# Shadowing in inelastic nucleon-nucleon cross section?

Kari J. Eskola<EMAIL_ADDRESS>Ilkka Helenius<EMAIL_ADDRESS>Mikko Kuha<EMAIL_ADDRESS>Hannu Paukkunen
<EMAIL_ADDRESS>University of Jyvaskyla, Department of Physics, P.O.
Box 35, FI-40014 University of Jyvaskyla, Finland  Helsinki Institute of
Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland

###### Abstract

Experimental results of inclusive hard-process cross sections in heavy-ion
collisions conventionally lean on a normalization computed from Glauber models
where the inelastic nucleon-nucleon cross section $\sigma_{\rm nn}^{\rm inel}$
– a crucial input parameter – is simply taken from proton-proton measurements.
In this letter, using the computed electro-weak boson production cross
sections in lead-lead collisions as a benchmark, we determine $\sigma_{\rm
nn}^{\rm inel}$ from the recent ATLAS data. We find a significantly suppressed
$\sigma_{\rm nn}^{\rm inel}$ relative to what is usually assumed, show the
consequences for the centrality dependence of the cross sections, and address
the phenomenon in an eikonal minijet model with nuclear shadowing.

##  I Introduction

In a high-energy nucleus-nucleus collision the produced particle multiplicity
correlates strongly with the collision geometry: the more central the
collision, typically the higher the multiplicity. Experimentally, the
centrality classification is obtained by organizing the events according to
their multiplicity (or transverse energy) into bins of equal fraction, say
10%, of all events. Conventionally, 0–10% (90–100%) centrality refers to the
events of highest (lowest) multiplicities, and 0–100% to all events, minimum
bias.

Inclusive hard processes in turn are rarer processes of a large momentum scale
whose cross sections in nucleus-nucleus collisions are traditionally obtained
by converting the measured per-event yields $N_{h}^{c}/{N_{\mathrm{evt}}^{c}}$
in a centrality class $c$ into hard nucleon-nucleon cross sections
$\sigma_{h}^{c}$ through

$\sigma_{h}^{c}=\frac{\sigma_{\mathrm{nn}}^{\mathrm{inel}}}{\langle N_{\rm
bin}\rangle_{c}}\frac{N_{h}^{c}}{N_{\mathrm{evt}}^{c}}\,,$ (1)

where $\langle N_{\rm bin}\rangle_{c}$ is the mean number of independent
inelastically interacting nucleon-nucleon pairs, binary collisions, in the
centrality class $c$, and $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ is the
inelastic nucleon-nucleon cross section. The model-dependent quantity $\langle
N_{\rm bin}\rangle_{c}$ here is obtained from the Monte-Carlo (MC) Glauber
model Miller  _et al._ (2007). The nuclear modification ratio $R_{AA}^{h,c}$
for the hard process, in the centrality class $c$, is then obtained by
dividing $\sigma_{h}^{c}$ by the corresponding minimum-bias cross-section in
proton-proton collisions.

This method of constructing hard cross sections is a routine procedure in the
heavy-ion measurements at RHIC and at the LHC and it has been used e.g. to
construct nuclear modification ratios for jets Abelev  _et al._ (2014);
Khachatryan  _et al._ (2017a); Aad  _et al._ (2013a); Aaboud  _et al._ (2019a)
and charged particles Sirunyan  _et al._ (2018); Khachatryan  _et al._
(2017b); Aad  _et al._ (2015a); Acharya  _et al._ (2019, 2018a); Adare  _et
al._ (2016); Adler  _et al._ (2004) which, in turn, are widely used in
theoretical studies of jet quenching Bjorken (1982); Qin and Wang (2015);
d’Enterria (2010) and partonic energy loss Arleo (2017); Andrés  _et al._
(2016); Chien  _et al._ (2016); Zapp  _et al._ (2013). In the same way, Eq.
(1) forms the basis for measuring centrality-dependent cross sections of
direct photon Aad  _et al._ (2016); Chatrchyan  _et al._ (2012a); Afanasiev
_et al._ (2012) and electro-weak (EW) boson Aad  _et al._ (2015b, 2013b);
Chatrchyan  _et al._ (2012b, 2011) production, which can be used to study e.g.
nuclear effects in parton distribution functions (PDFs) Paukkunen and Salgado
(2011); Helenius  _et al._ (2013).

The basic inputs of the Glauber model are the nuclear geometry and
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ Miller  _et al._ (2007). In the MC
Glauber model the positions of the nucleons are sampled event by event
according to the nuclear density profile, usually the Woods-Saxon distribution
Woods and Saxon (1954). The probability for an interaction between two
nucleons depends on their mutual distance and
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$. As a result, cross-section
measurements through Eq. (1) depend on $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$
in a non-trivial way. An established procedure is to take the value
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ and its energy dependence from proton-
proton measurements. However, at high-enough energies the particle production
becomes sensitive to QCD dynamics at small momentum fractions $x$ where some
suppression is expected due to gluon shadowing Qiu (1987); Armesto (2006);
Frankfurt  _et al._ (2012) or saturation phenomena Gribov  _et al._ (1983);
McLerran and Venugopalan (1994); Gelis  _et al._ (2010). Such effects become
more pronounced in heavy nuclei and towards lower scales so one could argue
that in collisions involving heavy ions the value of
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ should also be reduced relative to what
is measured in proton-proton collisions. Through Eq. (1), this would then
change the obtained hard cross sections and nuclear modification ratios, and
thereby affect all the subsequent analyses that take these measured cross
sections as an input. In this way, the value of
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ could be critical and have far-reaching
consequences e.g. for the precision studies of jet quenching and other related
phenomena. Thus, an alternative benchmark for
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ is called for.

As proposed in Ref. Paukkunen and Salgado (2011), the Glauber model and its
inputs could be tested by studying the production of well known ”standard
candles”, such as EW bosons, in Pb+Pb collisions at the LHC, but so far this
has been limited by the precision of the LHC Run-I measurements Aad  _et al._
(2013b); Chatrchyan  _et al._ (2012b); Aad  _et al._ (2015b); Chatrchyan  _et
al._ (2015). Thanks to the increased luminosity and collision energy of Run
II, the recent $W^{\pm}$\- and $Z$-boson measurements by ATLAS Aad  _et al._
(2019, 2020) have pushed the precision to a few-percent level enabling now a
more precise Glauber model calibration. In the present letter, we use these
ATLAS data to study the possible nuclear suppression of
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ in Pb+Pb collisions. Since the ALICE
measurement Acharya  _et al._ (2018b) is less precise and has no reference p+p
data we leave it out from the analysis. The idea is to first nail down the EW-
boson cross sections by using a next-to-next-to-leading order (NNLO)
perturbative QCD (pQCD) with state-of-the-art PDFs for protons and nuclei.
Using the theory prediction on the left-hand-side of Eq. (1), we can then
determine $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ within the same MC Glauber
implementation as in the experimental analyses. We find that the data favor a
significant suppression in $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$. We show
that this is compatible with predictions from an eikonal minijet model with
nuclear shadowing. We also demonstrate that the unexpected enhancement seen by
ATLAS in the ratios $R_{\mathrm{PbPb}}^{W^{\pm},Z}$ towards peripheral
collisions disappears with the found smaller value of
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$.

