# Identifying and diagnosing coherent associations and causalities between
multi-channels of the gravitational wave detector

Piljong Jung  National Institute for Mathematical Sciences, Daejeon 34047,
Republic of Korea  Sang Hoon Oh  National Institute for Mathematical Sciences,
Daejeon 34047, Republic of Korea  Young-Min Kim  Department of Physics, Ulsan
National Institute of Science and Technology, Ulsan 44919, South Korea  Edwin
J. Son  National Institute for Mathematical Sciences, Daejeon 34047, Republic
of Korea  Takaaki Yokozawa  Institute for Cosmic Ray Research (ICRR), KAGRA
Observatory, the University of Tokyo  Tatsuki Washimi  Gravitational Wave
Science Project (GWSP), Kamioka branch, National Astronomical Observatory of
Japan (NAOJ)  John J. Oh<EMAIL_ADDRESS>National Institute for
Mathematical Sciences, Daejeon 34047, Republic of Korea

###### Abstract

The gravitational-wave detector is a very complicated and sensitive collection
of advanced instruments, which is influenced not only by the mutual
interaction between mechanical/electronics systems but also by the surrounding
environment. Thus, it is necessary to categorize and reduce noises from many
channels interconnected by such instruments and environment for achieving the
detection of gravitational waves because it enhances to increase of a signal-
to-noise ratio and reduces false alarms from coincident loud events. For this
reason, it is of great importance to identify some coherent associations
between complicated channels. This study presents a way of identifying (non-)
linear couplings between interconnected channels by using some correlation
coefficients, which are applied to practical issues such as noises by hardware
injection test, lightning strokes, and air compressor vibrations
gravitational-wave detector.

##  I Introduction

Since the era of gravitational wave (GW) astronomy was opened by the detection
of GW emitted from the binary black hole coalescence (GW150914) Abbott  _et
al._ (2016a), it has entered the stage of a leap to pursuit new perspectives
on the structure of the Universe and the evolution of astrophysical objects
such as stars and galaxies. For more profound understanding and expansion of
the knowledge on the Universe, there still exist many challenges to be
overcome for GW physics and astronomy. Much more accurate sky localization for
GW sources and a much farther range of luminosity distance in GW observation
require the improved sensitivity of GW detectors, overcoming the present
limitation of technologies.

Detectors’ sensitivity confines the boundary of observations so that the
number of GW events within the boundary determines the detection rate that can
be achieved in the current GW detector. The currently operating ground-based
laser interferometric GW detectors such as Laser Interferometer Gravitational-
wave Observatory (LIGO), the Virgo, and the KAGRA have a similar design
sensitivity curve with a frequency band of $30-2000Hz$, which is characterized
by three main noise sources: photon shot noise by the laser system at high-
frequency range, thermal noise by test-mass mirrors at mid-frequency range,
and seismic noise by ground vibrations at low-frequency range Aasi  _et al._
(2015a); Acernese  _et al._ (2015); Akutsu  _et al._ (2019); Abbott  _et al._
(2018). Those GW detectors are now being operated for detecting GW signals and
planned for upgrades to improve their sensitivities using many engineering
challenges such as cryogenics, quantum squeezed light, and so on. Furthermore,
next-generation ground-based GW detectors with new conceptual designs are now
being planned Punturo  _et al._ (2010); Abbott  _et al._ (2017).

Besides, advanced methodologies and analysis algorithms should be required to
enhance the quality of data Aasi  _et al._ (2015b); Abbott  _et al._ (2016b);
Jung  _et al._ (2018) taken from the very sensitive instruments and isolate GW
signals from noises caused by the sensitive devices and the surrounding
environments efficiently because it improves the detection statistics such as
signal-to-noise ratio (SNR) to provide the reliable detection criterion. This,
consequently, also improves the detection range of GW detectors, which yields
better detection rates of GW sources. In this point of view, both developments
of advanced methodologies for data analysis and understanding the status of GW
detectors, as well as the most sensitive instruments, are of special
importance to achieve for detecting GW signals coming from the farthest GW
emitting objects. For this reason, many different tools for characterizing
noises have been developed and utilized for analyzing GW signals so far. The
first purpose of those tools is to categorize and classify
transient/continuous noises that are harmful to the main GW channel,
identifying the interconnected families of noisy channels with the causal
relationship. Then they should be mitigated with the help of various advanced
mathematical algorithms as is possible. Finally, if they are caused by some
instrumental defects with a possibility of repeated malfunctions, the causes
should be reported and amended for maintaining the consistent status of the
detector.

Many efforts for identifying and vetoing transient noises have been made so
far and is widely utilized for GW data analysis, such as a computation of
significance called hierarchical veto (Hveto) Nuttall  _et al._ (2015),
Q-transform based trigger generator Robinet  _et al._ (2020), Hilbert-Huang
transform-based method Son  _et al._ (2021), linear regressions Walker  _et
al._ (2018), machine learning algorithms Cuoco  _et al._ (2020), and so on.
These methods cover classification and a vetoing method as well as
identification of the influences between the main channel and other auxiliary
channels monitoring the environmental and/or instrumental status. This needs
consistent understandings of which channel effects can cause the transient
noises in the main channel and/or which can be useful to mitigate such
abnormalities to maintain the normal status of the detector. However, the GW
detector behaves with a highly non-stationary and non-linear nature, we still
face great challenges in dealing with noises of GW data.

For this reason, it is necessary to develop more advanced analysis tools for
noise hunting to improve the detector’s data quality. Here we focus on the
couplings and causalities propagated from a certain auxiliary channel to the
main GW channel. In Ref. Bose  _et al._ (2016), it has been shown that the
excess noises from bilinear and nonlinear couplings in GW interferometers can
be treated using the bilinear coupling veto (BCV) method Ajith  _et al._
(2014) with the trigger-based correlation coefficient. Along this line, we
suggest a new way of identifying coherent associations between GW and the
related auxiliary channels, in which we use three kinds of the correlation
coefficient: Pearson’s correlation coefficient (PCC) Pearson (1920) and
Kendall’s $\tau$ coefficient (Ktau) Kendall (1938) as a linear measure and
Maximal Information Coefficient (MIC) as a nonlinear measure Reshef  _et al._
(2011, 2016, 2018), respectively. In particular, MIC is an information-
theoretic measure to discriminate the nonlinear association between two random
variables. Together with these measures, we construct a systematic way of
identifying the noises from (non-)linear couplings and causalities propagating
from instrumental and/or environmental disturbances of GW detectors. Then we
apply the suggested method to the well-known issues of data analysis and noise
identification in LIGO and KAGRA detectors.

In this study, we present a way of identifying coherent associations between
GW and auxiliary noise channels by computing correlation measures. The method
we consider here is the PCC and Ktau as a linear measure, and MIC as a non-
linear measure. We construct a consistent way of discriminating the relevant
noise effect and its causality, applying to some issues in gravitational wave
detection. In Section II, we describe the methods to measure (non-)linear
associations and build an analysis process with proper statistical algorithms
between GW and many auxiliary channels. In Section III, we exhibit the
exemplary results based on some noise data taken from the KAGRA GW detector; a
lightning stroke, an air compressor noises, and the noises caused by wind
effect. Finally, we summarize and discuss our results in Section IV.

##  II Method and Workflow

We use the data with the main GW channel and many auxiliary channels taken
from the KAGRA GW detector. The KAGRA is a gravitational-wave detector with a
similar configuration of laser interferometry such as the LIGO/Virgo except
for the cryogenic test-mass mirrors and the underground installation Akutsu
et. al. (2020). The initial installation of the KAGRA has finished in 2019,
and after a year-long commissioning, it started its first observing run during
a month in 2020 and joined the O3 observing run together with the advanced
LIGO and the Advanced Virgo, recently Kokeyama (2020).

The main feature of the KAGRA is the cryogenic and underground GW detector,
which implies that the KAGRA detector has somewhat unique characteristics
caused by the nature of the underground cryogenic facility, producing the
relevant noise effects that are not reported in the ground-based GW detectors.
Therefore, it is of great importance to understand the noise characteristic
from the KAGRA detector and its environment.

###  II.1 Methods for Data Correlations

We introduce three methods of analysis for investigating the correlation of
two data samples; the PCC, the Ktau, and the MIC. Let us consider two time-
series data $X$ and $Y$ to be non-stationary and uni-variate data sets with
equal size of $n$. We assume that $(x_{i},y_{i})$ is a set of $i$-th bi-
variate data pair from the paired data $(X,Y)$. If we assume that the data set
$Y$ includes certain noises, being affected the sensitivity of GW detectors,
there exist a meaningful statistical association between two observed
variables $X$ and $Y$ because the noise propagates to another data set $X$.
With these assumptions, we analyze the correlations and the causalities of
noise propagation resulted from the instrumental anomalies and/or
environmental causes in gravitational wave detectors. In this section, we
describe three major methods to estimate the linear and non-linear
associations based on the time-series data recorded in each channel.

