# Neural Ray-Tracing:
Learning Surfaces and Reflectance for Relighting and View Synthesis

Julian Knodt1 Seung-Hwan Baek1 Felix Heide1,2
1Princeton University 2Algolux

###### Abstract

Recent neural rendering methods have demonstrated accurate view interpolation
by predicting volumetric density and color with a neural network. Although
such volumetric representations can be supervised on static and dynamic
scenes, existing methods implicitly bake the complete scene light transport
into a single neural network for a given scene, including surface modeling,
bidirectional scattering distribution functions, and indirect lighting
effects. In contrast to traditional rendering pipelines, this prohibits
editing surface reflectance, illumination, or composing other objects in the
scene.

In this work, we explicitly model the light transport between scene surfaces
and we rely on traditional integration schemes and the rendering equation to
reconstruct a scene. The proposed method allows reflectance recovery with
unknown light conditions and classic light transports such as pathtracing. By
learning decomposed transport with surface representations established in
conventional rendering methods, the method naturally facilitates editing
shape, reflectance, lighting and scene composition. The method outperforms
existing neural rendering methods for relighting under known lighting
conditions, and produces realistic reconstructions for relit and edited
scenes. We validate the proposed approach for scene editing, relighting and
reflectance estimation learned from synthetic and captured views on existing
datasets.

Figure 1:  We demonstrate the ability of our methodology to recover surfaces
from a set of RGB images. The first row is the recovered surface with learned
BSDFs and learned lighting, and the second row are the surface normals. We
show our approach on the NeRF [18] datasets, and the DTU [9] dataset. Instead
of using a volumetric representation like NeRF, we use a surface-based
representation with learned BSDFs in order to reconstruct scenes. We use a
traditional ray-tracing and integration approach, with an entirely
differentiable pipeline to find a solution to the inverse-rendering equation.

##  1 Introduction

View synthesis and scene reconstruction from a series of images is a
fundamental problem in computer vision and graphics. Recent advances in neural
scene rendering [5, 18, 32, 17] have achieved photo-realistic view synthesis
results by encoding the entire scene’s light transport into a learned
volumetric representation. In contrast, traditional rendering approaches
physically model light transport that is decomposed into illumination,
reflectance, and multi-path scene inter-reflections. However, while
physically-based differentiable rendering methods [19, 33, 19] allow for fine-
grained lighting calculations, these rendering methods have been restricted to
far more constrained settings than neural rendering approaches, solving for a
small set of parameters with known geometries and lighting. In this work, we
present a method that achieves high expressivity optimization with physically-
based transport, separating the learning of the scattering functions, surface,
and lighting, while providing the model capacity of volumetric
representations.

Previous works on implicit neural radiance and reflectance representations
[18, 5, 12, 17, 29, 35] do not represent scene components individually, but
rely directly on a continuous volumetric representation of scene radiance,
allowing for high reconstruction accuracy when interpolating views from a few
RGB images. Because of the implicit radiance representations that these works
rely on, light transport, surface representations, and scattering interactions
are entangled. This prevents any one component from being changed or modified
without fundamentally altering the others, prohibiting relighting and texture
modification. Recently, attempts to incorporate these textures and relighting
into existing neural pipelines have been proposed [1, 2, 28]. While these
approaches allow for relighting, they still rely on volumetric representations
that prohibit fine-grained, physically-based decomposition of transport
offered by conventional rendering approaches. Instead of further augmenting
reflectance fields, we move in an alternative direction, from a physically-
based rendering framework, adding neural networks into the middle of the
rendering pipeline to better approximate textures and lighting, providing an
understandable and learnable reconstruction. Through this ease of
understanding, we are able to modify our representation to handle relighting,
retexturing, and reshaping. As an added benefit, our work is more easily
incorporated into existing pipelines as compared to many neural models.

We present an approach for scene reconstruction with signed-distance fields
for surface representation, spatially-varying high-frequency scattering
functions, and parametrized light models, rendered using physically-based
integration and sphere-marching during evaluation, only requiring extra
raymarching for training. This partitions the optimization space into the
discrete problem of surface and normal estimation, and texture and lighting
estimation, allowing for interchanging of different components. We incorporate
multiple multilayer perceptrons (MLP) inside of the rendering pipeline to
mimic the universality demonstrated in volumetric neural representations [18,
1, 32], but in a principled fashion so as to maintain the understandability
and efficiency of prior pipelines. We validate the proposed method using
synthetic renderings and experimental data, where we outperform NeRV [28] and
other approaches on their point-light dataset. Our code is available at
https://github.com/princeton-computational-imaging/neural_raytracing.

Specifically, we make the following contributions in this work:

  * •

An end-to-end surface-based forward and inverse rendering framework for scene
reconstruction.

  * •

We introduce a surface-based scene representation for shape and reflectance,
that allows for multi-bounce transport.

  * •

We demonstrate relighting and highly-editable learned representations,
allowing for scene compositions, reflectance and surface editing.

  * •

Validate the proposed method using synthetic data and experimental data,
outperforming existing neural rendering methods that learn reflectance.

We note our method is not limited to RGB estimation, but can be used to
directly estimate other surface properties, such as normals, as our method is
not bound to the integration method.

##  2 Related Work

###  2.1 Neural Scene Representations

#### Monolithic Representations.

With the advent of deep learning and neural networks, a growing body of work
has explored methods for scene representations with convolutional neural
networks [14, 21, 27] and implicit neural networks [16, 22, 18]. Notably,
modeling the scene radiance with a neural implicit function, commonly in the
form of a MLP, has proven to be effective, as demonstrated in NeRF [18]. MLP-
based scene modeling integrates light radiance in a volumetric structure and
optimizes the volumetric radiance function as a MLP that accurately reproduces
a set of reference images, demonstrating learned view-dependant effects. While
previous works are effective at novel view synthesis, most of them entangle
reflectance, shape, and lighting in a single radiance volume, limiting its
usage for physically-accurate scene decomposition [18, 32]. In contrast, we
learn physically-based scene representations of shapes, reflectances, and
lighting. This enables high-quality, efficient rendering with
interpretability, and it allows for the same editability that conventional
rendering pipelines offer.

