# StyleTRF: Stylizing Tensorial Radiance Fields

Rahul Goel<EMAIL_ADDRESS>CVIT, KCIS, IIIT HyderabadIndia ,
Sirikonda Dhawal<EMAIL_ADDRESS>CVIT, KCIS, IIIT
HyderabadIndia ,  Saurabh Saini<EMAIL_ADDRESS>CVIT, KCIS,
IIIT HyderabadIndia and  P. J. Narayanan<EMAIL_ADDRESS>CVIT, KCIS, IIIT
HyderabadIndia

(2022; 2022)

###### Abstract.

Stylized view generation of scenes captured casually using a camera has
received much attention recently. The geometry and appearance of the scene are
typically captured as neural point sets or neural radiance fields in the
previous work. An image stylization method is used to stylize the captured
appearance by training its network jointly or iteratively with the structure
capture network. The state-of-the-art SNeRF (Nguyen-Phuoc et al., 2022) method
trains the NeRF and stylization network in an alternating manner. These
methods have high training time and require joint optimization. In this work,
we present StyleTRF, a compact, quick-to-optimize strategy for stylized view
generation using TensoRF (Chen et al., 2022). The appearance part is fine-
tuned using sparse stylized priors of a few views rendered using the TensoRF
representation for a few iterations. Our method thus effectively decouples
style-adaption from view capture and is much faster than the previous methods.
We show state-of-the-art results on several scenes used for this purpose.

NeRF, Content Stylization, Multi-view consistency, Fine-tuning, Fast
Adaptation

††copyright: rightsretained††isbn: 978-1-4503-9822-0/22/12††conference: 13th
Indian Conference on Computer Vision, Graphics and Image Processing; December
2022; Gandhinagar, India††journalyear: 2022††price: 15.00††doi:
10.1145/3571600.3571643††article: 43††journalyear: 2022††copyright:
acmcopyright††conference: Proceedings of the Thirteenth Indian Conference on
Computer Vision, Graphics and Image Processing; December 8–10, 2022;
Gandhinagar, India††booktitle: Proceedings of the Thirteenth Indian Conference
on Computer Vision, Graphics and Image Processing (ICVGIP’22), December 8–10,
2022, Gandhinagar, India††price: 15.00††doi: 10.1145/3571600.3571643††ccs:
Computing methodologies Appearance and texture representations††ccs: Computing
methodologies Computational photography††ccs: Computing methodologies
Volumetric models††ccs: Computing methodologies Non-photorealistic rendering

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Figure 1. _Stylization:_ We show results of stylization using our technique
presented in Sec. 3. We stylize the Play-Ground scene using two different
styles mosaic and mudbath. Our Pipeline adapts style in a nominal time of 40
sec on top of a pre-optimized TensoRF scene representation. Once the style is
adapted in accordance, stylized novel views can be generated with traditional
volumetric rendering techniques.

Figure 2. _System overview_ : The pipeline diagram presents an overview of the
strategy employed by our work. We first optimize Tensorial Radiance Fields for
representation of the scene as proposed by (Chen et al., 2022). Concurrently
we optimize stylization module utilizing Johnson et al. (2016) method on
COCO14-dataset(Lin et al., 2014). The stylization module is then used to
stylize a sparse set of novel views generated by the pre-optimized TensoRF.
These stylized views act as sparse style-priors and are used to fine-tune the
appearance of the previously optimized scene representation. It is to be noted
that we freeze the density terms of the TensoRF and only alter the appearance
vectors which retains geometric while adapting the novel style.

##  1\. Introduction

Stylizing a content image based on a reference style image has been of
interest to the community lately. With the development of 3D visual devices,
the demand for 3D content generation has grown. Stylizing entire 3D scenes has
applications in the world of augmented reality (AR) and virtual reality (VR).
With style transfer on 3D scenes, one can witness an entire world through a
painter’s eyes.

There have been several efforts to stylize image content (Gatys et al., 2016;
Johnson et al., 2016; Huang and Belongie, 2017; Sheng et al., 2018; Li et al.,
2017; Svoboda et al., 2020). Stylizing videos on the other hand is a harder
problem as it needs temporal consistency along with stylizing across the
frames of the video (Huang et al., 2017; Wang et al., 2020; Ruder et al.,
2016; Liu et al., 2021). Combining stylization with new view generation takes
the game one step further. We concentrate on stylized novel view generation in
this paper.

Radiance Fields have become the dominant method of capturing a scene using a
few images and then generating its views from other positions. NeRF
(Mildenhall et al., 2020) and its successors exploit the neural
representations to encode the whole scene and render novel views at test time.
They use a small MLP-based architecture to represent the radiance field
(typically using 8 layers of 256 neurons), which has a small memory footprint
of 5MB. They, however, need very high training time, ranging between 16hrs to
1 day per scene. Plenoxels (Fridovich-Keil and Yu et al., 2022) leverage the
traditional voxel space to optimize for the feature vectors associated per
voxel and rely on standard tri-linear interpolation to regress the radiance
field vectors for the continuous volumetric space. The voxel-based approach
benefits from the adaptive upsampling and pruning techniques, thereby reducing
the optimization times to about 15-minutes. Plenoxels has a large memory
footprint, often running into GBs. The recently introduced TensoRF method
(Chen et al., 2022) uses the Tensor decomposition based on BTD and provides a
way to reduce both memory and training times simultaneously. TensoRF can
optimize a scene in 10-20 minutes while maintaining a memory footprint of just
5-10MB.

