##  Appendix H Multi-layer perceptron results

Figure 35 shows the probabilities of the stars obtained by the MLP method in
CBD-magnitude-radius space, where the radius is the distance from Sgr A*.

Figure 35: Probabilities of the stars obtained by the MLP method in CBD-
magnitude-radius space, where the radius is the distance from Sgr A*.

##  Appendix I Gradient boosted trees results

Figure 36 shows the probabilities obtained by the random forest in CBD-
magnitude-radius space.

Figure 36: Probabilities obtained by the random forest in CBD-magnitude-radius
space, the three most important features.

##  Appendix J SEDs of early-type candidates

Due to discrepancies in the number of early-type candidates among the three
methods, particularly in the outer regions where the MLP method identifies a
higher number of candidates compared to the other two methods, this appendix
delves into the SEDs of stars detected exclusively with MLP. The goal is to
assess the reliability of the results. Figure 37 displays the SEDs of three
early candidates located in the corners of the image (see Fig. 13). The CBD
values, as well as the shapes of their SEDs, suggest characteristics typical
of early-type stars. Additionally, in the same regions, late-type candidates
exhibit distinct SED features, including a noticeable dip corresponding to the
CO band in wavelengths larger than 2.27. This observation leads us to the
conclusion that some of the stars detected only by the MLP method may indeed
correspond to genuine young stars.

Figure 37: SEDs for early-type candidates detected only with MLP method,
located in the upper left corner (upper panel), lower left corner (middle
panel), and upper right corner (bottom panel).

##  Appendix K On metallicity

We show in Fig. 38 the interaction of predicted metallicity and late
probability. Both need to be modelled at once for reliable results.

Figure 38: Predicted metallicity. Excluded are stars which are
spectroscopically early and stars outside the magnitude range where the
prediction works


# ParaNet: Deep Regular Representation for 3D Point Clouds

Qijian Zhang1 Junhui Hou1∗ Yue Qian1 Juyong Zhang2 Ying He3
1Department of Computer Science  City University of Hong Kong
2School of Mathematical Sciences  University of Science and Technology of
China
3School of Computer Science and Engineering  Nanyang Technological University
<EMAIL_ADDRESS>jh.hou<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Although convolutional neural networks have achieved remarkable success in
analyzing 2D images/videos, it is still non-trivial to apply the well-
developed 2D techniques in regular domains to the irregular 3D point cloud
data. To bridge this gap, we propose ParaNet, a novel end-to-end deep learning
framework, for representing 3D point clouds in a completely regular and nearly
lossless manner. To be specific, ParaNet converts an irregular 3D point cloud
into a regular 2D color image, named point geometry image (PGI), where each
pixel encodes the spatial coordinates of a point. In contrast to conventional
regular representation modalities based on multi-view projection and
voxelization, the proposed representation is differentiable and reversible.
Technically, ParaNet is composed of a surface embedding module, which
parameterizes 3D surface points onto a unit square, and a grid resampling
module, which resamples the embedded 2D manifold over regular dense grids.
Note that ParaNet is unsupervised, i.e., the training simply relies on
reference-free geometry constraints. The PGIs can be seamlessly coupled with a
task network established upon standard and mature techniques for 2D
images/videos to realize a specific task for 3D point clouds. We evaluate
ParaNet over shape classification and point cloud upsampling, in which our
solutions perform favorably against the existing state-of-the-art methods. We
believe such a paradigm will open up many possibilities to advance the
progress of deep learning-based point cloud processing and understanding.

11footnotetext: Corresponding author.

##  1 Introduction

Figure 1: Illustration of the pipeline of ParaNet. ParaNet maps an input 3D
point cloud (a) to a 2D domain (b) and adopts grid resampling to produce a 2D
color image named point geometry image (PGI) (c), which is a complete regular
structure that explicitly encodes 3D information into pixel colors and can
faithfully represent the geometry of the input shape. Thanks to the regular
structure, the generated geometry images enable the standard 2D CNNs for 3D
applications, such as classification (d) and upsampling (e). Note that ParaNet
is unsupervised, i.e., the ground-truth PGIs are not required during training.

The recent decade has witnessed remarkable success of deep convolutional
neural networks (CNNs) in analyzing regular visual modalities such as 2D
images/videos and 3D volumes [21, 17, 42, 48, 38, 30, 12, 43]. In the
meantime, with the advanced 3D sensing technology, 3D point clouds become a
popular scheme for representing 3D objects and scenes and have been used in a
wide range of applications, including immersive telepresence [34], 3D city
reconstruction [22, 33], autonomous driving [4, 23], and virtual/augmented
reality [13, 40]. However, due to the irregular and orderless nature of point
clouds, applying the well-developed CNNs is highly non-trivial.

In order to apply the well-developed CNNs to irregular 3D point clouds, there
are two main classes of approaches: rasterization-based methods and point-
based methods. The former converts the input model into either a regular 3D
volume via voxelization or a collection of multi-view 2D images using
projection. Though one can directly apply the standard 2D/3D CNNs to the
regularly rasterized data, their performances are limited by either
irreversible information loss or high computational complexity. The latter
takes 3D point clouds directly as input, and extracts features from each
point. For example, the pioneering work, PointNet [35], can be viewed as
$1\times 1$ convolution applied to each point. Many follow-up works aim at
designing convolution-like operators on point clouds for feature extraction
[45, 14, 24, 49, 26, 54]. Each point-based convolution operator has its own
merits and limitations. Currently, there is no unified framework adopted by
the community. Moreover, it is observed that in point-based networks most of
the computational resources are wasted on the process of neighborhood
construction (e.g., $k$-NN) and structurization, not on the actual feature
extraction.

Figure 2: Illustration of three classes of methods for applying the regular-
data-fed CNNs to 3D point clouds. (a) Projection-based methods take 3D point
clouds as a collection of multi-view images, and adopts multiple 2D CNNs with
additional viewpoint aggregation. (b) Voxelization-based methods quantize
point clouds as volumetric grids, which are processed by 3D CNNs. (c) ParaNet
converts the input model into a PGI and adopts 2D CNNs.

Instead of seeking for convolution-like operators on unstructured points, we
consider a different paradigm that converts an irregular 3D point cloud to a
completely regular 2D representation where standard 2D CNNs can be applied. In
the computer graphics community, there are extensive research on mesh
parameterization. When mapping a 3D surface to a 2D domain, distortion is
inevitable unless the surface is developable. Therefore, most of the existing
techniques formulate parameterization as an optimization problem, aiming at
finding a balance between distortion and parameterization quality (e.g., the
number of singularities, continuity along cuts, etc). Although the state-of-
the-art methods [3, 31] can compute globally smooth and seamless
parameterization with low angle and area distortions, these approaches highly
rely on the explicit connectivity information encoded in meshes, extending
them to point clouds is highly non-trivial. Moreover, they usually work for
shapes with simple geometry and topology, since the complexity and difficulty
in solving the optimization problem increases significantly with the
complexity of geometry and topology.

In this paper we propose an end-to-end deep learning-based framework, called
ParaNet, which is capable of transforming an arbitrary 3D point cloud into a
completely regular 2D structure dubbed as point geometry image (PGI), which
encodes the Cartesian coordinates of 3D points $(x,y,z)$ as the color values
of pixels $(r,g,b)$. See Fig. 1. Since the generated PGIs can faithfully
represent the input shape and naturally fit into the existing CNN pipelines,
it bridges the gap between 3D geometry and the powerful deep learning
techniques well-investigated for regular data structures. Technically, we
design ParaNet as a two-stage network and train it in an end-to-end
unsupervised fashion. In the first stage, the points of an input shape pass
through a surface embedding module and are parameterized into a unit square
while maintaining local proximity as much as possible. In the second stage,
the grid resampling module converts the 2D samples to a PGI. The ParaNet can
be seamlessly coupled with downstream task networks established upon standard
convolution techniques to boost various point cloud applications such as
classification and upsampling.

Summary of Contributions. ParaNet is the first end-to-end deep neural network
for representing an irregular 3D point cloud as a regular 2D image amenable to
conventional CNNs driven by the standard convolution. Compared with multi-view
projection and voxelization based approaches [46, 16, 36, 55, 32, 39, 50, 19],
our PGI-based representation modality is more accurate and maintains
reversible, which means that the original point clouds can be directly
restored without borrowing additional computations. Furthermore, as
illustrated in Fig. 2, the generation process of PGI is differentiable and can
be optimized by a deep learning framework, whereas rasterization-based methods
adopt non-differentiable quantization. To validate its potential in point
cloud analysis, we combine ParaNet with two downstream task networks,
classification and patch-based upsampling, and obtain favorable performance
compared with state-of-the-art methods. The proposed regular point cloud
representation modality enables effortless adaptation of the mature visual
processing techniques originally developed for 2D images/videos, making the
wide range of 2D computational tools available to the point cloud community.

##  2 Related Work

###  2.1 Rasterization-based Point Cloud Analysis

View projection [46, 16, 36] generates a set of 2D images by setting virtual
cameras around 3D shape. Although multi-view projection is able to utilize
well-studied 2D CNN for feature extraction, it suffers from information loss
[19] and is not easily extended for fine-grained 3D tasks like segmentation
[5]. Another rasterization-based scheme is to voxelize point clouds into
quantized volumetric grids followed by standard 3D CNNs [55, 32, 39, 50, 19].
However, voxelization suffers from inaccurate data representation, or heavy
computation and large memory expense, depending on the resolution.

###  2.2 Convolution on Point Clouds

To eliminate quantization loss, latest researches focus on designing deep
learning based architectures directly operated on point clouds without any
rasterization procedure as pre-processing. Qi et al. [35] developed the
pioneering point-based deep set architecture, i.e., PointNet, that employs
shared multi-layer perceptrons for point-wise feature embedding and achieves
permutation-invariance by max-pooling, which is improved in PointNet++ [37] by
incorporating neighbor information to model local and global geometry
hierarchically. Based on such point-wise feature learning, more methods [45,
14, 24, 49, 26, 54] were proposed with different design in local feature
aggregations and hierarchical learning strategies. In general, existing
studies devote to tailoring convolution-like operators and CNN-like
architectures for point cloud feature extraction. Compared with rasterization-
based methods, point-based methods does not have quantization issue and can
achieve better performance. Unlike image analysis in which 2D CNNs are
dominating, currently there is no standard framework for 3D point clouds.
Moreover, directly processing irregular points is computationally expensive,
due to extra resources for structurization and neighborhood construction [29].

###  2.3 Parameterization and Geometry Images

In digital geometry processing, parameterization is a typical way to convert
an irregular representation (e.g., polygonal mesh, point clouds) to a regular
one. Since a map from 3D surfaces to 2D domains inevitably has distortion
unless the surface is developable, conformal parameterization is popular
thanks to its angle preserving property [10]. It is well known that area
distortion of a conformal parameterization is closely related to the
singularities of parameterization, i.e., the number, their locations and
indices. Modern approaches often formulate conformal parameterization as a
mixed-integer programming [2], in which the locations of singularities are
continuous, and their numbers and indices are discrete. Once the
parameterization is computed, one can generate a multi-chart geometry image by
partitioning the model via a motorcycle graph [3]. These methods are
theoretically sound and elegant, however, they are not practical for surfaces
with complex geometry and/or topology due to high computational cost.
Moreover, these approaches work only for single-component manifold surfaces,
whereas many man-made models are non-manifolds with multiple components. Note
that the proposed point geometry images are fundamentally different from the
classic geometry images in terms of generation and application.

###  2.4 Deformation-based 3D Shape Representation

FoldingNet, proposed by Yang et al. [57], is a novel deep auto-encoder. The
folding-based decoder, which consists of shared multi-layer perceptrons,
deforms a predefined 2D regular grid to the input 3D point cloud. The idea of
deforming a predefined template in shape modeling is also adopted in many
other works. For example, AtlasNet [8] achieves better reconstruction quality
by deforming multiple 2D planes. Deprelle et al. [6] proposed to learn
adaptive elementary structures that better fit the whole training data, and
3D-Coded [7] predicts correspondence for meshes via deforming the human
template.