#### Decomposed Neural Scene Representations.

With the goal of extending radiance to reflectance, recent works have also
explored decomposed modelling of reflectance [1, 28] and volumetric density to
enable editable materials and lighting, similar to our approach. However,
these approaches are restricted by the nature of volumetric rendering which is
neither interpretable or editable, and makes multi-bounce rendering
impractical, requiring 128 TPUs for training [28].

###  2.2 Differentiable Rendering with Ray-Tracing

Ray-tracing has been extensively studied with the goal of reproducing
photorealistic imagery [23] using physically-based techniques. Inverting the
rendering function gives us a framework known as inverse rendering, which is
used to estimate scene compositions and parameters for a given set of input
images. Recently, differentiable rendering has gained significant interest
through frameworks implementing the forward rendering operations in a
differentiable manner, so that backpropagation can be applied to optimize over
some loss over the whole rendering pipeline [20, 19, 34]. While these efforts
demonstrate promising results, this inversion faces great challenges. One of
the central problems is the conventional representation of shape and
reflectance. Shapes are commonly represented as a mesh in existing rendering
approaches, imposing non-differentiable geometric discontinuity. While
different approaches have been proposed to tackle this problem [7], it is
still an open challenge to effectively and efficiently invert transport using
existing mesh representations. Similarly, analytic reflectance models do not
facilitate learning via gradient-based optimization, often due to
discontinuities over lighting or low-likelihood of sampling specular
highlights.

Figure 2:  Overview of the proposed neural ray-tracing pipeline. For all
integration schemes, we first take a cropped sample from UV space and
transform it to camera space to compute rays from the camera center. Then, we
perform sphere-marching to compute locations and normals of intersections.
Given these positions, we perform light sampling in order to compute an
incoming light direction and use the BSDF to compute an RGB value. If
pathtracing, we sample a direction given the BSDF to compute a light-bounce
ray and recurse for some given depth.

##  3 Inverting the Rendering Equation

The rendering equation has served as a firm ground for photo-realistic
rendering that includes rich light transport details encompassing both direct
and indirect lighting [11]. It describes the radiance coming from a 3D
position $\mathbf{x}$ in the direction $\omega_{o}$ as

$\displaystyle L(\mathbf{x},\omega_{o})=$ $\displaystyle
L_{e}(\mathbf{x},\omega_{o})+$
$\displaystyle\int_{\Omega}f(\mathbf{x},\omega_{i},\omega_{o})L_{i}(\mathbf{x},\omega_{i})(\omega_{i}\cdot\mathbf{n})d\omega_{i},$
(1)

where $L_{e}(\mathbf{x},\omega_{o})$ and $L_{i}(\mathbf{x},\omega_{i})$ are
the emitted and incident radiance for solid angle $\omega_{o}$ and
$\omega_{i}$ respectively. Contributions from other scene points are
integrated over the hemisphere $\Omega$. $f$ is the BRDF and $\mathbf{n}$ is
the surface normals of the scene geometry [23].

Solving this rendering equation [23] requires two key ingredients: (1)
representations of geometry, BSDF and lighting, and (2) an efficient
integration scheme. In forward rendering, this is commonly solved with meshes,
analytic reflectance and lighting models for the representations, and the
integration problem is solved by pathtracing with importance sampling. Ray-
tracing is commonly used for physically-based rendering, by directly modelling
light bounces we are able to reconstruct realistic lighting conditions. While
practical for forward rendering, this is problematic for inverse rendering as
some operations are non-differentiable and these models may not be able to
accurately model unknown conditions. For example, surfaces not directly
illuminated are not able to be differentiated with respect to their spectrum
in a direct rendering framework, and one-sided BSDFs with light hitting from
the opposite side are also not differentiable.

The proposed neural ray-tracing method aims to attain the rich light
simulation properties of forward ray-tracing while resolving the non-
differentiable nature of conventional representations by using fully-
differentiable neural representations of surfaces and reflectances, while
maintaining efficient light and surface intersection algorithms from
traditional rendering. We then propose a practical method of solving the
rendering equation following an analysis-by-synthesis principle using the
proposed neural representations.

##  4 Neural Ray-Tracing

We present a ray-tracing framework for rendering and optimizing them based on
analysis-by-synthesis principle, and introduce a forward rendering function
$\mathrm{Render}()$ from which the inverse rendering function
$\mathrm{Render}^{-1}()$ is derived. Both functions are differentiable thanks
to the neural shape and reflectance representations as described in later
sections, enabling us to learn scene properties from photographic
observations.

###  4.1 Forward Rendering

As in conventional pathtracing [23], we shoot a ray toward a camera pixel from
the camera center. We then identify a surface point where the ray intersects
with. With the shape as a neural SDF, we can efficiently implement this with
sphere-tracing [6], see Figure 3. Once the surface point is located, we sample
direct paths to light sources and multi-bounce paths using random sampling
with the neural BSDF representations. By iterating with this intersection
test, light and BSDF sampling we render an image captured by a camera,
constituting our rendering function

$I=\mathrm{Render}(\mathrm{SDF},f,L,C),$ (2)

where $I$ is the rendered image, $L$ is the lighting, and $C$ is the camera,
specified by its intrinsic and extrinsic parameters.

Lighting. When the lighting for a dataset is known, we model the lighting
explicitly with known models, as in traditional ray-tracing. Otherwise, we
assume static lighting conditions across images and model illumination
directly as a light field with a MLP. This lighting MLP takes a spatial
position, $\mathbf{x}$, and outputs the direction and intensity of incident
light with primal contribution to that spatial position.

