, our
method clearly reconstructs the geometry with more surface details compared to
all other baselines that adopt the surface prior formulated in NeuS, since our
method can dehaze different views more consistently thanks to the underlying
geometry that is optimized jointly.

Quantitative Evaluation. We measure the image quality using peak signal-to-
noise ratio (PSNR), structural similarity (SSIM), and perceptual similarity
(LPIPS). The geometry quality is measured using Chamfer Distances (CD) using
the DTU standard protocol. As shown in Tab. S.1, DehazeNeRF achieves the best
results compared to all baselines, with superior performance in PSNR, a metric
sensitive to low-frequency color shift. This indicates that while other
methods struggle to estimate the true air light and scattering coefficient
from either statistical or data priors, our method benefits from jointly
optimizing these quantities along with the scene appearance and geometry.

((a)) Without $\mathcal{L}_{\textrm{dcp}}$

((b)) With $\mathcal{L}_{\textrm{dcp}}$

((c)) Ground Truth

((d)) Without $\mathcal{L}_{\textrm{2D}}$

((e)) With $\mathcal{L}_{\textrm{2D}}$

((f)) Ground Truth

Figure 4: Ablation: $\mathcal{L}_{\textrm{dcp}}$ and
$\mathcal{L}_{\textrm{2D}}$ lead to more accurate clear-view color photometric
details.

Figure 5: Ablation on $\mathcal{L}_{\textrm{2D}}$. We compare the validation
results during training with and without $\mathcal{L}_{\textrm{2D}}$: With
$\mathcal{L}_{\textrm{2D}}$ the image quality continuously improve.

Method |  PSNR ($\uparrow$) |  SSIM ($\uparrow$) |  LPIPS ($\downarrow$) |  Chamfer ($\downarrow$)
---|---|---|---|---
NeuS | 16.722 | 0.879 | 0.087 | 2.635
COLMAP + NeuS | 16.750 | 0.785 | 0.225 | 5.971
ImDehaze[18] \+ NeuS | 17.887 | 0.899 | 0.078 | 2.298
VidDehaze[82] \+ NeuS | 18.324 | 0.901 | 0.087 | 2.349
DehazeNeRF | 25.702 | 0.920 | 0.052 | 2.066

Table 1: Quantitative comparison using synthetic data with heterogeneous haze.
DehazeNeRF yields better reconstruction both in image quality and geometry
accuracy.

###  4.3 Ablation Study

We conduct ablation studies for the optimization regularizers on 3 scenes and
report their average in Tab. 2.

As Tab. 2 shows, both $\mathcal{L}_{\textrm{2D}}$ and
$\mathcal{L}_{\textrm{dcp}}$ contribute positively to the image quality and
surface reconstruction accuracy. The $\mathcal{L}_{\textrm{dcp}}$ has a
prominent effect in reducing global color shift as indicated by the PSNR
value. Similarly, from the visual comparison in Figs. 4(a), 4(b) and 4(c), we
can observe that by adopting $\mathcal{L}_{\textrm{dcp}}$, the residual haze
can be suppressed effectively. Moreover, as shown in Figs. 4(d),
LABEL:sub@fig:ablation_cyle_ours and LABEL:sub@fig:ablation_cyle_gt, the
results optimized with $\mathcal{L}_{\textrm{2D}}$ show more accurate
structure (see orange bounding boxes). In addition, we plot the evolution of
validation PSNR and SSIM in Fig. 5. $\mathcal{L}_{\textrm{2D}}$ improves the
convergence behavior and yields better validation results.

Method | PSNR ($\uparrow$) | SSIM ($\uparrow$) | LPIPS ($\downarrow$) | Chamfer ($\downarrow$)
---|---|---|---|---
w/o $\mathcal{L}_{\textrm{2D}}+\mathcal{L}_{\textrm{dcp}}$ | 18.21 | 0.88 | 0.09 | 2.47
w $\mathcal{L}_{\textrm{2D}}$ | 18.53 | 0.89 | 0.09 | 2.44
w $\mathcal{L}_{\textrm{dcp}}$ | 24.02 | 0.90 | 0.06 | 2.44
DehazeNeRF | 25.23 | 0.91 | 0.05 | 2.41

Table 2: Ablations on regularization. Each of the proposed two regularizations can individually improve the image and geometry reconstruction, and the best quality is achieved with both. The results are averaged from 10 test scenes from the DTU dataset. NeuS |  COLMAP + NeuS |  ImDehaze + NeuS |  VidDehaze + NeuS |  DehazeNeRF |  Ground Truth
---|---|---|---|---|---
|  |  |  |  |
|  |  |  |  |
|  |  |  |  |
Figure 6: Qualitative comparison on captured data. Our method recovers the
haze-free scenes with more image details and colors closer to the ground
truth.

Method | PSNR ($\uparrow$) | SSIM ($\uparrow$) | LPIPS ($\downarrow$)
---|---|---|---
NeuS | 12.60 | 0.48 | 0.38
COLMAP + NeuS | 9.27 | 0.38 | 0.47
ImDehaze + NeuS | 13.35 | 0.49 | 0.36
VidDehaze + NeuS | 14.56 | 0.50 | 0.34
DehazeNeRF | 17.47 | 0.68 | 0.16

Table 3: Quantitative evaluation on experimentally captured data averaged over
3 scenes. Our method outperforms other methods by a large margin.

##  5 Experimentally Captured Results

###  5.1 Data Collection

We captured three indoor hazy scenes using two professional haze machines and
an iPhone 12 Pro. The scenes contain 38, 82, and 68 hazy images and 79, 47,
and 47 clear images for testing respectively. To ensure temporal consistency
(dynamic haze is out of scope for this paper), we capture the data after the
haze has settled and appears steady. After the capturing, we merge hazy images
and clear images under the same scene and adopt COLMAP to register camera
positions, which yields a consistent registration for the hazy and clear view
for ease of evaluation. We scale and translate the object such that it lies
inside the unit sphere and roughly at the center of the world coordinates.
More details on the data capture are described in the supplement.

###  5.2 Implementation details.

Several changes are applied to the model to facilitate the optimization for
captured data. First, we remove the mask loss ( Eq. 21). Second, similar to
NeuS [71], the region outside uses the parameterization of NeRF++[81], for
which we also applied the haze attenuation following Eqs. 8, 3.3 and 9.

###  5.3 Comparison

For evaluation, we adopt the same baselines described in Sec. 4.2. The 2D
photometric quality is evaluated using the same metrics, and geometry
reconstruction quality is omitted due to the lack of ground truth.

#### Quantitative Evaluation.

The results are reported in Tab. 3. One can observe that our method
outperforms other baselines in all metrics, demonstrating the robustness of
the proposed method in the real-world scenario.

#### Qualitative Evaluation.

As shown in Fig. 6, vanilla NeuS includes haze in the synthesized views. On
the other hand, adopting 2D image dehazing or video dehazing methods as pre-
processing also cannot generate desired clear results since these learning-
based models have limited generalization ability in real-world data. Although
using COLMAP as pre-processing can suppress residual haze, the dehazed results
tend to have the color distortion problem and the limited structure (see the
background at the $3^{rd}$ column of Fig. 6). Our method achieves
significantly better visual quality compared to other baselines.

##  6 Discussion

In this paper, we address challenges to apply NeRFs in hazy scenes. Our method
jointly learns the 3D structure and the plausible clear-view appearance of the
scene, as well as the haze properties (airlight and scattering coefficient)
from hazy observations. Our key technical contributions are: 1\. incorporating
the scattering phenomena in NeRF’s rendering equation; 2\. deploying
physically inspired inductive biases and haze-specific regularizers to
disambiguate haze and clear-view components. The proposed techniques take a
significant step towards adopting NeRF-related approaches in real-life
scenarios where adverse weather conditions are detrimental to 3D
reconstruction.

#### Relation to the Koschmieder Law.

Koschmieder model is a 2D image formation model that describes the
relationship between the clear-view and the hazy images. Assuming homogeneous
scattering, our formulation can be simplified to Koschmieder’s model, which
reduces to supervising with Eq. S.7 and $\mathcal{L}_{\textrm{dcp}}$. Our
$\mathcal{L}_{\textrm{2D}}$ exploits this relation to improve the convergence
as shown in Sec. 4.3. In Tab. 4, we examine this relation by comparing this
simplified method (denoted as Koschm) with a variant of DehazeNeRF that uses a
learnable scalar instead of the proposed MLP to model the scattering
coefficient (denoted as Scalar). For both synthetic and real data, the Scalar
and Koschm solutions are comparable, which validates the relation to
Koschmieder law; our full model shows a considerable advantage in both
scenarios thanks to its more general formulation.