The PCC produces the linear correlation score between two variables defined by
the ratio between the covariance and the product of standard deviation of each
variable. Let $\bar{x}=\sum_{i=1}^{n}x_{i}/n$ and
$\bar{y}=\sum_{i=1}^{n}y_{i}/n$ be the means of $X$ and $Y$, respectively.
Then, the PCC $\rho$ is defined as

$\rho(X,Y)=\frac{{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}}{{\sqrt{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\sum_{i=1}^{n}(y_{i}-\bar{y})^{2}}}},$
(1)

which determines the value between $-1$ and $1$. For independent two
variables, we have $|\rho|=0$. When $|\rho|>0.5$, it is typically interpreted
as a significant association. $|\rho|=1$ indicates that two variables have a
perfect linear correlation, where positive and negative signs imply increasing
and decreasing linear dependence, respectively. Here, we used the absolute
value of PCC for convenience.

Kendall’s $\tau$ correlation coefficient Kendall (1938) measures the strength
of the monotonicity of the relationship between two variables, which is
defined by

$\tau(X,Y)=\frac{(c-d)}{{}_{n}C_{2}},$ (2)

where $c$ and $d$ are the number of concordant and discordant pairs in
$(X,Y)$, respectively. Given two samples of observations from the combined
variable
$(X,Y)=\left\\{(x_{1},y_{1}),(x_{2},y_{2}),\cdots,(x_{n},y_{n})\right\\}$, if
two sampled pairs are either $x_{i}>x_{j}$ and $y_{i}>y_{j}$ or $x_{i}<x_{j}$
and $y_{i}<y_{j}$, it is called a concordant pair. On the other hand, if two
samples are either $x_{i}>x_{j}$ and $y_{i}<y_{j}$ or $x_{i}<x_{j}$ and
$y_{i}>y_{j}$, it is a disconcordant pair. Hence, Kendall’s $\tau$ coefficient
provides the ordinal association as the proportion of the difference between
ordered and disordered pairs to all the possible combinations. As with the
PCC, $\tau(X,Y)$ varies from $-1$ to $1$. If the order of two pairs is
randomly distributed, they are monotonically independent, yielding
$\tau(X,Y)=0$. If $Y$ values tend to change with increasing or decreasing $X$
values, the absolute value of Kendall’s $\tau$ becomes to one.

Meanwhile, mutual information (MI) can estimate and characterize the strength
of shared information between data sets. For given two discrete random
variables $A$ and $B$ with a joint probability mass function $p(a,b)$ and
marginal probability mass functions $p(a)$ and $p(b)$, MI is defined as:

$I(A;B)=\sum_{a\in A}\sum_{b\in B}p(a,b)\log_{2}{\frac{p(a,b)}{p(a)p(b)}}.$
(3)

Note that MI provides non-negative values, $I(A;B)\geq 0$. If the random
variable $B$ is a function of $A$, $B=f(A)$, $I(A;B)$ diverges. In addition,
if $A$ and $B$ have no shared information, $p(a,b)=p(a)p(b)$, then $I(A;B)$
vanishes clearly; they are statistically independent. Refer to more detailed
properties in Cover and Thomas (2006).

Suppose that the joint probability distribution $p_{XY}(i,j)$ is approximated
by the number of points falling into the $i$-by-$j$th bin on the partitioned
plane of scattered plots of two time-series data sets $X$ and $Y$. Then,
approximated MI is obtained by the occupied elements in each cell as:

$I(X;Y)_{k,l}=\sum_{i=1}^{k}\sum_{j=1}^{l}\frac{p_{XY}(i,j)}{\log_{2}{\min\left\\{k,l\right\\}}}\log_{2}{\frac{p_{XY}(i,j)}{p_{X}(i)p_{Y}(j)}},$
(4)

where $p_{X}(i)$ and $p_{Y}(j)$ are marginal distributions on $i$-th column
and $j$-th row. In addition, $k$ and $l$ indicate the partition size of $X$
column and $Y$ row, respectively. Because Jensen inequality, $0\leq
I(X;Y)_{k,l}\leq\log_{2}{\min\left\\{k,l\right\\}}$, is satisfied Cover and
Thomas (2006), the value in Eq. (4) can be normalized and spans between zero
and one.

To overcome a heuristic approach of maximizing MI value for all possible
resolution of $k$-by-$l$ grid Reshef  _et al._ (2011), three intrinsic
properties in Eq. (4) are utilized; they are 1) monotonic convex function, 2)
bounded set, and 3) uniformly continuous function, which allows computing Eq.
(4) more efficiently and effectively.

###  II.2 CAGMon: A Novel Tool for Identifying Correlations

Figure 1: A flowchart of the CAGMon tool. Figure 2: Exemplary plots provided
by the CAMon tool: a) correlation matrix plot of $\rho,~{}\tau$, and ${\rm
MIC}$ between primary and auxiliary channels b) scattered plot of both
channels (Each relative amplitude is rescaled by its median.) c) correlation
trend plot within a certain duration. Here, we demonstrate one of vibration
isolation system (VIS) channels among tens of auxiliary channels in the KAGRA
GW detector as a simple example.

Let $S$ be a $n$ ordered pair-set and $B(n)=n^{\alpha}$ and the parameter $c$
restricts the total grid size. Here, $\alpha$ is a dimensionless parameter
that controls the size of grids and $c$ is a controlling the coarseness of
maximizing the discrete grid search. Then, the empirical MIC called MICe is
defined as:

$MICe(X,Y,\alpha,c)=\max_{ab<B(n)}\Big{\\{}\frac{\max
I^{[*]}(S,k,l)}{\log_{2}\min\\{k,l\\}}\Big{\\}},$ (5)

where $I^{[*]}(S,k,l)$ denotes the maximized MI in $k\times l$ grids. Note
that the coefficient of MICe varies from zero to one. If two time-series data
have an clear association, MICe converges to one for sufficiently large data
sizes whereas it becomes to zero in the opposite condition. However, even
though MICe has a unique performance to extract the complex associations, it
was uncertain how reliable MICe values can be provided about their
correlations in the aspect of the parameter selection. To investigate its
reliability, we have determined the relevant parameters of MICe via the
statistical power to optimize them. Refer to Jung  _et al._ (2021) more
details on the optimized parameters of MICe.

We utilize the aforementioned correlation measure indices to identify the data
associations between the primary channel and auxiliary channels of GW
detectors. Two types of channels are considered as a primary channel to
witness of the sensitivity degradation by transient noises: the binary neutron
star (BNS) range channel and the strain channel of GW detectors. The BNS range
channel is a monitoring channel that represents the volume/orientation-
averaged distance based on the GW signals with a matched filtered signal-to-
noise ratio (SNR) of $8$ from the two $1.4M_{\odot}$ neutron stars in a GW
detector Finn and Chernoff (1993); Abbott  _et al._ (2018). The slowly-moving
behavior caused by long-duration noise disturbances in LIGO detector has been
investigated via the least absolute shrink-age and selection operator (LASSO)
regression, where the BNS range channel was used as a primary channel Walker
_et al._ (2018).

To monitor the chronological trend of association coefficients between two
datasets, we considered a minimal bin of data segment, called stride, which
represents a piece of an equal interval unit in the whole data segments. For
given data sets, aforementioned three correlation coefficients,
$\left\\{\rho(X_{m},Y_{m}),\tau(X_{m},Y_{m}),MICe(X_{m},Y_{m})\right\\}$, are
computed for divided subsets of $X_{m}$ and $Y_{m}$ by stride. By gathering
the sequential coefficients together, we can demonstrate a time-serial trend
of the association strength between two channels. In this section, we present
a novel tool for identifying and diagnosing the association of two datasets,
called CAGMon; the workflow and design of the algorithm.