###  4.2 Inverse Rendering

Inverse rendering aims to invert the rendering function,
$\mathrm{Render}^{-1}()$. In forward rendering ray-tracing, this involves
evaluating numerous integrations and sampling with careful consideration of
non-differentiable operations such as those meshes and reflectance. A key
component for our work is that the forward rendering and neural scene
representation are fully differentiable, and therefore they can be optimized
by backpropagation. Based on an analysis-by-synthesis principle, our
optimization function is defined as an iterative framework to find an optimal
solutions for the shape $\mathrm{SDF}$, reflectance $f$, and lighting $C$
parameters that reproduce the input images. Specifically, the inverse function
$\mathrm{Render}^{-1}()$ takes multiple images
$\mathbf{I}=\\{I_{1},...I_{K}\\}$ as inputs captured at camera positions
$\mathbf{C}=\\{C_{1},...C_{K}\\}$ and lighting configurations
$\mathbf{L}=\\{L_{1},...L_{K}\\}$. $K$ is the number of images. The inverse
rendering function is then written as the optimization

$\displaystyle\mathcal{L}_{p}(I_{\text{ref}},I)$
$\displaystyle=\frac{1}{3}(\lVert I_{\text{ref}}-I\rVert_{1}+\lVert
I_{\text{ref}}-I\rVert_{2}+\lVert I_{\text{ref}}-I\rVert_{2}^{\frac{1}{2}})$
$\displaystyle\mathcal{L}_{s}(M_{\text{ref}},M)=M_{\text{ref}}$
$\displaystyle\log(M)+(1-M_{\text{ref}})\log(1-M)$
$\displaystyle=\operatorname*{arg\,min}_{\mathrm{SDF},f}\sum_{i=1}^{K}$
$\displaystyle\\{\mathcal{L}_{p}\left(I_{i},\mathrm{Render}\left(\mathrm{SDF},f,L_{i},C_{i}\right)\right)$
$\displaystyle+\mathcal{L}_{s}\left(M_{i},\mathrm{Silhouette}\left(\mathrm{SDF},C_{i}\right)\right)\\},$
(3)

where $\mathcal{L}_{p},\mathcal{L}_{s}$ are the photometric and geometric loss
respectively. Specifically, the photometric loss we use is the sum of L1, L2,
and RMSE losses between the input images $I$ and the rendered images
$\mathrm{Render}\left(\mathrm{SDF},f,L_{i},C_{i}\right)$. The geometric loss
$\mathcal{L}_{s}$ is the binary cross-entropy loss between the ground-truth
silhouette and the estimated shape silhouettes [32], where the silhouette map
is defined as $\delta_{\text{surface visible}}$. To optimize over this
silhouette map, we ray-march from each camera pixel through a scene and
compute $\min_{t}\sigma(\mathrm{SDF}(r_{t}))$, the minimum SDF value along the
ray. We use the AdamW optimizer [15] to optimize our parameters with respect
to Equation (4.2). Our specific parameter set and learning rate are specified
in the Supplemental Material. If the lighting is unknown, its MLP
representation is also optimized in Equation (4.2).

##  5 Neural Shape and Reflectance Representations

We introduce our neural representations of shape and reflectance which are
necessary for our differentiable scene representation. Recent progress on
neural representations have fueled high-quality view synthesis [18] and
reflectance acquisition [28, 1]. In this work, we devise novel neural shape
and reflectance models suitable to be incorporated in ray-tracing where
evaluation, modelling, intersection computation, and training efficiency are
critical.

###  5.1 Two-level Shape Modeling

Recent methods use signed distance functions (SDF) [3] extensively as an
implicit shape representation that can be differentiated for inverse rendering
[30, 21]. SDFs takes a position in space, $\mathbf{x}$, and return the signed
distance to the closest surface. The surface can be retrieved from a SDF as
the zero level-set of the function. In order to learn a SDF, recent works show
promising results of using a MLP [30, 21]. However, it is computationally
expensive and unstable to use this MLP-based SDF representation in the ray-
tracing pipeline. Therefore, we devise a two-level SDF representation
consisting of spherical geometric sketches and detail refinement with a MLP as
shown in Figure 3.

Figure 3:  Our surface model: sphere-marching is used for computing surface
intersections, and ray-marching is used for learning the surface
representation. In order to inherit the flexibility of classical surface
models and the universality of neural networks, our model is the sum of a MLP
and a set of transformed spheres.  Figure 4:  We visualize the learned surface
through its surface normals. The Armadillo scene (left) has known point-source
lighting condition while the lighting is unknown for the Rabbit scene (right).
Scene data is from NeRV [28] and DTU [9, 32] respectively. We note that the
top of the bunny is intentionally cut off in the sampled view.

Geometric Sketches. We use spheres with parametrized radii and positions to
describe low-frequency components of a true SDF. This is inspired by SDF based
shape modelling as can be seen on Shadertoy [10]. The spheres are described as
center and radius values, $\mathbf{c}\in\mathbb{R}^{N\times 3}$ and
$\mathbf{r}\in\mathbb{R}^{N\times 1}$, where $N$ is the number of spheres set
as either $2^{6},2^{7}$ in our experiments, depending on the complexity of the
object. We blend the basis sphere functions with smooth-min interpolation,
resulting in the low-frequency SDF, $\mathrm{SDF}_{\mathrm{sketch}}$, as

$\displaystyle\mathrm{SDF}_{\mathrm{sketch}}(\mathbf{x})=-\frac{1}{k}\ln\sum_{i=1}^{N}e^{-ks(\mathbf{A}_{i}\mathbf{x},\mathbf{c}_{i},\mathbf{r}_{i})},$
$\displaystyle
s(\mathbf{x},\mathbf{c},r)=\lVert\mathbf{x}-\mathbf{c}\rVert_{2}-r,$ (4)

where $s(\mathbf{x},\mathbf{c},r)$ measures the signed distance between a
point $\mathbf{x}$ from a sphere surface and $k$ is a smoothing parameter.
Inspired by kernel regression, we define this SDF in kernel space for each
sphere by a general transformation $\mathbf{A}\in{\mathbb{R}^{3\times 3}}$.
This improves learning low frequency shapes, by adding more degrees of freedom
to approximate. The benefit of our geometric sketch with spheres is their
small number of learnable parameters ($\mathbf{A}$, $\mathbf{x}$,
$\mathbf{c}$), facilitating efficient learning, crucial for our ray-tracing
framework. This approach also guarantees the approximate correctness of our
surface representation as a SDF, because it is known to be correct. However,
high-frequency geometric detail cannot be effectively learned solely from this
approach.

Geometric Refinement. To compensate for lost details in the sketch, we add
geometric-refinement stage with an additional MLP. Specifically, we implement
the geometric SDF residual using a MLP with softplus activation. Combining
both geometric sketch and MLP-based refinement leads to an efficient and
accurate SDF representation as

$\mathrm{SDF}(\mathbf{x})=\mathrm{SDF}_{\mathrm{sketch}}(\mathbf{x})+\mathrm{MLP}(\mathbf{x}),$
(5)

This SDF representation combines the power of explicit and implicit shape
representations to enable efficient and effective learning for the
differentiable ray-tracing rendering.