Recent stylized new view generation methods use NeRF to represent the scene.
They jointly or iteratively optimize NeRF with a stylization technique while
rendering stylized new views (Chiang et al., 2022; Huang et al., 2022; Nguyen-
Phuoc et al., 2022). The latest method SNeRF proposed by Nguyen-Phuoc et al.
(2022) uses an iterative training strategy to optimize for each style using
Gatys et al. (2016). Adapting to a new style is thus a heavy process, taking
days of processing.

In this work, we build on and modify the ideas of SNeRF (Nguyen-Phuoc et al.,
2022) and present a simple yet powerful method for stylized new view
generation. Our method nearly decouples the scene capture step from the
stylization step. Each can be performed in respective pre-processing step. The
joint optimization is limited to a few iterations of appearance fine-tuning
that takes about 40-50 seconds to adapt to a new style. We use TensoRF (Chen
et al., 2022) to represent the geometry and appearance of the scene, which
comes with low optimization time and a small memory footprint. We use the
method by Johnson et al. (2016) for stylization. It can be trained for a
particular style image (typically in under 20 minutes) ahead of time and can
produce stylized images at a rate of 30 frames per sec. We generate a few
novel views using the learned TensoRF representation and stylize them using
Johnson’s method. We then fine-tune the TensoRF’s appearance branch while
freezing the density components for a few iterations to obtain a stylized
scene representation. Consistent, stylized novel views of the scene can be
generated using this fine-tuned TensoRF representation.

The main contribution of this work is using fast and light TensoRF
representation for stylized view generation and devising a fast style-
adaptation technique. We do this by fine-tuning the TensoRF appearance for a
small number of iterations, using as prior a sparse set of images rendered
from the optimized TensoRF and stylized using Johnson et al. (2016). This
effectively decouples style-adaption from view capture and makes it much
faster than methods like SNeRF.

##  2\. Related Work

Stylizing Images: Stylizing images has been a well-studied problem in the
vision community for a long time. The method proposed by Gatys et al. (2016)
optimizes white noise to match the content of one image while transferring the
style from the other. Johnson et al. (2016) proposed to use simple feed-
forward architecture, which produces stylized content in real-time. While
being quick at producing results, Johnson et al. (2016) require a separate
network for each style. Several works like (Huang and Belongie, 2017; Sheng et
al., 2018; Li et al., 2017) have addressed this issue of style-dependent
training. More recently, Svoboda et al. (2020) have been able to decompose an
image into its style and content codes. While these works have provided a
strong basis for stylizing 2D content, most of the aforementioned works do not
faithfully stylize temporal data, let alone stylizing 3D content.

Temporal Stylization of Videos: The stylization of temporal-data-like videos
necessitate temporal-consistent stylization. Though 2D stylization techniques
provide good stylization per frame, when stitched together, they often result
in flickering. This flicker is caused due to inconsistent temporal
stylization. To address these issues, various methods have tried incorporating
temporal-consistency losses across frames, namely Ruder et al. (2016). Further
works like Huang et al. (2017) concentrate on the real-time generation of
stylization in addition to temporal-consistent stylization. Recent work in
this line by Wang et al. (2020) has relaxed the objective function to be
optimized and derived a new regularization strategy. This allowed them to
stylize video content for any arbitrary given style without any re-training or
fine-tuning. Though these methods provide temporally-consistent stylization,
they can not produce novel stylized views. This requires scene understanding
at the geometric and as well as at the appearance levels. Consequently,
temporal consistency methods proposed by video stylization works cannot be
extended to novel view stylization.

Stylizing Geometric Appearance: As discussed earlier, obtaining stylized novel
views require an understanding of the geometric content of the scene. Methods
like Cao et al. (2020) have proposed stylizing geometry represented as
3D-point-cloud referenced on another point-cloud or image. However, stylizing
point-cloud do not fully address the problem of novel view stylization as they
are often accompanied by undesirable noise and missing surface information.
These noise points and missing geometries can only be filled up using neural
rendering-based frames works. Works like NPBG (Aliev et al., 2020) used U-Net-
based architecture to fill the missing gaps in geometry and alleviate the
noisy data. Moving in this direction Huang et al. (2021) introduced consistent
3D point cloud stylization through a learned linear transformation matrix to
change the appearance. They build up on NPBG (Aliev et al., 2020) to account
for the sparsity and noisy nature of point clouds for the rendering of
stylized novel views. Though the method proposed by Huang et al. (2021)
produces promising stylization of novel views, the geometric representation
and appearance captured by the underlying U-Net-based pipelines are not
accurate.

Novel View Synthesis(NVS) using Implicit Neural Representation: The Novel View
Synthesis(NVS) has been a challenging field that has been tackled for decades
with different approaches like Lumigraph (Gortler et al., 1996), Multi-Plane
Imaging (Tucker and Snavely, 2020), and Light Field Fusion (Mildenhall et al.,
2019). Recently the field of NVS has taken a large leap forward due to the
introduction of NeRF(Mildenhall et al., 2020). NeRF(Neural Radiance Fields)
proposed by Mildenhall et al. (2020), learns the radiance fields using simple
MLPs coupled with positionally encoded input features to regress radiance at
novel viewpoints, given a sparse set of posed images. Despite its great
representational power, it is accompanied by limitations like slow training
and rendering times, non-editable appearance, and lighting. These limitations
have paved the path for various extensions ranging from speeding up its
rendering process (Reiser et al., 2021; Hedman et al., 2021), edit-ability of
material (Boss et al., 2021, 2020; Zhang et al., 2021), relighting (Srinivasan
et al., 2021; Boss et al., 2021) and improving implicit geometry (Kellnhofer
et al., 2021). Recently HDR based NeRFs have also garnered great interest
(Mildenhall et al., 2022; Gera et al., 2022). Unlike these methods, we
concentrate on stylizing Radiance fields with reference style images.