##  II Nuclear suppression in $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$

The observables we exploit in this work to extract
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ are the rapidity-dependent nuclear
modification ratios for $W^{\pm}$ and $Z$ boson production in different
centrality classes. Experimentally these are defined as

$R_{\mathrm{PbPb}}^{\mathrm{exp}}(y)=\frac{1}{\langle
T_{AA}\rangle}\frac{\frac{1}{N_{\mathrm{evt}}}\mathrm{d}N^{W^{\pm},Z}_{\mathrm{PbPb}}/\mathrm{d}y}{\mathrm{d}\sigma^{W^{\pm},Z}_{\mathrm{pp}}/\mathrm{d}y},$
(2)

where the per-event yield is normalized into nucleon-nucleon cross section by
diving with the mean nuclear overlap $\langle T_{AA}\rangle=\langle N_{\rm
bin}\rangle_{c}/\sigma_{\rm nn}^{\rm inel}$ obtained from a MC Glauber model
calculation. For minimum-bias collisions the same quantity can be calculated
directly as a ratio between the cross sections in Pb+Pb and p+p collisions,

$R_{\mathrm{PbPb}}^{\mathrm{theor}}(y)=\frac{1}{(208)^{2}}\frac{\mathrm{d}\sigma^{W^{\pm},Z}_{\mathrm{PbPb}}/\mathrm{d}y}{\mathrm{d}\sigma^{W^{\pm},Z}_{\mathrm{pp}}/\mathrm{d}y}.$
(3)

We have calculated the cross sections in Eq. (3) at NNLO with the mcfm code
(version 8.3) Boughezal  _et al._ (2017). For the protons we use the recent
NNPDF3.1 PDFs Ball  _et al._ (2017) which provide an excellent agreement to
ATLAS data for $W^{\pm}$ and $Z$ boson production in p+p collisions at
$\sqrt{s}=5.02~{}\text{TeV}$ Aaboud  _et al._ (2019b). The nuclear
modifications for the PDFs are obtained from the EPPS16 NLO analysis Eskola
_et al._ (2017) which includes Run-I data for $W^{\pm}$ and $Z$ production in
p+Pb collisions at the LHC Khachatryan  _et al._ (2015, 2016); Aad  _et al._
(2015c) and provide an excellent description of the more recent Run-II data
Sirunyan  _et al._ (2020). The available NNLO nuclear PDFs Walt  _et al._
(2019); Abdul Khalek  _et al._ (2019) do not include any constraints beyond
deeply inelastic scattering, so the applied PDFs provide currently the most
accurately constrained setup for the considered observables. The factorization
and renormalization scales are fixed to the respective EW boson masses.

The ratios $R_{\mathrm{PbPb}}^{\mathrm{theor}}$ and
$R_{\mathrm{PbPb}}^{\mathrm{exp}}$ are compared in the upper panel of Fig. 1.
For $W^{\pm}$, $R_{\mathrm{PbPb}}^{\mathrm{exp}}$ is formed by diving the
normalized yield in Pb+Pb from Ref. Aad  _et al._ (2019) with the
corresponding cross section in p+p from Ref. Aaboud  _et al._ (2019b) adding
the uncertainties in quadrature. The plotted experimental uncertainties do not
include the uncertainty in $\langle T_{AA}\rangle$. The theoretical
uncertainties derive from the EPPS16 error sets and correspond to the 68%
confidence level. Note that the $W^{\pm}$ measurement is for 0–80% centrality
instead of full 0–100%. However, for rare processes like the EW bosons the
contribution from the 80–100% region is negligible so the comparison with the
minimum-bias calculations is justified. It is evident that with
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}=70\,{\rm mb}$ both the $W^{\pm}$ and the
$Z$ data tend to lie above the calculated result, which we will interpret as
an evidence of nuclear suppression in $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$
as explained below.

Figure 1: Nuclear modification ratios of $W^{\pm}$ and $Z$, computed from pQCD
(solid lines with error bands) and from ATLAS data Aad  _et al._ (2019, 2020)
with $\sigma_{\mathrm{nn}}^{\mathrm{inel}}=70\,{\rm mb}$ (upper panel) and
41.5 mb (lower panel).

By equating Eqs. (2) and (3) we can convert each data point to $\langle
T_{AA}\rangle$. The outcome is shown in the upper panel of Fig. 2. The
obtained values tend to be higher than the nominal $\langle
T_{AA}\rangle=5.605\,{\rm mb}^{-1}$ (0–100%) and $\langle
T_{AA}\rangle=6.993\,{\rm mb}^{-1}$ (0–80%) which assume
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}=70\,{\rm mb}$, see Table 1. The fact
that the preferred values of $\langle T_{AA}\rangle$ are independent of the
rapidity strongly suggests that the original mismatch in $R_{\mathrm{PbPb}}$
is a normalization issue – the nuclear PDFs predict the rapidity dependence
correctly.

Figure 2: Extracted values of the mean nuclear overlap functions (upper panel)
and $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ (lower panel). The dark-gray bands
show the values obtained by fitting $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ and
the dashed lines and the light-gray band corresponds to the the nominal
$\sigma_{\mathrm{pp}}^{\mathrm{inel}}$.

Since each $\langle T_{AA}\rangle$ maps to
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ through MC Glauber, we can also
directly convert $R_{\mathrm{PbPb}}^{\mathrm{exp}}$ to
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$. Here, we have used TGlauberMC (version
2.4) Loizides  _et al._ (2015) which is the same MC Glauber implementation as
in the considered ATLAS analyses. The centrality classification is done with a
two-component model, including negative binomial fluctuations111While
modifying $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$, the parameters of the two-
component model should be adjusted to maintain a good description of the
measured multiplicity or transverse energy distribution. However, as the
change in $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ can be accurately compensated
purely by increasing the mean of the negative binomial distribution not
affecting the resulting $\langle T_{AA}\rangle$, the presented results would
remain unmodified., similar to the ALICE prescription Abelev  _et al._ (2013)
with parameters from Ref. ALICE Collaboration (2018). The obtained values of
$\langle T_{AA}\rangle$ are in an excellent agreement with the ATLAS values in
Refs. Aad  _et al._ (2019, 2020) in all centrality classes when using the
nominal, unsuppressed, value
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}=70~{}\text{mb}$. The values of
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ extracted from each data point are
shown in Fig. 2. It is obvious that the data prefer a value of
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ which is less than the
$\sigma_{\mathrm{pp}}^{\mathrm{inel}}=70\,{\rm mb}$ obtained from p+p data.