###  5.2 Neural Basis Reflectance Model

Reflectance modelling has been extensively studied for decades and its history
goes back to the seminal analytic models of the Phong [24] model and the GGX
model [31]. Recently, researchers have attempted to model BSDFs with neural
networks for analysis [8] and inverse rendering [4]. However, this approach
faces great challenges because of the complex spatial distribution of BSDFs
over a limited set of views where per-point BSDF modelling becomes highly
inefficient. To overcome this challenge in our ray-tracing framework, we
combine recent MLP-based neural modeling and the concept of basis BSDFs and
weight maps [13]. Representing both basis BSDFs and weight maps with MLPs
allow us to model the spatially-varying BSDF of a scene in an efficient form
for our ray-tracing framework, see Figure 5 for an overview.

Figure 5:  Illustration of our BSDF model. We take the dot product of a
spatially-varying set of N weights with the spectrum output from a set of N
component BSDFs.  Figure 6:  Our learned basis BSDF representations
efficiently exploits spatial coherence in a scene, grouping surfaces with
similar spectrums for efficient gathering of BSDF samples. We show two primal
bases BSDFs and weights corresponding to the specular Armadillo and diffuse
ground as well as their reflection.

To represent each basis BSDF $b_{i}$, we use an MLP that takes incident and
exitant angles with respect to the surface normals and outputs trichromatic
reflectance as

$\displaystyle b_{i}(\phi_{d},\theta_{h},\theta_{d})$
$\displaystyle=\mathrm{MLP}(\phi_{d},\cos{\theta_{h}},\cos{\theta_{d}})$
$\displaystyle=r_{i},g_{i},b_{i},$ (6)

where $\phi_{d}$, $\theta_{h}$, $\theta_{d}$ are the Rusinkiwewicz angles of
azimuth, zenithal halfway angle, and incident halfway angle [26]. To avoid
numerical instability and encode periodicity, we parameterize the MLP inputs
by taking the cosine values for $\phi_{d},\theta_{h},\theta_{d}$. Given $M$
basis BSDFs, we define their spatial distribution as a MLP-based weight
function

$w(\mathbf{x})=\mathrm{softmax}\left(\mathrm{MLP}(\mathbf{x})\right)\in\mathbb{R}^{M\times
1},$ (7)

where $\mathrm{softmax}$ is the softmax function for normalization. Note that
each basis BSDF is defined by one respective MLP, the weight function over all
bases is learned by a separate, single MLP to exploit spatial correlation of
natural images. Combining the basis BSDFs and their corresponding weights, we
obtain the complete BSDF representation as a linear combination

$f(\mathbf{x},\phi_{d},\theta_{h},\theta_{d})=\sum_{i=1}^{M}w_{i}(\mathbf{x})b_{i}(\phi_{d},\theta_{h},\theta_{d}),$
(8)

where $w_{i}$ is the weight of the i-th BSDF. We are also feel free to add a
classical model instead of a neural representation.

Figure 7:  Our untangled representations of shape and reflectance allows us to
perform effective relighting. By adding in learned occlusion, we are able to
simulate soft-shadows cheaply, allowing us to learn shaded regions with cheap
single-order bounces.

##  6 Assessment

We assess the proposed model by analyzing the light transport decompositions
that our approach learns and applications of it, including relighting and the
editability of the entire scene rendering pipeline. We refer the reader to the
Supplemental Material for additional validation experiments that are not
covered here due to limited space.

###  6.1 Neural Scene Representation

#### Geometry.

We represent the shape as an implicit SDF with a sphere basis and a residual
MLP, training on a set of RGB images with known segmentation masks, and
corresponding camera positions. This allows for effective reconstruction of
smooth shapes in a scene as shown in Figure 4. It is worth noting that we
estimate accurate geometry under both known and unknown lighting conditions,
since our model for lighting and surfaces are fully distinct.

#### Reflectance.

Using the annotated RGB images, we reconstruct $K=8$ basis BSDFs in the scene,
as well as their weights at every point on the surface. We then qualitatively
examine them to see if they appear physically plausible. Figure 6 shows that
our method accurately reconstructs two components, which are approximately the
specular and the diffuse BSDFs of the object.

###  6.2 Applications

#### Relighting.

With the estimated geometry and reflectance, we are fully equipped to tackle
the applications of relighting and view synthesis. Figure 7 shows our rendered
images with three different point light source locations at a fixed viewpoint.
Note that both hard shadows and soft shadows are accurately predicted,
demonstrating the accuracy of our representations for relighting.

#### View Synthesis.

We also demonstrate view synthesis in Figure 8 by changing the viewpoints of
the camera with fixed illumination, by varying it according to the test-set
provided by NeRV [28]. Reflectances of the objects are consistent across the
views and the shadow is faithfully reproduced because of our physically-based
transport model.

Figure 8:  We perform free-viewpoint view synthesis leveraging our neural
shape and reflectance representations. Our learned BSDFs and shadows from the
learned surface are consistent across different viewpoints and maintain
highlights.

#### Scene Editing.

In order to demonstrate the editability of scenes and the rendering pipelines
following our method, we explore performing various transformations of the
BSDF, lighting and the surface, as shown in Figure 10. We explore these
transformations on two datasets from NeRF [18], and one dataset from DTU [9].
Specifically, we showcase BSDF editing and geometry deformation, which
consists of SDF transformations, and remapping portions of space to have a
different composition of BSDFs, as can be seen in Fig. 10. We find
implementing these changes to be close to that of a traditional forward
renderer, and, since it is written in Python, is amenable to fast
modifications, while being significantly faster than modifying a traditional
renderer.

Reference |  NeRV |  Ours + Occ |  Ours + HS
---|---|---|---

Figure 9:  We compare our method on NeRV [28]’s public point-light datasets.
Our method provides accurate RGB reproduction due to the disentanglement
between the learned surface and BSDF. Adding hard shadows to our method,
denoted as Ours+HS, enables geometry-based enhanced image contrast. Ours+Occ
is a learned soft-shadow approximation, used during training to allow for
differentiable approximations in regions of shadow, but can also be used at
test time.