Novel View Synthesis(NVS) using voxel grids: Apart from Implicit Neural
representation of radiance fields, recently, PlenOxels (Fridovich-Keil and Yu
et al., 2022) proposed the use of voxel grid with density and appearance
associated in the form SH-vectors stored at every voxel of the grid to
represent the underlying radiance field. They utilized custom Cuda
implementation to achieve a training time of around 20 minutes while obtaining
similar results as NeRF. However, their approach requires per-scene storage of
around 1GB. Addressing the issue of memory footprint and requirement of custom
Cuda kernels, TensoRF (Chen et al., 2022) has proposed a new VM-(Vector
Matrix) Decomposition (a special case of Block Term Decomposition) (De
Lathauwer, 2008; Giampouras et al., 2022). The usage of VM-decomposition
alleviated the issue of a large memory footprint while maintaining similar
scene optimization times as PlenOxels.

Stylizing Radiance Fields Recently, Chiang et al. (2022) extended NeRF to
generate stylized novel views. Similar to NeRF, they rely on simple MLPs to
regress radiance fields, compositing density and radiance. To achieve
stylization, they split the representation (1) volumetric density MLP and (2)
appearance MLP. They use a hypernetwork (Ha et al., 2017) trained on the style
to predict the weights of the appearance MLP which introduces style
information into the appearance of the scene. The losses involved in their
optimization framework are memory intensive. To circumvent this issue they
optimize the losses in a patch-based manner. Due to the usage of patch-based
loss, their method suffers from inconsistencies across the patch borders in a
single image frame, which is undesirable. Along with patch artifacts, due to
different MLP splits(density and appearance), Chiang et al. (2022) require
large training and rendering times.

Recent work StylizedNeRF (Huang et al., 2022) approaches the problem of
stylizing radiance fields by mutually optimizing 2D-3D consistency. For
incorporating style, they replace the appearance MLP of NeRF with a style
module. They also introduce consistency and mimic losses to train the style
module and a 2D stylization decoder simultaneously. However, their style
transfer is not accurate, and their approach involves nearly a day-long
training.

SNeRF (Nguyen-Phuoc et al., 2022) devises a new training strategy by
incrementally stylizing novel views. They first train a NeRF model on real-
world data and generate novel views using it. Using the approach by Gatys et
al. (2016), they generate a stylized version of these novel views based on a
reference style. Then the NeRF model is re-trained NeRF on the so-obtained
stylized views. Iteratively repeating the previous two steps stylizes a Neural
radiance field representation to incorporate style information. Once the scene
is stylized novel views can be generated using traditional volumetric
rendering techniques. However, this learning of radiance fields involves
optimization of the Neural implicit representation which comes with a heavy
training cost of 4-5 days.

We take inspiration from SNeRF and train a radiance field representation on
stylized images themselves. However, we realize the limitations that Gatys’s
method introduces when generalizing the 2D images to a 3D domain and propose
the usage of a different stylization module.

##  3\. Method

Given a set of posed images of a scene and a reference style image, our aim is
to render stylized novel views of the scene, which are consistent in
appearance and geometry across different frames. We achieve this by fine-
tuning the appearance of a TensoRF (Chen et al., 2022) using stylized images.

###  3.1. Preprocessing Stylization Module

In our method, we use the stylization method presented by Johnson et al.
(2016), which produces stylized content in real time. The approach requires a
per-style training of a CNN, for which it utilizes COCO2014 dataset (Lin et
al., 2014). The per-style optimization takes an approximate time of 20-minutes
and at the inference, it produces 30 images/sec. Since training per-style
takes only 20 minutes, we can simultaneously train multiple Johnson et al.
(2016) models for each desirable style independently and infer based on the
preference.

We choose Johnson et al. (2016) as our underlying stylization module over
Gatys et al. (2016), unlike others (Nguyen-Phuoc et al., 2022) because it
gives stable stylization for two closeby viewpoints. In the case of Gatys et
al. (2016), stylizing two closeby viewpoints that share a large amount of
image content usually results in drastically different stylization as shown in
Fig. 3. This is because the image developed using Gatys et al. (2016) start
with white-noise and try to converge the distribution of _reference-style_ and
_content_ images. This results in unstable stylization across nearby
viewpoints as the optimization depends on various factors such as the learning
rate, initialization, and type of optimizer. One of the reasons for the slow
multi-iter stylization in SNeRF was to alleviate the inconsistencies caused by
Gatys et al. (2016) based stylization.

On the other hand, once Johnson et al. (2016) is trained, the output image is
deterministic with respect to the input image. Furthermore, two close-by
camera viewpoints share a large amount of spatial information. Since CNN’s are
spatially invariant in the local region, our image transformation network
gives fairly stable stylization across closeby viewpoints Fig. 3.

Though Johnson et al. (2016) provide stable stylization across adjacent views,
it is _not temporally consistent_. Hence it is to be noted that, we do not
rely on temporally-consistent stylization while only relying on the nominally-
stable stylization of nearby viewpoints. The per-style training of our
Stylization Module is depicted in Fig. 2.