Table 1: Mean nuclear overlap functions $\langle T_{AA}\rangle~{}[1/\text{mb}]$ for ATLAS centrality classes with nominal and fitted $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$. $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ | $70.0~{}\text{mb}$ | $57.7~{}\text{mb}$ | $41.5~{}\text{mb}$ | $29.5~{}\text{mb}$
---|---|---|---|---
$0-2\%$ | 28.26 | 28.39 | 28.55 | 28.69
$2-4\%$ | 25.51 | 25.67 | 25.91 | 26.10
$4-6\%$ | 23.09 | 23.28 | 23.55 | 23.80
$6-8\%$ | 20.94 | 21.14 | 21.45 | 21.73
$\phantom{1}8-10\%$ | 19.00 | 19.23 | 19.56 | 19.86
$10-15\%$ | 16.08 | 16.31 | 16.67 | 17.02
$15-20\%$ | 12.58 | 12.83 | 13.22 | 13.59
$20-25\%$ | 9.762 | 10.01 | 10.40 | 10.78
$25-30\%$ | 7.487 | 7.722 | 8.102 | 8.469
$30-40\%$ | 4.933 | 5.138 | 5.474 | 5.808
$40-50\%$ | 2.628 | 2.780 | 3.036 | 3.300
$50-60\%$ | 1.281 | 1.378 | 1.550 | 1.733
$60-80\%$ | 0.395 | 0.435 | 0.510 | 0.595
$\phantom{1}80-100\%$ | 0.052 | 0.060 | 0.076 | 0.096
$\phantom{1}0-80\%$ | 6.993 | 7.143 | 7.385 | 7.624
$\phantom{10}0-100\%$ | 5.605 | 5.726 | 5.923 | 6.118

To quantify the optimal $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ we fit its
value by requiring a match between $R_{\mathrm{PbPb}}^{\mathrm{exp}}$ and
$R_{\mathrm{PbPb}}^{\mathrm{theor}}$ treating the EPPS16 uncertainties as
Gaussian correlated errors. In practice we define a $\chi^{2}$ function by

$\displaystyle\chi^{2}$ $\displaystyle=$
$\displaystyle\sum_{i}\left[\frac{\mathcal{N}_{i}R_{i}^{\mathrm{exp}}-R_{i}^{\mathrm{theor}}+\sum_{k}f_{k}\beta_{i}^{k}}{\mathcal{N}_{i}\delta_{i}^{\mathrm{exp}}}\right]^{2}+T\sum_{k}f_{k}^{2}$
$\displaystyle\mathcal{N}_{i}$ $\displaystyle=$ $\displaystyle{\langle
T_{AA}^{i}(\sigma_{\mathrm{pp}}^{\mathrm{inel}})\rangle}/{\langle
T_{AA}^{i}(\sigma_{\mathrm{nn}}^{\mathrm{inel}})\rangle}$ (4)

where $i$ runs over the data points and $k=1,\ldots,20$ over the number error-
set pairs in EPPS16. The factors $\mathcal{N}_{i}$ with
$\sigma_{\mathrm{pp}}^{\mathrm{inel}}=70~{}\text{mb}$ account for the shifted
normalizations when $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ changes. Also the
data uncertainties $\delta_{i}^{\mathrm{exp}}$ are scaled by this factor to
avoid D’Agostini bias D’Agostini (1994). The tolerance $T=1.645^{2}$ in the
penalty term takes into account scaling the 90% confidence limit uncertainties
of EPPS16 into 68% and
$\beta_{i}^{k}\equiv\left[R_{i}^{\mathrm{theor}}(S_{k}^{+})-R_{i}^{\mathrm{theor}}(S_{k}^{-})\right]/2,$where
$S_{k}^{+}$ and $S_{k}^{-}$ are the positive and negative variations,
respectively, of EPPS16 error sets. The $\chi^{2}$ is minimized with respect
to $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ and $f_{k}$ (1+20 parameters). We
find

$\sigma_{\mathrm{nn}}^{\mathrm{inel}}=41.5^{+16.2}_{-12.0}~{}\text{mb}\,,$

where the uncertainties follow from the $\Delta\chi^{2}=1$ criterion. The
resulting values for $\langle T_{AA}\rangle$ and
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ are compared to the data-extracted
values in Fig. 2, and the re-normalized data for $R_{\mathrm{PbPb}}$ are
compared with theoretical predictions in the lower panel of Fig. 1. It is
worth stressing that different final states prefer a very similar, suppressed
value of $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ and that a very good agreement
in $R_{\mathrm{PbPb}}$ is found when normalizing with $\langle T_{AA}\rangle$
calculated using the suppressed cross section in the MC Glauber calculation.

Figure 3: The centrality-dependent nuclear modification ratios for $W^{\pm}$
and $Z$ boson production in Pb+Pb collisions from ATLAS Aad  _et al._ (2019,
2020) compared to NNLO pQCD calculation with EPPS16 nuclear modification with
the nominal value of $\sigma_{\mathrm{nn}}^{\mathrm{inel}}=70.0~{}\text{mb}$
(left) and with the nuclear-suppressed value
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}=41.5~{}\text{mb}$ (right).

##  III Centrality dependence

Even the quite significant suppression in
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ leads to rather modest modifications in
$\langle T_{AA}\rangle$ for central and (close-to) minimum-bias collisions.
The impact, however, grows towards more peripheral centrality classes, see
Table 1. To illustrate this, Fig. 3 compares the centrality dependent
$R_{\mathrm{PbPb}}^{\mathrm{exp}}$ before and after rescaling the data by
${\langle T_{AA}(\sigma_{\mathrm{pp}}^{\mathrm{inel}})\rangle}/{\langle
T_{AA}(\sigma_{\mathrm{nn}}^{\mathrm{inel}})\rangle}$ using the fitted
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$. The left-hand panels show the original
ATLAS data including the quoted $\langle T_{AA}\rangle$ uncertainties, and in
the right-hand panels the data have been rescaled and the uncertainties follow
from the $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ fit. The striking effect is
that the mysterious rise towards more peripheral collisions in the original
data becomes compatible with a negligible centrality dependence, the central
values indicating perhaps a mildly decreasing trend towards peripheral bins.
As discussed e.g. in the ATLAS publications Aad  _et al._ (2019, 2020), such a
suppression could be expected from selection and geometrical biases associated
with the MC Glauber modeling Loizides and Morsch (2017). Also other effects
such as possible centrality dependence of
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ and the neutron-skin effect Paukkunen
(2015); Helenius  _et al._ (2017) may become relevant to explain the data
behaviour in the far periphery.