Original |  Changed BSDF |  Twisted & Phong |  Shifted Surface
---|---|---|---

Original |  Banded BSDF |  Flattened |  Subsection
---|---|---|---

Original |  Checkerboard BSDF |  Surface Normals |  Light Direction
---|---|---|---

Figure 10:  We demonstrate the editability of our methodology on two datasets
from NeRF [18] and one from DTU [9]. To modify the surface representations, we
perform SDF operations including intersection, subtraction, twisting, and
translation of subsets of the surface. To modify the BSDF representation, we
subsitute a diffuse red/orange BSDF over a subset of the image, such as adding
bands, or changing the BSDF for all points with a positive z value. We also
demonstrate the understandibility of our model, by showing surface normals and
incoming light directions along the surface, by modifying integrators. All
these changes took less than 15 LOC, and can be composited together to produce
interesting effects. The learned BSDFs with unknown lighting may not be
amenable to all transformations, due to unseen view positions far from the
observed.

###  6.3 Quantitative and Qualitative Validation

We compare our method to recent neural reflectance methods including NeRV
[28], Bi et al. [1], and a NeRF variant augmented with an illumination
representation. Specifically, following [28], we compare against NeRF+LE, a
NeRF variant, which combines the viewing direction with a light-embedding
created with PointNet [25], to approximate varying light conditions. Figure 9
shows that our method outperforms existing methods both quantitatively and
qualitatively, facilitated by our explicit surface model and smooth spatially-
varying BSDF. By ensuring that only the BSDF captures relighting, the surface
can be regularized to be smooth, whereas the other methods suffer from
entangled surface and reflectance representations. Unfortunately, public code
is not available for many recent methods, and, as such, it is necessary to
report results on two public datasets adopted from [28]. Is is worth noting
that Bi et al. [1] differs from our approach in that it requires a colocated
point light source to learn. For training, NeRV [28] requires significant
computing resources, e.g. 128 TPUs in their work whereas our method is memory-
efficient training on a single 16GB GPU and can be generalized to various
lighting configurations.

| Armadillo | Hotdog
---|---|---
| PSNR↑ | SSIM↑ | PSNR↑ | MS-SSIM↑
NeRF+LE | 20.35 | 0.883 | 19.96 | 0.868
Bi et al. | 22.35 | 0.894 | 23.74 | 0.862
NeRV | 22.14 | 0.897 | 23.93 | 0.863
Ours+Occ | 24.27 | 0.931 | 25.53 | 0.860
Ours+HS | 24.17 | 0.926 | 25.53 | 0.868
Table 1:  We compare results against NeRV [28] and other methodologies on
NeRV’s single point-light dataset, with additional results in the Supplemental
Document.

As the previous works of Bi et al. [1] and NeRV [28] do not yet provide public
implementations, we compare two NeRF variants with illumination information on
a new rendered datasets which has threes scenes including Plastic Bunny,
Diffuse Teapot, and Plastic Buddha. Since we only train on a singular point
light-source of fixed intensity and color, we compare against NeRF + PL, which
directly embeds the 3-dimensional position of the point light into NeRF. In
addition, we compare to NeRF + Env, which encodes the illumination along a
fixed set of points along the scene’s bounding sphere. Table 2 shows that our
method outperforms the two variants.

Method |  Plastic
Bunny |  Diffuse
Teapot |  Plastic
Buddha
---|---|---|---
| PSNR↑
NeRF+PL |  12.92 |  11.82 |  12.11
NeRF+Env |  12.59 |  11.60 |  13.12
Ours+HS |  16.58 |  17.96 |  15.53
| SSIM↑
NeRF+PL |  0.738 |  0.708 |  0.670
NeRF+Env |  0.743 |  0.700 |  0.652
Ours+HS |  0.746 |  0.777 |  0.736
Table 2:  We compare our method on a single-point relighting task to two NeRF
variants, NeRF + PL and NeRF + Env. We note that since there are no inter-
reflections in the scene, there are no soft-shadows and our hard-shadow model
performs significantly better. See the text for a description of these
compared methods, and the Supplement for additional benchmarks on these
datasets.

###  6.4 Multi-Bounce Pathtracing

We also perform multi-bounce pathtracing, to validate that it is possible with
our methodology. We find that it is prohibitive for learning low-noise
results, and that the cost incurred by random sampling outweights the benefits
of having a soft-shadow. We refer to the Supplemental Material for more
details.

##  7 Discussion

Our explicit surface modelling allows us to directly model the light
transport, and thus efficiently compute light-samples and surface
intersections, permitting liberal transformations, approximations of soft-
shadows, and quickly learn representations. Due to the prohibitive cost and
noise introduced through pathtracing, we explore an alternative to pathtracing
for soft-shadow based illumination, and find it to be effective.

Comparing to prior work on view-synthesis and scene reconstruction, our work
matches the performance of existing methods, while providing higher
editability and physical interpretation thanks to our decomposed scene
representations. Exploring more compact and easily-trainable decomposed scene
representations is one extension which could be explored in future work.

##  8 Conclusion

Our work bridges the gap between recent MLP-based approaches for neural
rendering which rely on a more complicated pipeline and traditional
physically-based rendering approaches. We show that it is possible to combine
concepts from conventional, physically-based, inverse-rendering methods with
high-fidelity scene representations via neural networks. We hope this work
motivates follow-up future work into physically-based rendering with embedded
neural components where current theory cannot fill in the gaps. We further
envision future neural rendering methods to depart from the volumetric models
that is central to a large body of existing methods in neural rendering,
making today’s methods inefficient, uninterpretable, and challenging to
integrate with the rich set of tools and methods developed in traditional
rendering approaches – limitations that the proposed method makes steps
towards lifting.