These methods demonstrate the ability for deep neural networks to deform one
shape to another. However, existing works require either pre-defined or pre-
generated templates (e.g., a 2D grid with fixed resolution, a learned set of
elementary structures, or a 3D triangle mesh with fixed connectivity) to
reconstruct the input point cloud or mesh. In this paper, we show that the
opposite direction, i.e., unfolding/deforming arbitrary 3D point clouds to 2D
regular grids of arbitrary resolution, is also applicable, which has never
been explored before.

Figure 3: ParaNet consists of feature extractor and two core modules, namely
surface embedding and grid resampling. The feature extractor computes a global
feature for the input 3D point cloud. Using the global feature, the surface
unfolding module maps each 3D point to a unique 2D point in a square. Then the
grid resampling module generates a geometry image, which can be fed into a
CNN-based task network. The PGI generation is unsupervised and only
regularized by geometric constraints denoted as the parameterization loss. It
can also be jointly optimized with the specific task loss. Figure 4: The
surface embedding module assigns each 3D point $\mathbf{p}_{i}$ a unique pair
of 2D coordinates $(u_{i},v_{i})$, where $0\leq u_{i},v_{i}\leq 1$.

##  3 Proposed Method

###  3.1 Overview

This work explores the potential to encode irregular 3D geometry in regular 2D
grids and then apply conventional CNN architectures to perform specific point
cloud processing tasks. Given a point cloud $\mathbf{P}\in\mathbb{R}^{N\times
3}$ with $N$ 3D points $\mathbf{p}_{i}=(x_{i},y_{i},z_{i})$, where
$i=1,2,...,N$, we construct a completely regular representation structure
$\mathbf{G}\in\mathbb{R}^{m\times m\times 3}$, dubbed as a point geometry
image (PGI), capturing the Cartesian coordinates of spatial points in the
color channels of a 2D image. The generated PGI can be directly reshaped into
a point cloud $\mathbf{Q}\in\mathbb{R}^{M\times 3}$, where $M=m\times m$. In
practice, we always require $M>N$ so that the original points are redundantly
represented in the PGI, reducing the loss of geometric information
effectively.

The overall workflow of ParaNet is illustrated in Fig. 3. We start by
extracting the global feature $\mathbf{c}\in\mathbb{R}^{D}$ from the input
point cloud $\mathbf{P}$, and then generate a PGI in two stages. In the first
stage, the surface embedding module maps each 3D spatial point to a 2D point
over a unit square domain. In the second stage, the grid resampling module
produces a regular point geometry image by either grid resampling or a
differentiable relaxation of the nearest grid neighbor resampling. In our
implementation, we employ the popular DGCNN [52] as the backbone point feature
extractor.

Thanks to the regular structure, PGIs can be fed into the subsequent CNN-based
downstream task networks. In the paper, we demonstrate the potential of PGIs
on classification (Sec. 4.2) and upsampling (Sec. 4.3). Please refer to the
Supplementary Material for the detailed discussions on the differences between
our PGI and the classic GI representation.

###  3.2 Surface Embedding Module

The surface embedding module applies the unfolding operator twice to map each
3D point to a unique location in a unit square. See Fig. 4. Specifically, we
first duplicate the global feature $N$ times and arrange them in a row-wise
matrix $\mathbf{C}\in\mathbb{R}^{N\times D}$. Concatenating $\bf C$ with the
input Cartesian coordinates $\mathbf{P}$, we obtain the initial embeddings
$\mathbf{H}_{1}\in\mathbb{R}^{N\times(D+3)}$. We then apply a four-layer
perceptron to the rows of $\mathbf{H}_{1}$ to generate intermediate 2D
embeddings $\mathbf{E}_{1}\in\mathbb{R}^{N\times 2}$.

Figure 5: The unfold operators can be applied multiple times like function
composition. (a) shows the input point cloud $\bf P$, and (b), (c) and (d)
show the 2D embedded results of unfolding once, twice and three times,
respectively.

We apply the second unfolding operator in a similar fashion: we concatenate
$\mathbf{E}_{1}$ with $\mathbf{C}$ to form
$\mathbf{H}_{2}\in\mathbb{R}^{N\times(D+2)}$, and then feed $\mathbf{H}_{2}$
into another four-layer perceptron yields the final 2D embeddings
$\mathbf{E}\in\mathbb{R}^{N\times 2}$.

Putting it all together, we can express the surface embedding procedure as

$\mathbf{E}=\sigma(f_{2}([f_{1}([\mathbf{P};\mathbf{C}]);\mathbf{C}])),$ (1)

where $[\cdot;\cdot]$ denotes channel-wise concatenation between two feature
matrices, $f_{1}(\cdot)$ and $f_{2}(\cdot)$ are two separate unfolding units,
and $\sigma$ is the sigmoid activation function to normalize the embedded 2D
points in the range of $[0,1]$. The 2D embeddings after various unfold
operators are visualized in Fig. 5. We observe that two-time unfolding leads
to satisfactory results.

For convenience, we denote the $i$-th point contained in $\mathbf{E}$ as
$\mathbf{e}_{i}=(u_{i},v_{i})$. It is worth mentioning that there exist one-
to-one correspondence between the input point cloud $\mathbf{P}$ and the 2D
embeddings $\mathbf{E}$, i.e., each 3D point $\mathbf{p}_{i}\in\mathbb{R}^{3}$
is mapped to a unique 2D position $\mathbf{e}_{i}\in\mathbb{R}^{2}$.

To prevent the embedded 2D points $\mathbf{e}_{i}$ from being highly
clustered, we design a simple repulsion loss to stretch the point distribution
by imposing punishment on clustered points until they are separated by a
distance threshold $T$. Mathematically, we formulate the repulsion loss
$\ell_{rep}(\cdot)$ for every point $\mathbf{e}_{i}$ as

$\begin{split}\ell_{rep}(\mathbf{e}_{i})=&\begin{cases}0,&\text{if~{}$d_{i}\geq
T$}\\\ -\log(d_{i}+1-T),&\text{otherwise},\\\ \end{cases}\\\ &\text{where
}d_{i}=\min_{\mathbf{e}_{j}\in\mathbf{E}\backslash\\{\mathbf{e}_{i}\\}}\|\mathbf{e}_{i}-\mathbf{e}_{j}\|_{2}.\end{split}$
(2)

In our implementation we empirically set the distance threshold
$T=\frac{1}{m-1}$.

We define the objective function for the surface embedding module as:

$\mathcal{L}_{rep}=\frac{1}{N}\sum_{i=1}^{N}\ell_{rep}(\mathbf{e}_{i}).$ (3)

We observe that by minimizing $\mathcal{L}_{rep}$, the surface embedding
module is capable of unfolding 3D surfaces onto a unit square. However, the
embedded 2D points are not likely on the grid positions of the regular image
lattice. Therefore, we need a resampling module (Sec. 3.3) to produce a
regular geometry image.

###  3.3 Grid Resampling Module

Figure 6: The grid resampling module converts 2D parameters
$\\{(u_{i},v_{i})\\}_{i=1}^{N}$ to an $m\times m$ geometry image. It supports
both hard and soft resampling. The former maps each 2D sample to the nearest
grid point and the latter adopts distance-weighted interpolation to assign the
values for grid points.

The grid resampling module consumes the irregularly-arranged 2D embeddings
$\mathbf{E}\in\mathbb{R}^{N\times 2}$ and produces a regular PGI
$\mathbf{G}\in\mathbb{R}^{m\times m\times 3}$ where $m$ is the user-specified
resolution.

There are two common strategies in resampling, as shown in Fig 6. One is to
drag each 2D sample $\mathbf{e}_{i}$ to the nearest pixel, and the other is to
compute the pixel value using bilinear interpolation. Here, we denote the two
strategies as hard grid resampling and soft grid resampling, respectively.

Hard Grid Resampling. We start by constructing a canonical 2D grid uniformly
defined on a unit square domain, denoted as $\mathbf{Z}\in\mathbb{R}^{M\times
2}$ containing $M$ points. Each row of matrix $\mathbf{Z}$ corresponds to a
certain 2D point $\mathbf{z}_{i}$ within an $m\times m$ gridding structure.
Suppose that, in the point set $\mathbf{E}$, the nearest neighbor of
$\mathbf{z}_{i}$ is $\mathbf{e}_{j}$. Based on the exact row-wise matching
relations between $\mathbf{P}$ and $\mathbf{E}$, we can deduce a 2D array
$\mathbf{Q}\in\mathbb{R}^{M\times 3}$ whose $i$-th entry
$\mathbf{q}_{i}\in\mathbb{R}^{3}$ is the $j$-th entry of $\mathbf{P}$. We
formulate the hard resampling as follows:

$\mathbf{q}_{i}=\mathbf{p}_{j^{*}}\text{, where
}j^{*}=\mathop{\arg\min}_{j\in\\{1,2,\cdots,N\\}}\|\mathbf{z}_{i}-\mathbf{e}_{j}\|_{2}.$
(4)

After that, we reshape the $M\times 3$ array $\mathbf{Q}$ into an $m\times
m\times 3$ PGI $\mathbf{G}$. As two mapped samples $\mathbf{e}_{i}$ and
$\mathbf{e}_{j}$ could be cluttered, there is a possibility that some 3D
points are not included in $\mathbf{G}$. This problem can be reduced by
increasing the PGI resolution $m$. With a sufficiently high resolution, PGI is
guaranteed to include all points of the input point cloud, resulting a
lossless representation.

Soft Grid Resampling. Notice that the hard grid resampling workflow is non-
differentiable, which hinders end-to-end optimization. To overcome this
limitation, we further provide an annealing strategy to produce PGIs in a
differentiable manner. In soft resampling, we model $\mathbf{q}_{i}$ as a
weighted average of the input points $\mathbf{P}$. For grid point
$\mathbf{z}_{i}$, we select its $K$ nearest neighbors from $\mathbf{E}$, which
are denoted as $\\{\mathbf{e}_{k}\\}_{k=1}^{K}$, and their corresponding 3D
points $\\{\mathbf{p}_{k}\\}_{k=1}^{K}$. We then determine a set of weights
using the distances between $\mathbf{z}_{i}$ and its $K$ neighbors:

$\varpi_{ik}=\frac{\exp(-d_{ik}/|\tau|)}{\sum_{j=1}^{K}\exp(-d_{ij}/|\tau|)},$
(5)

where $d_{ik}=\|\mathbf{z}_{i}-\mathbf{e}_{k}\|_{2}$, and $\tau$ is a
temperature coefficient. We express the $i$-th entry of $\mathbf{Q}$ as

$\mathbf{q}_{i}=\sum_{k=1}^{K}\varpi_{ik}\mathbf{p}_{k}.$ (6)

During annealing, the distribution of weights gradually converges to the
Kronecker delta function with $\tau\rightarrow 0$, approximating the nearest
neighbor point. We can easily achieve this by adding a parameter
regularization term:

$\mathcal{L}_{ann}=|\tau|$ (7)

Combining Eq. (3) and Eq. (7), we obtain the objective function for the
unsupervised ParaNet:

$\mathcal{L}_{para}=\alpha\mathcal{L}_{rep}+\beta\mathcal{L}_{ann}.$ (8)

Note that annealing is dropped in hard grid resampling.

Fig. 7 shows some visual examples by running ParaNet over input shapes after
sufficient iterations. We can observe that the PGIs accurately represent the
original point clouds while maintaining satisfactory smoothness.

Figure 7: PGI results. Row 1: 3D meshes. Row 2: Point clouds $\mathbf{P}$. Row
3: 2D embedded results $\mathbf{E}_{2}$. Row 4: Point geometry images of
resolution $128\times 128$. From left to right, the Chamfer distance errors
($10^{-8}$) are 1.71, 1.28, 1.89, 1.04, 1.72 and 1.78 respectively. See
Supplementary Material for more results.