###  3.2. Scene Representation using TensoRF

In order to generate a novel view, it is necessary to have a geometric
representation and appearance understanding. In order to address this issue,
we use Radiance fields($\mathcal{L}$). The radiance fields (Zhou et al., 2005)
($\mathcal{L}$) is given by:

(1) $\mathcal{L}:R^{3}\times S^{2}\rightarrow R^{3}$

where $R^{3}$ on the left is the scene’s world space; $S^{2}$ is the sphere of
directions about each point, and the $R^{3}$ on the right is radiance at the
point. Radiance fields in a way encode geometry and radiance in their mapped
representation. Though there exist various recent adaptions of the mapping
function presented in Eq.(1) like NeRF(Mildenhall et al., 2020),
PlenOxel(Fridovich-Keil and Yu et al., 2022), we chose TensoRF (Chen et al.,
2022) which is a compact, accurate, and fast-to-optimize representation of
Radiance Fields. We use a specifically _VM-48_ variant of TensoRF which
optimizes a scene within a timeframe of 15-20 minutes while maintaining a
memory footprint of 10-15MB.

The TensoRF representation utilizes a Tensor decomposition known as VM-
decomposition which in itself is a special case of BT-decomposition. This
reduces the voxel grid memory by order of $\mathcal{O}(n)$. This scene
optimization can be done independently for every scene irrespective of style.
In a later phase, we alter the appearance to adapt to the reference style in a
short period of 40-50 seconds.

Similar to TensoRF (Chen et al., 2022), we utilize $L1$ norm loss and total
variation (TV) loss (Eq. 2) for regularization. This helps our process to
avoid getting stuck in local minima and prevents overfitting. For scenes with
less number of captured images, TV loss is a better choice to obtain good
results. The equation for TV loss is given by:

(2)
$\mathcal{L}_{TV}=\dfrac{1}{N}\sum\left(\sqrt{\Delta^{2}T_{\sigma}}+0.1\sqrt{\Delta^{2}T_{C}}\right)$

Here, $\Delta^{2}$ is the squared difference between the neighboring values in
our tensors, $N$ is the total number of parameters across our TensoRF
representation $T$. $T_{\sigma}$ represents the density value and $T_{C}$
represent the appearance value in the TensoRF representation. They are
weighted in the ratio of $10:1$ respectively. More details about TV Loss can
be found in the work by Chen et al. (2022).

###  3.3. Stylizing TensoRF representation

####  3.3.1. Novel View stylization

Upon optimizing the radiance fields which encode the geometry and radiance of
the scene as discussed in the Sec.3.2, we render a sparse set of $20-30$ novel
views in a simple trajectory (spiral). We stylize these novel renders using
the pre-trained Stylization Module discussed in 3.1. Fig. 2 (top-right) shows
the generation stylization of these renders utilizing the Stylization Module.

####  3.3.2. Stylizing Appearance of TensoRF

We utilize the sparse set of stylized novel views generated using the per-
style optimized Johnson et al. (2016) module and optimize the appearance
vectors of the TensoRF. During the process of optimization, we ensure that the
density terms are frozen, and only the appearance is altered. We explicitly
chose to freeze density as we have observed that stylizing looks pleasing and
free from artifacts when density is kept frozen. This fine-tuning only takes a
nominal time of downwards of 40 secs. Once the fine-tuning is done, we obtain
a geometric scene represented as Tensorial Radiance fields, which can be used
to render _stylized novel views_ with consistent appearance across the
viewpoints. The rendering of each image having a resolution of $800\times 800$
takes an approximate time of 4-5 seconds. Fig. 2 (bottom-right) shows the
appearance modification from a sparse set of inputs and novel view generation.

##  4\. Implementation Details

###  4.1. Optimizing TensoRF

The training/optimization of radiance fields requires information of the
camera poses from which an image is captured. In the case of real scenes, we
rely on COLMAP (Schönberger and Frahm, 2016)to obtain this information and in
the case of synthetic scenes, we use the data obtained from Blender. We
optimize TensoRF (VM-Split-48) on the input images for $15k$ iterations. In
each iteration, we shoot $4096$ rays into the voxel grid. We obtain the
radiance using volumetric rendering (Eq. 3) and optimize the grid iteratively.

(3)
$\mathcal{C}=\sum_{q=1}^{Q}\tau_{q}\left(1-exp\left(-\sigma_{q}\Delta_{q}\right)\right)c_{q},\
\tau_{q}=exp\left(-\sum_{p=1}^{q-1}\sigma_{p}\Delta_{p}\right)$

We use Adam optimizer (Kingma and Ba, 2014) which is initialized to a learning
rate of $0.02$ and is re-initialized to $0.02$ after upsampling. The voxel
grid is initialized with an effective resolution of $128^{3}$ and iteratively
upsampled every $1000$ iterations, first upsampling starting $2000$ until
$5000$ iterations are reached. We finally reach an effective voxel-grid
resolution of $640^{3}$ for real-world scenes and $300^{3}$ for synthetic
scenes. It is to be noted that the resolution mentioned here is _effective_
but not exact, as the VM-decomposition provided by TensoRF presents a compact
representation of voxel grid. Similar to TensoRF, we bi-linearly interpolate
the matrix and linearly interpolate the vector in the VM decomposed
representation during upsampling. This (bi-linear + linear) interpolation is
similar to the tri-linear interpolation of the _full-voxel_ grid. We perform
such interpolations with neighboring voxels during the evaluation of a ray
query. This enables us to obtain continuity in our rendered images. This pre-
optimization phase requires 10-15 minutes. Once optimized it can be used to
generate new views with various camera poses which post-stylization act as
priors to our fine-tuning phase.