The workflow of CAGMon tool is depicted in Fig. 1, which comprises four
different stages: i) reading initial configuration, ii) loading time-series
data and pre-processing, iii) computing each coefficient and the relevant
statistics, and iv) plotting results and building a result summary page.
First, the configuration file consists of user-defined parameters, pre-
processing options, and general feature of data such as sampling rate,
start/end times, and stride. Because each coefficient is computed for each
stride of time-series datasets, the tool can only investigate the similar
timing coincidence within the same stride bin. Hence, there has a limitation
of identifying association for transient noises correlated to noise with a
specific frequency. To overcome this limitation, we provide several customized
options of data pre-processing, such as high-pass, low-pass, and bandpass
filters, for a variety of scalable analyses. In addition, general feature of
data comprises a primary channel name, the path of data files, an auxiliary
channel list file, segment file, output file type and its save path, and so
on. When the time-series data is retrieved from GW frame files, the
configuration option refers to the initial configuration file to apply the
pre-processing. If time-series data contain either blank or NaN, zero arrays
would be padded properly to avoid a computational error. Then, data are re-
scaled to have the same data size using the resampling algorithm. The detailed
discussion and analyses on the reliability of MICe values, data sample size,
and optimal parameters of MICe has been presented in Jung  _et al._ (2021). In
this study, we select a set of optimal parameters of MICe presented in Table
1.

primary channel | stride(s) | sampling rate(Hz) | data size | $\alpha$ | $c$
---|---|---|---|---|---
GW strain | 2 | 4096 | 8192 | 0.5 | 0.7
BNS range | 512 | 16 | 8192 | 0.5 | 7.0
Table 1: The primary channel information and parameters of MICe used in this
study: these values are selected by the empirical optimized parameter
selection in Jung  _et al._ (2021).

To monitor the long duration behavior during a few hours or a day, it is
reasonable to select the BNS range channel as a primary channel to set up the
sampling frequency of 16Hz. Then, it is crucial to perform a pre-processing
via high-pass/bandpass filters to focus on targeted frequency band. Whereas if
we focus on the short-duration transient noises between GW and auxiliary
channels, it is plausible to select two seconds of stride with 4096Hz. Most of
all, it is essential to maintain a consistent method for interpreting the
result in the whole process of analysis. After computation, a result summary
page is built and published automatically, containing the initial
configurations, distribution trend plots for each coefficient, ranked summary
table, and detailed trend plots for each channel.

##  III Application to Gravitational-Wave Data

CAGMon tool was designed to identify and diagnose associated auxiliary
channels that influence the primary channel for GW detection, producing a
time-sequential correlation trend propagated by instrumental or environmental
disturbances. We can estimate a significant association as an excess from the
long duration trend of its median value. In this section, we apply CAGMon tool
to the KAGRA GW data for characterizing and identifying associations between
channels caused from well-known environmental and instrumental events such as
a lightning stroke and an air compressor noises.

###  III.1 Magnetic Field Transients from Lightning stroke

Figure 3: Omega scan plots of GW interferometer channel (top) and magnetometer
at the beam splitter (BS) station (bottom).

First, CAGMon tool can be applied to investigate a magnetic field transient
noise caused by a lightning stroke event observed by the KAGRA collaboration
that was verified by a clear causal relationship between two channels in
Washimi  _et al._ (2021). A lightning stroke is a locally enormous release of
electromagnetic waves, which clearly affects to GW detector by a sudden
variance of magnetic fields. KAGRA GW detector was influenced by a lightning
stroke at 02:38:40.38 on March 21, 2020 UTC, which was recorded in the GW
strain channel and magnetometers. This detection was the first evidence
observed by the KAGRA GW detector that the lightning strokes in the atmosphere
would be able to affect the underground-based GW detector within the detection
range. The data recorded in magnetometer can propagate to the primary channel
of the KAGRA GW detector, producing a coincident transient noise in the
primary channel with a significant association. Hence, we can infer that there
exist a significant causal relationship between the primary channel and the
magnetometer-related channels. The omega-scan spectrogram plots based on
Q-transform are presented in Fig. 3.

Figure 4: Correlation trend plot of association between GW interferometer channel and magnetometer associated channels at BS station at the lightning event time. The solid-line and the dashed-line indicate the strength of correlation coefficient and median value, respectively. Here, the star-mark represents the maximum value of $({\rm MICe},\rho,\tau)$. Event time(GPS) |  Associated auxiliary channels111All exhibited channels here are related to magnetometers installed in several areas nearby the KAGRA GW detector. | MICe |  Med(MICe)222The median value was computed for four minutes duration segment. ($10^{-2}$) | $\rho$ |  Med($\rho$)($10^{-2}$) | $\tau$ |  Med($\tau$)($10^{-2}$)
---|---|---|---|---|---|---|---
March 22, 2020 02:38:39-41UTC (1268879937.38 -1268879939.38) |  K1:PEM333PEM stands for the physical environment monitor that is related to subsystems for sensing environmental changes such as seismomter, magnetometer, accelerometer, and so on Akutsu et. al. (2021).-MAG_BS_BOOTH_BS_Z_OUT_DQ | 0.079 | 2.210 | 0.021 | 0.212 | 0.022 | 0.172
K1:PEM-MAG_BS_BOOTH_BS_Y_OUT_DQ | 0.050 | 2.210 | 0.052 | 0.250 | 0.015 | 0.275
K1:PEM-MAG_BS_BOOTH_BS_X_OUT_DQ | 0.026 | 2.188 | 0.040 | 0.266 | 0.001 | 0.197
K1:PEM-MAG_EXC_BOOTH_EXC_X_OUT_DQ | 0.021 | 1.499 | 0.064 | 0.423 | 0.047 | 0.521
K1:PEM-MAG_EYC_BOOTH_EYC_Z_OUT_DQ | 0.069 | 2.309 | 0.141 | 0.709 | 0.045 | 0.474
K1:PEM-MAG_SR_BOOTH_SR_Z_OUT_DQ | 0.022 | 2.271 | 0.052 | 1.595 | 0.040 | 1.458
Table 2: Correlation values of MICe, $\rho$, and $\tau$ at the lightning
stroke event time and some associated channels with the magnetometer in the
KAGRA GW detector.

Here, we used the GW interferometer channel as a primary channel and set two
seconds of stride with $8192$ data size to investigate a correlation between
short-duration data segments. The mutual correlations between the GW
interferometer channel and the magnetometer related channels detects a
meaningful signal at the event time of the lightning stroke. Here, correlation
trend plots are depicted in Fig. 4, yielding a clear peak signal at the event
time of the lightning stroke. It is inferred that the magnetic field noise
from the lightning stroke event can propagate and affect the GW interferometer
channel. Interestingly, the aspect of each correlation exhibited in Fig. 4
shows that they are linearly or non-linearly correlated with each other for
the individual subsystems. The values with significant correlations are much
greater than the median values of four minutes duration data segment. In
addition, we list the magnetometer associated subsystems with a significant
correlation of the lightning stroke event in Table. 2.

###  III.2 Periodic Noises from Air compressors

Figure 5: Correlation trend plots between the BNS range and microphone
channels installed in the BS Station POP room on April 9, 2020 (top) and April
11, 2020 (bottom); Relatively strong MICe correlations were repeated for every
2.58 hours. Figure 6: The BNS range curve experiences a sudden drop as much as
about 50kpc owing to the air compressor vibration from OMC station on April 9,
2020 (top) and April 11, 2020 (bottom). Figure 7: Correlation trend plots
between the BNS range channel and microphone channel installed the initial
x-arm tank(top-left), the test mass optical leverage at BS VIS channel
(bottom-left), and the y-axis seismometer at BS (top-right). And, the plot of
wind speed defined as the average wind speed over two minutes at the entrance
of KAGRA (Atotsu) on April 11, 2020 (bottom-right)

We applied the CAGMon tool to the observing run of KAGRA (O3GK run) from April
7, 2020, to April 21, 2020 Abbott  _et al._ (2022); Abe  _et al._ (2022).
Unlike the aforementioned analysis for transient noise, we used the BNS range
channel as a primary channel and investigated internal or external influences
on the detector’s sensitivity during the long period of observational mode.
Because data record a minute trend of sensitivity range with around three
minutes latency, we set 512 seconds of stride and 8192 of data size. With this
configuration, significant correlations were periodically repeated between the
BNS range channel and microphone channels at the BS station. These correlated
peak signals have a harmonic frequency of 26.5Hz and repeat with a periodicity
of 2.58 hours every day. Figure 5 exhibits the correlation trend plots of this
periodic behavior using the CAGMon tool. In addition, it has been found that
the air compressor at the output mode cleaner(OMC) produces periodic
vibrations that correlated through nearby instruments to result in the sudden
drop of the BNS range of KAGRA sensitivity curve. The BNS range drop owing to
the periodical air compressor noises is demonstrated in Fig. 6. This
phenomenon was previously reported during the KAGRA engineering operation and
reduced significantly via the installation of the vibration isolation and the
soundproof devices in the air compressors. However, the CAGMon tool caught
this effect clearly even if it had reduced. Consequently, this example
indicates that association may exist if the trend of the coefficients is
changed regardless of their strength.