Method | Synthetic Data (Scene24) | Real Data (“Elephant” Scene)
---|---|---
PSNR |  SSIM |  LPIPS |  Chamfer |  PSNR |  SSIM |  LPIPS
Koschm |  18.62 |  0.79 |  0.13 |  1.44 |  16.84 |  0.69 |  0.18
Scalar |  18.50 |  0.79 |  0.13 |  1.46 |  16.80 |  0.69 |  0.18
DehazeNeRF |  22.78 |  0.82 |  0.09 |  1.22 |  17.87 |  0.73 |  0.15

Table 4: Relation to the 2D Koschmieder Law. When using a scalar to model the
scattering coefficient (Scalar), our method can simplify to supervising with
the 2D Koschmieder law (Koschm). Our full model using an MLP to model the
spatially varying scattering coefficient has a clear advantage for
heterogeneous haze.

#### Limitations and future work.

While we have demonstrated the effectiveness of our method to capture data,
certain scenarios are not addressed in the scope of this paper and are
research directions for future work. These include severe haze scenes, where
camera registration fails due to the lack of discriminative image features,
and dynamic haze, where the haze distribution change not only spatially but
also temporally. Nonetheless, we believe that our method is an important step
towards anticipated application scenarios, such as autonomous driving or
underwater imaging. The principle of incorporating physics into the neural
rendering is also applicable to other ill-posed low-level vision tasks, such
as image denoising, image brightening, and super-resolution.

## Acknowledgement

This project was in part supported by Samsung, Stanford HAI, a PECASE from the
ARO, SNF Postdoc.mobility fellowship, and Google PhD Fellowship. We thank to
National Center for High-performance Computing (NCHC) for providing
computational and storage resources.