##  IV Minijets with shadowing

To study the plausibility of the obtained suppression in
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$, we calculate its value in an eikonal
model for minijet production with nuclear shadowing. The model is based on a
similar setup as in Ref. Wang (1991) but in the eikonal function we include
only the contribution from the hard minijet cross section
$\sigma_{\mathrm{jet}}(\sqrt{s_{\mathrm{nn}}},p_{0},[Q])$, calculated at
leading order in pQCD. The transverse-momentum cutoff $p_{0}$ (which depends
on $\sqrt{s_{\mathrm{nn}}}$, scale choice $Q$ and the proton thickness) and
the width of the assumed Gaussian proton thickness function we fix so that the
model reproduces $\sigma_{\mathrm{pp}}^{\mathrm{inel}}=70$ mb matching the
COMPETE analysis Cudell  _et al._ (2002) at $\sqrt{s}=5.02~{}\text{GeV}$. The
free proton PDFs are here CT14lo Dulat  _et al._ (2016), and we take the
nuclear PDF modifications from the EPPS16 Eskola  _et al._ (2017) and nCTEQ15
Kovarik  _et al._ (2016) analyses. The results for
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$, obtained with $p_{0}$ and proton
thickness function width fixed to the the p+p case, are shown in Fig. 4. The
error bars are again from the nuclear PDFs scaled to the 68% confidence level.
As expected at the few-GeV scales, the predicted
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ depends strongly on the
factorization/renormalization scale $Q$, but within the uncertainties the
nuclear suppression obtained from the fits to the ATLAS $W^{\pm}$ and $Z$ data
seems compatible with the eikonal model predictions with both nuclear PDFs.

Figure 4:  Predictions for $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ from an
eikonal minijet model with the EPPS16 and nCTEQ15 nuclear PDFs for three scale
choices. The fitted $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ is indicated with
the dark-gray band and the nominal $\sigma_{\mathrm{pp}}^{\mathrm{inel}}$ with
the light-gray band.

##  V Summary

In the canonical approach the normalization for the measured per-event yields
in nuclear collisions is obtained from the Glauber model taking the value of
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ from proton-proton measurements.
Contrary to this, our strategy was to compare the state-of-the-art pQCD
calculations with the measured $W^{\pm}$ and $Z$ boson $R_{\mathrm{PbPb}}$ and
thereby unfold the value for $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$ at
$\sqrt{s_{\mathrm{nn}}}=5.02~{}\text{TeV}$. We find that the recent high-
precision ATLAS data from Run II prefer the value
$\sigma_{\mathrm{nn}}^{\mathrm{inel}}=41.5^{+16.2}_{-12.0}~{}\text{mb}\,,$
which is significantly lower than $\sigma_{\mathrm{pp}}^{\mathrm{inel}}=70\pm
5~{}\text{mb}$. Such a suppression is in line with the expectations from an
eikonal minijet model including nuclear shadowing. Remarkably, when using the
fitted value for $\sigma_{\mathrm{nn}}^{\mathrm{inel}}$, the unexpected
enhancements of $R_{\mathrm{PbPb}}$ in peripheral collisions disappear and the
results become compatible with no centrality dependence. A possible hint of a
slight decreasing trend toward peripheral collisions is observed which would
be qualitatively in line with possible selection and geometrical biases. Our
results thus suggest that the standard paradigm of using
$\sigma_{\mathrm{pp}}^{\mathrm{inel}}$ as an input to Glauber modeling
potentially leads to a misinterpretation of the experimental data.

###### Acknowledgements.

We thank Mirta Dumancic for discussion concerning the ATLAS measurements. We
acknowledge the Academy of Finland, projects 297058 (K. J. E.) and 308301 (H.
P. and I. H.), and the Väisälä Foundation (M. K.) for financial support. The
Finnish IT Center for Science (CSC) is acknowledged for the computing time
through the project jyy2580.

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# Multi-Features Guidance Network for partial-to-partial point cloud
registration

Hongyuan Wang
<EMAIL_ADDRESS>
Xiang Liu
<EMAIL_ADDRESS>
Wen Kang
<EMAIL_ADDRESS>
Zhiqiang Yan
<EMAIL_ADDRESS>
Bingwen Wang
<EMAIL_ADDRESS>
Qianhao Ning
<EMAIL_ADDRESS>

###### Abstract

To eliminate the problems of large dimensional differences, big semantic gap,
and mutual interference caused by hybrid features, in this paper, we propose a
novel Multi-Features Guidance Network for partial-to-partial point cloud
registration(MFG). The proposed network mainly includes four parts: keypoints’
feature extraction, correspondences searching, correspondences credibility
computation, and SVD, among which correspondences searching and correspondence
credibility computation are the cores of the network. Unlike the previous
work, we utilize the shape features and the spatial coordinates to guide
correspondences search independently and fusing the matching results to obtain
the final matching matrix. In the correspondences credibility computation
module, based on the conflicted relationship between the features matching
matrix and the coordinates matching matrix, we score the reliability for each
correspondence, which can reduce the impact of mismatched or non-matched
points. Experimental results show that our network outperforms the current
state-of-the-art while maintaining computational efficiency.

##  1 Introduction

With the rapid development of sensor technology, there are more and more
diversified methods to obtain 3d point cloud of the target scene (e.g., LiDAR,
RGB-D camera, Stereo-camera, etc.), and 3D point cloud data become ubiquitous.
Due to the point clouds collected from different views and distance, it is
necessary to align two or more point clouds by estimating the rigid
transformation between them, which play a vital role in 3d reconstruction,
autonomous robot positioning, pose estimation, and other applications.

Sample Consensus Initial Alignment(SAC-IA) and Iterative Closest Point(ICP)
[2] are widely used to solve the rigid transformation between point clouds.
SAC-IA uses FPFH [15] features to search for point correspondences, eliminates
unreliable pairwise using RANSAC [4], and finally compute the rigid
transformation between point clouds. SAC-IA uses the informative feature
descriptors to guide the matching points assignment, making the algorithm
insensitive to point cloud initialization, whereas its performance is poorer
than local registration algorithms. ICP iteratively assigns each point in one
point cloud to the spatially closest point in another point cloud and computes
the least-square rigid transformation between correspondences. However, owing
to using spatial coordinates to guide correspondences search merely, ICP is
significantly sensitive to initialization and outlier points, which hinders
its use in practical applications. Therefore, many registration frameworks use
a combination of feature descriptors and the point cloud’s spatial position,
using feature descriptors for a rough initial pre-alignment, followed by a
more precise fine registration step based on the spatial position.