###  III.3 Glimpse of Acoustic Noise from Winds

Another interesting glimpse via CAGMon tool was observed by a clear
correlation between the BNS range channel and the microphone/vibration
isolation channels. The correlation trend plot exhibits a meaningful
association from morning(9 A.M, JST) to evening (7 P.M, JST) everyday in
spring season, which is inferred by a relationship with a strong wind power
during this time (Fig. 7). Because the KAGRA GW detector is installed in the
underground tunnel in the Mt. Ikeno, we infer that this is responsible for the
propagation of an acoustic wave noise owing to strong winds between Mt. Ikeno.
Fig. 7 exhibits an interesting feature of MICe because this association only
was detected by MIC rather than PCC and Ktau; one can infer that the effect of
winds can affect non-linearly the KAGRA detector. A possibility of influences
by the fluctuating gravity gradient noises in the ground and air around the
detector has been reported in Creighton (2008). For this reason, it is worth
while studying because the seismic coupling and up-conversion effect may
affect the GW detection band for the underground detector.

A potential scenario of this effect is considered by a simple simulation of
the elastic wave equation in the underground like the KAGRA detector. The wall
of the L-shape tunnel in the KAGRA detector consists of the concrete material
and air inside the tunnel. The $y$-arm of the KAGRA detector is parallel to
the slop of the valley in Mt. Ikeno whereas the $x$-arm is located from the
slope to deep inside the mountain. The seismic incident wave generated from
the strong winds striking the slope vibrates and propagates then transforms
into an acoustic wave inside the tunnel. The acoustic vibration produces sound
pressure noise inside the tunnel. Because of the location of both arms in the
KAGRA detector, the sound pressure noise level in the $y$-arm direction is
much more severe than that in the $x$-arm direction due to the attenuation
inside the deeper region of the tunnel.

The finite element method (FEM) simulation of the seismic and acoustic waves
with a multi-physics configuration and the sound pressure level along $x$-arm
direction in diverse frequencies are shown in Fig. 8.444The simulation has
been performed by COMSOL-Multiphysics 5.6 with somewhat ad hoc parameters and
assumptions in order to check the glimpse of the scenario. Consequently, the
seismic vibration from Mt. slope can propagate to air fluctuation in the
tunnel of the KAGRA, which might produce acoustic noises inside the tunnel,
which needs to be verified by more accurate simulation and instrumental
measurement.

Figure 8: The simulation of seismic and acoustic wave propagation in the
underground tunnel of KAGRA and its $x$-arm sound pressure level in diverse
frequencies. The perfectly matched layer (PML) was applied in the end region
of $x$-arm.

##  IV Discussions

We developed a novel tool for identifying and diagnosing data association
between two variables to investigate presumably correlated events between
multi-channels of the GW detectors, called CAGMon. In this tool, three linear
and non-linear measures, PCC, Ktau, and MIC, were adopted and the optimized
parameter selection for MIC was referred in Jung  _et al._ (2021).

We applied this tool for the transient and periodic noises caused by a
lightning stroke event and an acoustic noise caused by an air compressor in
the KAGRA GW detector. Consequently, we verified that several magnetometer
associated subsystems influenced by the lightning stroke event have a
significant association with the GW strain channel. On the other hand, we
found that the sudden detection range reductions in the BNS range curve of the
KAGRA detector is associated with the periodic noise in every 2.58 hours
appeared in the microphone channels. The noise was identified with a harmonic
frequency of 26.5Hz and the evident cause was confirmed by acoustic noises
from the air compressor nearby the BS station. Finally, it is observed a
glimpse of the wind effect to the underground detector via CAGMon tool. In the
daytime, air nearby mountains becomes heated and makes relatively strong wind
between mountains. This wind hits the ground surface where the detector
locates, yielding and propagating microseismic noise and infrasound waves
toward the underground detector Chimonas (1977). Thus, the detector
experiences the acoustic and seismic vibrations during the windy time. This
scenario seems plausible but needs to verify in various aspects.

Potentially, this study will help to overcome several limitations in the KAGRA
detector, and thereby not only improve the noise reduction study but also
develop advanced tools and interfaces will contribute to next-generation
gravitational-wave telescopes. Furthermore, this tool will help scientists in
the GW detection as well as other fields of sciences.

###### Acknowledgements.

The authors would like to thank Kyujin Kwak, Kyungmin Kim, and Whansun Kim for
their helpful discussions and comments. This work was supported by the Basic
Science Research Program through a National Research Foundation of Korea (NRF)
funded by the Ministry of Education (NRF-2020R1I1A2054376). Furthermore, this
work is partially supported by the NRF grant funded by the Korean government’s
Ministry of Science and ICT (No. 2019R1A2C2006787, No. 2016R1A5A1013277, No.
2019R1C1C1010571, and No. 2021R1A2C1093059).

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  *[GW]: gravitational wave
  *[LIGO]: Laser Interferometer Gravitational-wave Observatory
  *[SNR]: signal-to-noise ratio
  *[PCC]: Pearson’s correlation coefficient
  *[Ktau]: Kendall’s τ coefficient
  *[MIC]: Maximal Information Coefficient
  *[MI]: mutual information


# NeCGS: Neural Compression for 3D Geometry Sets

Siyu Ren
City University of Hong Kong
<EMAIL_ADDRESS>
&Juhui Hou
City University of Hong Kong
<EMAIL_ADDRESS>
&Wenping Wang
Texas A&M University
<EMAIL_ADDRESS>
Corresponding author

###### Abstract

This paper explores the problem of effectively compressing 3D geometry sets
containing diverse categories. We make the first attempt to tackle this
fundamental and challenging problem and propose NeCGS, a neural compression
paradigm, which can compress hundreds of detailed and diverse 3D mesh models
($\sim$684 MB) by about 900 times (0.76 MB) with high accuracy and
preservation of detailed geometric details. Specifically, we first represent
each irregular mesh model/shape in a regular representation that implicitly
describes the geometry structure of the model using a 4D regular volume,
called TSDF-Def volume. Such a regular representation can not only capture
local surfaces more effectively but also facilitate the subsequent process.
Then we construct a quantization-aware auto-decoder network architecture to
regress these 4D volumes, which can summarize the similarity of local
geometric structures within a model and across different models for redundancy
elimination, resulting in more compact representations, including an embedded
feature of a smaller size associated with each model and a network parameter
set shared by all models. We finally encode the resulting features and network
parameters into bitstreams through entropy coding. After decompressing the
features and network parameters, we can reconstruct the TSDF-Def volumes,
where the 3D surfaces can be extracted through the deformable marching cubes.
Extensive experiments and ablation studies demonstrate the significant
advantages of our NeCGS over state-of-the-art methods both quantitatively and
qualitatively.

##  1 Introduction

3D mesh models/shapes are widely used in various fields, such as computer
graphics, virtual reality, robotics, and autonomous driving. As geometric data
becomes increasingly complex and voluminous, effective compression techniques
have become critical for efficient storage and transmission. Moreover, current
geometry compression methods primarily focus on individual 3D models or
sequences of 3D models that are temporally correlated, but struggle to handle
more general data sets, such as compressing large numbers of unrelated 3D
shapes.

Unlike images and videos represented as regular 2D or 3D volumes, mesh models
are commonly represented as triangle meshes, which are irregular and
challenging to compress. Thus, a natural idea is to structure the mesh models
and then leverage image or video compression techniques to compress them.
Converting mesh models into voxelized point clouds is a common practice, and
the mesh models can be recovered from the point clouds via surface
reconstruction methods [22, 24]. Based on this, in recent years, MPEG has
developed two types of 3D point cloud compression (PCC) standards [46, 28]:
geometry-based PCC (GPCC) for static models and video-based PCC (VPCC) for
sequential models. And with advancements in deep learning, numerous learning-
based PCC methods [41, 14, 55, 19, 54] have emerged, enhancing compression
efficiency. However, the voxelized point clouds require a high resolution
(typically $2^{10}$ or more) to accurately represent geometry data, which is
redundancy, limiting the compression efficiency.

Another regular representation involves utilizing implicit fields of mesh
models, such as signed distance fields (SDF) and truncated signed distance
fields (TSDF). This is achieved by calculating the value of the implicit field
at each uniformly distributed grid point, resulting in a regular volume. And
the mesh models can be recovered from the implicit fields through Matching
Cubes [32] or its variants [15, 45]. Compared with point clouds, the implicit
volume could represent the mesh models in a relatively small resolution.
Recently proposed methods, such as DeepSDF [36], utilize multilayer
perceptrons (MLPs) to regress the SDFs of any given query points. While this
representation achieves high accuracy for single or similar models (e.g.,
chairs, tables), the limited receptive field of MLPs makes it challenging to
represent large numbers of models in different categories, which is a more
common scenario in practice.

We propose NeCGS, a novel framework for compressing large sets of geometric
models. Our NeCGS framework consists of two stages: regular geometry
representation and compact neural compression. In the first stage, each model
is converted into a regular 4D volumetric format, called the TSDF-Def volume,
which can be considered a 3D ‘image’. In the second stage, we use an auto-
decoder to regress these 4D volumes. The embedded features and decoder
parameters represent these models, and compressing these components allows us
to compress the entire geometry set. We conducted extensive experiments on
various datasets, demonstrating that our NeCGS framework achieves higher
compression efficiency compared to existing geometry compression methods when
handling large numbers of models. Our NeCGS can achieve a compression ratio of
nearly 900 on some datasets, compressing hundreds or even thousands of
different models into 1$\sim$2 MB while preserving detailed structures.