## References

  * [1] Codruta O Ancuti, Cosmin Ancuti, Radu Timofte, Luc Van Gool, Lei Zhang, and Ming-Hsuan Yang.  Ntire 2019 image dehazing challenge report.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 0–0, 2019.
  * [2] Dana Berman, Shai Avidan, et al.  Non-local image dehazing.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1674–1682, 2016.
  * [3] Jacopo Bertolotti, Elbert G Van Putten, Christian Blum, Ad Lagendijk, Willem L Vos, and Allard P Mosk.  Non-invasive imaging through opaque scattering layers.  Nature, 2012.
  * [4] David A Boas, Dana H Brooks, Eric L Miller, Charles A DiMarzio, Misha Kilmer, Richard J Gaudette, and Quan Zhang.  Imaging the body with diffuse optical tomography.  IEEE signal processing magazine, 18(6):57–75, 2001.
  * [5] Trung Minh Bui and Wonha Kim.  Single image dehazing using color ellipsoid prior.  IEEE Transactions on Image Processing, 27(2):999–1009, 2017.
  * [6] Subrahmanyan Chandrasekhar.  Radiative transfer.  Courier Corporation, 2013.
  * [7] Wei-Ting Chen, Jian-Jiun Ding, and Sy-Yen Kuo.  Pms-net: Robust haze removal based on patch map for single images.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11681–11689, 2019.
  * [8] Xingyu Chen, Qi Zhang, Xiaoyu Li, Yue Chen, Ying Feng, Xuan Wang, and Jue Wang.  Hallucinated neural radiance fields in the wild.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12943–12952, 2022.
  * [9] Won Young Chung, Sun Young Kim, and Chang Ho Kang.  Image dehazing using lidar generated grayscale depth prior.  Sensors, 22(3):1199, 2022.
  * [10] Hang Dong, Jinshan Pan, Lei Xiang, Zhe Hu, Xinyi Zhang, Fei Wang, and Ming-Hsuan Yang.  Multi-scale boosted dehazing network with dense feature fusion.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 2157–2167, 2020.
  * [11] C Dunsby and PMW French.  Techniques for depth-resolved imaging through turbid media including coherence-gated imaging.  Journal of Physics D: Applied Physics, 36(14):R207, 2003.
  * [12] Daniele Faccio, Andreas Velten, and Gordon Wetzstein.  Non-line-of-sight imaging.  Nature Reviews Physics, 2(6):318–327, 2020.
  * [13] Raanan Fattal.  Dehazing using color-lines.  ACM transactions on graphics (TOG), 34(1):1–14, 2014.
  * [14] Yuki Fujimura, Motoharu Sonogashira, and Masaaki Iiyama.  Dehazing cost volume for deep multi-view stereo in scattering media.  In Proceedings of the Asian Conference on Computer Vision, 2020\.
  * [15] Adam Gibson and Hamid Dehghani.  Diffuse optical imaging.  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367(1900):3055–3072, 2009.
  * [16] Amos Gropp, Lior Yariv, Niv Haim, Matan Atzmon, and Yaron Lipman.  Implicit geometric regularization for learning shapes.  arXiv preprint arXiv:2002.10099, 2020.
  * [17] Zhenfei Gu, Mingye Ju, and Dengyin Zhang.  A single image dehazing method using average saturation prior.  Mathematical Problems in Engineering, 2017, 2017.
  * [18] Chun-Le Guo, Qixin Yan, Saeed Anwar, Runmin Cong, Wenqi Ren, and Chongyi Li.  Image dehazing transformer with transmission-aware 3d position embedding.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5812–5820, 2022.
  * [19] Yuan-Chen Guo, Di Kang, Linchao Bao, Yu He, and Song-Hai Zhang.  Nerfren: Neural radiance fields with reflections.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 18409–18418, 2022.
  * [20] Kaiming He, Jian Sun, and Xiaoou Tang.  Single image haze removal using dark channel prior.  IEEE transactions on pattern analysis and machine intelligence, 33(12):2341–2353, 2010.
  * [21] Roarke Horstmeyer, Haowen Ruan, and Changhuei Yang.  Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue.  Nature photonics, 9(9):563–571, 2015.
  * [22] Xin Huang, Qi Zhang, Ying Feng, Hongdong Li, Xuan Wang, and Qing Wang.  Hdr-nerf: High dynamic range neural radiance fields.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 18398–18408, 2022.
  * [23] Guy Indebetouw and Prapong Klysubun.  Imaging through scattering media with depth resolution by use of low-coherence gating in spatiotemporal digital holography.  Optics Letters, 25(4):212–214, 2000.
  * [24] Hans Israël and Fritz Kasten.  Koschmieders theorie der horizontalen sichtweite.  In Die Sichtweite im Nebel und die Möglichkeiten ihrer künstlichen Beeinflussung, pages 7–10. Springer, 1959.
  * [25] Rasmus Jensen, Anders Dahl, George Vogiatzis, Engil Tola, and Henrik Aanæs.  Large scale multi-view stereopsis evaluation.  In 2014 IEEE Conference on Computer Vision and Pattern Recognition, pages 406–413. IEEE, 2014.
  * [26] Mingye Ju, Can Ding, Y Jay Guo, and Dengyin Zhang.  Idgcp: Image dehazing based on gamma correction prior.  IEEE Transactions on Image Processing, 29:3104–3118, 2019.
  * [27] James T Kajiya and Brian P Von Herzen.  Ray tracing volume densities.  ACM SIGGRAPH computer graphics, 18(3):165–174, 1984.
  * [28] Sungsam Kang, Seungwon Jeong, Wonjun Choi, Hakseok Ko, Taeseok D Yang, Jang Ho Joo, Jae-Seung Lee, Yong-Sik Lim, Q-Han Park, and Wonshik Choi.  Imaging deep within a scattering medium using collective accumulation of single-scattered waves.  Nature Photonics, 9(4):253–258, 2015.
  * [29] Ori Katz, Pierre Heidmann, Mathias Fink, and Sylvain Gigan.  Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations.  Nature photonics, 8(10):784–790, 2014.
  * [30] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  arXiv preprint arXiv:1412.6980, 2014.
  * [31] Deborah Levy, Amit Peleg, Derya Akkaynak, Naama Pearl, Dan Rosenbaum, Simon Korman, and Tali Treibitz.  Seathru-nerf: Neural radiance fields in scattering media.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023.
  * [32] Boyi Li, Wenqi Ren, Dengpan Fu, Dacheng Tao, Dan Feng, Wenjun Zeng, and Zhangyang Wang.  Benchmarking single-image dehazing and beyond.  IEEE Transactions on Image Processing, 28(1):492–505, 2018.
  * [33] Zhuwen Li, Ping Tan, Robby T Tan, Danping Zou, Steven Zhiying Zhou, and Loong-Fah Cheong.  Simultaneous video defogging and stereo reconstruction.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4988–4997, 2015.
  * [34] David B Lindell, Dave Van Veen, Jeong Joon Park, and Gordon Wetzstein.  Bacon: Band-limited coordinate networks for multiscale scene representation.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 16252–16262, 2022.
  * [35] David B Lindell and Gordon Wetzstein.  Three-dimensional imaging through scattering media based on confocal diffuse tomography.  Nat. Commun., 11(4517), 2020.
  * [36] David B Lindell, Gordon Wetzstein, and Matthew O’Toole.  Wave-based non-line-of-sight imaging using fast fk migration.  ACM Transactions on Graphics (ToG), 38(4):1–13, 2019.
  * [37] Xiaochun Liu, Sebastian Bauer, and Andreas Velten.  Phasor field diffraction based reconstruction for fast non-line-of-sight imaging systems.  Nature communications, 11(1):1–13, 2020.
  * [38] Xiaochun Liu, Ibón Guillén, Marco La Manna, Ji Hyun Nam, Syed Azer Reza, Toan Huu Le, Adrian Jarabo, Diego Gutierrez, and Andreas Velten.  Non-line-of-sight imaging using phasor-field virtual wave optics.  Nature, 572(7771):620–623, 2019.
  * [39] Yang Liu, Jinshan Pan, Jimmy Ren, and Zhixun Su.  Learning deep priors for image dehazing.  In Proceedings of the IEEE/CVF international conference on computer vision, pages 2492–2500, 2019.
  * [40] Li Ma, Xiaoyu Li, Jing Liao, Qi Zhang, Xuan Wang, Jue Wang, and Pedro V Sander.  Deblur-nerf: Neural radiance fields from blurry images.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12861–12870, 2022.
  * [41] Ricardo Martin-Brualla, Noha Radwan, Mehdi SM Sajjadi, Jonathan T Barron, Alexey Dosovitskiy, and Daniel Duckworth.  Nerf in the wild: Neural radiance fields for unconstrained photo collections.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7210–7219, 2021.
  * [42] Nelson Max.  Optical models for direct volume rendering.  IEEE Transactions on Visualization and Computer Graphics, 1(2):99–108, 1995.
  * [43] Armin Mehri, Parichehr B Ardakani, and Angel D Sappa.  Mprnet: Multi-path residual network for lightweight image super resolution.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 2704–2713, 2021.
  * [44] William Edgar Knowles Middleton.  Vision through the atmosphere.  In geophysik ii/geophysics ii, pages 254–287. Springer, 1957.
  * [45] Ben Mildenhall, Peter Hedman, Ricardo Martin-Brualla, Pratul P Srinivasan, and Jonathan T Barron.  Nerf in the dark: High dynamic range view synthesis from noisy raw images.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 16190–16199, 2022.
  * [46] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng.  Nerf: Representing scenes as neural radiance fields for view synthesis.  In European conference on computer vision, pages 405–421. Springer, 2020.
  * [47] Srinivasa G Narasimhan.  Models and algorithms for vision through the atmosphere.  Columbia University, 2004.
  * [48] Srinivasa G. Narasimhan and Shree K. Nayar.  Contrast restoration of weather degraded images.  IEEE transactions on pattern analysis and machine intelligence, 25(6):713–724, 2003.
  * [49] Michael Niemeyer, Jonathan T Barron, Ben Mildenhall, Mehdi SM Sajjadi, Andreas Geiger, and Noha Radwan.  Regnerf: Regularizing neural radiance fields for view synthesis from sparse inputs.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5480–5490, 2022.
  * [50] Michael Oechsle, Songyou Peng, and Andreas Geiger.  Unisurf: Unifying neural implicit surfaces and radiance fields for multi-view reconstruction.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 5589–5599, 2021.
  * [51] Matthew O’Toole, David B Lindell, and Gordon Wetzstein.  Confocal non-line-of-sight imaging based on the light-cone transform.  Nature, 555(7696):338–341, 2018.
  * [52] Naama Pearl, Tali Treibitz, and Simon Korman.  Nan: Noise-aware nerfs for burst-denoising.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12672–12681, 2022.
  * [53] Matt Pharr, Wenzel Jakob, and Greg Humphreys.  Physically based rendering: From theory to implementation.  Morgan Kaufmann, 2016.
  * [54] Sébastien Popoff, Geoffroy Lerosey, Mathias Fink, Albert Claude Boccara, and Sylvain Gigan.  Image transmission through an opaque material.  Nature communications, 1(1):1–5, 2010.
  * [55] Yanyun Qu, Yizi Chen, Jingying Huang, and Yuan Xie.  Enhanced pix2pix dehazing network.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
  * [56] Albert Redo-Sanchez, Barmak Heshmat, Alireza Aghasi, Salman Naqvi, Mingjie Zhang, Justin Romberg, and Ramesh Raskar.  Terahertz time-gated spectral imaging for content extraction through layered structures.  Nature communications, 7(1):1–7, 2016.
  * [57] Wenqi Ren, Jingang Zhang, Xiangyu Xu, Lin Ma, Xiaochun Cao, Gaofeng Meng, and Wei Liu.  Deep video dehazing with semantic segmentation.  IEEE transactions on image processing, 28(4):1895–1908, 2018.
  * [58] Yoav Y Schechner, Srinivasa G Narasimhan, and Shree K Nayar.  Instant dehazing of images using polarization.  In Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001, volume 1, pages I–I. IEEE, 2001.
  * [59] Johannes Lutz Schönberger and Jan-Michael Frahm.  Structure-from-motion revisited.  In Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
  * [60] Johannes Lutz Schönberger, Enliang Zheng, Marc Pollefeys, and Jan-Michael Frahm.  Pixelwise view selection for unstructured multi-view stereo.  In European Conference on Computer Vision (ECCV), 2016.
  * [61] Yuanjie Shao, Lerenhan Li, Wenqi Ren, Changxin Gao, and Nong Sang.  Domain adaptation for image dehazing.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2808–2817, 2020.
  * [62] Dilbag Singh and Vijay Kumar.  A comprehensive review of computational dehazing techniques.  Archives of Computational Methods in Engineering, 26(5):1395–1413, 2019.
  * [63] Andrea Tagliasacchi and Ben Mildenhall.  Volume rendering digest (for nerf).  arXiv preprint arXiv:2209.02417, 2022.
  * [64] Ayush Tewari, Justus Thies, Ben Mildenhall, Pratul Srinivasan, Edgar Tretschk, W Yifan, Christoph Lassner, Vincent Sitzmann, Ricardo Martin-Brualla, Stephen Lombardi, et al.  Advances in neural rendering.  In Computer Graphics Forum, volume 41, pages 703–735. Wiley Online Library, 2022.
  * [65] MCW van van Rossum and Th M Nieuwenhuizen.  Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion.  Reviews of Modern Physics, 71(1):313, 1999.
  * [66] Ivo M Vellekoop and AP Mosk.  Focusing coherent light through opaque strongly scattering media.  Optics letters, 32(16):2309–2311, 2007.
  * [67] Andreas Velten, Thomas Willwacher, Otkrist Gupta, Ashok Veeraraghavan, Moungi G Bawendi, and Ramesh Raskar.  Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging.  Nature communications, 3(1):1–8, 2012.
  * [68] Cong Wang, Yan Huang, Yuexian Zou, and Yong Xu.  Fully non-homogeneous atmospheric scattering modeling with convolutional neural networks for single image dehazing.  arXiv preprint arXiv:2108.11292, 2021.
  * [69] Chen Wang, Xian Wu, Yuan-Chen Guo, Song-Hai Zhang, Yu-Wing Tai, and Shi-Min Hu.  Nerf-sr: High quality neural radiance fields using supersampling.  In Proceedings of the 30th ACM International Conference on Multimedia, pages 6445–6454, 2022.
  * [70] L Wang, PP Ho, C Liu, G Zhang, and RR Alfano.  Ballistic 2-d imaging through scattering walls using an ultrafast optical kerr gate.  Science, 253(5021):769–771, 1991.
  * [71] Peng Wang, Lingjie Liu, Yuan Liu, Christian Theobalt, Taku Komura, and Wenping Wang.  Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction.  Advances in Neural Information Processing Systems, 34:27171–27183, 2021.
  * [72] Xintao Wang, Kelvin CK Chan, Ke Yu, Chao Dong, and Chen Change Loy.  Edvr: Video restoration with enhanced deformable convolutional networks.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 0–0, 2019.
  * [73] Yiqun Wang, Ivan Skorokhodov, and Peter Wonka.  Hf-neus: Improved surface reconstruction using high-frequency details.  In Advances in Neural Information Processing Systems.
  * [74] Yang Yang, Chaoyue Wang, Risheng Liu, Lin Zhang, Xiaojie Guo, and Dacheng Tao.  Self-augmented unpaired image dehazing via density and depth decomposition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2037–2046, 2022.
  * [75] Lior Yariv, Jiatao Gu, Yoni Kasten, and Yaron Lipman.  Volume rendering of neural implicit surfaces.  Advances in Neural Information Processing Systems, 34:4805–4815, 2021.
  * [76] Lior Yariv, Yoni Kasten, Dror Moran, Meirav Galun, Matan Atzmon, Basri Ronen, and Yaron Lipman.  Multiview neural surface reconstruction by disentangling geometry and appearance.  Advances in Neural Information Processing Systems, 33:2492–2502, 2020.
  * [77] Sean I Young, David B Lindell, Bernd Girod, David Taubman, and Gordon Wetzstein.  Non-line-of-sight surface reconstruction using the directional light-cone transform.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1407–1416, 2020.
  * [78] Zehao Yu, Songyou Peng, Michael Niemeyer, Torsten Sattler, and Andreas Geiger.  Monosdf: Exploring monocular geometric cues for neural implicit surface reconstruction.  arXiv preprint arXiv:2206.00665, 2022.
  * [79] Syed Waqas Zamir, Aditya Arora, Salman Khan, Munawar Hayat, Fahad Shahbaz Khan, Ming-Hsuan Yang, and Ling Shao.  Multi-stage progressive image restoration.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 14821–14831, 2021.
  * [80] Jiawan Zhang, Liang Li, Yi Zhang, Guoqiang Yang, Xiaochun Cao, and Jizhou Sun.  Video dehazing with spatial and temporal coherence.  The Visual Computer, 27(6):749–757, 2011.
  * [81] Kai Zhang, Gernot Riegler, Noah Snavely, and Vladlen Koltun.  Nerf++: Analyzing and improving neural radiance fields.  arXiv preprint arXiv:2010.07492, 2020.
  * [82] Xinyi Zhang, Hang Dong, Jinshan Pan, Chao Zhu, Ying Tai, Chengjie Wang, Jilin Li, Feiyue Huang, and Fei Wang.  Learning to restore hazy video: A new real-world dataset and a new method.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9239–9248, 2021.
  * [83] Qingsong Zhu, Jiaming Mai, and Ling Shao.  A fast single image haze removal algorithm using color attenuation prior.  IEEE transactions on image processing, 24(11):3522–3533, 2015.