##  4 Experimental Results

###  4.1 Training Strategies

As an unsupervised learning approach, ParaNet enables flexible and customized
optimization schemes. In order to fit the specific characteristics and
requirements of different application scenarios, we investigated three types
of training strategies: 1) joint optimization; 2) offline independent
optimization; and 3) online independent optimization. Note that we employ soft
grid resampling for joint optimization, and hard grid resampling for
independent optimization.

Joint Optimization (JO). ParaNet and the downstream task network are jointly
trained with a combined objective function involving the parameterization loss
$\mathcal{L}_{para}$ and the task-specific loss $\mathcal{L}_{task}$. We pre-
train the ParaNet before connecting it with the task network to stabilize and
speed up the joint training process. The JO strategy benefits from the task-
oriented learning paradigm. Thanks to the implicit guidance of the task
network, the generated PGIs can be automatically adjusted to suit downstream
applications.

Offline Independent Optimization (Off-IO). We start by pre-training ParaNet on
the whole training set, and then freeze its parameters before connecting it
with the downstream task network. Accordingly, we only exploit the task-
specific loss $\mathcal{L}_{task}$ to supervise the overall framework. We
observe that the independent optimization strategy usually leads to PGIs with
higher representation accuracy, while sacrificing the possibility of
adaptively adjusting the geometry parameterization process.

Online Independent Optimization (On-IO). We separately optimize ParaNet for
every input shape with adequate iterations to obtain a nearly lossless
representation during both training and testing. We can achieve this by
generating an accurate PGI for every single input shape over the training and
testing sets. The generated PGIs are cached and further fed into the
downstream task network. Obviously, the On-IO strategy is much slower than the
offline strategy, but it produces highly-precise representation of input point
clouds with little information loss. In practice, On-IO can be greatly
accelerated by feeding batch-wise inputs.

###  4.2 Shape Classification

Figure 8: ParaNet fed into a classification network.

Task network. Image classification has been adequately addressed in numerous
CNN-based architectures. In our shape classification task, we adopted the
commonly-used ResNet-18 [12] architecture, which consists of five convolution
layers, as illustrated in Fig. 8. Considering that the original point cloud is
redundantly represented in the PGI, we applied three spatial attention modules
(SAM) to highlight informative regions, and another two channel attention
modules (CAM) were equipped to boost feature learning. Afterwards, the
extracted feature maps were squeezed into a one-dimensional feature vector by
global average pooling, and passed through a fully-connected layer followed by
Softmax to predict the class probabilities. We employed cross-entropy as the
task loss to supervise the task network. Notice that we did not utilize the
ImageNet [21] pretrained parameters for fair comparisons.

Table 1: Classification accuracy on ModelNet40. Method | Input | Accuracy (%)
---|---|---
VoxNet [32] | voxels | 85.9
KD-Net [19] | 1024 points | 91.8
PointNet [35] | 1024 points | 89.2
PointNet++ [37] | 5000 points + normal | 91.9
3D-GCN [27] | 1024 points | 92.1
SpiderCNN [56] | 1024 points + normal | 92.4
PointConv [54] | 1024 points + normal | 92.5
A-CNN [20] | 1024 points | 92.6
KPConv [47] | 6800 points | 92.9
DGCNN [52] | 1024 points | 92.9
ParaNet-SC (JO) | 1024 points | 93.1
ParaNet-SC (Off-IO) | 1024 points | 92.7
ParaNet-SC (On-IO) | 1024 points | 92.9

Development protocols. We evaluated the shape classification performance of
our framework over the ModelNet40 [55] benchmark dataset, which contains
$12311$ CAD models from $40$ man-made object categories. Following the
official train-test partition, we employed $9843$ shapes for training and the
rest $2468$ for testing. We used Poisson Disk Sampling to generate $1024$
points from the original mesh. Common data augmentation techniques including
random point dropout and jittering point positions by Gaussian noises were
employed to enrich the training data and boost the generalization ability of
the learning model. For parameters in Eq. 8, we empirically set the parameters
$\alpha=1$ and $\beta=0.1$.

Comparison with state-of-the-art methods. We report the performances of the
proposed framework under different training strategies in Table 1, which
includes $10$ classic and latest state-of-the-art competitors for comparisons.
As shown in Table 1, our ParaNet model under the JO strategy achieves the
highest classification accuracy compared with other rasterization- and point-
based frameworks. An interesting observation is that, although the Off-IO and
On-IO schemes typically produce more accurate PGI representations, the JO
scheme shows superiority benefiting from the end-to-end task-oriented learning
paradigm.

Ablation studies. We conducted ablation studies to explore the effectiveness
and potential of the proposed PGI representation. The results of different
model variants of the ParaNet-JO are reported in Table 2. To demonstrate the
necessity of representing raw point clouds as PGIs, we removed the ParaNet
from the full pipeline and directly and randomly reshape the input point cloud
from a 2D array to the form of a color image, denoted as ParaNet-SC-random.
Under this circumstance, the performance significantly drops from $93.1\%$ to
$45.6\%$, which reveals that it is non-trivial to apply standard convolutions
to process irregular and unstructured point clouds. Then, we designed a naive
baseline CNN architecture serving as the downstream task network to form the
method ParaNet-SC-baseline. Details of the baseline design can be found in the
Supplemental Material. This simple variant also achieves favorable performance
of $91.9\%$. When we replaced the backbone network in the baseline with a more
powerful ResNet18 [12] encoder, the classification accuracy for ParaNet-SC-
ResNet18 increases to $92.6\%$. Moreover, we explored whether the success of
image attention mechanisms can be extended to our task setting. Empirically,
we observe performance decrease when the SAM or the CAM modules are removed
from the full pipeline, which implies that our method enables direct
adaptation of existing visual techniques that have been proven to be effective
in image domain applications. We can expect to achieve further performance
gains as more advanced 2D techniques are introduced to cooperate with ParaNet.

We further explored the impact of generating the PGIs with different
resolutions. As discussed before, the original point cloud is redundantly
represented in image pixels. Predictably, the information loss will decrease
as the image resolution increases. Thus, we are able to flexibly adjust the
representation accuracy at the cost of the representation redundancy. As
reported in Table 3, low resolution of the parameterized PGI leads to degraded
performance, since a noticeable number of points may be lost during grid
resampling. When the image resolution is sufficiently large to accurately
represent the point cloud, the performance gain vanishes accordingly.

Table 2: Quantitative comparisons among different variants under the JO strategy over the ModelNet40 dataset. Method | Accuracy (%)
---|---
ParaNet-SC-random | 45.6
ParaNet-SC-baseline | 91.9
ParaNet-SC-ResNet18 | 92.6
ParaNet-SC w/o CAM | 92.8
ParaNet-SC w/o SAM | 93.0
ParaNet-SC (full) | 93.1
Table 3: Impacts of different resolution of PGIs. Resolution | Accuracy (%) | CD Errors
---|---|---
$32\times 32$ | 91.6 | $1.14\times 10^{-3}$
$48\times 48$ | 92.5 | $2.12\times 10^{-4}$
$64\times 64$ | 93.0 | $4.49\times 10^{-5}$
$96\times 96$ | 93.1 | $7.38\times 10^{-6}$
$128\times 128$ | 93.1 | $\mathbf{2.32\times 10^{-6}}$

###  4.3 Point Cloud Upsampling

Figure 9: ParaNet fed into a upsampling network. Figure 10: Point cloud
upsampling with a factor $4\times$. To ease checking the result quality, we
show the triangle meshes reconstructed from the upsampled point clouds. See
Supplementary Material for more results.

Task network. Extensive studies have been conducted for image super-
resolution, which provide valuable insights for point cloud upsampling. We
followed the common patch-based processing pipeline [25] to achieve $4\times$
upsampling for non-uniform inputs. As illustrated in Fig. 9, we first
converted input patches into PGIs, which passed through the subsequent
convolutional layers for feature embedding. The back projection mechanism [11]
was introduced to fully exploit the mutual dependencies of low- and high-
resolution contents, and self-attention operations [51] were introduced to
encode global prior of the input patch. We utilized the Earth Mover’s Distance
(EMD) to supervise point cloud generation, and employed a uniformity
constraint to enhance point distribution. Different from shape classification,
the upsampling task requires highly-accurate representations with little
information loss. Therefore, we employed the On-IO scheme for training.

Development protocols. Following previous study [25], we collected $147$ 3D
mesh models from the released datasets of PU-Net [59] and MPU [58] as well as
Visionair repository [1], among which $120$ for training and $27$ for testing.
We used Poisson disk sampling to sample $8192$ points as ground-truths, and
randomly selected $2048$ points as inputs. The size of extracted patches is
$256$, and the resolution of converted PGIs is $128\times 128$. We employed
three evaluation metrics: 1) point-to-surface (P2F) distance, 2) Chamfer
distance (CD), and 3) Hausdorff distance (HD).

Comparison with state-of-the-art methods. We compared our pipeline, i.e.,
ParaNet-PU, with three deep point upsampling models by retraining their models
on our dataset. Note that PU-GAN [25] additionally adopts adversarial
learning, hence, we also report the performance of its point generator, i.e.,
PU-GAN w/o D. As reported in Table 4, our PGI-driven framework outperforms
these point-based architectures. We provided some visual comparisons of
upsampled point clouds and the reconstructed surfaces using [18] in Fig. 10,
from which we can observe that our results better restore detailed shape
geometry and contain less noises that deviating from the underlying surface.

Table 4: Quantitative comparison of different methods for $4\times$ point cloud upsampling. The best results are highlighted in bold. Method | P2F | CD | HD
---|---|---|---
| $(10^{-3})$ | $(10^{-3})$ | $(10^{-3})$
PU-Net [59] | 6.97 | 0.72 | 8.93
PU-GAN w/o D [25] | 4.75 | 0.59 | 7.44
MPU [58] | 3.93 | 0.49 | 6.06
PU-GAN [25] | 2.40 | 0.29 | 4.75
ParaNet-PU | 2.26 | 0.28 | 4.44

##  5 Conclusion

We proposed an end-to-end deep neural network, i.e., ParaNet, to convert
irregular 3D point clouds into a completely regular 2D representation
modality, which is an ideal input of standard 2D convolutional neural
networks. ParaNet can be trained either in an unsupervised manner or jointly
with the specific downstream task network. We demonstrated the effectiveness
of ParaNet on shape classification and point cloud upsampling. Extensive
experiments demonstrate the superiority of our point cloud processing
pipelines. Our method naturally bridges the gap between 2D CNNs and 3D point
cloud processing. We believe it will open a door to apply well-developed 2D
deep learning tools to 3D applications. In future studies, we would like to
explore richer applications based on point geometry image for point cloud
analysis.