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# Magnetic and magnetoresistive behavior of the ferromagnetic heavy fermion
YbNi_2

O. Olicón  Instituto de Investigaciones en Materiales, Universidad Nacional
Autónoma de México, Ciudad de México, 04510, México  R. Escamilla  Instituto
de Investigaciones en Materiales, Universidad Nacional Autónoma de México,
Ciudad de México, 04510, México  A. Conde-Gallardo  Departamento de Física,
Centro de Investigación y de Estudios Avanzados del Instituto Politécnico
Nacional, A.P. 14740, C.P. 07300, Ciudad de México, México  F. Morales
Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de
México, Ciudad de México, 04510, México

###### Abstract

We present a study on the magnetic susceptibility $\chi(T)$ and electrical
resistance, as a function of temperature and magnetic field $R(T,H)$, of the
ferromagnetic heavy fermion YbNi_2. The X-ray diffraction analysis shows that
the synthesized polycrystalline samples crystallizes in the cubic Laves phase
structure C15, with a spatial group $Fd\overline{3}m$. The magnetic
measurements indicate a ferromagnetic behavior with transition temperature at
9 K. The electrical resistance is metallic-like at high temperatures and no
signature of Kondo effect was observed. In the ferromagnetic state, the
electrical resistance can be justified by electron-magnon scattering
considering the existence of an energy gap in the magnonic spectrum. The
energy gap was determined for various applied magnetic fields.
Magnetoresistance as a function of applied magnetic field, subtracted from the
$R(T,H)$ curves at several temperatures, is negative from 2 K until about 40 K
for all applied magnetic fields. The negative magnetoresistance originates
from the suppression of magnetic disorder by the magnetic field.

Heavy fermion and YbNi_2 alloy and Ferromagnetism and Magnetoresistance

##  I Introduction

The Yb and Ce-based alloys and compounds have been a subject of interest due
to its physical properties like heavy fermion behavior, mixed valence state
and Kondo lattice behavior Stewart (2001); Wirth and Steglich (2016); Stewart
(1984); Bauer (1991); Varma (1976). These properties are associated to the
hybridization between conduction-electron band and $f$-electron band. The
interaction is mediated via the polarization of the conduction electrons,
which is known as RKKY (Ruderman-Kittel-Kasuya-Yosida) interaction Ruderman
and Kittel (1954); Yosida (1957). In the last years the study of quantum
criticality in heavy fermion (HF) systems have constituted a subject of
interest in the condensed matter physics, due to the phenomena related to
magnetic critical points at low temperatures, where quantum fluctuations
compete with classical thermal fluctuations Gegenwart  _et al._ (2008); Si and
Steglich (2010). Examples of such phenomena are unconventional
superconductivity and non-Fermi liquid (NFL) behavior. Usually, the search for
the existence of a quantum critical point (QCP) is through the modification of
the system using a non-thermal parameter, such as pressure, magnetic field or
chemical doping.

The ferromagnetic (FM) order in HF systems is rare compared with the
antiferromagnetic (AFM) order or superconductivity. In the Laves phases the
$\rm{RNi_{2}}$ alloys, where R is a rare-earth element, the magnetic
properties are associated to the rare-earth element. It was proposed that in
these alloys the $d$-shell of Ni atoms is full, $3d^{10}$ configuration
Skrabek and Wallace (1963); Wallace  _et al._ (1977), consequently Ni does not
carry a magnetic moment and does not contribute to the magnetic properties.
The exchange interactions responsible of magnetism must take place between the
$4f$ electron magnetic moments of the rare-earth element mediated by the
conduction electrons.

In particular, the $\rm{YbNi_{2}}$ alloy has been classified as a heavy
fermion because the Sommerfeld coefficient determined at low temperatures is
$\gamma=573$ mJ/mol K Rojas  _et al._ (2012). At low temperatures YbNi_2 is
ferromagnetic below a transition temperature $T_{C}$=10.5 K Rojas  _et al._
(2012). Calculations of the electronic density of states show that the
electrons of the conduction band are hybridized with the Yb $4f$-electrons,
this hybridization confirms the heavy fermion character of YbNi_2 Yan and Wu
(2014). However, there is a lack of studies on the electronic transport
properties. The present work studies the magnetic field effects on the
electrical resistance of the YbNi_2 alloy. We found that in the ferromagnetic
state the temperature dependence of the electrical resistance is produced by
the scattering of the conduction electrons by magnons. Furthermore, an energy
gap in the magnonic spectrum was inferred.

##  II Experimental details

Polycrystalline samples of YbNi_2, were synthesized weighting stoichiometry
quantities of high purity Yb (99.9% Sigma-Aldrich) and Ni (99.99% Sigma-
Aldrich) powders. An excess of Yb 10% wt was added to compensate the Yb loss
because its low melting point. The powders were mixed and pelletized into a
plastic bag with Ar atmosphere. Afterward, the pellet was melted in high-
purity Ar atmosphere in an arc furnace. The produced button was turned and
remelted several times in order to obtain an homogeneous sample. The
structural characterization was performed with the X-ray powder method using a
Bruker diffractometer model D8 Advanced, with a Cu-$\rm{K_{\alpha}}$
radiation. The diffraction patterns were collected at room temperature and
over a $2\theta$ range of $20-120^{\circ}$ with a step size $0.02^{\circ}$.
The Rietveld refinement of the X-ray pattern was performed using the MAUD
software Lutterotti and Gialanella (1997). DC-magnetization measurements were
made in a SQUID based magnetometer (Quantum Design, MPMS-5) in the temperature
range of 2-300 K with an applied magnetic field of 200 Oe. Electrical
resistance, as a function of temperature and magnetic field, was measured
through the conventional AC four-probe method. For this purpose the sample was
connected using Cu wires glued with Ag paint. The measurements were performed
in a Physical Property Measurement System (PPMS, Quantum Design) at
temperatures ranging from 2 K to 300 K and magnetic fields ranging from 0 up
to 90 kOe. The magnetic field was applied perpendicular to the electrical
current applied to the sample.

Figure 1:  X-ray diffraction pattern and Rietveld fitting of YbNi_2. The
$R$-factors of refinement are: $R_{wp}=5.2,\,R_{b}=3.9,\,R_{exp}=4.1$ and
$\sigma^{2}=1.2$. The vertical red bars indicate the Bragg reflections of
YbNi_2. The continuous line at the bottom is the difference between the
experimental data and the fit.