Figure 1: Our NeCGeS can compress geometry data with hundreds or even
thousands of shapes into 1~2 MB while preserving details. Left: Original
Geometry Data. Right: Decompressed Geometry Data.  Zoom in for details.

##  2 Related Work

###  2.1 Geometry Representation

In general, the representation of geometry data is divided into two main
categories, explicit representation and implicit representation, and they
could be transformed into another.

Explicit Representation. Among the explicit representations, voxelization [7]
is the most intuitive. In this method, geometry models are represented by
regularly distributed grids, effectively converting them into 3D ‘images’.
While this approach simplifies the processing of geometry models using image
processing techniques, it requires a high resolution to accurately represent
the models, which demands substantial memory and limits its application.
Another widely used geometry representation method is the point cloud, which
consists of discrete points sampled from the surfaces of models. This method
has become a predominant approach for surface representation [2, 39, 40].
However, the discrete nature of the points imposes constraints on its use in
downstream tasks such as rendering and editing. Triangle meshes offer a more
precise and efficient geometry representation. By approximating surfaces with
numerous triangles, they achieve higher accuracy and efficiency for certain
downstream tasks.

Implicit Representation. Implicit representations use the isosurface of a
function or field to represent surfaces. The most widely used implicit
representations include Binary Occupancy Field (BOF) [22, 35], Signed Distance
Field (SDF) [36, 29], and Truncated Signed Distance Field (TSDF) [11], from
which the model’s surface can be easily extracted. However, these methods are
limited to representing watertight models. The Unsigned Distance Field (UDF)
[8], which is the absolute value of the SDF, can represent more general
models, not just watertight ones. Despite this advantage, extracting surfaces
from UDF is challenging, which limits its application.

Conversion between Geometry Representations. Geometry representations can be
converted between explicit and implicit forms. Various methods [21, 22, 24, 6,
35, 29, 45] are available for calculating the implicit field from given
models. Conversely, when converting from implicit to explicit forms, Marching
Cubes [32] and its derivatives [48, 49, 15, 45] can reconstruct continuous
surfaces from various implicit fields.

###  2.2 3D Geometry Data Compression

Single 3D Geometric Model Compression. In recent decades, compression
techniques for images and videos have rapidly advanced [51, 34, 59, 5, 4].
However, the irregular nature of geometry data makes it more challenging to
compress compared to images and video, which are represented as volumetric
data. A natural approach is to convert geometry data into voxelized point
clouds, treating them as 3D ‘images’, and then applying image and video
compression techniques to them. Following this intuition, MPEG developed the
GPCC standards [13, 28, 47], where triangle meshes or triangle soup
approximates the surfaces of 3D models, enabling the compression of models
with more complex structures. Subsequently, several improved methods [37, 60,
53, 62] and learning-based methods [18, 43, 10, 9, 3, 42, 54] have been
proposed to further enhance compression performance. However, these methods
rely on voxelized point clouds to represent geometry models, which is
inefficient and memory-intensive, limiting their compression efficiency. In
contrast to the previously mentioned methods, Draco [12] uses a kd-tree-based
coding method to compress vertices and employs the EdgeBreaker algorithm to
encode the topological relationships of the geometry data. Draco utilizes
uniform quantization to control the compression ratio, but its performance
decreases at higher compression ratios.

Multiple Model Compression. Compared to compressing single 3D geometric
models, compressing multiple objects is significantly more challenging. SLRMA
[17] addresses this by using a low-rank matrix to approximate vertex matrices,
thus compressing sequential models. Mekuria et al. [33] proposed the first
codec for compressing sequential point clouds, where each frame is coded using
Octree subdivision through an 8-bit occupancy code. Building on this concept,
MPEG developed the VPCC standards [13, 28, 47], which utilize 3D-to-2D
projection and encode time-varying projected planes, depth maps, and other
data using video codecs. Several improved methods [57, 26, 1, 44] have been
proposed to enhance the compression of sequential models. Recently, shape
priors like SMPL [31] and SMAL [63] have been introduced, allowing the pose
and shape of a template frame to be altered using only a few parameters. Pose-
driven geometry compression methods [16, 58, 56] leverage this approach to
achieve high compression efficiency. However, these methods are limited to
sequences of corresponding geometry data and cannot handle sets of unrelated
geometry data, which is more common in practice.

##  3 Proposed Method

Overview. Given a set of $N$ 3D mesh models containing diverse categories,
denoted as $\mathcal{S}=\\{\mathbf{S}_{i}\\}_{i=1}^{N}$, we aim to compress
them into a bitstream while maintaining the quality of the decompressed models
as much as possible. To this end, we propose a neural compression paradigm
called NeCGS. As shown in Fig. 2, NeCGS consists of two main modules, i.e.,
Regular Geometry Representation (RGR) and Compact Neural Representation (CNR).
Specifically, RGR first represents each irregular mesh model within
$\mathcal{S}$ into a regular 4D volume, namely TSDF-Def volume that mplicitly
describes the geometry structure of the model, via a rendering-based
optimization, thus leading to a set of 4D volumes
$\mathcal{V}:=\\{\mathbf{V}_{i}\\}_{i=1}^{N}$ with $\mathbf{V}_{i}$
corresponding to $\mathbf{S}_{i}$. Then CNR further obtains a more compact
neural representation of $\mathcal{V}$, where a quantization-aware auto-
decoder-based network is constructed to regress these volumes, producing an
embedded feature for each volume. Finally, the embedded features along with
the network parameters are encoded into a bitstream through a typical entropy
coding method to achieve compression. We also want to note that NeCGS can also
be applied to compress 3D geometry sets represented in 3D point clouds, where
one can either reconstruct from the given point clouds 3D surfaces through a
typical surface reconstruction method or adopt a pre-trained network for SDF
estimation from point clouds, e.g., SPSR [22] or IMLS [24], to bridge the gap
between 3D mesh and point cloud models. In what follows, we will detail NeCGS.

Figure 2: The pipeline of NeCGS. It first represents original meshes regularly
into TSDF-Def volumes, and an auto-decoder network is utilized to regress
these volume. Then the embedded features and decoder parameters are compressed
into bitstreams through entropy coding. When decompressing the models, the
decompressed embedded features are fed into the decoder with the decompressed
parameters from the bitstreams, reconstructing the TSDF-Def volumes, and the
models can be extracted from them.

###  3.1  Regular Geometry Representation

Figure 3: 2D visual illustration of DMC. The blue points refer to the
deformable grid points, the green points refer to the vertices of the
extracted surfaces, and the orange lines refer to the faces of the extracted
surfaces. Left: The original grid points. Right: The surface extraction.

Unlike 2D images and videos, where pixels are uniformly distributed on 2D
regular girds, the irregular characteristic of 3D mesh models makes it
challenging to compress them efficiently and effectively. We propose to
convert each 3D mesh model to a 4D regular volume called TSDF-Def volume,
which implicitly represents the geometry structure of the model. Such a
regular representation can describe the model precisely, and its regular
nature proves beneficial for compression in the subsequent stage.

TSDF-Def Volume. Although 3D regular SDF or TSDF volumes are widely used for
representing 3D geometry models, they may introduce distortions when the
volume resolution is relatively limited. Inspired by recent shape extracting
methods [48, 49], we propose TSDF-Def, which extends the regular TSDF volume
by introducing an additional deformation for each grid point to adjust the
detailed structure during the extraction of models, as shown in Fig. 3.
Accordingly, we develop the differentiable Deformable Marching Cubes (DMC),
the variant of the Marching Cubes method [32], for surface extraction from a
TSDF-Def volume. Consequently, each shape $\mathbf{S}$ is represented as a 4D
TSDF-Def volume, denoted as $\mathbf{V}\in\mathbb{R}^{\scriptscriptstyle
K\times K\times K\times 4}$, where $K$ is the volume resolution. More
specifically, the value of the grid point located at $(u,v,w)$ is
$\mathbf{V}(u,v,w):=[\mathtt{TSDF}(u,v,w),\Delta u,\Delta v,\Delta w]$, where
$(\Delta u,\Delta v,\Delta w)$ are the deformation for the grid point and
$1\leq u,v,w\leq K$. TSDF-Def enhances representation accuracy, particularly
when the grid resolution is relatively low.