## References

  * [1] Visionair.  http://www.infra-visionair.eu/, 2019-07.
  * [2] David Bommes, Henrik Zimmer, and Leif Kobbelt.  Mixed-integer quadrangulation.  ACM Trans. Graph., 28(3):77, 2009.
  * [3] Marcel Campen, David Bommes, and Leif Kobbelt.  Quantized global parametrization.  ACM Trans. Graph., 34(6):192:1–192:12, 2015.
  * [4] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia.  Multi-view 3d object detection network for autonomous driving.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1907–1915, 2017.
  * [5] Angela Dai, Angel X Chang, Manolis Savva, Maciej Halber, Thomas Funkhouser, and Matthias Nießner.  Scannet: Richly-annotated 3d reconstructions of indoor scenes.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5828–5839, 2017.
  * [6] Theo Deprelle, Thibault Groueix, Matthew Fisher, Vladimir Kim, Bryan Russell, and Mathieu Aubry.  Learning elementary structures for 3d shape generation and matching.  In Advances in Neural Information Processing Systems, pages 7435–7445, 2019.
  * [7] Thibault Groueix, Matthew Fisher, Vladimir G Kim, Bryan C Russell, and Mathieu Aubry.  3d-coded: 3d correspondences by deep deformation.  In Proceedings of the European Conference on Computer Vision (ECCV), pages 230–246, 2018.
  * [8] Thibault Groueix, Matthew Fisher, Vladimir G Kim, Bryan C Russell, and Mathieu Aubry.  A papier-mâché approach to learning 3d surface generation.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 216–224, 2018.
  * [9] Xianfeng Gu, Steven J. Gortler, and Hugues Hoppe.  Geometry images.  In Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques, pages 355–361, 2002.
  * [10] Xianfeng Gu and Shing-Tung Yau.  Global conformal parameterization.  In Leif Kobbelt, Peter Schröder, and Hugues Hoppe, editors, First Eurographics Symposium on Geometry Processing, volume 43, pages 127–137, 2003.
  * [11] Muhammad Haris, Gregory Shakhnarovich, and Norimichi Ukita.  Deep back-projection networks for super-resolution.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1664–1673, 2018.
  * [12] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
  * [13] Robert Held, Ankit Gupta, Brian Curless, and Maneesh Agrawala.  3d puppetry: a kinect-based interface for 3d animation.  In UIST, pages 423–434. Citeseer, 2012.
  * [14] Varun Jampani, Martin Kiefel, and Peter V Gehler.  Learning sparse high dimensional filters: Image filtering, dense crfs and bilateral neural networks.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4452–4461, 2016.
  * [15] Miao Jin, Junho Kim, Feng Luo, and Xianfeng Gu.  Discrete surface ricci flow.  IEEE Trans. Vis. Comput. Graph., 14(5):1030–1043, 2008.
  * [16] Evangelos Kalogerakis, Melinos Averkiou, Subhransu Maji, and Siddhartha Chaudhuri.  3d shape segmentation with projective convolutional networks.  In proceedings of the IEEE conference on computer vision and pattern recognition, pages 3779–3788, 2017.
  * [17] Andrej Karpathy, George Toderici, Sanketh Shetty, Thomas Leung, Rahul Sukthankar, and Li Fei-Fei.  Large-scale video classification with convolutional neural networks.  In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 1725–1732, 2014.
  * [18] Michael Kazhdan and Hugues Hoppe.  Screened poisson surface reconstruction.  ACM Transactions on Graphics (ToG), 32(3):1–13, 2013.
  * [19] Roman Klokov and Victor Lempitsky.  Escape from cells: Deep kd-networks for the recognition of 3d point cloud models.  In Proceedings of the IEEE International Conference on Computer Vision, pages 863–872, 2017.
  * [20] Artem Komarichev, Zichun Zhong, and Jing Hua.  A-cnn: Annularly convolutional neural networks on point clouds.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7421–7430, 2019.
  * [21] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.  Imagenet classification with deep convolutional neural networks.  In Advances in neural information processing systems, pages 1097–1105, 2012.
  * [22] Florent Lafarge and Clément Mallet.  Creating large-scale city models from 3d-point clouds: a robust approach with hybrid representation.  International Journal of Computer Vision, 99(1):69–85, 2012.
  * [23] Bo Li.  3d fully convolutional network for vehicle detection in point cloud.  In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 1513–1518. IEEE, 2017.
  * [24] Jiaxin Li, Ben M Chen, and Gim Hee Lee.  So-net: Self-organizing network for point cloud analysis.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 9397–9406, 2018.
  * [25] Ruihui Li, Xianzhi Li, Chi-Wing Fu, Daniel Cohen-Or, and Pheng-Ann Heng.  Pu-gan: a point cloud upsampling adversarial network.  In Proceedings of the IEEE International Conference on Computer Vision, pages 7203–7212, 2019.
  * [26] Yangyan Li, Rui Bu, Mingchao Sun, Wei Wu, Xinhan Di, and Baoquan Chen.  Pointcnn: Convolution on x-transformed points.  In Advances in neural information processing systems, pages 820–830, 2018.
  * [27] Zhi-Hao Lin, Sheng-Yu Huang, and Yu-Chiang Frank Wang.  Convolution in the cloud: Learning deformable kernels in 3d graph convolution networks for point cloud analysis.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1800–1809, 2020.
  * [28] Ligang Liu, Lei Zhang, Yin Xu, Craig Gotsman, and Steven J. Gortler.  A local/global approach to mesh parameterization.  Comput. Graph. Forum, 27(5):1495–1504, 2008.
  * [29] Zhijian Liu, Haotian Tang, Yujun Lin, and Song Han.  Point-voxel cnn for efficient 3d deep learning.  In Advances in Neural Information Processing Systems, pages 965–975, 2019.
  * [30] Jonathan Long, Evan Shelhamer, and Trevor Darrell.  Fully convolutional networks for semantic segmentation.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3431–3440, 2015.
  * [31] Max Lyon, Marcel Campen, David Bommes, and Leif Kobbelt.  Parametrization quantization with free boundaries for trimmed quad meshing.  ACM Trans. Graph., 38(4):51:1–51:14, 2019.
  * [32] Daniel Maturana and Sebastian Scherer.  Voxnet: A 3d convolutional neural network for real-time object recognition.  In 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 922–928, 2015.
  * [33] Przemyslaw Musialski, Peter Wonka, Daniel G Aliaga, Michael Wimmer, Luc Van Gool, and Werner Purgathofer.  A survey of urban reconstruction.  In Computer Graphics Forum, volume 32, pages 146–177. Wiley Online Library, 2013.
  * [34] Sergio Orts-Escolano, Christoph Rhemann, Sean Fanello, Wayne Chang, Adarsh Kowdle, Yury Degtyarev, David Kim, Philip L Davidson, Sameh Khamis, Mingsong Dou, et al.  Holoportation: Virtual 3d teleportation in real-time.  In Proceedings of the 29th Annual Symposium on User Interface Software and Technology, pages 741–754. ACM, 2016.
  * [35] Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas.  Pointnet: Deep learning on point sets for 3d classification and segmentation.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 652–660, 2017.
  * [36] Charles R Qi, Hao Su, Matthias Nießner, Angela Dai, Mengyuan Yan, and Leonidas J Guibas.  Volumetric and multi-view cnns for object classification on 3d data.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5648–5656, 2016.
  * [37] Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J Guibas.  Pointnet++: Deep hierarchical feature learning on point sets in a metric space.  In Advances in neural information processing systems, pages 5099–5108, 2017.
  * [38] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun.  Faster r-cnn: Towards real-time object detection with region proposal networks.  In Advances in neural information processing systems, pages 91–99, 2015.
  * [39] Gernot Riegler, Ali Osman Ulusoy, and Andreas Geiger.  Octnet: Learning deep 3d representations at high resolutions.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3577–3586, 2017.
  * [40] José Miguel Santana, Jochen Wendel, Agustín Trujillo, José Pablo Suárez, Alexander Simons, and Andreas Koch.  Multimodal location based services—semantic 3d city data as virtual and augmented reality.  In Progress in Location-based Services 2016, pages 329–353. Springer, 2017.
  * [41] Alla Sheffer, Bruno Lévy, Maxim Mogilnitsky, and Alexander Bogomyakov.  ABF++: fast and robust angle based flattening.  ACM Trans. Graph., 24(2):311–330, 2005.
  * [42] Karen Simonyan and Andrew Zisserman.  Two-stream convolutional networks for action recognition in videos.  In Advances in neural information processing systems, pages 568–576, 2014.
  * [43] Shuran Song, Fisher Yu, Andy Zeng, Angel X Chang, Manolis Savva, and Thomas Funkhouser.  Semantic scene completion from a single depth image.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1746–1754, 2017.
  * [44] Boris Springborn, Peter Schröder, and Ulrich Pinkall.  Conformal equivalence of triangle meshes.  ACM Trans. Graph., 27(3):77, 2008.
  * [45] Hang Su, Varun Jampani, Deqing Sun, Subhransu Maji, Evangelos Kalogerakis, Ming-Hsuan Yang, and Jan Kautz.  Splatnet: Sparse lattice networks for point cloud processing.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2530–2539, 2018.
  * [46] Hang Su, Subhransu Maji, Evangelos Kalogerakis, and Erik Learned-Miller.  Multi-view convolutional neural networks for 3d shape recognition.  In Proceedings of the IEEE International Conference on Computer Vision (ICCV), pages 945–953, 2015.
  * [47] Hugues Thomas, Charles R Qi, Jean-Emmanuel Deschaud, Beatriz Marcotegui, François Goulette, and Leonidas J Guibas.  Kpconv: Flexible and deformable convolution for point clouds.  In Proceedings of the IEEE International Conference on Computer Vision, pages 6411–6420, 2019.
  * [48] Du Tran, Lubomir Bourdev, Rob Fergus, Lorenzo Torresani, and Manohar Paluri.  Learning spatiotemporal features with 3d convolutional networks.  In Proceedings of the IEEE international conference on computer vision, pages 4489–4497, 2015.
  * [49] Nitika Verma, Edmond Boyer, and Jakob Verbeek.  Feastnet: Feature-steered graph convolutions for 3d shape analysis.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2598–2606, 2018.
  * [50] Peng-Shuai Wang, Yang Liu, Yu-Xiao Guo, Chun-Yu Sun, and Xin Tong.  O-cnn: Octree-based convolutional neural networks for 3d shape analysis.  ACM Transactions on Graphics (TOG), 36(4):1–11, 2017.
  * [51] Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He.  Non-local neural networks.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7794–7803, 2018.
  * [52] Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E Sarma, Michael M Bronstein, and Justin M Solomon.  Dynamic graph cnn for learning on point clouds.  Acm Transactions On Graphics (tog), 38(5):1–12, 2019.
  * [53] Sanghyun Woo, Jongchan Park, Joon-Young Lee, and In So Kweon.  Cbam: Convolutional block attention module.  In Proceedings of the European conference on computer vision (ECCV), pages 3–19, 2018.
  * [54] Wenxuan Wu, Zhongang Qi, and Li Fuxin.  Pointconv: Deep convolutional networks on 3d point clouds.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 9621–9630, 2019.
  * [55] Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and Jianxiong Xiao.  3d shapenets: A deep representation for volumetric shapes.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1912–1920, 2015.
  * [56] Yifan Xu, Tianqi Fan, Mingye Xu, Long Zeng, and Yu Qiao.  Spidercnn: Deep learning on point sets with parameterized convolutional filters.  In Proceedings of the European Conference on Computer Vision (ECCV), pages 87–102, 2018.
  * [57] Yaoqing Yang, Chen Feng, Yiru Shen, and Dong Tian.  Foldingnet: Point cloud auto-encoder via deep grid deformation.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 206–215, 2018.
  * [58] Wang Yifan, Shihao Wu, Hui Huang, Daniel Cohen-Or, and Olga Sorkine-Hornung.  Patch-based progressive 3d point set upsampling.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5958–5967, 2019.
  * [59] Lequan Yu, Xianzhi Li, Chi-Wing Fu, Daniel Cohen-Or, and Pheng-Ann Heng.  Pu-net: Point cloud upsampling network.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2790–2799, 2018.
  * [60] Hui Zhao, Kehua Su, Chenchen Li, Boyu Zhang, Lei Yang, Na Lei, Xiaoling Wang, Steven J. Gortler, and Xianfeng Gu.  Mesh parametrization driven by unit normal flow.  Comput. Graph. Forum, 39(1):34–49, 2020.

Supplementary Materials

## 1\. Differences Between the Proposed PGIs and the Classic GIs

Both the proposed point geometry images (PGIs) and the classic geometry images
(GIs) [9] are able to represent 3D shapes with a completely regular structure,
i.e., a color image. When the image resolution is sufficiently high, both can
encode 3D geometry in a nearly lossless manner. Besides the above two common
characteristics, the proposed PGIs are fundamentally different from the
classic GIs in terms of generation and application.