Benefiting from the rise of deep learning technology, researchers begin to
study the use of data-driven point cloud processing methods. However, unlike
traditional two-dimensional images, the point cloud is composed of a series of
points, and the characteristics of unstructured make them problematic for use
in deep learning architectures. PointNet [14] is a pioneering framework
applying deep learning techniques directly to unordered point cloud data.
Inspired by PointNet, several deep learning architectures for point cloud
registration tasks have been proposed and show that learning-based
registration can be more robust and effective than traditional algorithms.
Most of them [1, 21, 22, 17, 10] design a feature extraction network to
compute local or global features and then regress correspondences and
transformation between point clouds. Only a few methods [11, 25] extract
hybrid features to guide the correspondence assignment and transformation
prediction, which improves registration performance to a certain extent.
Nevertheless, using hybrid features also has several intrinsic flaws: 1) The
dimension of spatial coordinates and geometric features varies greatly, and
concatenate them directly is not conducive to exploiting point clouds’ spatial
position. 2) there are large semantic gaps between the spatial position and
the shape features extracted from the feature extractor.3) The spatial
coordinates and the shape features may interfere with each other and cannot
give full play to their respective advantages.

To solve these problems mentioned above, and inspired by the combination of
the coarse pre-alignment stage and the refinement stage, we propose the Multi-
Features Guidance Network(MFGNet) for point cloud registration, which uses
spatial coordinates and local features to guide the correspondences search
jointly. Unlike approaches using hybrid features, MFGNet contains two
different matching matrix computing branches: the coordinates matching matrix
computing branch and the feature matching matrix computing branch. These two
branches learn to assign correct correspondence independently, and then we
fuse these two matching matrices to obtain the final matching matrix, followed
by a differentiable singular value decomposition(SVD) layer to extract the
rigid transformation. The main contributions of this paper are as follows:

1) A novel correspondences search module is proposed in this paper, which
utilizes the shape features and the spatial coordinates to compute the
matching matrix independently and finally fusing the matching results to
obtain the final matching matrix.

2) We propose to use the conflicted relationship between the coordinates
matching matrix and the features matching matrix to score the reliability for
correspondences, which reduce the influence of mismatched or non-matched
points.

3) Extensive analysis and tests of MFGNet are present, and experiments show
that it achieves state-of-the-art performance on the ModelNet40 dataset.

4) The code and the pre-trained weight of MFGNet will be released to
facilitate reproducibility and future research.

Figure 1: MFGNet overall Architecture.

##  2 Related Work

###  2.1 Traditional registration methods

Over the last decades, numerous algorithms for point cloud registration have
been proposed, which can be divided into two categories: coarse registration
and fine registration. The coarse registration algorithm can generally be
divided into three steps: 1) extract the structural features of points; 2)
search the correspondences; 3) calculate the optimal transformation matrix.
Among them, structural features extracting is vital for this type of method.
Generally speaking, feature descriptors map neighboring points’ features into
histograms according to the number of neighboring points [20], Euclidean
distance [16], the difference of normal [15], surface curvature [3], etc.

Among the fine registration methods, ICP [2] is the most representative
algorithm, which assigns each point in one point cloud to the spatially
closest point in another point cloud and computes the least-square rigid
transformation between correspondences iteratively. However, ICP is extremely
sensitive to the initialization and outliers, and it is easy to fall into a
local minimum. Therefore, many variants attempt to optimize the standard ICP
by aggregating sensor uncertainty [18], using other optimization [7], or
weighting matching points [5], but most variants still strongly depend on the
initialization. Therefore, many registration frameworks combine coarse
registration with fine registration to avoid converging into local optimal.

###  2.2 Learning-based registration methods

Motivated by the excellent performance of deep learning in image processing,
people begin to study learning-based point cloud registration methods.
PointNetLK [1] utilizes PointNet [14] to extract global features for point
clouds and then uses the modified Lucas-Kanade [12] algorithm to optimizes the
transformation iteratively. PCRNet [17] compares the features extracted from
PointNet to find the transformation between point clouds. DCP [21] improves
the initial features extracted from DGCNN [23] by incorporating the local and
global information, enhancing the ability to approximate combinatorial
matching. PRNet [22] detects keypoints by comparing the L2 norms of features
and uses Gumbel-Softmax [8] with a straight-through gradient estimator to
sample keypoints correspondence, which tackles a general partial-to-partial
registration problem. IDAM [11] utilizes hybrid features to search
correspondences, which incorporates the shape features information and the
spatial position information. Furthermore, a mutual-supervision loss is
proposed to train the two-stage point elimination without extra annotations.
Our work is similar to IDAM, but we independently use shape features and
spatial coordinates to guide the correspondences search and fuse them finally.
Besides, we calculate the credibility of matching points based on the matching
results. RPM-Net [25] uses the differentiable Sinkhorn [19] layer and
annealing to get soft assignment matrix from hybrid features, which solves the
problem of big difference in dimensions between spatial coordinates and local
geometry to some extent, but it requires additional normal information.
CorsNet [10] concatenates the local features with the global features and
regresses correspondences between point clouds, followed by SVD to estimate
the rigid transformation.

##  3 Problem Statement

The registration between two rigid point clouds refers to the task of finding
a transformation to align different point clouds into one coordinate system.
Given a source point cloud $\left\\{X:x_{i}\mid i=1,\ldots
N_{1}\right\\}\subset\mathbb{R}^{3}$ and a target point cloud
$\left\\{Y:y_{j}\mid j=1,\ldots N_{2}\right\\}\subset\mathbb{R}^{3}$, our
objective is to find rigid transformation $\left\\{R,t\right\\}$ to align two
point clouds, where $R\in SO(3)$ is a rotation matrix and $t\in\mathbb{R}^{3}$
is a translation vector. Different from the most learning-based registration
approach, the one-to-one correspondence between points is not required in our
method, and $XY$ can cover different extents, which means $N_{1}\neq N_{2}$in
most cases. In this situation, we define a matching matrix
$\left\\{M:m_{ij}\mid i=1,\ldots N_{1};j=1,\ldots N_{2}\right\\}$ to represent
whether $x_{i}$ corresponds to $y_{j}$ . Our objective is to find
$\left\\{R,t\right\\}$ to minimize the mapping error between correspondences :

$\displaystyle\textbf{$\min_{R,t}\sum_{j=1}^{N_{2}}\sum_{i=1}^{N_{1}}m_{ij}\left\|Rx_{i}+t-y_{j}\right\|_{2}^{2}$}.$
(1)

Where $m_{ij}\in\\{0,1\\}$ is the $i_{th}$ row and $j_{th}$ column element of
the matching matrix $M$, $m_{ij}=1$ if point $x_{i}$ corresponds to $y_{j}$
and $m_{ij}=0$ otherwise.