Supplemental Materials

Data | Method | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | Chamfer $\downarrow$
---|---|---|---|---|---
Scan 24 | NeuS | 13.93 | 0.72 | 0.19 | 1.51
COLMAP+NeuS | 11.82 | 0.41 | 0.54 | 6.22
ImDehaze[18]+NeuS | 16.83 | 0.80 | 0.18 | 1.60
VidDehaze[82]+NeuS | 16.73 | 0.77 | 0.19 | 1.59
DehazeNeRF | 22.78 | 0.82 | 0.09 | 1.22
Scan 37 | NeuS | 20.64 | 0.88 | 0.06 | 1.44
COLMAP+NeuS | 13.34 | 0.75 | 0.23 | 3.04
ImDehaze[18]+NeuS | 21.55 | 0.90 | 0.06 | 1.90
VidDehaze[82]+NeuS | 20.90 | 0.88 | 0.08 | 2.16
DehazeNeRF | 21.61 | 0.90 | 0.06 | 1.43
Scan 83 | NeuS | 20.25 | 0.94 | 0.05 | 7.03
COLMAP+NeuS | 17.23 | 0.92 | 0.08 | 7.10
ImDehaze[18]+NeuS | 22.89 | 0.94 | 0.05 | 7.01
VidDehaze[82]+NeuS | 21.12 | 0.95 | 0.05 | 7.03
DehazeNeRF | 28.46 | 0.95 | 0.03 | 6.91
Scan 97 | NeuS | 17.32 | 0.91 | 0.07 | 2.08
COLMAP+NeuS | 16.93 | 0.77 | 0.25 | 2.96
ImDehaze[18]+NeuS | 18.18 | 0.93 | 0.07 | 2.02
VidDehaze[82]+NeuS | 18.61 | 0.93 | 0.07 | 2.05
DehazeNeRF | 24.41 | 0.92 | 0.05 | 1.89
Scan 105 | NeuS | 18.98 | 0.93 | 0.07 | 2.65
COLMAP+NeuS | 15.65 | 0.77 | 0.33 | 3.78
ImDehaze[18]+NeuS | 18.84 | 0.94 | 0.06 | 2.64
VidDehaze[82]+NeuS | 17.74 | 0.92 | 0.09 | 2.95
DehazeNeRF | 26.18 | 0.95 | 0.04 | 2.65
Scan 106 | NeuS | 14.57 | 0.84 | 0.11 | 4.24
COLMAP+NeuS | 17.03 | 0.78 | 0.22 | 2.29
ImDehaze[18]+NeuS | 15.45 | 0.86 | 0.09 | 1.34
VidDehaze[82]+NeuS | 17.20 | 0.89 | 0.11 | 1.67
DehazeNeRF | 28.41 | 0.92 | 0.07 | 1.23
Scan 110 | NeuS | 15.21 | 0.86 | 0.13 | 1.90
COLMAP+NeuS | 19.01 | 0.84 | 0.20 | 2.88
ImDehaze[18]+NeuS | 17.53 | 0.89 | 0.10 | 2.30
VidDehaze[82]+NeuS | 15.69 | 0.86 | 0.13 | 2.11
DehazeNeRF | 23.28 | 0.93 | 0.06 | 1.76
Scan 114 | NeuS | 16.88 | 0.90 | 0.07 | 0.86
COLMAP+NeuS | 15.60 | 0.82 | 0.17 | 1.19
ImDehaze[18]+NeuS | 17.39 | 0.92 | 0.06 | 0.80
VidDehaze[82]+NeuS | 19.96 | 0.93 | 0.06 | 0.87
DehazeNeRF | 22.73 | 0.91 | 0.06 | 0.84
Scan 118 | NeuS | 14.47 | 0.86 | 0.09 | 2.64
COLMAP+NeuS | 18.28 | 0.82 | 0.17 | 1.90
ImDehaze[18]+NeuS | 15.14 | 0.88 | 0.07 | 1.66
VidDehaze[82]+NeuS | 17.57 | 0.91 | 0.08 | 1.53
DehazeNeRF | 28.28 | 0.92 | 0.05 | 1.25
Scan 122 | NeuS | 15.35 | 0.90 | 0.07 | 1.23
COLMAP+NeuS | 19.83 | 0.85 | 0.17 | 1.57
ImDehaze[18]+NeuS | 15.95 | 0.92 | 0.06 | 1.35
VidDehaze[82]+NeuS | 18.05 | 0.92 | 0.08 | 1.22
DehazeNeRF | 28.55 | 0.94 | 0.06 | 0.99
Average | NeuS | 16.72 | 0.88 | 0.09 | 2.64
COLMAP+NeuS | 16.75 | 0.79 | 0.23 | 5.97
ImDehaze[18]+NeuS | 17.89 | 0.90 | 0.08 | 2.30
VidDehaze[82]+NeuS | 18.32 | 0.90 | 0.09 | 2.35
DehazeNeRF | 25.70 | 0.92 | 0.05 | 2.07

Table S.1: Quantitative evaluation for synthetic haze dataset.

##  Appendix A Data generation

###  A.1 Synthetic Data

An example of our synthesized results are shown in Fig. S.1. The scattering
coefficient is modeled using the sum of 4 scaled Gaussian blobs located inside
the spatial bounding box with a standard deviation uniformly sampled from 1.0
to 3.0; the 3-D atmospheric light is sampled from a uniform distribution in
the range $[0.7,0.9]$.

Figure S.1: An example of the synthetic data. Right: original input image.
Left: synthetic hazy input with heterogeneous scattering.

###  A.2 Real Data Collection

We use two professional haze machines to generate a dense vapor. These haze
generators adopt cast or platen-type aluminum heat exchangers, which make
evaporation of the water-based haze liquid. We capture our scenes in a
hermetic chamber. First, we captured clear reference images from different
view directions. Then, we employed 2 minutes for both haze machines and waited
for another 2 minutes until the haze was temporally stable. We then captured
hazy images from different viewpoints. An iPhone 12 was used as our camera.
The camera’s settings including exposure, iso, and focal length were the same
for all captures.

After the capturing, we merge hazy images and clear images under the same
scene and run COLMAP [59, 60] to register camera positions.