GIs are generated through mesh-based parameterization algorithms, which are
usually formulated as either a non-linear optimization problem that minimizes
area and/or angle distortions [2, 28, 44, 41] or a non-linear PDE that
iteratively deforms the input 3D shape to a target shape with constant
Gaussian curvature [60, 15]. GIs are typically judged by the overall angle and
area distortion, and applied to GPU-accelerated rendering and texture mapping
(e.g., in movies and video games). The mesh-based algorithms are able to
compute high-quality GIs for 3D models with simple geometry and topology.
However, increasing the complexity of geometry/topology often poses
significant challenges to these methods. Furthermore, they only work for the
manifold meshes with a single component, whereas the non-manifold models with
multiple connected components are common in real-world applications. Last but
not least, extending the mesh-based parameterization algorithms to point
clouds is non-trivial due to their high reliance on the connectivity
information provided by meshes.

Our method is data-driven and can work for arbitrary manifold and non-manifold
models with single or multiple components. Since ParaNet makes no use of
connectivity, it is ideal for processing point clouds. As an end-to-end
learning framework, it is trained in an unsupervised manner. Moreover, it can
be combined and jointly optimized with downstream task networks. Since PGIs
can be seamlessly coupled with conventional 2D CNN architectures, they are
particularly useful for visual analysis tasks, such as classification,
segmentation, super-resolution and recognition, in which 2D CNNs have already
demonstrated great potentials.

Due to these fundamental differences, we do not naïvely follow conventional GI
metrics to measure PGIs. In our paper, we evaluate PGIs in terms of: 1) the
information loss between the input point cloud and the PGI-induced point
cloud; and 2) the performance of specific downstream applications when
applying 2D CNNs to the generated PGIs.

In Figure 11, we provide more examples of the 2D embeddings and PGIs. The
displayed models belong to the ModelNet40 [55] repository. We observe that
ParaNet works well for diverse object categories.

Figure 11: More results of the 2D embeddings and PGIs produced by ParaNet.

## 2\. Details of Network Architectures

### 2.1 ParaNet

In our implementation, we employed the popular DGCNN [52] architecture for
feature extraction of input point clouds. The extracted global feature for an
input point cloud is a $512$-dimensional feature vector, i.e.,
$\mathbf{c}\in\mathbb{R}^{512}$. In the surface embedding module, each
unfolding operator was implemented as a shared four-layer perceptron. The
dimensions of the first and the second unfolding operator are
$\\{512+3,256,128,64,2\\}$ and $\\{512+2,256,128,64,2\\}$, respectively. For
the annealing regularization term in soft grid resampling, we initiated the
learnable temperature coefficient as $\tau=10^{-5}$. Besides, the number of
nearest neighbors selected for each grid point was set as $K=5$. In practice,
we discover that the soft grid resampling module is not sensitive to the value
of $K$ due to the effective annealing process.

### 2.2 ParaNet-SC: Downstream Task Network for 3D Shape Classification

Table 5: Components and outputs of ParaNet-SC. Components | Outputs ([Channels, Height, Width])
---|---
PGI | [3, 128, 128]
Conv-1 | [64, 64, 64]
SAM-1 | [64, 64, 64]
Conv-2 | [64, 64, 64]
SAM-2 | [64, 64, 64]
Conv-3 | [128, 32, 32]
SAM-3 | [128, 32, 32]
Conv-4 | [256, 16, 16]
CAM-4 | [256, 16, 16]
Conv-5 | [512, 8, 8]
CAM-5 | [512, 8, 8]
Global Average Pooling | [512, 1, 1]
Fully Connected Layer | [40]

The ParaNet-SC framework tailored for 3D shape classification adopted the
ResNet18 [12] architecture, which has been widely applied to a variety of
image domain visual applications, as the backbone feature extractor. Following
common notations, the standard ResNet18 feature encoder can be partitioned and
packed into five sequential convolution stages, denoted as: $\\{$Conv-1,
Conv-2, Conv-3, Conv-4, Conv-5$\\}$. Note that we dropped the max-pooling
layer in the Conv-1 stage to enlarge the feature resolution.

Considering that the generated PGIs usually represent the original point
clouds in a highly-redundant manner, it is necessary to deal with the
information redundancy issue. Thus, we introduced the popular attention
mechanism, as proposed in [53], which has already gained remarkable success in
numerous image/video processing tasks. In [53], a spatial attention module
(SAM) is applied to aggregate inter-spatial feature dependency and suppress
the less informative image parts, and a channel attention module (CAM) is
applied to exploit inter-channel relationships and emphasize the most
discriminative feature subspace. Intuitively, the combination of SAM and CAM
enables the network to learn where and what to emphasize or suppress,
effectively enhancing the representation capability of the learned
convolutional feature maps. Please refer to the original paper [53] for
complete and detailed descriptions of SAM and CAM. To further clarify the
structure composition of ParaNet-SC, we list all the included major components
as well as the corresponding output dimensions in Table 5.

### 2.3 ParaNet-PU: Downstream Task Network for Point Cloud Upsampling

The ParaNet-PU framework is designed to achieve $4\times$ point cloud
upsampling following a patch-based processing pipeline. Specifically, we tend
to upsample a $256$-point sparse patch to a $1024$-point dense patch through
ParaNet-PU. As described in the paper, we explored three different types of
training strategies (i.e., JO, Off-IO, and On-IO) for connecting ParaNet with
a specific downstream task network. For the shape classification task, we can
observe that the JO scheme outperforms the other two choices. Although joint
optimization always results in higher information loss, i.e., more points may
be lost in the generated PGI, it benefits from end-to-end training in which
the PGI representation can be adaptively adjusted to some extent, boosting the
overall classification performance. Different from shape classification that
is relatively less sensitive to information loss, point cloud upsampling
highly relies on the original geometry and topology information provided by
the input sparse point set. Even if only a few critical points are missing, it
would be rather difficult to accurately infer the underlying surface. Thus, we
adopted the On-IO scheme for training our ParaNet-PU framework, generating
PGIs with the resolution of $128\times 128$ for all training and testing
patches with $256$ points.

Technically, we started by feeding the input PGI into a series of
convolutional layers with large sized filters followed by max-pooling
operations to reduce feature resolution. We introduced self-attention, as
proposed in [51], to build long-range point dependency and encode the global
shape prior of the input patch. After that, we employed the back projection
mechanism [11] to fully exploit the mutual relationships of low- and high-
resolution feature maps. As investigated in [11], the back projection workflow
involves the up-projection units (UPU) and the down-projection units (DPU).
Following the original design, we sequentially organized the two types of
projection units, alternating between the UPU and the DPU, which can be
considered as a self-correcting learning procedure providing a beneficial
error feedback mechanism. In Table 6, we summarize the major components and
the corresponding output dimensions. As suggested, we concatenated the up-
sampled feature maps produced by the three UPUs and then applied $1\times 1$
convolutions to generate 3D Cartesian coordinates. In the end, we could easily
reshape the predictions into a 2D array, representing the upsampled point
cloud patch.

During training, we employed the Earth Mover’s Distance (EMD) to measure the
difference between upsampling results and ground truths. To promote the
uniformity of the generated patches, we additionally imposed a point
distribution constraint, expecting that the nearest neighbors are neither too
close nor too far from each other. To achieve this, we adopted a modified
repulsion loss, which was originally proposed in PU-Net [59]. Given an
upsampled patch $\mathbf{X}=\\{{x_{i}}\\}_{i=1}^{U}$ and the corresponding
ground truth patch $\mathbf{Y}=\\{{y_{i}}\\}_{i=1}^{U}$, we formulated the
uniformity constraint in a fast-decaying manner as follows

$\mathcal{L}_{uni}=\gamma\cdot\frac{1}{U}\sum_{i=1}^{U}-r(x_{i})\cdot\exp(-\frac{{r(x_{i})}^{2}}{2h^{2}}),$
(9)

where $r(x_{i})$ represents the Euclidean distance between $x_{i}$ and its
nearest neighbor, and $\gamma$ controls the contribution of
$\mathcal{L}_{uni}$ when combined with the EMD loss function for point
generation supervision. Empirically, we set the parameter $\gamma$ to be
$0.01$. Actually, the above constraint tends to push the nearest neighbor
distance $r(x_{i})$ to a fixed value of $h$, which is chosen as the average
nearest neighbor distance in patch $\mathbf{Y}$ as follows

$h=\frac{1}{U}\sum_{i=1}^{U}r(y_{i}).$ (10)

To further boost the generation of uniformly distributed point clouds, we
introduced side supervisions to enhance the learned intermediate feature
representations. In our implementation, we applied additional $1\times 1$
convolutions to the outputs of DPU-1 and DPU-2, predicting sparse patches with
$256$ points. Then we used farthest point sampling to downsample the
corresponding ground truth patch with $1024$ points into a uniform $256$-point
patch, which serves as the intermediate supervision information to guide the
predicted side outputs using the weighted combination of EMD and uniformity
constraint.

Table 6: Components and outputs of ParaNet-PU. Components | Outputs ([Channels, Height, Width])
---|---
PGI | [3, 128, 128]
Conv ($12\times 12$) | [64, 64, 64]
Conv ($8\times 8$) | [128, 32, 32]
Self-Att | [128, 32, 32]
Conv ($6\times 6$) | [256, 16, 16]
Self-Att | [256, 16, 16]
UPU-1 | [256, 32, 32]
DPU-1 | [256, 16, 16]
UPU-2 | [256, 32, 32]
DPU-2 | [256, 16, 16]
UPU-3 | [256, 32, 32]
Up-Concat | [768, 32, 32]
Conv ($1\times 1$) | [3, 32, 32]
Reshape | [3, 1024]

## 3\. Backbone Point Feature Extractor

Table 7: Comparison of Different Point Feature Extractors for 3D Shape Classification. Method | Rep Acc (CD) | Cls Acc (%)
---|---|---
ParaNet-SC-DGCNN (JO) | $2.32\times 10^{-6}$ | 93.1
ParaNet-SC-PointNet (JO) | $9.61\times 10^{-6}$ | 93.1
Table 8: Comparison of Different Point Feature Extractors for Point Cloud Upsampling. Method | Rep Acc (CD) | P2F | CD | HD
---|---|---|---|---
| $(10^{-8})$ | $(10^{-3})$ | $(10^{-3})$ | $(10^{-3})$
ParaNet-PU-DGCNN (On-IO) | 2.09 | 2.26 | 0.28 | 4.44
ParaNet-PU-PointNet (On-IO) | 7.15 | 2.27 | 0.28 | 4.49

In our paper, we employed DGCNN [52] as the backbone point feature extractor
of ParaNet. Actually, we argue that DGCNN can be seamlessly replaced by any
other kind of deep set architecture, and the overall performance over a
specific downstream application is dominated by the task network and less
sensitive to the point feature extractor. To empirically demonstrate this
point, we replaced DGCNN [52] with PointNet [35] in the ParaNet for generating
PGIs while maintaining the subsequent components. In Tables 7 and 8, we
compared the data representation accuracy (Rep Acc) under the Chamfer distance
(CD) metric and the overall task performance of DGCNN-driven (i.e., ParaNet-
SC/PU-DGCNN) and PointNet-driven (i.e., ParaNet-SC/PU-PointNet) processing
pipelines. We can observe that although PointNet, a backbone point encoder
which is not as powerful as DGCNN, results in some decrease in the data
representation accuracy, the overall performances of the downstream tasks are
basically not affected, which implies that the proposed PGI-driven point cloud
processing framework is not limited by a specific type of point feature
extractor. In practice, we can expect to focus on exploiting and designing
generic CNN architectures to promote various point cloud applications.