##  III Results and discussion

Figure 1 shows the powder X-ray diffraction pattern of YbNi_2. The analysis of
these data indicates that the crystalline structure of the sample corresponds
to YbNi_2 (JCPDS No 03-065-5017), however a faint trace of Yb_2O_3 (JCPDS No
00-041-1106) are observed. This oxide is indicated by an asterisk in the
diffraction pattern. The X-ray diffraction pattern was Rietveld-fitted using a
space group $Fd\overline{3}m$ (No 227), considering the presence of Yb_2O_3.
The value obtained for the lattice parameter $a=7.0965(3)$ Å is in agreement
with the values reported previously Buschow (1972); Palenzona and Cirafici
(1973); Rojas  _et al._ (2012).

Figure 2:  Magnetic susceptibility $\chi=M/H$ (circles) and $\chi^{-1}$
(triangles), as a function of temperature measured under 200 Oe. The
continuous line is a linear fit of $\chi^{-1}(T)$ data from 150 K to room
temperature. The inset shows the determination of $T_{C}$ like the minimum in
$d\chi/dT$.

Figure 2 shows the temperature dependence of the magnetic susceptibility,
$\chi(T)$, and the inverse magnetic susceptibility, $\chi^{-1}(T)$, measured
in a magnetic field of 200 Oe. The magnetic susceptibility between 2 K and 300
K shows a fast increase at low temperatures, which characterize the
ferromagnetic behavior. The inset in this figure shows the $\chi(T)$ and
$d\chi/dT$ curves at low temperatures, here the ferromagnetic behavior is
clearly noted and the minimum in $d\chi/dT$ is defined as the ferromagnetic
transition temperature $T_{C}=9$ K. This value is lower than $T_{C}=10.5$ K
earlier reported Ivanshin  _et al._ (2017); Rojas  _et al._ (2012). In spite
of the tiny amount of Yb_2O_3 in our sample, its influence on the magnetic
properties of YbNi_2 was not considered, since it has an antiferromagnetic
transition temperature at 2.1 K Adachi  _et al._ (2005).

To analyze the paramagnetic state, $\chi^{-1}(T)$ is plotted in Fig. 2 (right
scale). The continuous line is a linear fit of the data from 150 K to 300 K.
The linear behavior can be described by the Curie-Weiss law,
$\chi(T)=C/(T-\theta_{CW})$, where $C$ is the Curie constant and $\theta_{CW}$
is the Curie-Weiss temperature. The effective magnetic moment, $\mu_{eff}$,
can be determined from $C$. In the present case, the values of these
parameters obtained from the fitting were; $\theta_{CW}=7.86$ K and $C=2.71$
emu K/mol Oe which gives $\mu_{eff}=4.65\mu_{B}$. This effective magnetic
moment is near to the theoretical value, $\mu_{eff}=g_{J}[J(J+1)]^{1/2}=4.53$
$\mu_{B}$, reported for the Yb3+ ion Blundell (2001).

Rojas et al. Rojas  _et al._ (2012) calculated the magnetic entropy of YbNi_2
based on specific heat measurements. The entropy saturates around 130 K, which
is congruent with a full population of the crystalline electric field (CEF)
effects. As observed in Fig. 2, there is a deviation of $\chi^{-1}(T)$ from
the linear behavior, this deviation could be related to the CEF effect on the
magnetization below 150 K.

Figure 3:  Electrical resistance as a function of temperature and applied
magnetic field, measured between 2 K and 100 K. The inset shows the electrical
resistance, without applied magnetic field, at temperatures between 2 K and
300 K.

The electrical resistance of YbNi_2 as a function of temperature in zero
magnetic field is shown in the inset of Fig. 3. The curve displays a metallic-
like behavior and shows a sharp decrease about the ferromagnetic transition at
low temperatures. However, for temperatures between $T_{C}$ and 15 K the
electrical resistance shows a $T^{2}$ dependence indicative of a Fermi liquid
behavior. The onset of the sharp resistance decrease is about $T_{on}=8$ K,
this value is near to the $\theta_{CW}$ determined from the magnetic
measurements. It is noteworthy the absence of a logarithmic increase of
$R(T)$, signature of a Kondo effect, this behavior suggest that the Kondo
effect is weak and that the RKKY interaction dominates. The Kondo temperature
has been reported as $T_{K}=27$ K, determined from the jump of the specific
heat at the ferromagnetic transition Rojas  _et al._ (2012). Taking the
approximation $|\theta_{CW}|=2T_{K}$ Blanco  _et al._ (1994); Yamauchi and
Fukamichi (2000) we obtain $T_{K}=3.93$ K. This $T_{K}$ value is lower than
reported, but congruent to the Kondo signature fault in $R(T)$.

The residual resistance ratio ($RRR$) is defined by the relation $RRR=R$(300
K)$/R$(2 K), this ratio provides qualitative information about the electron
scattering by structural defects. For YbNi_2 was obtained $RRR=$2.7, which
indicates that our samples have defects, probably included Yb vacancies
because of the low melting point of Yb. The $RRR$ value is similar to the
already observed values in other $\rm{RNi_{2}}$ alloys Kotur  _et al._ (2010);
Gratz  _et al._ (1996).

The main panel of Fig. 3 shows the $R(T)$ curves measured under different
applied magnetic fields. Above 40 K the curves are similar to the one measured
at zero magnetic field. The changes occur at low temperature, the electrical
resistance decreases as the magnetic field increases. At first sight, the
application of a magnetic field smooth out the magnetic transition and moves
up in temperature as the magnetic field is increased; this behavior is
characteristic of a ferromagnetic ordering. The reduction of $R(T)$ with
magnetic field suggest a negative magnetoresistance $(MR)$, as expected for a
ferromagnetically ordered system. Qualitatively, it is observed that the $MR$
magnitude increases as the temperature increases from 2 K, it reaches a
maximum and goes to zero as the temperature increases. A negative MR indicates
that the magnetic fluctuations decreases as the magnetic field increases
Moriya (1985).