###  4.2. Stylization

Independently, we train the stylization module Johnson et al. (2016) with the
various reference style image on the COCO-14 Dataset (Lin et al., 2014) which
takes 20 minutes. We train multiple such Johnson et al. (2016) models for each
desired style as the time required to train is quite less. We use this to
create stylized priors from the views generated in the previous phase.

While optimizing for the style we freeze the density parameters in the TensoRF
representation and optimize only the appearance parameters for a small number
of iterations($1k$ iterations) with the stylized prior. This style adaption
only takes a nominal time of 40-50 seconds. The style-adapted TensoRF
representation obtained in the previous step can be thus used to generate
novel stylized views using traditional voxel rendering techniques. The
generation of each view takes around 4-5 seconds.

(a) _Gatys_ View 1 (b) _Gatys_ View 2

(c) _Johnson_ View 1 (d) _Johnson_ View 2

Figure 3.  _Gatys vs Johnson Stylized Priors_ : The figures shows the output
of the stylization modules proposed by Gatys et al. (2016) and Johnson et al.
(2016) on Horns stylized based on udnie. Gatys et al. (2016) produces
inconsistencies in the style generated across near-by views which provide a
poor prior to optimize appearance for our module. Johnson et al. (2016)
provides stable stylization across close-by views as seen in the figure.
Figure 4.  _Comparison of the different optimization strategies:_ Strategy
_S1_ optimized directly on stylized images produces massive artifacts due to
the loss in geometry. Unconstrained optimization _S2_ on the other hand
achieves results compared to _S1_ while struggling to capture micro-details as
seen in the insets. Our StyleTRF on the other hand freezes density while only
optimizing for appearance, capturing the style while maintaining the geometric
detail of the original scene.

##  5\. Experiments

In this section, we present the various experiments with the proposed
StyleTRF. We conduct all the experiments mentioned throughout the paper
including the comparisons, on a workstation PC equipped with an AMD
Ryzen-5800x and an NVidia RTX-3090 GPU. In Sec. 5.1 we discuss the choice of
stylization module used to stylize the sparse prior used in our approach. We
also experimentally compare and contrast between temporal-stylization of
smooth trajectories of ground truth 3D content vs Actual 3D-Stylization in
Sec. 5.2. Finally, in Sec. 5.3 we show the effect of different optimization
strategies used to stylize the 3D content.

###  5.1. Stylization Module

For stylizing the sparse prior required by our method we use Johnson et al.
(2016). The concurrent work SNeRF(Nguyen-Phuoc et al., 2022) uses Gatys et al.
(2016) method to iteratively stylize the radiance fields. We choose Johnson et
al. (2016) over Gatys et al. (2016) because the latter depends on several
factors such as the initialization, learning rate, and the optimization method
which make it unreliable to obtain stable stylization across close-by views
let alone temporal consistency. This can be observed in Fig. 3. This is one of
the reasons, Nguyen-Phuoc et al. (2022) requires multiple epochs to reflect
the style in appearance.

On the other hand, Johnson et al. (2016) use a fixed CNN-based architecture to
infer the stylized image. Due to the spatial consistency of CNNs, two close-by
views sharing significant spatial content lead to stable stylized close-by
views.

###  5.2. Video Stylization v/s 3D Stylization

It can be argued that instead of stylizing 3D content, one can generate novel
views on a camera trajectory and use temporally consistent stylization
frameworks like ReReVST (Wang et al., 2020) to obtain stylized novel views.

However, we have observed that though temporal stylization is maintained, the
work of ReReVST fails to fully capture the style information. The same can not
be said for the stylization of 3D content. In Fig. 5 it can be seen that both
StylizedNeRF and our method capture better style compared to ReReVST.

###  5.3. Optimization Strategies for Style Adaptation

For the stylization of Radiance fields we present three strategies:

  1. (1)

_S1:_ Optimizing a TensoRF from scratch directly using the stylized priors.

  2. (2)

_S2:_ Pre-optimizing a TensoRF on the original ground truth and adapting for
style using sparse stylized priors _without freezing any parameters (both
density and appearance)_.

  3. (3)

_S3:_ Pre-optimizing a TensoRF on the original ground truth data and adapting
for style using sparse stylized priors while freezing the density.

When following the _S1_ we observed geometric artifacts. This is because the
stylized priors generated may not share the exact geometry with the ground
truth. As stated above in Sec. 3.1, Johnson et al. (2016) does not provide
temporally consistent stylization which might also affect the geometry so-
optimized in the stylized views in the case of _S1_.

Another strategy _S2_ produces considerably better results compared to _S1_ as
seen in Fig. 4. This is because most of the geometric prior is learned from
the pre-optimization phase which uses ground-truth images to obtain scene
properties.

Though _S2_ has produced good stylization at a micro-level, fuzzy geometry can
be observed which reduces the appeal of the stylization. Specifically, thin
geometric structures suffer from these undesirable fuzzy geometric changes.
The limit-free optimization strategy of _S2_ fiddles with the geometry
components and leads to these artifacts. This can be observed in the insets
provided in Fig. 4.