##  4 Method

In this section, we describe the proposed MFGNet point cloud registration
model. Fig. 1 shows the architecture of MFGNet in detail. The model mainly
consists of four components: 1) Keypoints’ Feature Extraction, 2)
Correspondences search, 3) Correspondences Credibility Computation, 4) SVD.
The details of each component are described in the following sections.

###  4.1 Keypoints’ Feature Extraction

The first stage is to extract the local feature descriptor from raw point
clouds, which is essential to the subsequent correspondences search. A
discriminative feature can well represent the local geometric structure and
can find its correspondence easily. MFGNet can be compatible with both hand-
crafted [15, 16, 6] and learning-based [14, 13, 23] point cloud feature
extraction methods, but this paper only uses the GNN network feature
extraction layer to extract point cloud features. In our network, the
dimension of the output features is 64.

Figure 2: Network architecture for the keypoints selection module.

It’s excessive-memory-required and time-consuming to use all points for
correspondences search. Meanwhile, undiscriminating features will also lead to
ambiguous matching results, which increase the difficulty of correspondence
assignment. To avoid these problems mentioned above, we introduce the
keypoints selection module to preserve discriminative features and eliminate
undiscriminating features [11]. Fig. 2 illustrates the pipeline of the
keypoints selection module. Given the extracted features for each point, a
multi-layer perceptron is applied to feature descriptors, and the significance
score of features is obtained. A high significance score means a more
discriminative feature, which is beneficial to the matching point search. We
preserve the $K$ points with the highest significance scores and discard the
remaining points. We denote the keypoints preserved in the source and target
point cloud as $P_{S}$ and $P_{T}$ and denote corresponding features of them
as $F_{S}$ and $F_{T}$ . All of them will be used to guide the subsequent
correspondence assignment.

###  4.2 Correspondences search

In this section, we elaborate on our correspondence matching strategy. Most of
the existing learning-based methods merely use features extracted by MLP to
guide correspondences search. However, many point cloud structures are
repetitive, and it is challenging to obtain correct matching points without
considering location information. To alleviating this problem, some networks
use hybrid features learning from both shape feature and spatial coordinates
to guide matching point assignment, but stack the features with different
dimensions and semantics directly, making them interfere with each other and
hindering the network from making full use of the advantages of different
information. Inspired by the combined registration frameworks of coarse
registration and fine registration, we argue that the shape features and the
spatial coordinates can guide correspondences search independently, and fusing
them adaptively at different registration stages can improve the registration
accuracy.

We propose the correspondences search strategy guided jointly by shape
features and spatial coordinates based on the above intuition. We obtain the
keypoints’ spatial coordinates and corresponding shape features from the point
cloud in the last section. Given keypoints’ shape features $f_{S}(i)\in F_{S}$
and $f_{T}(j)\in F_{T}$ for $p_{S}(i)\in P_{S}$ and $p_{T}(j)\in P_{T}$ , then
we can form shape feature tensor and spatial coordinate tensor at iteration
$n$ as

$\displaystyle\begin{split}f^{(n)}(i,j)=\left[\left\|f_{S}(i)-f_{T}(j)\right\|;\frac{f_{S}(i)-f_{T}(j)}{\left\|f_{S}(i)-f_{T}(j)\right\|}\right].\end{split}$
(2)
$\displaystyle\begin{split}s^{(n)}(i,j)=\left[\left\|p_{S}(i)-p_{T}(j)\right\|;\frac{p_{S}(i)-p_{T}(j)}{\left\|p_{S}(i)-p_{T}(j)\right\|};p_{S}(i)\right]\end{split}$
(3)

Where $\left[;\right]$ means concatenate operation and $\left\|\cdot\right\|$
denotes modulus of vectors. Unlike IDAM, which concatenates $f_{S}(i)$ and
$f_{T}(j)$ directly, we use the distance between shape features to
characterize the features’ similarity. The shape feature tensor also contains
the feature’s relative direction, representing the difference between the
features. The spatial coordinate tensor includes the distance, the direction
between points, and the spatial coordinate of $p_{S}(i)$ .

In order to obtain the similarity of each point pair, a Multi-Layer
Perceptron(MLP) is applied to the shape feature tensor and spatial coordinate
tensor, respectively. In our network, we implement MLP with fully
connected(FC) layers, each followed by ReLU activation and batch normation(BN)
except the last layer. It is noted that the two MLPs are not weight shared,
which allows them to guide the correspondence matching independently. The two
MLPs output a coordinate matching matrix and a feature matching matrix
respectively, each of which can be regarded as the correspondence matching
result between point clouds, and a high response means a high matching
probability, while a low response value means that the point pair is unlikely
to be a correspondence. Logically, we add the two matrices to obtain the final
matching matrix M, which enhances the response of the correct match point and
suppresses the wrong match result. Obviously, different matching matrices have
different credibility in different registration stages. Our network can learn
this well and give them appropriate response values, which can be verified in
fig. 3. We can see that the highest response ratio between the feature
matching matrix and the coordinate matching matrix is approximately 1:1, 1:3,
1:4, and 1:5 in the four iterations.

(a) The Matching Matrix under the first iteration.

(b) The Matching Matrix under the second iteration

(c) The Matching Matrix under the third iteration

(d) The Matching Matrix under the fourth iteration

Figure 3: The Feature Matching Matrix and The Coordinate Matching Matrix
results in different registration stages.

###  4.3 Correspondences Credibility Computation

The previous section adopted the correspondences search strategy guided
jointly by shape features and spatial coordinates, generating a shape feature
matching matrix and a spatial coordinate matching matrix. In this section,
these two matching matrices will be used to calculate the matching results’
credibility. Intuitively, when the matching result of shape features is
consistent with the matching result of spatial coordinates, it is reasonable
to believe the matching results are correct in high probability. In contrast,
when two matching results conflict, the degree of conflict between them and
the registration stage should be considered to judge whether the matching
result is credible. Therefore, to reduce the impact of incorrect matching
points, we introduce a credibility calculation module to evaluate matching
results’ reliability.

Fig.4 shows the specific structure of the correspondences credibility
calculation module. To obtain the conflicted relationship between different
matching matrices, we concatenate the shape feature matching matrix and the
spatial coordinate matching matrix and map it to the high-dimensional feature
space with a $1\times 1$ convolution operation. Then the element-wise max
aggregation and MLP are used to obtain the matching results’ credibility. A
high score means high credibility and has a higher weight in the subsequent
weighted SVD layer, while a low score has little or no influence in the
subsequent transformation solution process. We define the weight of the
$i_{th}$ correspondence result as

$\displaystyle\omega_{i}=\frac{c(i)\cdot\mathds{1}\llbracket
c(i)\geq\operatorname{median}(c)\rrbracket}{\left.\sum_{i}c(i)\cdot\mathds{1}\llbracket
c(i)\geq\operatorname{median}(c)\right\rrbracket}$ (4)

where $c(i)$ denotes the credibility of the matching point, and
$\mathds{1}\llbracket\rrbracket$ is an indicator function to judge whether the
matching result’s credibility is greater than the median of scores. Once the
correspondence assignment and its credibility are obtained, the final step is
to estimate the rigid transformation ${R,t}$. Like most existing methods, we
use the weighted SVD to solve the transformation matrix, which can be defined
as Eq. 5. The slight difference between Eq. 1 and Eq. 5 is that $\omega_{i}$
is a decimal between $0\sim 1$ while $m_{ij}\in\\{0,1\\}$.