# Effect of Bose-Einstein condensation on the viscosity of a hot pion gas

Kshitish Kumar Pradhan  Dushmanta Sahu  Captain R. Singh  Raghunath
Sahoo111Corresponding author Email<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Indore, Simrol, Indore 453552, India

###### Abstract

Pion condensation in an ultra-relativistic collision is an open puzzle yet to
be verified experimentally. However, it is not trivial to have a conclusive
result of pion condensation due to various experimental constraints.
Nevertheless, numerous theoretical approaches can help in understanding the
high-temperature Bose-Einstein condensation (BEC) in a better way. This work
uses the shear ($\eta$) and bulk viscosity ($\zeta$) phenomena for a hot and
dense pion gas to investigate the effect of possible Bose-Einstein
condensation in such an environment. This study considers the relaxation time
approximation under the Boltzmann transport equation to estimate $\eta$ and
$\zeta$. Further, it explores the significance of the temperature and chemical
potential on the number of particles in the condensate and viscosities. We
find a substantial change in the viscosity when the pion gas is under BEC. In
addition, we also study the speed of sound ($c_{\rm s}$) for the pion gas
under BEC.

##  I Introduction

The formation of hot and dense matter in ultra-relativistic collisions at the
Relativistic Heavy Ion Collider (RHIC) at BNL and the Large Hadron Collider
(LHC) at CERN leads to numerous exciting consequences. The lightest hadron,
pions, are the most abundant particles in the final state of such collisions.
Pions being mesons, have an integral spin (bosons) and thus follow Bose-
Einstein (BE) statistics, named after Satyendra Nath Bose and Albert Einstein.
In their 1924 paper, they argued that a system of bosons has a unique property
to condense in the ground energy state at temperatures $T\sim 0^{o}$ K and
when their momentum almost becomes zero Bose ; Einstein . This phenomenon is
known as Bose-Einstein condensation (BEC). Since bosons have symmetric wave
functions, a large number of bosons can occupy the ground state where their
wave functions can interfere with each other. This effect is then observed
macroscopically. This phenomenon eluded physicists for a long time, as
achieving the conditions required for BEC formation is tricky. However, in
1995 BEC in cold Rubidium atoms was observed experimentally, confirming the
decades-long predictions to be correct Anderson:1995gf . Since then, BEC has
been one of the most exciting research topics in various fields, from
condensed matter to astrophysics and high-energy physics Begun:2006gj ;
Begun:2008hq ; Strinati:2018wdg ; Nozieres:1985zz ; Funaki:2008gb ;
Chavanis:2011cz ; Mishustin:2019otg ; Padilla:2019fju .

High pion density in the hadronic medium offers a possible critical point for
transition into the BEC phase. Such critical behavior has been explored under
a chemical non-equilibrium model where the pion abundances are characterized
by non-zero chemical potential Letessier:1998sz , which might indicate an
onset for BEC at LHC energies. Nevertheless, due to their high density in the
hadronic phase of the evolution in a collision, we may expect the formation of
the Bose-Einstein Condensation of pions. However, the question arises: how can
a violently expanding system such as one formed at the end of an ultra-
relativistic collision uphold a BEC-like phenomenon? As we know, the relative
momentum of the particles must be near or equal to zero for the formation of
BEC, which is a necessary and sometimes sufficient condition for the BEC phase
to exist. In ultra-relativistic collisions, the energy and particle density
are so high that the particles can exist very close to each other in the phase
space and have zero relative momentum. The small volume of the systems
produced in high-energy collisions and the strong interactions involved may
provide such an environment for BEC to exist even at high temperatures. The
possibility of high-temperature BEC in ultra-relativistic collisions is
fascinating and has been studied previously Begun:2015ifa ; Begun:2010ec ;
Begun:2008hq ; Deb:2021gzg . To explore the onset of BEC in a pion gas, the
interaction between them is considered in Ref. Savchuk:2020yxc . In addition,
in a system of interacting pions, the location of a critical point has also
been explored Kuznietsov:2021lax . The available literature has also reported
that pions could form condensate in the $pp$ and peripheral Pb-Pb collisions
at the LHC Deb:2021gzg . Though, the formation of BEC in ultra-relativistic
collisions is still a subject of verification through the experiments, the
proposal to find BEC experimentally was based on the concept of an increase in
the yields at the lower transverse momentum region of the pion $p_{\rm
T}$-spectra. However, no prominent signal has been observed yet as the pion
spectra get contaminated by the resonance decays and measurements at very low
$p_{\rm T}$ are not yet available.

In high-energy heavy-ion collisions, the possibility of coherent pion
production has been considered previously. In recent works at ALICE, analysis
of two- and three-pion correlation measurements show almost 23$\%$ coherent
fraction in charged pion emission ALICE:2013uhj . The pion radiation from a
BEC is expected to be coherent and thus suppress Bose-Einstein correlations
Fowler:1978us . Effective suppression of three-pion correlation compared to
two-pion correlation at low triplet momentum has been observed, which is
consistent with the formation of a Bose-Einstein condensate ALICE:2013uhj . In
addition, the measurement of particle number fluctuation can also act as a
signature of BEC. Thus, it is fascinating to see the outcome of such a
phenomenon on the system dynamics.

From the elliptic flow measurements at RHIC, a minimal value for shear
viscosity to entropy density ratio ($\eta/s$) has been observed STAR:2005gfr ,
approaching the KSS bound limit ($\eta/s=1/4\pi$) estimated from the AdS/CFT
calculations Kovtun:2004de . $\eta/s$, which shows minima near the QCD
critical phase transition. The $\eta/s$ is crucial to understanding the
behavior of the matter formed in ultra-relativistic energies. Many studies
have been done to estimate shear viscosity in hot QCD matter and in hadronic
matter. However, unlike $\eta/s$, the bulk viscosity to entropy density ratio
($\zeta/s$) is poorly explored at very high temperatures, where conformal
symmetry in the QCD matter holds; $\zeta/s\to$ 0\. Nevertheless, at the QCD
critical temperature, the conformal symmetry breaks, which results in the
observation of a peak in the trace anomaly ($(\epsilon-3P)/T^{4}$). Thus, we
expect similar behavior in $\zeta/s$ near the critical temperature.

The dissipative properties, such as shear viscosity and bulk viscosity, are
essential to study the behavior of the matter formed in ultra-relativistic
collisions. In Ref. Chen:2018mwr , the authors have taken a gluon plasma and
studied the effect of BEC on the shear viscosity of the system with the Green-
Kubo method. They observed the shear viscosity to be decreasing under BEC.
Given this, it would be interesting to have a study on the pion gas and the
effect of pion condensation on the viscosity. In this work, we study the BEC
of pions and further estimate the shear ($\eta$) and bulk viscosity ($\zeta$)
of a pion gas under condensation. We employ the Boltzmann transport equation
under relaxation time approximation to study the viscosities of a pion gas
under BEC. We have also studied the speed of sound in the pion gas under the
effect of BEC to understand the nature of interaction in the medium. This
discussion is justified for two main reasons. Firstly, pions are the most
abundant particle species in the final state of the ultra-relativistic
collision. Furthermore, secondly, as suggested by the chiral perturbation
theory earlier Son:2000xc , recent results from lattice QCD support the
existence of pion BEC at a finite isospin chemical potential Brandt:2017oyy .
Moreover, studying pion BEC has many applications apart from the systems
formed in ultra-relativistic collisions, such as hypothetical pion stars
Brandt:2018bwq ; Andersen:2018nzq  and cosmic trajectory in the early universe
Brandt:2017oyy ; Abuki:2009hx .

The article is organized as follows: a detailed formulation of the shear and
bulk viscosity and the speed of sound estimation for a hot pion gas under BEC
formulation is characterized in Sec. II. In Sec. III, a detailed discussion is
carried out on the obtained results and their implications. Finally, in Sec.
IV, we summarize our findings.

##  II Formulation

From basic statistical mechanics, the Bose-Einstein distribution function is
given by KHuang ,

$f=\left[{\rm exp}\left(\frac{E-\mu}{T}\right)-1\right]^{-1}$ (1)

Here, $E=\sqrt{{\bf p}^{2}+m^{2}}$ is the energy of the particle, ${\bf p}$ is
the momentum, $m$ is the mass of the particle, $\mu$ is the chemical potential
and $T$ is the temperature of the system. The particle multiplicities can be
calculated from the equation Begun:2015ifa ,

$N=g\int\frac{d^{3}xd^{3}p}{h^{3}}\left[{\rm exp}\left(\frac{\sqrt{{\bf
p}^{2}+m^{2}}-\mu}{T}\right)-1\right]^{-1}$ $\simeq\frac{gV}{(2\pi)^{3}}\int
d^{3}p\left[{\rm exp}\left(\frac{\sqrt{{\bf
p}^{2}+m^{2}}-\mu}{T}\right)-1\right]^{-1}$ (2)

where $g$ is the degeneracy of the particle, $h$ is Planck’s constant, and $V$
is the volume of the system.

The integrand in the first line of Eq. II stays constant for ${\bf p}$ $\to$
0, if $\mu\to m$. This is due to the fact that the singularity of the
denominator is canceled by the integration measure $d^{3}\it{p}$. However,
summing over quantum levels, the first term becomes infinite for $\mu\to m$ at
p $=0$, i.e.,

$N_{\rm cond}\simeq\frac{g}{{\rm
exp}(\frac{m-\mu}{T})-1}\to\infty~{}\rm{for}~{}\mu\to m$ (3)

Thus, when $\mu\to m$, marking the onset of BEC, integration of Eq. II should
begin for p $>$ 0 while keeping the summation over low momentum. In the
thermodynamic limit Begun:2008hq , $V\to\infty$, one can write Eq. II with
separate terms for $p$ = 0 and $p$ $>$ 0, as:

$N\simeq\frac{g}{{\rm
exp}(\frac{m-\mu}{T})-1}+V\int\frac{d^{3}p}{(2\pi)^{3}}\frac{g}{{\rm
exp}\bigg{(}\frac{\sqrt{{\bf p}^{2}+m^{2}}-\mu}{T}\bigg{)}-1}\\\ $
$\Rightarrow N_{\rm total}=N_{\rm cond}+N_{\rm ex}.$ (4)

Here, $N_{\rm cond}$ and $N_{\rm ex}$ are the number of particles in the
condensate and the number of particles in the excited states, respectively.

Thus, the modified number density becomes,

$n\simeq\frac{g}{V({\rm
exp}(\frac{m-\mu}{T})-1)}+\int\frac{d^{3}p}{(2\pi)^{3}}\frac{g}{{\rm
exp}\bigg{(}\frac{\sqrt{{\bf p}^{2}+m^{2}}-\mu}{T}\bigg{)}-1}\\\ $
$\Rightarrow n=n_{\rm cond}+n_{\rm ex}.$ (5)

Figure 1: (Color Online) Number of pions in the condensate as a function of
temperature and chemical potential.

As the chemical potential approaches closer to the mass of the particle,
$\mu\to m$, the $N_{\rm cond}$ becomes dominant and tends to $N_{\rm total}$
at $\mu=m$. However, at the LHC in $pp$ collisions, the chemical potential is
minimal and can be taken as zero Cleymans:2005xv .