Our StyleTRF (_S3_) approach on the other hand alleviates these issues by
freezing the geometric components of scene representation. Our approach
generates crispier results compared to the rest of the aforementioned
strategies _S1, S2_. This behavior is consistent across all the scenes.

We experiment with a different number of stylized priors to stylize the
underlying scene. As shown in Fig. 7, we find that good stylization is
obtained when all the priors cover the entirety of the scene. In most cases,
30-40 randomly sampled camera positions around the object suffice.

Scene Method | ReReVST(Wang et al., 2020) | StylizedNeRF(Huang et al., 2022) | Ours
---|---|---|---
short-term | long-term | short-term | long-term | short-term | long-term
Horns | 0.0046 | 0.0137 | 0.0229 | 0.0239 | 0.0040 | 0.0120
Fern | 0.0028 | 0.0080 | 0.0100 | 0.0168 | 0.0020 | 0.0069
Flower | 0.0039 | 0.0106 | 0.0020 | 0.0277 | 0.0030 | 0.0089
Table 1. _Consistency Metrics:_ We show short-term and long-term consistency
metrics across a smooth trajectory generated using our appearance stylized
scene representation. We have found that we obtain better short & long-term
consistency compared to the SOTA 3D-Stylization technique StylizedNeRF(Huang
et al., 2022), while maintaining better style transfer compared to temporal
stylizing methods like ReReVST(Wang et al., 2020).

lb

lb

lb

(a) GT Unstylized (b) ReReVST

(c) StylizedNeRF (d) StyleTRF (Ours)

Figure 5.  _Qualitative Comparison_ : [Row 1: santamaria, Row 2: udnie, Row 3:
mediterranean] Here we show the comparisons Un-stylized frame in column-(a),
temporally consistent stylization of ReReVST (Wang et al., 2020)in column-(b),
StylizedNeRF (Huang et al., 2022) in column-(c) and stylization using our
pipeline in column-(d). It can be observed that due to the usage of combined
neural representation density and radiance, the style adaptation is affecting
in the case of StylizedNeRF. Observe noisy geometric structures in mediter
applied onto Fern

Figure 6.  _View Consistency Across Frames_ : The figure shows stylized novel
views of a simple trajectory. To keep it simple we named the frames in $t_{i}$
with increasing order of $i$ from left to right. It can be observed clearly
that our stylization is multi-view consistent both in the case of Stove and
Horns .

(a) 5 Priors (b) 10 Priors

(c) 20 Priors (d) 40 Priors

Figure 7.  _Number of Stylized Priors_ : The images in this figure are
obtained by fine-tuning StyleTRF using a different number of priors on the
kitchen scene from the MipNeRF360 (Barron et al., 2022) dataset which is an
unbounded 360 degree scene. Depending on priors quality improves and saturates
at 35-40 priors.

##  6\. Results

In this section, we compare our results against both 3D-stylization techniques
and temporally-consistent stylization techniques both quantitatively and
qualitatively.

###  6.1. Qualitative Results

For comparing qualitative results, we can use smooth trajectories which are
similar to a video. This similarity enables us to compare with temporally-
consistent stylization techniques alongside 3D-Stylization. For 3D stylized
novel view synthesis, though there exists Huang et al. (2021), they do not
hold an accurate representation of geometry and rely on explicit geometry as
input. Although other methods like Chiang et al. (2022) do not rely on
explicit geometry input, they suffer from patch border artifacts as discussed
in (Huang et al., 2022) and Sec. 2. Hence, we compare our results with the
latest 3D radiance field-based stylization method StylizedNeRF (Huang et al.,
2022) and temporally-consistent stylization technique of ReReVST Wang et al.
(2020), on similar lines as (Huang et al., 2022).

Comparison amongst different stylization Techniques: We have observed that
though the novel views generated using StylizedNeRF are reasonably consistent,
the geometry after stylization in the case of StylizedNeRF has been greatly
affected as seen in Fig. 5(c). It produces blurry geometry in the case of TRex
stylized using udnie along with the missing greenish tints present in the
udnie, and produces extremely noisy results for mediterranean applied on Fern
. Contrary to this, our method distinctively transfers different colors to
different parts of the scene as seen in TRex stylized using udnie and captures
the complete color palette in mediterranean adapted onto Fern . Our method
also consistently preserves geometry across all scenes and styles. The
geometric noise in the StylizedNeRF can partly be attributed to the combined
density and appearance encoded of a single MLP, which hinders the
disentanglement of geometry and radiance as observed by (Nguyen-Phuoc et al.,
2022).

In the case of the temporal-consistent style transfer technique ReReVST, we
have observed that although the method is robust to adapt to new styles on the
fly and produces results in real-time, it lacks proper capture of style
information. This can be observed Fig. 5(b), ReReVST fails to capture the
prevalent blue tint in santamaria in the case of Flower , this could also be
observed in the case of StylizedNeRF and vaguely captures the color palette of
mediterranean in Fern . For udnie applied on TRex , it does not account for
distinct colors present in the style and the resultant image contains an
unappealing blend of colors throughout the image.

View Consistency: To qualitatively show view consistency, we render views
across a smooth trajectory and show different views across it. In Fig. 6 we
show our renders for udnie and mosiac styles adapted onto Stove and Horns
respectively. It can be observed from the insets of Fig. 6 that stylized
radiance across the frame is consistent, validating our claim that
3D-stylization-based novel view generation can produce multi-view consistent
stylized content.