Optimization of TSDF-Def Volumes. To obtain the optimal TSDF-Def volume
$\mathbf{V}$ for a given model $\mathbf{S}$, after initializing the
deformations of each grid to zero and computing the TSDF value for each grid
we optimize the following problem:

$\min_{\mathbf{V}}\mathcal{E}_{\rm Rec}(\texttt{DMC}(\mathbf{V}),\mathbf{S}),$
(1)

where $\texttt{DMC}(\cdot)$ refers to the differentiable DMC process for
extracting surfaces from TSDF-Def volumes, and the $\mathcal{E}_{\rm
Reg}(\cdot,~{}\cdot)$ measures the differences between the rendered depth and
silhouette images of two mesh models through the differentiable rasterization
[25]. Algorithm 1 summarizes the whole optimization process. More details can
be found in Sec. A.2 of the subsequent Appendix.

Input: 3D mesh model $\mathbf{S}$; the maximum number of iterations maxIter.

Output: The optimal TSDF-Def volume
$\mathbf{V}\in\mathbb{R}^{\scriptscriptstyle K\times K\times K\times 4}$.

1 Place uniformly distributed grids in the cube of $\mathbf{S}$, denoted as
$\mathbf{G}\in\mathbb{R}^{\scriptscriptstyle K\times K\times K\times 3}$;

2 Initialize $\mathbf{V}[...,0]$ as the ground truth TSDF of $\mathbf{S}$ at
the location of $\mathbf{G}$, the deformation $\mathbf{V}[...,1:]$=0, and the
current iteration $\texttt{Iter}=0$;

3 while  _$\mathtt{{Iter}} <\mathtt{maxIter}$_ do

4 Recover shape from $\mathbf{V}$ according to DMC,
$\texttt{DMC}(\mathbf{V})$;

5 Calculate the reconstruction error,
$\mathcal{E}_{\text{Rec}}(\texttt{DMC}(\mathbf{V}),\mathbf{S})$;

6 Optimize $\mathbf{V}$ using ADAM optimizer based on the reconstruction
error;

7 Iter:=Iter+1;

8

9 end while

10return $\mathbf{V}$;

Algorithm 1 Optimization of TSDF-Def Volumes

###  3.2 Compact Neural Representation

Observing the similarity of local geometric structures within a typical 3D
model and across different models, i.e., redundancy, we further propose a
quantization-aware neural representation process to summarize the similarity
within $\mathcal{V}$, leading to more compact representations with redundancy
removed.

Network Architecture. We construct an auto-decoder network architecture to
regress these 4D TSDF-Def volumes. Specifically, it is composed of a head
layer, which increases the channel of its input, and $L$ cascaded upsampling
modules, which progressively upscale the feature volume. We also utilize the
PixelShuffle technique [50] between the convolution and activation layers to
achieve upscaling. We refer reviewers to Sec. B of Appendix for more details.
For TSDF-Def volume $\mathbf{V}_{i}$, the corresponding input to the auto-
decoder is the embedded feature, denoted as
$\mathbf{F}_{i}\in\mathbb{R}^{\scriptscriptstyle K^{\prime}\times
K^{\prime}\times K^{\prime}\times C}$, where $K^{\prime}$ is the resolution
satisfying $K^{\prime}\ll K$ and $C$ is the number of channels. Moreover, we
integrate differentiable quantization to the embedded features and network
parameters in the process, which can efficiently reduce the quantization
error. In all, the compact neural representation process can be written as

$\widehat{\mathbf{V}}_{i}=\mathcal{D}_{\mathcal{Q}(\bm{{\Theta}})}(\mathcal{Q}(\mathbf{F}_{i})).$
(2)

where $\mathcal{Q}(\cdot)$ stands for the differentiable quantization
operator, and $\widehat{\mathbf{V}}_{i}$ is the regressed TSDF-Def.

Loss Function. We employ a joint loss function comprising Mean Absolute Error
(MAE) and Structural Similarity Index (SSIM) to simultaneously optimize the
embedded features $\\{\mathbf{F}_{i}\\}$ and the network parameters
$\bm{{\Theta}}$. In computing the MAE between the predicted and ground truth
TSDF-Def volumes, we concentrate more on the grids close to the surface. These
surface grids crucially determine the surfaces through their TSDFs and
deformations; hence we assign them higher weights during optimization than the
grids farther away from the surface. The overall loss function for the $i$-th
model is written as

$\mathcal{L}(\widehat{\mathbf{V}}_{i},\mathbf{V}_{i})=\|\widehat{\mathbf{V}}_{i}-\mathbf{V}_{i}\|_{1}+\lambda_{1}\|\mathbf{M}_{i}\odot(\widehat{\mathbf{V}}_{i}-\mathbf{V}_{i})\|_{1}+\lambda_{2}(1-\texttt{SSIM}(\widehat{\mathbf{V}}_{i},\mathbf{V}_{i})),$
(3)

where $\mathbf{M}_{i}=\mathbbm{1}(|\mathbf{V}_{i}[...,0])|<\tau)$ is the mask,
indicating whether a grid is near the surface, i.e., its TSDF is less than the
threshold $\tau$, while $\lambda_{1}$ and $\lambda_{2}$ are the weights to
balance each term of the loss function.

Entropy Coding. After obtaining the quantized features
$\\{\widetilde{\mathbf{F}}_{i}=\mathcal{Q}(\mathbf{F}_{i})\\}$ and quantized
network parameters $\widetilde{\bm{\Theta}}=\mathcal{Q}(\bm{\Theta})$, we
adopt the Huffman Codec [20] to further compress them into a bitstream. More
advanced entropy coding methods can be employed to further improve compression
performance.

###  3.3 Decompression

To obtain the 3D mesh models from the bitstream, we first decompress the
bitstream to derive the embedded features, $\\{\widetilde{\mathbf{F}}_{i}\\}$
and the decoder parameter, $\widetilde{\bm{\Theta}}$. Then, for each
$\widetilde{\mathbf{F}}_{i}$, we feed it to the decoder
$\mathcal{D}_{\widetilde{\bm{\Theta}}}(\cdot)$ to generate its corresponding
TSDF-Def volume

$\widehat{\mathbf{V}}_{i}=\mathcal{D}_{\widetilde{\bm{\Theta}}}(\widetilde{\mathbf{F}}_{i}).$
(4)

Finally, we utilize DMC to recover each shape from $\widehat{\mathbf{V}}_{i}$,
$\widehat{\mathbf{S}}_{i}=\texttt{DMC}(\widehat{\mathbf{V}}_{i})$, forming the
set of decompressed geometry data,
$\widehat{\mathcal{S}}=\\{\widehat{\mathbf{S}}_{i}\\}_{i=1}^{N}$.

##  4 Experiment

###  4.1 Experimental Setting

Implementation details. In the process of optimizing TSDF-Def volumes, we
employed the ADAM optimizer [23] for 500 iterations per shape, using a
learning rate of 0.01. The resolution of TSDF-Def volumes was $K=128$. The
resolution and the number of channels of the embedded features were
$K^{\prime}=4$ and $C=16$, respectively. And the decoder is composed of $L=5$
upsampling modules with an up-scaling factor of 2. During the optimization, we
set $\lambda_{1}=5$ and $\lambda_{2}=10$, and the embedded features and
decoder parameters were optimized by the ADAM optimizer for 400 epochs, with a
learning rate of 1e-3. We achieved different compression efficiencies by
adjusting decoder sizes. We conducted all experiments on an NVIDIA RTX 3090
GPU with Intel(R) Xeon(R) CPU.

Table 1:  Details of the selected datasets222The original geometry data is kept as triangle meshes, so the storage size is much less than the voxelized point clouds.. Dataset | Original Size (MB) | # Models
---|---|---
AMA | 378.41 | 500
DT4D | 683.80 | 500
Thingi10K | 335.92 | 1000
Mixed | 496.16 | 600

Datasets. We tested our NeCGS on various types of datasets, including humans,
animals, and CAD models. For human models, we randomly selected 500 shapes
from the AMA dataset [52]. For animal models, we randomly selected 500 shapes
from the DT4D dataset [27]. For the CAD models, we randomly selected 1000
shapes from the Thingi10K dataset [61]. Besides, we randomly selected 200
models from each dataset, forming a more challenging dataset, denoted as
Mixed. The details about the selected datasets are shown in Table 1. In all
experiments, we scaled all models in a cube with a range of $[-1,1]^{3}$ to
ensure they are in the same scale.

Methods under Comparison. In terms of traditional geometry codecs, we chose
the three most impactful geometry coding standards with released codes,
G-PCC333https://github.com/MPEGGroup/mpeg-pcc-tmc13 and
V-PCC444https://github.com/MPEGGroup/mpeg-pcc-tmc2 from MPEG (see more details
about them in [13, 28, 47]), and Draco 555https://github.com/google/draco from
Google as the baseline methods. Additionally, we compared our approach with
state-of-the-art deep learning-based compression methods, specifically PCGCv2
[54]. Furthermore, we adapted DeepSDF [36] with quantization to serve as
another baseline method, denoted as QuantDeepSDF. It is worth noting that
while some of the chosen baseline methods were originally designed for point
cloud compression, we utilized voxel sampling and SPSR [22] to convert them
between the forms of point cloud and surface. More details can be found in
Sec. C.2 appendix.