##  Appendix B Evaluation

#### Quantitative Evaluation on Synthetic Dataset

We present the complete quantitative results on DTU dataset [25] in Tab. S.1.
One can see that the proposed DehazeNeRF outperforms other baselines in most
of the scenes in terms of image quality and geometric reconstruction.

Surface Prior |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  LPIPS$\downarrow$ |  Chamfer$\downarrow$
---|---|---|---|---
None | 21.331 | 0.901 | 0.071 | 4.013
via NeuS | 26.039 | 0.921 | 0.056 | 2.863
Table S.2: The effect of NeuS surface prior evaluated on 5 test scenes.

#### Additional Ablation

Here, we present an additional ablation to demonstrate the effect of using the
NeuS surface prior. To this end, we replace the NeuS backbone with a NeRF [46]
backbone and compare the image quality and geometry reconstruction
performance. As shown in Tab. S.2, by adopting a surface prior, implemented
via the surface-based parameterization of the volume density NeuS, the model
can converge to a better solution in terms of haze-surface disambiguation
leading to significantly better image and geometry reconstruction quality.

##  Appendix C Implementation Detail

#### Network architecture

We adopt a similar network architecture as HF-NeuS [73]. It contains two MLPs
to encode surface SDF and surface color respectively. The SDF MLP consists of
8 hidden layers with hidden size of 256. A skip connection is applied from the
input to the fourth hidden layer. The color MLP contains 4 hidden layers with
size of 256. It takes the spatial location p, the view direction d, the normal
vector of SDF, and a 256-dimensional feature vector from the SDF MLP.
Positional encoding is applied to spatial location and view direction with 6
and 4 frequencies, respectively.

For the spatially variant scattering coefficient, we adopt band-limited
coordinate networks [34] using 3 hidden layers, where each layer has 32
channels and the frequency parameter set to 10.

#### Training and inference details

We train the proposed DehazeNeRF by using the Adam optimizer [30]. We set the
learning rate linearly from 0 to $5\times 10^{-4}$ by warmed up strategy in
the first 5k iterations. Then, we reduce the learning rate by the cosine decay
schedule to the minimum learning rate $2.5\times 10^{-5}$. We train our model
for 24 hours (for the ‘w/ mask’ setting) and 26 hours (for the ‘w/o mask’
setting) on a single Nvidia Tesla V100 GPU with batch size 512, respectively.
For the first 300k iterations, we set the scattering coefficients and
atmospheric light as zero and train the model based on the vanilla HF-NeuS. In
this stage, we disable the Koschmieder Consistency and dark channel losses.
After 300k iterations, the scattering coefficients and atmospheric light are
introduced and optimized with the Koschmieder Consistency and dark channel
losses. We train our network for 800k and 50k iterations for synthetic and
captured data, respectively. The scale factors $\lambda$, $\alpha$, and
$\beta$ in the loss function are set to 0.1, 5000, and 0.01 for synthetic data
and 0.1, 1.0, and 0.01 for the captured dataset, respectively. The scale
factor of mask loss is set to 0.1. For the inference stage, it takes 200
seconds to render an image in resolution of $1600\times 1200$.

##  Appendix D Deriving DehazeNeRF

###  D.1 Radiative Transfer Equation

Radiative transfer equation (RTE) [6, 65] describes the behaviour of light in
a medium that absorbs, scatters and emits radiation. Assuming, a ray
$\mathbf{r}\left(t\right)=\mathbf{o}+t\mathbf{d}$ hits a surface point at
$\mathbf{r}\left(t_{0}\right)$, the incident radiance at the near image plane
$t_{n}$ can be divided into three parts [53]:

$\displaystyle\begin{split}C(\mathbf{r},\mathbf{d})=&\underbrace{\int_{t_{n}}^{t_{0}}\epsilon\left(\mathbf{r}\left(t\right),\mathbf{d}\right)T_{\sigma_{t}}\left(t\right)dt}_{\textrm{emission}}\\\
&+\underbrace{\int_{t_{n}}^{t_{0}}c_{\textrm{s}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\sigma_{s}\left(\mathbf{r}\left(t\right)\right)T_{\sigma_{t}}\left(t\right)dt}_{\textrm{in-
scattering}}\\\
&+\underbrace{C_{e}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)T_{\sigma_{t}}\left(t_{0}\right)}_{\textrm{surface
reflection}},\end{split}$ (S.1)

where $\epsilon$ is the emission, $C_{e}$ is the outgoing radiance at the
surface intersection,
$c_{\textrm{s}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)$ is the in-
scattered light and $\sigma_{s}$ is the scattering coefficient. In particular
the transmittance here is computed from the attenuation coefficient
$\sigma_{t}$, _i.e_.,
$T_{\sigma_{t}}\left(t\right)=\exp\left(-\int_{t_{n}}^{t}\sigma_{t}(t^{\prime})dt^{\prime}\right)$,
where $\sigma_{t}=\sigma_{a}+\sigma_{s}$ including the absorption and out-
scattering effect.

Based on this equation, we then connect it to the neural radiance field (NeRF)
and hazy image formation model.

###  D.2 From RTE to NeRF

For the volume rendering equation adopted in NeRF, there are two assumptions:
(i) no scattering and (ii) only absorption and emission are considered. The
volume rendering equation can be derived by the RTE equation by dropping the
in-scattering term and combining the surface reflection term into the emission
term. It can be presented as: [42, 71, 75]

$\displaystyle\begin{split}C(\mathbf{r},\mathbf{d})=&\underbrace{\int_{t_{n}}^{t_{0}}\epsilon\left(\mathbf{r}\left(t\right),\mathbf{d}\right)T_{\sigma_{a}}\left(t\right)dt}_{\textrm{emission}}\\\
&+\underbrace{C_{e}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)T_{\sigma_{a}}\left(t_{0}\right)}_{\textrm{surface
reflection}}.\end{split}$ (S.2)

Since NeRF treats each point as an emission point, the radiance of the surface
can be considered as emission at $t_{0}$. Thus, the surface reflection in Eq.
S.2 becomes

$C_{e}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)T_{\sigma_{a}}\left(t_{0}\right)=\int_{t_{n}}^{t_{0}}\epsilon_{o}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)\delta(t-t_{0})T_{\sigma_{a}}\left(t\right)dt.$

(S.3)

Therefore Eq. S.2 can be written as:

$\displaystyle\begin{split}C(\mathbf{r},\mathbf{d})&=\int_{t_{n}}^{t_{0}}\epsilon_{\textrm{combined}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)T_{\sigma_{a}}\left(t\right)dt,\end{split}$
(S.4)

where
$\epsilon_{\textrm{combined}}=\epsilon\left(\mathbf{r}\left(t\right),\mathbf{d}\right)+\epsilon_{o}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)\delta(t-t_{0})$.

By further setting
$\epsilon_{\textrm{combined}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)=c(\mathbf{r}(t),\mathbf{d})\sigma(\mathbf{r}(t))$
and $\sigma_{a}=\sigma$, we can see Eq. S.4 is the same as famous NeRF
rendering equation below:

$\displaystyle\begin{split}{C}(\mathbf{r},\mathbf{d})=\int_{t_{n}}^{t_{f}}c(\mathbf{r}(t),\mathbf{d})\sigma(\mathbf{r}(t))T(t)\
dt.\end{split}$ (S.5)

In practice, absorption coefficient $\sigma_{a}(\mathbf{r}(t))$ describes the
probability of a photon being absorbed at position $\mathbf{r}\left(t\right)$,
while $\sigma(\mathbf{r}(t))$ (called volume density in NeRF) illustrates the
probability of a ray terminating at location $\mathbf{r}\left(t\right)$. That
is, $\sigma_{a}$ is the same meaning as $\sigma$ physically since NeRF only
considers the absorption. Moreover, generally, we can assume that when the ray
hits the object, the attenuation term (accumulative transmittance) may become
zero. That is, $T(t)\cong 0$ when $t>t_{0}$. Thus, the integral from $t_{n}$
to $t_{f}$ is approximately equal to that from $t_{n}$ to $t_{0}$.