We calculate the dissipative properties, shear and bulk viscosities from the
kinetic theory using the relativistic Boltzmann Transport equation (BTE) given
by Gavin

$\frac{\partial f_{p}}{\partial t}+v_{p}^{i}\frac{\partial f_{p}}{\partial
x^{i}}=I\\{f_{p}\\},$ (6)

where, $\vec{v}_{p}=\vec{p}/E_{p}$ is single particle velocity and the
collision integral $I\\{f_{p}\\}$ gives the rate of change of the non-
equilibrium distribution function $f_{p}$ due to collisions. Initially, when
the system is slightly away from the equilibrium, subsequent collision among
the constituent pions will try to restore the equilibrium after a certain
time. And according to the relaxation time approximation, the collisions bring
the system to the local equilibrium exponentially with relaxation time, which
is the order of collision time. Therefore the collision integral under the RTA
takes the form

$I\\{f_{p}\\}\backsimeq-\frac{(f_{p}-f_{p}^{0})}{\tau(E_{p})},$ (7)

where $\tau(E_{p})$ is called relaxation time which is of the order of
collision time. The equilibrium distribution function, denoted by $f_{p}^{0}$,
is given by

$f_{p}^{0}=\frac{1}{exp\bigg{(}\frac{E_{p}-\mu}{T}\bigg{)}\pm 1}$ (8)

where the $\pm$ corresponds to fermions and bosons, respectively. The stress-
energy tensor, $T^{\mu\nu}$, for a system can be considered the sum of an
ideal part ($T^{\mu\nu}_{0}$) and a dissipative part ($T^{\mu\nu}_{dissi}$)
and can be written as,

$T^{\mu\nu}=T^{\mu\nu}_{0}+T^{\mu\nu}_{dissi}.$ (9)

In a hydrodynamical description of the QCD matter, the shear and bulk
viscosities enter the dissipative part ($T_{dissi}^{\mu\nu}$) of the stress-
energy tensor. In the local Lorentz frame, it can be written as Gavin ,

$T_{dissi}^{ij}=-\eta\bigg{(}\frac{\partial u^{i}}{\partial
x^{j}}+\frac{\partial u^{j}}{\partial
x^{i}}\bigg{)}-(\zeta-\frac{2}{3}\eta)\frac{\partial u^{i}}{\partial
x^{j}}\delta^{ij}$ (10)

In terms of distribution function, the $T_{dissi}^{ij}$ can be expressed as

$T_{dissi}^{ij}=\int\frac{d^{3}p}{(2\pi)^{3}p^{0}}p^{i}p^{j}\delta f_{p}$ (11)

where $\delta f_{p}$ is a measure of the deviation of the distribution
function from equilibrium and can be obtained from Eq. (6) and Eq. (7) as,

$\delta f_{p}=-\tau(E_{p})\bigg{(}\frac{\partial f_{p}^{0}}{\partial
t}+v_{p}^{i}\frac{\partial f_{p}^{0}}{\partial x^{i}}\bigg{)}.$ (12)

To calculate the shear viscosity, consider a one-dimensional steady flow of
the form $u^{i}=(u_{x}(y),0,0)$ and a space-time independent temperature, then
Eq. (10) reduces to $T^{xy}=-\eta\partial u_{x}/\partial y$. The time
derivative in Eq. (12) vanishes for such a flow and hence From Eq. (11) using
Eq. ( 12) we get,

$T^{xy}=\bigg{\\{}-\frac{1}{T}\int\frac{d^{3}p}{(2\pi)^{3}}\tau(E_{p})\bigg{(}\frac{p_{x}p_{y}}{E_{p}}\bigg{)}^{2}f_{p}^{0}\bigg{\\}}\frac{\partial
u_{x}}{\partial y}$ (13)

Thus the expression for the coefficient of shear viscosity for a single
component of the hadronic matter is finally given by Kadam:2015xsa ,

$\eta=\frac{1}{15T}\int\frac{d^{3}p}{(2\pi)^{3}}\tau(E_{p})\frac{p^{4}}{E_{p}^{2}}f_{p}^{0}$
(14)

Now the departure of the system from equilibrium when it is compressed
uniformly is expressed through the dissipative parameter called Bulk
viscosity. Taking trace of Eq. (10), we get

$(T_{dissi})^{ii}=-3\zeta\frac{\partial u^{i}}{\partial x^{i}}$ (15)

Also from Eq. (11) and Eq. (12) we get

$(T_{dissi})^{ii}=-\int\frac{d^{3}p}{(2\pi)^{3}}\tau(E_{p})\frac{p^{2}}{E_{p}}\bigg{(}\frac{\partial
f_{p}^{0}}{\partial t}+v_{p}^{i}\frac{\partial f_{p}^{0}}{\partial
x^{i}}\bigg{)}$ (16)

In the local rest frame, the energy-momentum conservation implies
$\partial_{\mu}T^{\mu\nu}=0$, and together with Eq. (15) and Eq. (16) one can
arrive at Gavin

$\zeta=\frac{1}{T}\int\frac{d^{3}p}{(2\pi)^{3}}\tau(E_{p})f^{0}_{p}\bigg{[}E_{p}c_{s}^{2}-\frac{p^{2}}{3E_{p}}\bigg{]}^{2}$
(17)

where $E_{p}^{2}=p^{2}+m_{p}^{2}$, and $c_{s}^{2}=\frac{\partial
P}{\partial\varepsilon}$, is the square of the speed of sound at constant
baryon density.

The Energy dependent average relaxation time is defined by the expression

$\tau^{-1}(E_{a})=\sum_{bcd}\int\frac{d^{3}p_{b}}{(2\pi)^{3}}\frac{d^{3}p_{c}}{(2\pi)^{3}}\frac{d^{3}p_{d}}{(2\pi)^{3}}W(a,b\rightarrow
c,d)f_{b}^{0}$ (18)

where the transition rate $W(a,b\rightarrow c,d)$ is defined by

$W(a,b\rightarrow
c,d)=\frac{(2\pi)^{4}\delta(p_{a}+p_{b}-p_{c}-p_{d})}{2E_{a}2E_{b}2E_{c}2E_{d}}\mid\mathcal{M}\mid^{2}$
(19)

with transition amplitude $\mid\mathcal{M}\mid$. In the center of mass frame
Eq. ( 18) can be simplified as

$\tau^{-1}(E_{a})=\sum_{b}\int\frac{d^{3}p_{b}}{(2\pi)^{3}}\sigma_{ab}\frac{\sqrt{s-4m^{2}}}{2E_{a}2E_{b}}f_{b}^{0}$
$\equiv\sum_{b}\int\frac{d^{3}p_{b}}{(2\pi)^{3}}\sigma_{ab}v_{ab}f_{b}^{0}$
(20)

where $v_{ab}$ is relative velocity, $\sigma_{ab}$ is the total scattering
cross section, and $\sqrt{s}$ is center of mass energy.

For simplicity, the $\tau(E_{a})$ can be approximated to averaged relaxation
time ($\tilde{\tau}$), which one can obtain by averaging Eq. (20) over the
distribution function, $f_{a}^{0}$ as,

$\displaystyle\begin{split}{\tilde{\tau}}_{a}^{-1}&=\frac{\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\tau^{-1}(E_{a})f_{a}^{0}}{\int\frac{d^{3}p_{a}}{(2\pi)^{3}}f_{a}^{0}}\\\
&=\sum_{b}\frac{\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\frac{d^{3}p_{b}}{(2\pi)^{3}}\sigma_{ab}v_{ab}f_{a}^{0}f_{b}^{0}}{\int\frac{d^{3}p_{a}}{(2\pi)^{3}}f_{a}^{0}}\\\
&=\sum_{b}n_{b}\langle\sigma_{ab}v_{ab}\rangle\end{split},$ (21)

and for a single component of pion gas, the above equation can be written as,

$\tilde{\tau}_{a}^{-1}=n_{b}\langle\sigma_{ab}v_{ab}\rangle,$ (22)

where $n_{b}=\int\frac{d^{3}p_{b}}{(2\pi)^{3}}f_{b}^{0}$ is the number density
of pions.

In this work, for pion gas we will be using the equilibrium Bose-Einstein
distribution (in the local rest frame), which is given by

$f_{a}^{0}=\Bigg{[}exp\bigg{(}\frac{E_{a}-\mu}{T}\bigg{)}-1\Bigg{]}^{-1}$ (23)

$\langle\sigma v\rangle$ is the thermal average cross-section multiplied by
the relative velocity. Here the scattering particles are considered hard
spheres having a constant cross-section, $\sigma$. Thus, $\langle\sigma
v\rangle$ can be calculated as follows Cannoni ; Gondolo ,

$\langle\sigma_{ab}v_{ab}\rangle=\frac{\sigma\int
d^{3}p_{a}d^{3}p_{b}v_{ab}f_{a}^{0}f_{b}^{0}}{\int
d^{3}p_{a}d^{3}p_{b}f_{a}^{0}f_{b}^{0}}$ (24)

The momentum space volume elements can be written as,

$d^{3}p_{a}d^{3}p_{b}=8\pi^{2}p_{a}p_{b}dE_{a}dE_{b}~{}dcos\theta.$ (25)

On solving, Eq. 24 can be further written as,

$\langle\sigma_{ab}v_{ab}\rangle=\frac{\sigma\int
8\pi^{2}p_{a}p_{b}dE_{a}dE_{b}~{}dcos\theta
f_{a}^{0}f_{b}^{0}\frac{\sqrt{(E_{a}E_{b}-p_{a}p_{b}cos\theta)^{2}-(m_{a}m_{b})^{2}}}{{E_{a}E{{}_{b}}-p_{a}p_{b}cos\theta}}}{\int
8\pi^{2}p_{a}p_{b}dE_{a}dE_{b}~{}dcos\theta f_{a}^{0}f_{b}^{0}}.$ (26)

$\sigma$ is the hadronic collision cross-section. We have used a constant
value of 11.3 mb Tiwari:2017aon ; Kadam:2015xsa  in our calculations for the
radius of pion taken to be 0.3 fm.

Now, we move towards the calculation of viscosities in a pion gas which is
under BEC. In such a situation, two types of collisions will occur; collisions
between the excited particles and the collisions between the excited particles
and condensate particles. Thus, the reciprocal of relaxation time can be
modified as the sum of the contributions from the above two types of
collisions ($\tilde{\tau}_{12}^{-1}+\tilde{\tau}_{22}^{-1}$) .

The modified average relaxation time can be written as,

${\tilde{\tau}}^{-1}=\bigg{[}n_{1}\langle\sigma_{12}v_{12}\rangle+n_{2}\langle\sigma_{22}v_{22}\rangle\bigg{]},$
(27)

$\langle\sigma_{12}v_{12}\rangle=\frac{\sigma\int{}d^{3}p_{1}d^{3}p_{2}v_{12}f_{1}^{0}f_{2}^{0}}{\int{}d^{3}p_{1}d^{3}p_{2}f_{1}^{0}f_{2}^{0}}.$
(28)

$\langle\sigma_{22}v_{22}\rangle=\frac{\sigma\int
8\pi^{2}p_{2}p_{2}dE_{2}dE_{2}~{}dcos\theta
f_{1}^{0}f_{2}^{0}\frac{\sqrt{(E_{2}E_{2}-p_{2}p_{2}cos\theta)^{2}-(m_{2}m_{2})^{2}}}{{E_{2}E_{2}-p_{2}p_{2}cos\theta}}}{\int
8\pi^{2}p_{2}p_{2}dE_{2}dE_{2}~{}dcos\theta f_{1}^{0}f_{2}^{0}}.$ (29)

In thermodynamics, the speed of sound in matter is expressed as,

$c_{\rm s}^{2}(T,\mu)=\frac{\partial P}{\partial\varepsilon}\bigg{|}_{s/n}.$
(30)

This can be further simplified as,

$c_{\rm s}^{2}(T,\mu)=\frac{\frac{\partial P}{\partial T}+\frac{\partial
P}{\partial\mu}\frac{d\mu}{dT}}{\frac{\partial\varepsilon}{\partial
T}+\frac{\partial\varepsilon}{\partial\mu}\frac{d\mu}{dT}},$ (31)

where,

$\frac{d\mu}{dT}=\frac{s\frac{\partial n}{\partial T}-n\frac{\partial
s}{\partial T}}{n\frac{\partial s}{\partial\mu}-s\frac{\partial
n}{\partial\mu}}.$ (32)

For a system under BEC, the pressure and energy density of the system changes
accordingly;

$\displaystyle P=$ $\displaystyle-\frac{g}{V}\ln{\bigg{(}1-\rm
exp\bigg{(}\frac{m-\mu}{T}\bigg{)}\bigg{)}}$ (33)
$\displaystyle-\frac{Tg}{2\pi^{2}}\int p^{2}dp\ln{\bigg{(}1-\rm
exp\bigg{(}\frac{E-\mu}{T}\bigg{)}\bigg{)}}$ (34)

$\varepsilon=\frac{gm}{V({\rm
exp}(\frac{m-\mu}{T})-1)}+\int\frac{d^{3}p}{(2\pi)^{3}}\frac{gE}{{\rm
exp}\bigg{(}\frac{\sqrt{{\bf p}^{2}+m^{2}}-\mu}{T}\bigg{)}-1}\\\ $

By substituting the above equations in Eq.30, we can estimate the speed of
sound of a pion gas under BEC.