To analyze the electrical resistance of YbNi_2 below the ferromagnetic
transition temperature we assume that the main scattering process is electron-
magnon. With this assumption, we use the Andersen and Smith model Andersen and
Smith (1979). According to this model, the magnetic resistance ($R_{mag}$) can
be describe by:

$R_{mag}=AT\Delta
e^{(-\Delta/T)}\left[1+2\frac{T}{\Delta}+\frac{1}{2}e^{(-\Delta/T)}+\cdots\right],$
(1)

where $\Delta$ is an energy gap in the magnonic density of states and $A$ is a
constant dependent on the material Andersen and Smith (1979). In the case of
$\Delta=0$ we deal with an isotropic ferromagnet and then $R_{mag}\propto
T^{2}$. However, fitting the experimental data to a $T^{2}$ dependence the
results were not acceptable. Taking in account Eq. 1, just with the first
term, and considering the residual resistance the data fit is good. Then the
electrical resistance is given by:

$R(T)=R_{0}+AT\Delta e^{(-\Delta/T)},$ (2)

where $R_{0}$ is the residual resistance. We fit the $R(T,H)$ curves at
temperatures between 2 K and approximately 7.5 K. The phonon contribution is
considered negligible because the Curie temperature $T_{C}=9$ K is small
compared to the Debye temperature reported, $\theta_{D}=272$ K Rojas  _et al._
(2012). Since the $R(T,H)$ curves below $T_{C}$ can be fitted with Eq. 2, we
conclude that the same scattering mechanism persists for the applied fields,
i.e., the electrical resistance is produced by the electron-magnon scattering.
Figure 4 shows the temperature dependence of the electrical resistance in
YbNi_2 at different magnetic fields in a temperature range between 2 K and 16
K. In this figure the continuous lines are the best fit obtained with Eq. 2.
Table 1 presents the parameters obtained from the fitting.

Figure 4:  Electrical resistance of YbNi_2 as a function of temperature at different magnetic fields indicated in the plot. The continuous lines are data fit using Eq. (2).  Table 1: Parameters obtained from fits using Eq. 2. $R_{0}$ is the residual resistance, A is a constant related to material and $\Delta$ is the magnonic energy gap. H(kOe) |  $R_{0}$(m$\Omega$) | A(m$\Omega$/K meV) |  $\Delta$ (meV)
---|---|---|---
0 | 0.2817(5) | 0.0089(1) | 0.31(1)
10 | 0.2690(2) | 0.0080(7) | 0.21(2)
30 | 0.2730(6) | 0.0035(1) | 0.39(3)
60 | 0.2701(6) | 0.0021(1) | 0.51(4)
90 | 0.2691(6) | 0.0014(1) | 0.62(8)
Figure 5: Magnonic energy gap $\Delta$ as a function of the applied magnetic
field. $\Delta$ was obtained from the fits of $R(T,H)$ curves of YbNi_2. The
continuous line is a linear fit of data.

It is well known that the magnonic energy gap is modified in the presence of
an external magnetic field as a consequence of the Zeeman effect. The magnonic
energy gap as a function of a magnetic field is given by
$\Delta=\Delta_{0}+g_{J}\mu_{B}\mu_{0}H$ Fontes  _et al._ (1999), where
$\Delta_{0}$ is the magnonic energy gap at zero field, $g_{J}=1.14$ is the
Landé factor of $\rm{Yb^{3+}}$ ion and $g_{J}\mu_{B}=0.065$ meVT-1. The
$g_{J}\mu_{B}$ value is the factor that determines the change of $\Delta$ as a
function of the magnetic field. Figure 5 shows the magnonic energy gap as a
function of the applied magnetic field, as expected, $\Delta$ increases as the
magnetic field is increased. The best linear fit of $\Delta(H)$ gives that
$\Delta(H)=0.24(4)+0.0043(6)H$, the slope must be equivalent to
$g_{J}\mu_{B}$, but its value is 0.043 meVT-1 lower than the calculated for
the Yb^3+ ion. In addition, it was noted that the parameter $A$ depends on
magnetic field.

Figure 6:  Magnetoresistance $MR$ of YbNi_2 as a function of applied magnetic
field at various temperatures. The data were extracted from curves of Fig. 3.
The lines are a guide for the eye.

Finally, the magnetoresistance as a function of applied magnetic field at
several temperatures was extracted from the $R(T,H)$ curves (Fig. 3). The
magnetoresistance was defined as $MR=[R(H)-R(0)]/R(0)$, where $R(0)$ is the
electrical resistance measured without applied magnetic field and $R(H)$ is
the electrical resistance measured with an applied magnetic field $H$. Figure
6 shows the plot of $MR(H)$ at 2, 5, 10, 15, 20 and 50 K. The maximum $MR$
observed in YbNi_2 is about -20% at 5 K and 10 K (near $T_{C}$) under 90 kOe.
At temperatures below 50 K, the $MR$ is negative above and below $T_{C}$ for
all applied magnetic fields. At fixed magnetic field the $MR$ magnitude
increases with temperature until about $T_{C}$, for temperatures above $T_{C}$
it diminishes. This behavior has been observed in other ferromagnetic Yb-based
HF such as YbPtGe Katoh  _et al._ (2010) and YbPdSi Tsujii  _et al._ (2016). A
negative MR for $T<T_{C}$ can be associated with a reduction of the spin-
disorder suppressed by the magnetic field. For $T>T_{C}$, a negative MR could
be associated with the existence of ferromagnetic short-range correlations or
spin fluctuations that preceded the onset of long-range magnetic order.
Further experiments on YbNi_2, under magnetic field and/or pressure, must be
performed to get more insight in their physical properties.

##  IV Conclusions

We have synthesized polycrystalline samples of YbNi_2, the structure type and
cell parameter were determined using X-ray diffraction analysis. Magnetic
measurement as a function of temperature shows a ferromagnetic transition at 9
K. The effective magnetic moment determined from the magnetic measurements
suggest a localized magnetism associated to Yb^3+ ions. Electrical resistance
as a function of temperature shows metallic-like behavior at high
temperatures. From this measurement, we infer that the RKKY is the dominant
interaction regard to Kondo effect. Below the ferromagnetic transition
temperature, the electrical resistance results from the electron-magnon
scattering, even in the presence of an external magnetic field. In addition,
the electrical resistance analysis indicates the presence of an energy gap in
the magnonic spectrum indicating that the ferromagnetic state is anisotropic.
The increment of the energy gap can be associated to the Zeeman effect.

###### Acknowledgements.

F.M. acknowledges the financial support from DGAPA-UNAM project IN105917 and
to R. Escudero by facilities to perform the magnetic measurements. O.O. tanks
the CONACyT scholarship.

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