###  6.2. Quantitative Results

In order to check the consistency of our stylized content, we generate a
smooth trajectory similar to SNeRF (Nguyen-Phuoc et al., 2022). The rendered
views along the trajectory are used to evaluate the consistency. Since the
smooth trajectory replicates the behavior of a video we can comfortably
compare our method with the temporal-consistent stylization techniques like
ReReVST(Wang et al., 2020) alongside 3D-stylization methods like
StylizedNeRF(Huang et al., 2022). For this comparison, we chose Fern , Flower
, and Horns scenes and report the respective metrics in Tbl. 1. We obtained
better metrics compared to both temporal stylization (Wang et al., 2020) and
implicit geometric stylization (Huang et al., 2022). We estimated the
consistency by calculating the optical flow $\smash{\mathcal{O}}$ between the
non-stylized frames $\smash{\mathcal{F}^{real}_{i}}$ and
$\smash{\mathcal{F}^{real}_{i+\delta}}$ rendered using TensoRF.

(4)
$\mathcal{O}:opticalflow\left(\mathcal{F}^{real}_{i},\mathcal{F}^{real}_{i+\delta}\right)$

Using the so-obtained optical-flow $\smash{O}$, we warp the stylized frame
$\smash{\mathcal{F}^{style}_{i}}$ to
$\smash{\hat{\mathcal{F}}^{style}_{i+\delta}}$.

(5) $\mathcal{W}:\hat{\mathcal{F}}^{style}_{i+\delta}\leftarrow
Warp(\mathcal{F}^{style}_{i},\mathcal{O})$

For the calculation of optical flow, we use RAFT (Teed and Deng, 2021) similar
to SNeRF.

We then calculate the pixel-averaged $L_{2}$ loss between the warped frame
${\hat{\mathcal{F}}^{style}_{i+\delta}}$ and the actual stylized frame
${\mathcal{F}^{style}_{i+\delta}}$. We aggregate this loss across the frame-
combinations and report the metrics in Tbl. 1. For calculation of short-term
consistency we chose $\delta=1$ and for long-term $\delta=5$. It is to be
noted that $\delta$ here represents the change in the camera position along
the trajectory. We have observed that though the temporal-cosistent
stylization techniques like Wang et al. (2020) come close to our results, they
struggle to capture the reference-style (as seen in Fig. 5(b)).

| GT Training | Style Training | Style Adaptation
---|---|---|---
StyleImp |  $\approx 12$ hrs | NA |  $\geq 5$ hrs
SNeRF |  $\approx 12$ hrs | NA | 3-4 days
StylizedNeRF |  $\approx 12$ hrs | NA |  $\geq 3$ hrs
Ours | 20 mins | 20 mins | 45 secs
Table 2. _Training Times:_ We compare our training times with StyleImp (Chiang
et al., 2022), SNeRF (Nguyen-Phuoc et al., 2022), StylizedNeRF (Huang et al.,
2022). Due to our compact representation, we get a training time of $\approx
20$ mins compared to at least $12$ hrs for others. The GT training and style
training can be performed in parallel in our method. We disentangle the style
training process from the style adaptation process which enables us to quickly
adapt to any scene in under a minute.

##  7\. Conclusions

In this paper, we presented StyleTRF, a compact and quick-to-optimize
stylization technique which can generate stylized novel views of a scene. We
have shown that our method can efficiently and faithfully incorporate style
into a radiance field representation of a casually captured scene. We have
qualitatively and quantitatively compared our method with the previous
stylization methods. Our qualitative results and quantitative metrics
demonstrate that StyleTRF is consistent across the views, having stylized the
underlying 3D representation. We also reported the _short and long-term_
consistency metrics which are better in most cases compared to the present 3D
stylization methods. Concurrently, Zhang et al. (Zhang et al., 2022) presented
stylized novel view synthesis using voxel-based grids, partly similar to our
method. But they chose PlenOxels (Fridovich-Keil and Yu et al., 2022) to
represent the radiance fields. They focussed more on obtaining brush strokes
while ours concentrates on geometric preserved fast-style adaptation.

Acknowledgements:  Sirikonda Dhawal was partially supported by DST. We also
acknowledge support from the TCS Foundation through the Kohli Center on
Intelligent Systems (KCIS).

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# An optimal lower curvature bound for convex hypersurfaces in Riemannian
manifolds

Stephanie Alexander  Department of Mathematics, University of Illinois at
Urbana-Champaign, IL 61801<EMAIL_ADDRESS>www.math.uiuc.edu/ sba ,  Vitali
Kapovitch  Department of Mathematics, University of Toronto, Toronto, ON, M5S
2E4<EMAIL_ADDRESS>www.math.toronto.edu/ vtk/ and  Anton Petrunin
Department of Mathematics, Penn State University, State College, PA 16802
<EMAIL_ADDRESS>www.math.psu.edu/petrunin/

###### Abstract.

It is proved that a convex hypersurface in a Riemannian manifold of sectional
curvature $\geqslant\kappa$ is an Alexandrov’s space of curvature
$\geqslant\kappa$. This theorem provides an optimal lower curvature bound for
an older theorem of Buyalo.

###### 1991 Mathematics Subject Classification:

Primary 53C20, 53B25; Secondary 53C23, 53C45

We thank the Mathematical Research Institute Oberwolfach (Research in Pairs
programme) for a valuable visit, of which this note is an offshoot. The third-
named author was supported in part by the National Science Foundation under
grant DMS-0406482.