###  D.3 From RTE to Koschmieder Model

For the conventional haze formation model, there are three assumptions[44,
47]: (i) no emission, (ii) only consider single scattering, and (iii) only
consider scattering since absorption in atmospheric particles is relatively
small. Therefore, Eq. S.1 simplifies to

$\displaystyle C(\mathbf{r},\mathbf{d})=$
$\displaystyle\underbrace{\int_{t_{n}}^{t_{0}}c_{\textrm{s}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\sigma_{s}\left(\mathbf{r}\left(t\right)\right)T_{\sigma_{s}}\left(t\right)dt}_{\textrm{in-
scattering}}$
$\displaystyle+\underbrace{C_{e}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)T_{\sigma_{s}}\left(t_{0}\right)}_{\textrm{surface
reflection}}.$ (S.6)

Assuming the scattering coefficient and atmospheric light are isotropic and
homogeneous, _i.e_.,
$\sigma_{s}\left(\mathbf{r}\left(t\right)\right)\equiv\bar{\sigma}_{s}$ and
$c_{s}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\equiv\bar{c}_{s}$,
setting $t_{n}=0$ and
$C_{e}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)=C_{\textrm{clear}}$,
we can rewrite Eq. S.6 to the Koschmieder model [24], namely

$\begin{split}C(\mathbf{r},\mathbf{d})&={\bar{c}_{\textrm{s}}\int_{0}^{t_{0}}\bar{\sigma}_{s}\underbrace{e^{-\int_{0}^{t}\bar{\sigma}_{s}dt^{{}^{\prime}}}}_{T_{\bar{\sigma}_{s}}\left(t\right)}dt}+{C_{e}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)\underbrace{e^{-\int_{0}^{t}\bar{\sigma}_{s}dt^{{}^{\prime}}}}_{T_{\bar{\sigma}_{s}}\left(t\right)}\left(t_{0}\right)}\\\
&=\bar{c}_{s}(1-\exp(-\bar{\sigma}_{s}t_{0}))+C_{\textrm{clear}}\exp(-\bar{\sigma}_{s}t_{0}).\end{split}$

(S.7)

###  D.4 3D Haze Formation in DehazeNeRF

In this part, we bring Eq. S.6 to a similar form as NeRF so that we can learn
the geometry and color through samples in the 3D space.

First, we bring back volume density $\sigma$, _i.e_., replace $\sigma_{s}$
with $\sigma_{t}=\sigma_{s}+\sigma$:

$\begin{split}C(\mathbf{r},\mathbf{d})=\int_{t_{n}}^{t_{0}}c_{\textrm{s}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\sigma_{s}\left(\mathbf{r}\left(t\right)\right)T_{\sigma_{t}}\left(t\right)dt\\\
+C_{e}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)T_{\sigma_{t}}\left(t_{0}\right),\end{split}$
(S.8)

which corresponds to Eq. (3) in the main paper. Using the same trick in Eq.
S.3, we can combine the surface reflection term into the integral and obtain
the rendering equation used in DehazeNeRF (Eq. (4) in the main paper):

$\displaystyle C(\mathbf{r},\mathbf{d})=$
$\displaystyle\int_{t_{n}}^{t_{0}}c_{\textrm{s}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\sigma_{s}\left(\mathbf{r}\left(t\right)\right)T_{\sigma_{t}}\left(t\right)dt$
$\displaystyle+\int_{t_{n}}^{t_{0}}\underbrace{\epsilon_{\textrm{surface}}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)}_{\epsilon_{o}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)\delta(t-t_{0})}T_{\sigma_{t}}\left(t\right)dt$
$\displaystyle=$
$\displaystyle{\int_{t_{n}}^{t_{0}}c_{\textrm{s}}\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\sigma_{s}\left(\mathbf{r}\left(t\right)\right)T_{\sigma_{t}}\left(t\right)dt}$
$\displaystyle+{\int_{t_{n}}^{t_{0}}\underbrace{c\left(\mathbf{r}\left(t\right),\mathbf{d}\right)\sigma\left(\mathbf{r}\left(t\right)\right)}_{\epsilon_{\textrm{surface}}\left(\mathbf{r}\left(t_{0}\right),\mathbf{d}\right)}T_{\sigma_{t}}\left(t\right)dt}.$
(S.9)

The first term (denoted as $C_{\text{haze}}$) captures the radiance caused by
haze scattering. The second term (denoted as $C_{\textrm{surface}}$) is
equivalent to NeRF for a scene with only solid surfaces.