$\displaystyle
R,t=\underset{R,t}{\operatorname{argmin}}\sum_{i}\omega_{i}\left\|Rp_{S}(i)+t-p_{T}\left(j^{*}\right)\right\|$
(5)

Figure 4: Network architecture for the correspondences credibility computation
module.

(a) Qualitative registration expamles on unseen shapes

(b) Qualitative registration expamles on unseen categories

(c) Qualitative registration expamles on noisy unseen shapes

Figure 5: Qualitative registration expamles under different experimental
setups

###  4.4 Loss Functions

The loss function of our network consists of three parts: the keypoints
selection loss $L_{keypoint}$, the correspondences search loss $L_{matching}$,
and the correspondences credibility computation loss $L_{credibility}$. Since
we do not have access to keypoints annotation directly, we use Mutual-
Supervision Loss [11] to train our network. The principle behind it is that
key points tend to have a low entropy because they are confident in matching.
The loss of keypoints selection is defined as

$\displaystyle L_{\text{keypoint
}}=\frac{1}{K}\sum_{i=1}^{K}\left(s(i)-\sum_{j=1}^{K}M(i,j)\log(M(i,j))\right)^{2}$
(6)

Where $s(i)$ is the significance scores for $p_{S}(i)$, and we only use the
keypoints selection loss for the first iteration.

The loss of correspondence matching loss for the $n_{th}$ iteration is defined
as

$\displaystyle
L_{\text{matching}}^{(n)}=\frac{1}{K}\sum_{i=1}^{K}-\hat{y}_{i}\log\left(M^{(n)}\left(i,j^{*}\right)\right)$
(7)

Where $j^{\ast}$ is the index of the point that is closest to $p_{S}(i)$ under
ground truth transformation, and $\hat{y}_{i}$ is the label to judge whether
the distance between $p_{S}(i)$ and $p_{T}(j^{\ast})$ is less than the
distance threshold $th$.

The loss of correspondences credibility computation for the $n_{th}$ iteration
is defined as

$\displaystyle L_{\text{credibility
}}^{(n)}=\frac{1}{K}\sum_{i=1}^{K}-\hat{m}_{i}\log(c(i))-\left(1-\hat{m}_{i}\right)\log(1-c(i))$
(8)

Where $\hat{m}_{i}$ is the label whether the correspondence distance under
ground truth transformation is less than the distance threshold $th$.

The overall loss is the weighted sum of the three losses

$\displaystyle L_{\text{total}}=\sum_{n}\left(\lambda(n)L_{\text{keypoint
}}+L_{\text{matching }}^{(n)}+L_{\text{credibility }}^{(n)}\right)$ (9)

Where

$\displaystyle\lambda(n)=\left\\{\begin{array}[]{ll}1&n=1\\\ 0&n\neq
1\end{array}\right.$ (12)

Table 1: Results for testing on unseen point clouds Model | RMSE(R) | MAE(R) | RMSE(t) | MAE(t)
---|---|---|---|---
ICP | 34.99 | 25.46 | 0.296 | 0.253
FGR | 8.72 | 1.86 | 0.017 | 0.005
SAC-IA | 6.04 | 1.90 | 0.016 | 0.012
SAC-IA+ICP | 6.64 | 0.71 | 0.022 | 0.005
PointNetLK | 16.74 | 7.55 | 0.045 | 0.025
DCP | 6.71 | 4.45 | 0.027 | 0.020
PRNet | 3.20 | 1.45 | 0.016 | 0.010
IDAM | 2.95 | 0.76 | 0.021 | 0.005
MFGNet | $\bf{1.56}$ | $\bf{0.39}$ | $\bf{0.006}$ | $\bf{0.002}$

##  5 Experiment

###  5.1 Dataset and Experimental Setup

We evaluate our model on the ModelNet40 dataset, which is widely used to train
point cloud registration networks. ModelNet40 [24] contains 12311 CAD models
from 40 human-made categories, split officially into two parts: the training
set containing 9843 models and the testing set containing 2468 models. We use
the processed data from [14], which sample 2048 points randomly from the
models and normalize them into a unit sphere. For consistency, a point cloud
with 1024 points is sampled, and a rigid transformation is randomly generated
(rotations within [0,45] along each axis and translation within [-0.5, 0.5]).
Following [22, 11], we fix a random point and preserve 768 points closest to
this point for each point cloud and generate partially overlap point clouds.

In the experiment, the top 128 points are preserved as key points, the
distance threshold th is set to 0.05, and the number of iterations is set to
4. We randomly sample 64 ”positive” points that have correspondence in the
target point cloud and 64 ”negative” points that do not have correspondence
during training. We train our model using Adam [9] optimizer for 50 epochs,
with an initial learning rate of 0.0001 and a weight decay of 0.001. We divide
the learning rate by 10 at 40 epochs.

We compare our model to classic registration methods(including SAC-IA [15],
ICP [2], FGR [26],SAC-IA+ICP) and recently learning-based registration
methods(including PointNetLK [1], DCP [21], PRNet [22],IDAM [11]). All the
learning-based methods are trained on the same training set, and we provide
the root mean square error(RMSE) and the mean absolute errors(MAE) for
rotation matrix and translation vectors to evaluate all methods. For the
rotation matrix, the unit of mean absolute error is the degree.

###  5.2 Experimental Results

####  5.2.1 Full Dataset Train & Test

In the first experiment, we evaluate our model on the full official
training/testing sets. Both the training set and test set contain point clouds
from all the 40 categories. We list the quantitative results for all methods
in Table 1. As we can see from the results, our method outperforms all the
other registration algorithms, both traditional and learning-based methods.

We can see that the ICP algorithm performs poorest due to the lack of a good
initial position, which means that only using spatial coordinates to guide the
registration is insufficient. FGR and SAC-IA obtain better results for using
shape features to guide registration. Besides, SAC-IA+ICP, which utilizes both
local shape features and spatial coordinates, performs best in traditional
methods, even better than learning-based methods. Inspired by SAC-IA+ICP, we
design a new network called MFGNet, which takes full advantage of point
clouds’ spatial coordinates and shape features. Experimental results show that
MFGNet is superior to other methods in all indicators.