The purpose of this paper is to provide a reference for the following theorem:

######  Theorem 1.

Let $M$ be a Riemannian manifold with sectional curvature $\geqslant\kappa$.
Then any convex hypersurface $F\subset M$ equipped with the induced intrinsic
metric is an Alexandrov’s space with curvature $\geqslant\kappa$.

Here is a slightly weaker statement:

######  Theorem 2.

[Buyalo] If $M$ is a Riemannian manifold, then any convex hypersurface
$F\subset M$ equipped with the induced intrinsic metric is locally an
Alexandrov’s space.

In the proof of Theorem 2 in [Buyalo], the (local) lower curvature bound
depends on (local) upper as well as lower curvature bounds of $M$. We show
that the approach in [Buyalo] can be modified to give Theorem 1.

######  Definition 3.

A locally Lipschitz function $f$ on an open subset of a Riemannian manifold is
called _$\lambda$ -concave_ ($\lambda\in\mathbb{R}$) if for any unit-speed
geodesic $\gamma$, the function

$f\circ\gamma(t)-\lambda t^{2}/2$

is concave.

######  Lemma 4.

Let $f:\Omega\to\mathbb{R}$ be a $\lambda$-concave function on an open subset
$\Omega$ of a Riemannian manifold. Then there is a sequence of nested open
domains $\Omega_{i}$, with $\Omega_{i}\subset\Omega_{j}$ for $i<j$ and
$\cup_{i}\Omega_{i}=\Omega$, and a sequence of smooth $\lambda_{i}$-concave
functions $f_{i}:\Omega_{i}\to\mathbb{R}$ such that

  1. (i)

on any compact subset $K\subset\Omega$, $f_{i}$ converges uniformly to $f$;

  2. (ii)

$\lambda_{i}\to\lambda$ as $i\to\infty$.

This lemma is a slight generalization of [Greene–Wu, Theorem 2] and can be
proved exactly the same way.

Proof of Theorem 1. Without loss of generality one can assume that

  1. (a)

$\kappa\geqslant-1$,

  2. (b)

$F$ bounds a compact convex set $C$ in $M$,

  3. (c)

there is a $(-2)$-concave function $\mu$ defined in a neighborhood of $C$ and
$|\mu(x)|<1/10$ for any $x\in C$,

  4. (d)

there is unique minimal geodesic between any two points in $C$.

(If not, rescale and pass to the boundary of the convex piece cut by $F$ from
a small convex ball centered at $x\in F$, taking
$\mu=-10\operatorname{dist}_{x}^{2}$.)

Consider the function $f=\operatorname{dist}_{F}$. By Rauch comparison (as in
[Petersen, 11.4.8]), for any unit-speed geodesic $\gamma$ in the interior of
$C$, $(f\circ\gamma)^{\prime\prime}$ is bounded in the support sense by the
corresponding value in the model case (where $M=\mathbb{H}^{2}$ and $F$ is a
geodesic). In particular,

$(f\circ\gamma)^{\prime\prime}\leqslant f\circ\gamma.$

Therefore $f+\varepsilon\mu$ is $(-\varepsilon)$-concave in
$\Omega_{\varepsilon}=C\cap f^{-1}((0,\varepsilon))$. Take
$K_{\varepsilon}=f^{-1}([\frac{1}{3}\varepsilon,\frac{2}{3}\varepsilon])\cap
C$. Applying lemma 4, we can find a smooth $(-\tfrac{\varepsilon}{2})$-concave
function $f_{\varepsilon}$ which is arbitrarily close to $f+\varepsilon\mu$ on
$K_{\varepsilon}$ and which is defined on a neighborhood of $K_{\varepsilon}$.
Take a regular value $\vartheta_{\varepsilon}\approx\tfrac{1}{2}\varepsilon$
of $f_{\varepsilon}$. (In fact one can take
$\vartheta_{\varepsilon}=\tfrac{1}{2}\varepsilon$, but it requires a little
work.) Since $|\mu|_{C}|<1/10$, the level set
$F_{\varepsilon}=f_{\varepsilon}^{-1}(\vartheta_{\varepsilon})$ will lie
entirely in $K_{\varepsilon}$. Therefore $F_{\varepsilon}$ forms a smooth
closed convex hypersurface. By the Gauss formula, the sectional curvature of
the induced intrinsic metric of $F_{\varepsilon}$ is $\geqslant\kappa$.
$F_{\varepsilon}$ bounds a compact convex set $C_{\varepsilon}$, where
$F_{\varepsilon}\to F$, $C_{\varepsilon}\to C$ in Hausdorff sense as
$\varepsilon\to 0$. By property (d), the restricted metrics from $M$ to
$C,C_{\varepsilon}$ are intrinsic, and so $C_{\varepsilon}$ is an Alexandrov
space with $F_{\varepsilon}$ as boundary, that converges in Gromov–Hausdorff
sense to $C$. It follows from [Petrunin, Theorem 1.2] (compare [Buyalo,
Theorem 1]) that $F_{\varepsilon}$ equipped with its intrinsic metric
converges in Gromov–Hausdorff sense to $F$ equipped with its intrinsic metric.
Therefore $F$ is an Alexandrov space with curvature $\geqslant\kappa$.∎

######  Remark 5.

We are not aware of any proof of theorem 1 which is not based on the Gauss
formula. (Although if $M$ is Euclidean space, there is a beautiful purely
synthetic proof in [Milka].) Finding such a proof would be interesting on its
own, and also could lead to the generalization of theorem 1 to the case when
$M$ is an Alexandrov space.

## References

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