(a) AMA

(b) DT4D

(c) Thingi10K

(d) Mixed

Figure 4: Quantitative comparisons of different methods on four 3D geometry
sets.

Evaluation Metrics. Following previous reconstruction methods [35, 38], we
utilize Chamfer Distance (CD), Normal Consistency (NC), F-Score with the
thresholds of 0.005 and 0.01 (F1-0.005 and F1-0.01) as the evaluation metrics.
Furthermore, to comprehensively compare the compression efficiency of
different methods, we use Rate-Distortion (RD) curves. These curves illustrate
the distortions at various compression ratios, with CD and F1-0.005
specifically describing the distortion of the decompressed models. Our goal is
to minimize distortion, indicated by a low CD and a high F1-Score, while
maximizing the compression ratio. Therefore, for the RD curve representing CD,
optimal compression performance is achieved when the curve is closest to the
lower right corner. Similarly, for the RD curve representing the F1-Score, the
ideal compression performance is when the curve is nearest to the upper right
corner. Their detailed definition can be found in Sec. C.1 of appendix.

###  4.2 Results

The RD curves of different compression methods under different datasets are
shown in Fig. 4. As the compression ratio increases, the distortion also
becomes larger. It is obvious that our NeCGS can achieve much better
compression performance than the baseline methods when the compression ratio
is high, even in the challenging Mixed dataset. In particular, our NeCGS
achieves a minimum compression ratio of 300, and on the DT4D dataset, the
compression ratio even reaches nearly 900, with minimal distortion. Due to the
larger model differences within the Thingi10K and Mixed datasets compared to
the other two datasets, the compression performance on these two datasets is
inferior.

(a) GPCC (b) VPCC (c) PCGCv2 (d) Draco (e) QuantDeepSDF (f) Ours (g) Ori.

AMA

DT4D

Thingi10K

Mixed

312.8341.95256.01123.20272.91307.79148.5049.81103.9096.52165.47166.79457.39244.38402.70153.45409.21455.25299.13165.75267.4299.94224.17362.80
Figure 5: Visual comparisons of different compression methods. All numbers in
corners represent the compression ratio.  Zoom in for details.

(a) Ori. (b) 455.25

(c) 651.85 (d) 899.73

Figure 6: Decompressed models under different compression ratios.

The visual results of different compression methods are shown in Fig. 5.
Compared to other methods, models compressed using our approach occupy a
larger compression ratio and retain more details after decompression. Fig. 6
illustrates the decompressed models under different compression ratio. Even
when the compression ratio reaches nearly 900, our method can still retain the
details of the models.

###  4.3 Ablation Study

In order to illustrate the efficiency of each design of our NeCGS, we
conducted extensive ablation study about them on the Mixed dataset.

Figure 7: Models recovered from different regular geometry representations
under various volume resolutions. From Left to Right: Original, TSDF with
$K=64$, TSDF with $K=128$, TSDF-Def with $K=64$, and TSDF-Def with $K=128$.

Necessity of the Deformation of Grids. We utilize TSDF-Def volumes to as the
regular geometry representation, instead of TSDF volumes like previous
methods. Compared with models recovered from TSDF volumes through MC, the
models recovered from TSDF-Def volumes through DMC preserve more details of
the thin structures, especially when the volume resolutions are relatively
small, as shown in Fig. 7. We also conducted a numerical comparison of the
decompressed models on the AMA dataset under these two settings, and the
results are shown in Table. 2, demonstrating its advantages.

Table 2: Quantitative comparisons of different RGRs.  RGR | Size (MB) | Com. Ratio | CD ($\times 10^{-3}$) $\downarrow$ | NC $\uparrow$ | F1-0.005 $\uparrow$ | F1-0.01 $\uparrow$
---|---|---|---|---|---|---
TSDF | 1.631 | 304.20 | 5.015 | 0.944 | 0.662 | 0.936
TSDF-Def | 1.612 | 307.79 | 4.913 | 0.947 | 0.674 | 0.943

Neural Representation Structure. To illustrate the superiority of auto-decoder
framework, we utilize an auto-encoder to regress the TSDF-Def volume.
Technically, we used a ConvNeXt block [30] as the encoder by replacing 2D
convolutions with 3D convolutions. Under the auto-encoder framework, we
optimize the parameters of the encoder to change the embedded features. The RD
curves about these two structures are shown in Fig. 8, demonstrating
rationality of our decoder structure.

Figure 8: (a) RD curves of different neural representation structures. (b) RD
curves of different regression losses.

(a) Original (b) w/o SSIM

(c) w/ SSIM

Figure 9: Visual comparison of regression loss w/ and w/o SSIM item.

SSIM Loss. Compared to MAE, which focuses on one-to-one errors between
predicted and ground truth volumes, the SSIM item in Eq. 3 emphasizes more on
the local similarity between volumes, increasing the regression accuracy. To
verify this, we removed the SSIM item and kept others unchanged. Their RD
curves are shown in Fig. 8, and it is obvious that the SSIM item in the
regression loss increases the compression performance. The visual comparison
is shown in Fig. 9, and without SSIM, there are floating parts around the
decompressed models.

(a) Ori. (b) 64

(c) 128 (d) 256

Figure 10: Visual comparison under different resolutions of TSDF-Def volume.

Resolution of TSDF-Def Volumes. We tested the compression performance at
different resolutions of TSDF-Def volumes by adjusting the decoder layers
accordingly. Specifically, we removed the last layer for a resolution of 64
and added an extra layer for a resolution of 256. The quantitative and
numerical comparisons are shown in Table 3 and Fig. 10, respectively.
Obviously, increasing the volume resolution can enhance the compression
effectiveness, resulting in more detailed structures preserved after
decompression. However, the optimization and inference time also increase
accordingly due to more layers involved.

Table 3: Quantitative comparisons of different resolutions of TSDF-Def
volumes.

Res. | Size (MB) | Com. Ratio | CD ($\times 10^{-3}$) $\downarrow$ | NC $\uparrow$ | F1-0.005 $\uparrow$ | F1-0.01 $\uparrow$ | Opt Time (h) | Infer. Time (ms)
---|---|---|---|---|---|---|---|---
64 | 1.408 | 268.75 | 4.271 | 0.927 | 0.721 | 0.966 | 2.16 | 38.97
128 | 1.493 | 253.45 | 3.436 | 0.952 | 0.842 | 0.991 | 16.32 | 98.95
256 | 1.627 | 232.58 | 3.234 | 0.962 | 0.870 | 0.995 | 94.50 | 421.94

##  5 Conclusion and Discussion

We have presented NeCGS, a highly effective neural compression scheme for 3D
geometry sets. NeCGS has achieved remarkable compression performance on
various datasets with diverse and detailed shapes, outperforming state-of-the-
art compression methods to a large extent. These advantages are attributed to
our regular geometry representation and the compression accomplished by a
convolution-based auto-decoder. We believe our NeCGS framework will inspire
further advancements in the field of geometry compression.

However, our method still suffers from the following two limitations. One is
that it requires more than 15 hours to regress the TSDF-Def volumes, and the
other one is that the usage of 3D convolution layers limits the inference
speed. Our future work will focus on addressing these challenges by
accelerating the optimization process and incorporating more efficient network
modules.

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Appendix

##  Appendix A Regular Geometry Representation

###  A.1 Tensor Quantization

Denoted $\mathbf{x}$ is a tensor, we quantize it in a fixed interval, $[a,b]$,
at $(2^{N}+1)$ levels666We partition the interval $[a,b]$ into $(2^{N}+1)$
levels, rather than $2^{N}$ levels, to ensure the inclusion of the value 0. by

$\mathcal{Q}(\mathbf{x})=\texttt{Round}\left(\frac{\texttt{Clamp}(\mathbf{x},a,b)-a}{s}\right)\times
s+a,$ (5)

where $s=(b-a)/2^{N}$. In our experiment, we set $a=-1$ and $b=1$.