* [50] Aaron Z. Goldberg, Ilaria Gianani, Marco Barbieri, Fabio Sciarrino, Aephraim M. Steinberg and Nicolò Spagnolo  “Multiphase estimation without a reference mode”  In _Phys. Rev. A_ 102 American Physical Society, 2020, pp. 022230  DOI: 10.1103/PhysRevA.102.022230
  * [51] Wojciech Górecki and Rafał Demkowicz-Dobrzański  “Multiparameter quantum metrology in the Heisenberg limit regime: Many-repetition scenario versus full optimization”  In _Phys. Rev. A_ 106 American Physical Society, 2022, pp. 012424  DOI: 10.1103/PhysRevA.106.012424
  * [52] Wojciech Górecki and Rafał Demkowicz-Dobrzański  “Multiple-Phase Quantum Interferometry: Real and Apparent Gains of Measuring All the Phases Simultaneously”  In _Phys. Rev. Lett._ 128 American Physical Society, 2022, pp. 040504  DOI: 10.1103/PhysRevLett.128.040504
  * [53] Wojciech Górecki, Rafał Demkowicz-Dobrzański, Howard M. Wiseman and Dominic W. Berry  “$\pi$-Corrected Heisenberg Limit”  In _Phys. Rev. Lett._ 124 American Physical Society, 2020, pp. 030501  DOI: 10.1103/PhysRevLett.124.030501
  * [54] Wojciech Górecki, Alberto Riccardi and Lorenzo Maccone  “Quantum Metrology of Noisy Spreading Channels”  In _Phys. Rev. Lett._ 129 American Physical Society, 2022, pp. 240503  DOI: 10.1103/PhysRevLett.129.240503
  * [55] Wojciech Górecki, Sisi Zhou, Liang Jiang and Rafał Demkowicz-Dobrzański  “Optimal probes and error-correction schemes in multi-parameter quantum metrology”  In _Quantum_ 4 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2020, pp. 288  DOI: 10.22331/q-2020-07-02-288
  * [56] Gregory Gundersen  “Asymptotic Normality of Maximum Likelihood Estimators” Accessed: 2023-04-12, https://gregorygundersen.com/blog/2019/11/28/asymptotic-normality-mle/, 2019
  * [57] Mădălin Guţă, Bas Janssens and Jonas Kahn  “Optimal estimation of qubit states with continuous time measurements”  In _Communications in Mathematical Physics_ 277 Springer, 2008, pp. 127–160  DOI: 10.1007/s00220-007-0357-5
  * [58] Mădălin Guţă and Jonas Kahn  “Local asymptotic normality for qubit states”  In _Phys. Rev. A_ 73 American Physical Society, 2006, pp. 052108  DOI: 10.1103/PhysRevA.73.052108
  * [59] Jeongwan Haah, Aram W. Harrow, Zhengfeng Ji, Xiaodi Wu and Nengkun Yu  “Sample-Optimal Tomography of Quantum States”  In _IEEE Transactions on Information Theory_ 63.9, 2017, pp. 5628–5641  DOI: 10.1109/TIT.2017.2719044
  * [60] Jaroslav Häjek  “Local asymptotic minimax and admissibility in estimation”  In _Theory of Statistics_ University of California Press, 1972, pp. 175–194
  * [61] Michael J.. Hall, Dominic W. Berry, Marcin Zwierz and Howard M. Wiseman  “Universality of the Heisenberg limit for estimates of random phase shifts”  In _Phys. Rev. A_ 85 American Physical Society, 2012, pp. 041802  DOI: 10.1103/PhysRevA.85.041802
  * [62] Michael J.. Hall and Howard M. Wiseman  “Does Nonlinear Metrology Offer Improved Resolution? Answers from Quantum Information Theory”  In _Phys. Rev. X_ 2 American Physical Society, 2012, pp. 041006  DOI: 10.1103/PhysRevX.2.041006
  * [63] Masahito Hayashi  “Parallel treatment of estimation of SU(2) and phase estimation”  In _Physics Letters A_ 354.3, 2006, pp. 183–189  DOI: 10.1016/j.physleta.2006.01.043
  * [64] Masahito Hayashi  “Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation”  In _Communications in mathematical physics_ 304.3 Springer, 2011, pp. 689–709  DOI: 10.1007/s00220-011-1239-4
  * [65] Masahito Hayashi  “Fourier analytic approach to quantum estimation of group action”  In _Communications in Mathematical Physics_ 347.1 Springer, 2016, pp. 3–82  DOI: 10.1007/s00220-016-2738-0
  * [66] W Heitler  “The Quantum Theory of Radiation, 3rd edn, Clarendon”  Oxford, 1954
  * [67] BL Higgins, DW Berry, SD Bartlett, MW Mitchell, HM Wiseman and GJ Pryde  “Demonstrating Heisenberg-limited unambiguous phase estimation without adaptive measurements”  In _New Journal of Physics_ 11.7 IOP Publishing, 2009, pp. 073023  DOI: 10.1088/1367-2630/11/7/073023
  * [68] Brendon L Higgins, Dominic W Berry, Stephen D Bartlett, Howard M Wiseman and Geoff J Pryde  “Entanglement-free Heisenberg-limited phase estimation”  In _Nature_ 450.7168 Nature Publishing Group, 2007, pp. 393–396  DOI: 10.1038/nature06257
  * [69] Le Bin Ho, Hideaki Hakoshima, Yuichiro Matsuzaki, Masayuki Matsuzaki and Yasushi Kondo  “Multiparameter quantum estimation under dephasing noise”  In _Phys. Rev. A_ 102 American Physical Society, 2020, pp. 022602  DOI: 10.1103/PhysRevA.102.022602
  * [70] Wassily Hoeffding  “Probability Inequalities for Sums of Bounded Random Variables”  In _Journal of the American Statistical Association_ 58.301 Taylor & Francis, 1963, pp. 13–30  DOI: 10.1080/01621459.1963.10500830
  * [71] A.. Holevo  “Probabilistic and Statistical Aspects of Quantum Theory”  North Holland, Amsterdam, 1982  DOI: 10.1007/978-88-7642-378-9
  * [72] M.. Holland and K. Burnett  “Interferometric detection of optical phase shifts at the Heisenberg limit”  In _Phys. Rev. Lett._ 71 American Physical Society, 1993, pp. 1355–1358  DOI: 10.1103/PhysRevLett.71.1355
  * [73] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki and Karol Horodecki  “Quantum entanglement”  In _Reviews of modern physics_ 81.2 APS, 2009, pp. 865  DOI: 10.1103/RevModPhys.81.865
  * [74] Matthias Hübner  “Explicit computation of the Bures distance for density matrices”  In _Physics Letters A_ 163.4, 1992, pp. 239–242  DOI: 10.1016/0375-9601(92)91004-B
  * [75] Peter C Humphreys, Marco Barbieri, Animesh Datta and Ian A Walmsley  “Quantum Enhanced Multiple Phase Estimation”  In _Phys. Rev. Lett._ 111.7 APS, 2013, pp. 070403  DOI: 10.1103/PhysRevLett.111.070403
  * [76] Hiroshi Imai and Masahito Hayashi  “Fourier analytic approach to phase estimation in quantum systems”  In _New Journal of Physics_ 11.4 IOP Publishing, 2009, pp. 043034  DOI: 10.1088/1367-2630/11/4/043034
  * [77] Yumang Jing, Matteo Fadel, Valentin Ivannikov and Tim Byrnes  “Split spin-squeezed Bose–Einstein condensates”  In _New Journal of Physics_ 21.9 IOP Publishing, 2019, pp. 093038  DOI: 10.1088/1367-2630/ab3fcf
  * [78] Tomasz Kaftal and Rafał Demkowicz-Dobrzański  “Usefulness of an enhanced Kitaev phase-estimation algorithm in quantum metrology and computation”  In _Phys. Rev. A_ 90 American Physical Society, 2014, pp. 062313  DOI: 10.1103/PhysRevA.90.062313
  * [79] Jonas Kahn  “Fast rate estimation of a unitary operation in $\mathrm{SU}(d)$”  In _Phys. Rev. A_ 75 American Physical Society, 2007, pp. 022326  DOI: 10.1103/PhysRevA.75.022326
  * [80] Jonas Kahn and Mădălin Gutţă  “Local asymptotic normality and optimal estimation for d-dimensional quantum systems”  In _Quantum stochastics and information: statistics, filtering and control_ World Scientific, 2008, pp. 300–322  DOI: 10.1142/9789812832962_0014
  * [81] J. Kaiser and R. Schafer  “On the use of the I0-sinh window for spectrum analysis”  In _IEEE Transactions on Acoustics, Speech, and Signal Processing_ 28.1, 1980, pp. 105–107  DOI: 10.1109/TASSP.1980.1163349
  * [82] Piotr Kolenderski and Rafal Demkowicz-Dobrzanski  “Optimal state for keeping reference frames aligned and the platonic solids”  In _Phys. Rev. A_ 78 American Physical Society, 2008, pp. 052333  DOI: 10.1103/PhysRevA.78.052333
  * [83] B. Kraus  “Local Unitary Equivalence of Multipartite Pure States”  In _Phys. Rev. Lett._ 104 American Physical Society, 2010, pp. 020504  DOI: 10.1103/PhysRevLett.104.020504
  * [84] H R Krishnamurthy, H S Mani and H C Verma  “Exact solution of the Schrodinger equation for a particle in a tetrahedral box”  In _Journal of Physics A: Mathematical and General_ 15.7 IOP Publishing, 1982, pp. 2131–2137  DOI: 10.1088/0305-4470/15/7/024
  * [85] Naoto Kura and Masahito Ueda  “Finite-error metrological bounds on multiparameter Hamiltonian estimation”  In _Phys. Rev. A_ 97 American Physical Society, 2018, pp. 012101  DOI: 10.1103/PhysRevA.97.012101
  * [86] Stanislaw Kurdzialek, Wojciech Gorecki, Francesco Albarelli and Rafal Demkowicz-Dobrzanski  “Using adaptiveness and causal superpositions against noise in quantum metrology”  In _arXiv preprint arXiv:2212.08106_ , 2022
  * [87] Wai-Kee Li  “A particle in an isoceles right triangle”  In _Journal of Chemical Education_ 61.12 ACS Publications, 1984, pp. 1034  DOI: 10.1021/ed061p1034
  * [88] The LIGO Scientific Collaboration LIGO  “A gravitational wave observatory operating beyond the quantum shot-noise limit”  In _Nature Physics_ 7.12 Nature Publishing Group UK London, 2011, pp. 962–965  DOI: 10.1038/nphys2083
  * [89] Jing Liu and Haidong Yuan  “Control-enhanced multiparameter quantum estimation”  In _Phys. Rev. A_ 96 American Physical Society, 2017, pp. 042114  DOI: 10.1103/PhysRevA.96.042114