####  5.2.2 Category Split

To test the generalizability of different methods, we conducted the second
experiment, which trains models on the first 20 categories in the training set
while evaluating them on the other 20 categories in the test set. The
quantitative results are summarized in Table 2.

Table 2: Results for testing on unseen categories Model | RMSE(R) | MAE(R) | RMSE(t) | MAE(t)
---|---|---|---|---
ICP | 34.27 | 25.63 | 0.294 | 0.250
FGR | 8.32 | 1.21 | 0.013 | 0.004
SAC-IA | 3.89 | 1.71 | 0.019 | 0.013
SAC-IA+ICP | 6.01 | 0.52 | 0.014 | 0.004
PointNetLK | 22.94 | 9.66 | 0.061 | 0.033
DCP | 9.77 | 6.95 | 0.034 | 0.025
PRNet | 4.99 | 2.33 | 0.021 | 0.015
IDAM | 3.42 | 0.93 | 0.022 | 0.005
MFGNet | $\bf{1.476}$ | $\bf{0.43}$ | $\bf{0.008}$ | $\bf{0.002}$
Table 3: Results for testing on point clouds with Gaussian noise Model | RMSE(R) | MAE(R) | RMSE(t) | MAE(t)
---|---|---|---|---
ICP | 33.86 | 25.07 | 0.292 | 0.249
FGR | 27.13 | 12.06 | 0.064 | 0.036
SAC-IA | 11.95 | 3.32 | 0.027 | 0.016
SAC-IA+ICP | 12.53 | 2.44 | 0.022 | $\bf{0.008}$
PointNetLK | 19.94 | 9.08 | 0.057 | 0.032
DCP | 6.88 | 4.53 | 0.028 | 0.021
PRNet | 4.32 | 2.05 | $\bf{0.017}$ | 0.012
IDAM | 3.72 | 1.85 | 0.023 | 0.011
MFGNet | $\bf{3.56}$ | $\bf{1.52}$ | 0.019 | $\bf{0.008}$

Compared with Table 1, it can be found that the invisible categories have a
little side effect on the traditional methods because they do not need to use
the training set to learn parameters. However, all the learning-based methods’
performance has a certain degree of degradation for lacking the prior
knowledge of test categories. Nevertheless, benefiting from the combination of
shape features and location features, MFGNet has less performance degradation
and still outperforms all other methods.

####  5.2.3 Gaussian Noise

In practical applications, the point cloud inevitably contains outliers and
noise. To verify the algorithm’s effectiveness under noise, we conducted the
last point cloud registration experiment by adding Gaussian noise. In this
experiment, we add Gaussian noise with a standard deviation 0.01 to all point
clouds and clip the noise to $\left[-0.5,0.5\right]$. We follow the first
experiment’s other settings (full dateset train & test) and summarize the
quantitative results in Table 3. As shown in Table 3, since Gaussian noise has
an immense impact on shape features and spatial coordinates, all registration
methods’ performance degrades greatly. Nevertheless, MFGNet has the best
performance among all registration algorithms. We show the results of our
model under different experimental setting in fig. 5.

####  5.2.4 Efficiency

To compare the efficiency of different models, we calculate the average
inference time for each model. We use the implementations of ICP, SAC-IA, FGR
in Intel Open3D [27], and the official implementation of PointLK, DCP, PRNet,
and IDAM released by the authors. In our experiment, we set the maximum number
of ICP iterations to 2000, the maximum number of RANSAC iterations to 4×106,
and the maximum number of validation steps to 500. The experiments are
performed on a 3.0GHz Intel i7-9700 CPU and an Nvidia GeForce RTX 2080 SUPER
GPU. We test the efficiency of the models on points with 512, 1024, and 2048
points, and use a batch size of 1 for all learning-based methods during
testing. The results are summarized in Table 4. It can be seen that MFGNet is
significantly faster than SAC-IA+ICP, which has comparable performance as our
method. At the same time, MFGNet have similar efficiency with other learning-
based registration networks.

Table 4: Average inference time required for registering point clouds of various sizes(in milliseconds) | ICP | FGR | SAC-IA | SAC-IA+ICP | PointNetLK | DCP | PRNet | IDAM | MFGNet
---|---|---|---|---|---|---|---|---|---
512 points | 8.3 | 14.1 | 92.1 | 86.9 | 118.5 | 13.2 | 83.4 | 34.0 | 44.9
1024 points | 10.7 | 27.8 | 96.7 | 98.1 | 120.3 | 22.5 | 134.3 | 34.7 | 45.8
2048 points | 16.5 | 54.6 | 120.5 | 149.6 | 124.6 | 56.25 | 219.1 | 36.6 | 47.3
Table 5: the quantitative results under different iteration times $n$ | RMSE(R) | MAE(R) | RMSE(t) | MAE(t) | Inference time
---|---|---|---|---|---
2 | 3.99 | 1.99 | 0.023 | 0.012 | 31.8
3 | 3.64 | 1.62 | 0.019 | 0.009 | 38.3
4 | 3.56 | 1.52 | 0.019 | 0.008 | 45.8
5 | 3.80 | 1.57 | 0.020 | 0.009 | 52.2
6 | 3.67 | 1.46 | 0.024 | 0.009 | 59.6

####  5.2.5 Ablation Study

In this section, we compare the performance of the MFGNet algorithm under
different iteration times $n$. We follow the third experiment settings
(training point clouds with Gaussian noise) and set $n$ to 2, 3, 4, 5, and 6,
respectively. Table 5 shows the quantitative results under different iteration
times. It can be seen that when $n=2$, the MFGNet can achieve similar
performance as PRNet but 3 times faster than PRNet. When $n=3$, the
performance of MFGNet has already surpassed the IDAM. When $n>4$, there is no
obvious improvement in performance. To balance the efficiency and the
performance, we set $n$ to 4 in all experiments conducted above.

##  6 Conclusion

In this paper, we propose a novel network for point cloud registration, called
MFGNet. Inspired by the combination of coarse registration and fine
registration, we propose a strategy that utilizes shape features and spatial
coordinates to guide the correspondences search, enhancing the network’s
ability to find correct correspondences. We also proposed a correspondences
credibility computation module based on the feature matching matrix and the
coordinate matching matrix, which scores the correspondences’ reliability and
reduces the wrong matching results’ side impact. The experimental results show
that our algorithm achieves state-of-the-art performance on the ModelNet40
dataset under different experimental settings.

## Acknowledgement

This preprint has not undergone peer review or any post-submission
improvements or corrections. The Final Version of this article is published in
Neural Computing and Applications, and is available online at
https://doi.org/10.1007/s00521-021-06464-y.

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