###  A.2 Optimization of TSDF-deformation Volumes

We set a series of camera pose, $\mathcal{T}=\\{\mathbf{T}_{i}\\}_{i=1}^{E}$,
around the meshes. Let
$\mathbf{I}^{\scriptscriptstyle\rm{D}}_{1}(\mathbf{T}_{i})$ and
$\mathbf{I}^{\scriptscriptstyle\rm{D}}_{2}(\mathbf{T}_{i})$ represent the
depth images obtained from the reconstructed mesh $\texttt{DMC}(\mathbf{V})$
and the given mesh $\mathbf{S}$ at the pose $\mathbf{T}_{i}$ respectively.
Similarly, let $\mathbf{I}^{\scriptscriptstyle\rm{M}}_{1}(\mathbf{T}_{i})$ and
$\mathbf{I}^{\scriptscriptstyle\rm{M}}_{2}(\mathbf{T}_{i})$ denote their
respective silhouette images at pose $\mathbf{T}_{i}$. The reconstruction
error produced by silhouette and depth images at all pose are

$\mathcal{E}_{\rm
M}(\texttt{DMC}(\mathbf{V}),\mathbf{S})=\sum_{\mathcal{T}_{i}\in\mathcal{T}}\|\mathbf{I}^{\scriptscriptstyle\rm{M}}_{1}(\mathbf{T}_{i})-\mathbf{I}^{\scriptscriptstyle\rm{M}}_{2}(\mathbf{T}_{i})\|_{1}$
(6)

and

$\mathcal{E}_{\rm
D}(\texttt{DMC}(\mathbf{V}),\mathbf{S})=\sum_{\mathcal{T}_{i}\in\mathcal{T}}\|\left(\mathbf{I}^{\scriptscriptstyle\rm{D}}_{1}(\mathbf{T}_{i})-\mathbf{I}^{\scriptscriptstyle\rm{D}}_{2}(\mathbf{T}_{i})\right)*\mathbf{I}^{\scriptscriptstyle\rm{M}}_{2}(\mathbf{T}_{i})\|_{1}.$
(7)

Then the reconstruction error is defined as

$\mathcal{E}_{\rm Rec}(\texttt{DMC}(\mathbf{V}),\mathbf{S})=\mathcal{E}_{\rm
M}(\texttt{DMC}(\mathbf{V}),\mathbf{S})+\lambda_{\rm rec}\mathcal{E}_{\rm
D}(\texttt{DMC}(\mathbf{V}),\mathbf{S}),$ (8)

where $E=4$ and $\lambda_{\rm rec}=10$ in our experiment.

##  Appendix B Auto-decoder-based Neural Compression

###  B.1 Upsampling Module

In each upsampling module, we utilize a PixelShuffle layer between the
convolution and activation layers to upscale the input, as shown in Fig. 11.
The input feature volume has dimensions $(N_{\rm in},N_{\rm in},N_{\rm
in},C_{\rm in})$, with an upsampling scale of $s$ and an output channel count
of $C_{\rm out}$.

Figure 11: Upsampling Module.

##  Appendix C Experiment

###  C.1 Evaluation Metric

Let $\mathbf{S}_{\rm Rec}$ and $\mathbf{S}_{\rm GT}$ denote the reconstructed
and ground-truth 3D shapes, respectively. We then randomly sample $N_{\rm
eval}=10^{5}$ points on them, obtaining two point clouds, $\mathbf{P}_{\rm
Rec}$ and $\mathbf{P}_{\rm GT}$. For each point of $\mathbf{P}_{\rm Rec}$ and
$\mathbf{P}_{\rm GT}$, the normal of the triangle face where it is sampled is
considered to be its normal vector, and the normal sets of $\mathbf{P}_{\rm
Rec}$ and $\mathbf{P}_{\rm GT}$ are denoted as $\mathbf{N}_{\rm Rec}$ and
$\mathbf{N}_{\rm GT}$, respectively. Let
$\mathtt{NN\\_Point}(\mathbf{x},\mathbf{P})$ be the operator that returns the
nearest point of $\mathbf{x}$ in the point cloud $\mathbf{P}$. The CD between
them is defined as

$\displaystyle\texttt{CD}(\mathbf{S}_{\rm Rec},\mathbf{S}_{\rm GT})=$
$\displaystyle\frac{1}{2N_{\rm eval}}\sum_{\mathbf{x}\in\mathbf{P}_{\rm
Rec}}\|\mathbf{x}-\mathtt{NN\\_Point}(\mathbf{x},\mathbf{P}_{\rm GT})\|_{2}$
(9) $\displaystyle+$ $\displaystyle\frac{1}{2N_{\rm
eval}}\sum_{\mathbf{x}\in\mathbf{P}_{\rm
GT}}\|\mathbf{x}-\mathtt{NN\\_Point}(\mathbf{x},\mathbf{P}_{\rm Rec})\|_{2}.$

Let $\mathtt{NN\\_Normal}(\mathbf{x},\mathbf{P})$ be the operator that returns
the normal vector of the point $\mathbf{x}$’s nearest point in the point cloud
$\mathbf{P}$. The NC is defined as

$\displaystyle\texttt{NC}(\mathbf{S}_{\rm Rec},\mathbf{S}_{\rm GT})=$
$\displaystyle\frac{1}{2N_{\rm eval}}\sum_{\mathbf{x}\in\mathbf{P}_{\rm
Rec}}|\mathbf{N}_{\rm
Rec}(\mathbf{x})\cdot\mathtt{NN\\_Normal}(\mathbf{x},\mathbf{P}_{\rm GT})|$
(10) $\displaystyle+$ $\displaystyle\frac{1}{2N_{\rm
eval}}\sum_{\mathbf{x}\in\mathbf{P}_{\rm GT}}|\mathbf{N}_{\rm
GT}(\mathbf{x})\cdot\mathtt{NN\\_Normal}(\mathbf{x},\mathbf{P}_{\rm Rec})|.$

F-Score is defined as the harmonic mean between the precision and the recall
of points that lie within a certain distance threshold $\epsilon$ between
$\mathbf{S}_{\rm Rec}$ and $\mathbf{S}_{\rm GT}$,

$\mathtt{F-Score}(\mathbf{S}_{\rm Rec},\mathbf{S}_{\rm
GT},\epsilon)=\frac{2\cdot\mathtt{Recall}\cdot\mathtt{Precision}}{\mathtt{Recall}+\mathtt{Precision}},$
(11)

where

$\displaystyle\mathtt{Recall}(\mathbf{S}_{\rm Rec},\mathbf{S}_{\rm
GT},\epsilon)$ $\displaystyle=\left|\left\\{\mathbf{x}_{1}\in\mathbf{P}_{\rm
Rec},\rm{s.t.}\min_{\mathbf{x}_{2}\in\mathbf{P}_{\rm
GT}}\|\mathbf{x}_{1}-\mathbf{x}_{2}\|_{2}<\epsilon\right\\}\right|,$ (12)
$\displaystyle\mathtt{Precision}(\mathbf{S}_{\rm Rec},\mathbf{S}_{\rm
GT},\epsilon)$ $\displaystyle=\left|\left\\{\mathbf{x}_{2}\in\mathbf{P}_{\rm
GT},\rm{s.t.}\min_{\mathbf{x}_{1}\in\mathbf{P}_{\rm
Rec}}\|\mathbf{x}_{1}-\mathbf{x}_{2}\|_{2}<\epsilon\right\\}\right|.$

Figure 12: Pipeline of QuantDeepSDF.

###  C.2 QuantDeepSDF

Compared to DeepSDF, our QuantDeepSDF incorporates the following two
modifications:

  * •

The decoder parameters are quantized to enhance compression efficiency.

  * •

To maintain consistency with our NeCGS, the points sampled during training are
drawn from TSDF-Def volumes.

The pipeline of QuantDeepSDF is shown in Fig. 12. Specifically, the decoder is
an MLP, where the input is the concatenated vector of coordinate
$\mathbf{x}\in\mathbb{R}^{3}$ and the $i$-th embedded feature vector
$\mathbf{F}_{i}\in\mathbb{R}^{C}$, and the output is the corresponding TSDF-
Def value. In our experiment, the decoder consists of 8 layers, and the
compression ratio is controled by changing the width of each layer.

###  C.3 Auto-Encoder in Ablation Study

Different from the auto-encoder used in our framework, where the embed
features are directly optimized, auto-encoder utilizes an encoder to produce
the embedded features, where the inputs are the TSDF-Def volumes. And the
decoder is kept the same as our framework. During the optimization, the
parameters of encoder and decoder are optimized. Once optimized, the embedded
features produced by the encoder and decoder parameters are compressed into
bitstreams.

###  C.4 More Visual Results

Fig. 13 depicts the visual results of the decompresed models from the AMA
dataset, DT4D dataset, and Thingi10K dataset under various compression ratios,
respectively. With the compression ratio increasing, the decompressed models
still preserve the detailed structures, without large distortion.

Ori. 253.45 362.80 500.54

Ori. 455.26 651.85 899.73

Ori. 166.79 219.84 273.32

Figure 13: Visual results of the decompressed models under different
compression ratios. From Top to Bottom: AMA, DT4D, and Thingi10K.  Zoom in for
details.