##  III Results and Discussion

Figure 2: Shear viscosity to entropy density ratio (left panel) and bulk
viscosity to entropy density ratio (right panel) of a pion gas under BEC as
functions of temperature for various system sizes. The black dashed line in
the left panel is the KSS bound limit .

Figure 3: Shear viscosity to entropy density ratio (left panel) and bulk
viscosity to entropy density ratio (right panel) of a pion gas under BEC as
functions of temperature and chemical potential.

Figure 4: Shear viscosity (left panel) and bulk viscosity (right panel) as
functions of ratio of number of pions in the condensate to the total number of
pions.

Figure 5: Speed of sound as a function of temperature (left panel) and ratio
of number of pions in the condensate to the total number of pions (right
panel).

Before discussing the viscosity of the system, we first estimate the number of
pions in the condensate and excited states by using the eq. 4. In this
equation, $N$ gives the total number of pions in a system of volume $V$. The
first term of the equation, estimated at $p\to 0$, gives the number of pions
in the condensate, whereas the second term gives the number of pions in the
excited states. We have taken the degeneracy, $g$ = 3 for three isospin states
of a pion. For simplicity, mass is taken to be 139 MeV, and chemical potential
is considered up to 120 MeV to avoid unnecessary divergences. Here, the pion
chemical potential will come from the charge chemical potential or isospin
chemical potential component.

We use the Boltzmann transport equation under relaxation time approximation to
estimate the shear and bulk viscosities of a pion gas under BEC. This becomes
a little tricky as, under BEC, the pion gas will have two kinds of subsystems;
excited pions with finite momentum and pions at the ground state with relative
momentum zero. Due to this, two types of collisions occur in the system; one
between two excited state pions and one between an excited state pion with a
pion in BEC. Thus, the conventional relaxation time approximation formalism
has to be modified accordingly.

First, we study the number of particles in the condensate as a function of
temperature and chemical potential in fig. 1. In a normal pion gas, the number
of pions increases with increasing temperature due to the Boltzmann-type
factor in the distribution function. When we introduce BEC to the system, the
number of pions in the condensate also increases with an increase in
temperature. Furthermore, with the addition of chemical potential, we observe
a rise in the number of particles in the condensate for a high $\mu$ value.
The study of BEC, considering a finite chemical potential, is extremely
significant, as we can understand from the plot. In addition, in ref
Liu:2017spl , the authors have studied charged pion condensation under
rotation at an external constant magnetic field. They report that with the
increase in rotation, the pion condensation increases. Interestingly, when we
introduce rotation to the system, we introduce a different type of chemical
potential, with the term $\Omega$ acting as the rotational chemical potential.
Similarly, we can also notice from the fundamental Euler’s thermodynamic
relation that introducing an external magnetic field to the system introduces
a magnetic chemical potential. Thus, we can easily infer that any addition of
chemical potential will increase the number of particles in the condensate.

To better understand the system’s behavior under BEC, we have plotted the
shear viscosity to entropy density ratio and bulk viscosity to entropy density
ratio as functions of temperature for various system sizes in fig. 2. For a
system with a smaller radius, both the viscosities are significantly lower at
a low-temperature regime. However, at a higher temperature regime, the system
dependency becomes lesser. A system with a smaller radius will have a higher
density and; thus, a higher number of interactions, resulting in lower
$\eta/s$ and $\zeta/s$. For all the system sizes, the trend of both $\eta/s$
and $\zeta/s$ are universal and decrease as functions of temperature, although
the value of $\zeta/s$ is higher in magnitude as compared to the value of
$\eta/s$. We observe that BEC affects the viscosity of the system
considerably.

To study the effect of chemical potential on the shear and bulk viscosity, we
have plotted $\eta/s$ and $\zeta/s$ as functions of temperature and chemical
potential on the left and right panels of fig. 3 respectively. We have fixed
the system size by taking the radius, $R$ = 5 fm. With an increase in
temperature, the viscosities decrease as expected. In addition, at a higher
chemical potential, the shear and bulk viscosities are lower. Both high
temperature and high chemical potential results in an increase in interaction
in the system, which decreases the viscosities.

In fig. 4, we have plotted the shear and bulk viscosity as functions of the
ratio $N_{\rm cond}/N_{\rm total}$ for various system sizes. This is helpful
to understand how the viscosity changes if the system has most of its pions in
the condensate. As one observes, the system’s viscosity is higher if most of
the pions are in the excited states. On the other hand, if most of the pions
in the system are in the condensate, the system’s viscosity is lower. It
approaches zero, suggesting that a system with all the pions in the condensate
becomes inviscid. This aligns with the finding of low-temperature cold helium
atom BEC, which is generally regarded as a superfluid due to its extremely low
viscosity. Moreover, it is interesting to note that a system size dependency
can also be observed. The decreasing trend of $\eta$ and $\zeta$ is steeper
for higher system size; however, the trend for all system sizes become the
same at a higher ratio of $N_{\rm cond}/N_{\rm total}$.

Furthermore, we have explored the effect of BEC on the speed of sound in a
pion gas. The speed of sound is a very important observable that tells us how
interactive the system is. In the left panel of fig. 5, we have plotted the
squared speed of sound of the hot pion gas under BEC. It is observed that when
the system is in BEC, the $c_{\rm s}^{2}$ is higher than the regular pion gas.
This means the system becomes less interactive in the presence of BEC.
Additionally, at a specific temperature, $c_{\rm s}^{2}$ is higher for lower
system size. On the right-hand side panel of fig. 5, we have plotted $c_{\rm
s}^{2}$ as a function of the ratio $N_{\rm cond}/N_{\rm total}$ for various
system sizes. A decreasing trend in $c_{\rm s}^{2}$ is observed as the system
gets populated with pions in the condensate. Here also, the rate of decrease
in the trend depends on the system size, however for all system sizes, the
value of $c_{\rm s}^{2}$ approaches to be nearly the same at $N_{\rm
cond}/N_{\rm total}$ $\simeq$ 1.

##  IV Summary

We have studied a hot pion gas under the possibility of Bose-Einstein
condensation and estimated the shear and bulk viscosity along with the speed
of sound in the medium. We observe a decrease in viscosity if there is BEC of
pions; however, the speed of sound of the medium increases under this
condition. We have also explored the dependency of chemical potential on BEC
and the viscosities. The higher the chemical potential, the higher the number
of particles in the condensate and the lower the viscosity.

Previously, the systems produced in heavy-ion collisions have been studied
from hydrodynamics. To explain the elliptic flow data with hydrodynamics
simulations, a clear necessity of $\eta/s$ appears. In view of our results, we
can infer that due to a decrease in viscosity in the presence of a BEC, an
increase in the elliptic flow of the pions can be expected, although a much
thorough study in this direction is necessary.

## Acknowledgement

KP acknowledges the doctoral fellowship from UGC, Government of India. The
authors gratefully acknowledge the DAE-DST, Govt. of India funding under the
mega-science project – “Indian participation in the ALICE experiment at CERN”
bearing Project No. SR/MF/PS-02/2021-IITI (E-37123).

## References

  * (1) S. N. Bose, Z. Phys. 26, 178 (1924). https://doi.org/10.1007/BF01327326
  * (2) Quantentheorie des einatomigen idealen Gases. https://doi.org/10.1002/3527608958.ch27
  * (3) M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science 269, 198 (1995).
  * (4) V. V. Begun and M. I. Gorenstein, Phys. Lett. B 653, 190 (2007).
  * (5) V. V. Begun and M. I. Gorenstein, Phys. Rev. C 77, 064903 (2008).
  * (6) G. C. Strinati, P. Pieri, G. Röpke, P. Schuck and M. Urban, Phys. Rept. 738, 1 (2018).
  * (7) P. Nozieres and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985).
  * (8) Y. Funaki, T. Yamada, H. Horiuchi, G. Ropke, P. Schuck and A. Tohsaki, Phys. Rev. Lett. 101, 082502 (2008).
  * (9) P. H. Chavanis and T. Harko, Phys. Rev. D 86, 064011 (2012).
  * (10) I. N. Mishustin, D. V. Anchishkin, L. M. Satarov, O. S. Stashko and H. Stoecker, Phys. Rev. C 100, 022201 (2019).
  * (11) L. E. Padilla, J. A. Vázquez, T. Matos and G. Germán, JCAP 05, 056 (2019).
  * (12) J. Letessier and J. Rafelski, Phys. Rev. C 59, 947 (1999).
  * (13) V. Begun and W. Florkowski, Phys. Rev. C 91, 054909 (2015).
  * (14) V. V. Begun, M. I. Gorenstein and O. A. Mogilevsky, Phys. Rev. C 82, 024904 (2010).
  * (15) S. Deb, D. Sahu, R. Sahoo and A. K. Pradhan, Eur. Phys. J. A 57, 158 (2021).
  * (16) O. Savchuk, Y. Bondar, O. Stashko, R. V. Poberezhnyuk, V. Vovchenko, M. I. Gorenstein and H. Stoecker, Phys. Rev. C 102, 035202 (2020).
  * (17) V. A. Kuznietsov, O. S. Stashko, O. V. Savchuk and M. I. Gorenstein, Phys. Rev. C 104, 055202 (2021).
  * (18) B. B. Abelev et al. [ALICE], Phys. Rev. C 89, no.2, 024911 (2014).
  * (19) G. N. Fowler, N. Stelte and R. M. Weiner, Nucl. Phys. A 319, 349-363 (1979).
  * (20) J. Adams et al. [STAR], Nucl. Phys. A 757, 102 (2005).
  * (21) P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005).
  * (22) Z. Chen, C. Greiner, Z. Xu and P. Zhuang, Phys. Rev. C 100, 014906 (2019).
  * (23) D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 86, 592 (2001).
  * (24) B. B. Brandt, G. Endrodi and S. Schmalzbauer, Phys. Rev. D 97, 054514 (2018).
  * (25) B. B. Brandt, G. Endrodi, E. S. Fraga, M. Hippert, J. Schaffner-Bielich and S. Schmalzbauer, Phys. Rev. D 98, 094510 (2018).
  * (26) J. O. Andersen and P. Kneschke, arXiv:1807.08951.
  * (27) H. Abuki, T. Brauner and H. J. Warringa, Eur. Phys. J. C 64, 123 (2009).
  * (28) Huang, Kerson. (1987). Statistical mechanics. New York : Wiley
  * (29) J. Cleymans, H. Oeschler, K. Redlich and S. Wheaton, Phys. Rev. C 73, 034905 (2006).
  * (30) S. Gavin, Nucl. Phys. A 435 826, (1985).
  * (31) S. K. Tiwari, S. Tripathy, R. Sahoo and N. Kakati, Eur. Phys. J. C 78, 938 (2018).
  * (32) G. P. Kadam and H. Mishra, Phys. Rev. C 92, 035203 (2015).
  * (33) M. Cannoni, Phys. Rev. D 89, 103533 (2014)
  * (34) P. Gondolo and G. Gelmini, Nucl. Phys. B 360, 145 (1991).
  * (35) M. A. Shahzamanian and H. Yavary, Ann. Phys. 321, 1063 (2006).
  * (36) Y. Liu and I. Zahed, Phys. Rev. Lett. 120, 032001 (2018).