  * [90] Qiushi Liu, Zihao Hu, Haidong Yuan and Yuxiang Yang  “Optimal Strategies of Quantum Metrology with a Strict Hierarchy”  In _Phys. Rev. Lett._ 130 American Physical Society, 2023, pp. 070803  DOI: 10.1103/PhysRevLett.130.070803
  * [91] A. Luis and J. Pe řina  “Optimum phase-shift estimation and the quantum description of the phase difference”  In _Phys. Rev. A_ 54 American Physical Society, 1996, pp. 4564–4570  DOI: 10.1103/PhysRevA.54.4564
  * [92] Katarzyna Macieszczak, Martin Fraas and Rafał Demkowicz-Dobrzański  “Bayesian quantum frequency estimation in presence of collective dephasing”  In _New Journal of Physics_ 16.11 IOP Publishing, 2014, pp. 113002  DOI: 10.1088/1367-2630/16/11/113002
  * [93] L. Mandelstam and Ig. Tamm  “The Uncertainty Relation Between Energy and Time in Non-relativistic Quantum Mechanics”  In _Selected Papers_ Berlin, Heidelberg: Springer Berlin Heidelberg, 1991, pp. 115–123  DOI: 10.1007/978-3-642-74626-0_8
  * [94] Norman Margolus and Lev B. Levitin  “The maximum speed of dynamical evolution” Proceedings of the Fourth Workshop on Physics and Consumption  In _Physica D: Nonlinear Phenomena_ 120.1, 1998, pp. 188–195  DOI: 10.1016/S0167-2789(98)00054-2
  * [95] Keiji Matsumoto  “A new approach to the Cramér-Rao-type bound of the pure-state model”  In _J. Phys. A.: Math. Theor._ 35.13 IOP Publishing, 2002, pp. 3111  DOI: 10.1088/0305-4470/35/13/307
  * [96] Morgan W Mitchell, Jeff S Lundeen and Aephraem M Steinberg  “Super-resolving phase measurements with a multiphoton entangled state”  In _Nature_ 429.6988 Nature Publishing Group, 2004, pp. 161–164  DOI: 10.1038/nature02493
  * [97] H Nagaoka and M Hayashi  “Asymptotic Theory Of Quantum Statistical Inference: Selected Papers”  World Scientific Singapore, 2005  DOI: 10.1142/5630
  * [98] Whitney K Newey and Daniel McFadden  “Chapter 36 large sample estimation and hypothesis testing. volume 4 of Handbook of Econometrics”  In _Elsevier_ 12, 1994, pp. 2111–2245  DOI: 10.1016/S1573-4412(05)80005-4
  * [99] Rosanna Nichols, Pietro Liuzzo-Scorpo, Paul A. Knott and Gerardo Adesso  “Multiparameter Gaussian quantum metrology”  In _Phys. Rev. A_ 98 American Physical Society, 2018, pp. 012114  DOI: 10.1103/PhysRevA.98.012114
  * [100] Artur Niezgoda, Dariusz Kajtoch, Joanna Dziekańska and Emilia Witkowska  “Optimal quantum interferometry robust to detection noise using spin-1 atomic condensates”  In _New Journal of Physics_ 21.9 IOP Publishing, 2019, pp. 093037  DOI: 10.1088/1367-2630/ab4099
  * [101] Mattteo G.. Paris  “Quantum Estimation for Quantum Technologies”  In _Int. J. Quantum Inf._ 07, 2009, pp. 125–137  DOI: 10.1142/S0219749909004839
  * [102] Luca Pezzé  “Sub-Heisenberg phase uncertainties”  In _Phys. Rev. A_ 88 American Physical Society, 2013, pp. 060101  DOI: 10.1103/PhysRevA.88.060101
  * [103] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied and Philipp Treutlein  “Quantum metrology with nonclassical states of atomic ensembles”  In _Rev. Mod. Phys._ 90 American Physical Society, 2018, pp. 035005  DOI: 10.1103/RevModPhys.90.035005
  * [104] S. Pirandola, B.. Bardhan, T. Gehring, C. Weedbrook and S. Lloyd  “Advances in photonic quantum sensing”  In _Nat. Photonics_ 12.12, 2018, pp. 724–733  DOI: 10.1038/s41566-018-0301-6
  * [105] Sammy Ragy, Marcin Jarzyna and Rafał Demkowicz-Dobrzański  “Compatibility in multiparameter quantum metrology”  In _Phys. Rev. A_ 94 American Physical Society, 2016, pp. 052108  DOI: 10.1103/PhysRevA.94.052108
  * [106] Angel Rivas and Alfredo Luis  “Sub-Heisenberg estimation of non-random phase shifts”  In _New Journal of Physics_ 14.9 IOP Publishing, 2012, pp. 093052  DOI: 10.1088/1367-2630/14/9/093052
  * [107] H.. Robertson  “An Indeterminacy Relation for Several Observables and Its Classical Interpretation”  In _Phys. Rev._ 46 American Physical Society, 1934, pp. 794–801  DOI: 10.1103/PhysRev.46.794
  * [108] R. Schnabel  “Squeezed states of light and their applications in laser interferometers”  In _Phys. Rep._ 684, 2017, pp. 1–51  DOI: 10.1016/j.physrep.2017.04.001
  * [109] Erwin Schrödinger  “Zum heisenbergschen unschärfeprinzip”  Akademie der Wissenschaften, 1930
  * [110] Hans-Jürgen Sommers and Karol Zyczkowski  “Bures volume of the set of mixed quantum states”  In _Journal of Physics A: Mathematical and General_ 36.39, 2003, pp. 10083  DOI: 10.1088/0305-4470/36/39/308
  * [111] G.S. Summy and D.T. Pegg  “Phase optimized quantum states of light”  In _Optics Communications_ 77.1, 1990, pp. 75–79  DOI: 10.1016/0030-4018(90)90464-5
  * [112] Jun Suzuki  “Nuisance parameter problem in quantum estimation theory: Tradeoff relation and qubit examples”  In _Journal of Physics A: Mathematical and Theoretical_ 53.26 IOP Publishing, 2020, pp. 264001  DOI: 10.1088/1751-8121/ab8672
  * [113] Jun Suzuki, Yuxiang Yang and Masahito Hayashi  “Quantum state estimation with nuisance parameters”  In _Journal of Physics A: Mathematical and Theoretical_ 53.45 IOP Publishing, 2020, pp. 453001  DOI: 10.1088/1751-8121/ab8b78
  * [114] IE Tamm, L Mandelstam and Ig Tamm  “The uncertainty relation between energy and time in non-relativistic quantum mechanics”  In _Selected papers_ Springer, 1991, pp. 115–123  DOI: 10.1007/978-3-642-74626-0_8
  * [115] Geza Toth and Iagoba Apellaniz  “Quantum metrology from a quantum information science perspective”  In _J. Phys. A: Math. Theor._ 47.42, 2014, pp. 424006  DOI: 10.1088/1751-8113/47/42/424006
  * [116] Mankei Tsang, Ranjith Nair and Xiao-Ming Lu  “Quantum theory of superresolution for two incoherent optical point sources”  In _Phys. Rev. X_ 6.3 APS, 2016, pp. 031033  DOI: 10.1103/PhysRevX.6.031033
  * [117] Mankei Tsang, Howard M. Wiseman and Carlton M. Caves  “Fundamental Quantum Limit to Waveform Estimation”  In _Phys. Rev. Lett._ 106 American Physical Society, 2011, pp. 090401  DOI: 10.1103/PhysRevLett.106.090401
  * [118] J W Turner  “On the quantum particle in a polyhedral box”  In _Journal of Physics A: Mathematical and General_ 17.14 IOP Publishing, 1984, pp. 2791–2797  DOI: 10.1088/0305-4470/17/14/022
  * [119] A.. der Vaart  “Asymptotic Statistics”, Cambridge Series in Statistical and Probabilistic Mathematics  Cambridge University Press, 1998  DOI: 10.1017/CBO9780511802256
  * [120] Fabian Wolf, Chunyan Shi, Jan C Heip, Manuel Gessner, Luca Pezzè, Augusto Smerzi, Marius Schulte, Klemens Hammerer and Piet O Schmidt  “Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions”  In _Nature communications_ 10.1 Nature Publishing Group, 2019, pp. 1–8  DOI: 10.1038/s41467-019-10576-4
  * [121] Yuxiang Yang, Giulio Chiribella and Masahito Hayashi  “Attaining the ultimate precision limit in quantum state estimation”  In _Communications in Mathematical Physics_ 368.1 Springer, 2019, pp. 223–293  DOI: 10.1007/s00220-019-03433-4
  * [122] Ying-Wang, Li-Yun Hu, Zhi-Ming Zhang and Chao-Ping Wei  “Simultaneously estimating two phases in three-mode nonlinear interferometer”  In _Optics Communications_ 427, 2018, pp. 250–256  DOI: https://doi.org/10.1016/j.optcom.2018.06.072
  * [123] Rozhin Yousefjani, Rosanna Nichols, Shahriar Salimi and Gerardo Adesso  “Estimating phase with a random generator: Strategies and resources in multiparameter quantum metrology”  In _Phys. Rev. A_ 95 American Physical Society, 2017, pp. 062307  DOI: 10.1103/PhysRevA.95.062307
  * [124] Haidong Yuan  “Sequential Feedback Scheme Outperforms the Parallel Scheme for Hamiltonian Parameter Estimation”  In _Phys. Rev. Lett._ 117 American Physical Society, 2016, pp. 160801  DOI: 10.1103/PhysRevLett.117.160801
  * [125] Xiaobin Zhao, Yuxiang Yang and Giulio Chiribella  “Quantum Metrology with Indefinite Causal Order”  In _Phys. Rev. Lett._ 124 American Physical Society, 2020, pp. 190503  DOI: 10.1103/PhysRevLett.124.190503
  * [126] Sisi Zhou and Liang Jiang  “An exact correspondence between the quantum Fisher information and the Bures metric”  In _arXiv preprint arXiv:1910.08473_ , 2019  DOI: 10.48550/arXiv.1910.08473
  * [127] Sisi Zhou and Liang Jiang  “Asymptotic Theory of Quantum Channel Estimation”  In _PRX Quantum_ 2 American Physical Society, 2021, pp. 010343  DOI: 10.1103/PRXQuantum.2.010343
  * [128] Marcin Zwierz, Carlos A. Pérez-Delgado and Pieter Kok  “General Optimality of the Heisenberg Limit for Quantum Metrology”  In _Phys. Rev. Lett._ 105 American Physical Society, 2010, pp. 180402  DOI: 10.1103/PhysRevLett.105.180402
  * [129] Marcin Zwierz, Carlos A. Pérez-Delgado and Pieter Kok  “Erratum: General Optimality of the Heisenberg Limit for Quantum Metrology [Phys. Rev. Lett. 105, 180402 (2010)]”  In _Phys. Rev. Lett._ 107 American Physical Society, 2011, pp. 059904  DOI: 10.1103/PhysRevLett.107.059904