2022

[2]A. Knecht

1]Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland
2]Paul Scherrer Institut, Villigen, Switzerland 3]Institut für Teilchen- und
Astrophysik, ETH Zürich, Zürich, Switzerland 4]Institute of Nuclear Physics,
Johannes Gutenberg University Mainz, Mainz, Germany 5]PRISMA+ Cluster of
Excellence, Johannes Gutenberg University Mainz, Mainz Germany 6]KU Leuven,
Instituut voor Kern- en Stralingfysica, Leuven, Belgium 7]Department of
Chemistry - TRIGA Site, Johannes Gutenberg University Mainz, Mainz, Germany
8]GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany
9]Helmholtz Institute Mainz, Mainz, Germany 10]Institut für Kernphysik,
Universität zu Köln, Köln, Germany 11]Laboratoire Kastler Brossel, Sorbonne
Université, CNRS, ENS-PSL Research University, Collège de France, Case 74; 4,
place Jussieu, F-75005 Paris, France 12]Institute of Physics, Johannes
Gutenberg Universität Mainz, Mainz, Germany 13]Department of Physics,
Universitá di Pisa, Pisa, Italy 14]Université Grenoble Alpes, CNRS, Grenoble
INP, LPSC-IN2P3, 38026 Grenoble, France

# Muonic atom spectroscopy with microgram target material

A. Adamczak  A. Antognini  N. Berger  T. E. Cocolios  N. Deokar  Ch. E.
Düllmann  A. Eggenberger  R. Eichler  M. Heines  H. Hess  P.
Indelicato${}^{\textrm{11}}$ K. Kirch<EMAIL_ADDRESS>J.J. Krauth  J. Nuber
A. Ouf  A. Papa  R. Pohl  E. Rapisarda  P. Reiter  N. Ritjoho  S. Roccia  M.
Seidlitz  N. Severijns  K. von Schoeler  A. Skawran  S. M. Vogiatzi  N. Warr
F. Wauters  [  [  [  [  [  [  [  [  [  [  [  [  [  [

###### Abstract

Muonic atom spectroscopy – the measurement of the x rays emitted during the
formation process of a muonic atom – has a long standing history in probing
the shape and size of nuclei. In fact, almost all stable elements have been
subject to muonic atom spectroscopy measurements and the absolute charge radii
extracted from these measurements typically offer the highest accuracy
available. However, so far only targets of at least a few hundred milligram
could be used as it required to stop a muon beam directly in the target to
form the muonic atom. We have developed a new method relying on repeated
transfer reactions taking place inside a 100 bar hydrogen gas cell with an
admixture of 0.25% deuterium that allows us to drastically reduce the amount
of target material needed while still offering an adequate efficiency.
Detailed simulations of the transfer reactions match the measured data,
suggesting good understanding of the processes taking place inside the gas
mixture. As a proof of principle we demonstrate the method with a measurement
of the 2p-1s muonic x rays from a $5\text{\,}\mathrm{\SIUnitSymbolMicro g}$
gold target.

###### keywords:

Muonic atoms, muonic atom spectroscopy, transfer process, nuclear charge
radius

##  1 Introduction

Muonic atoms are a type of exotic atoms where a negative muon ($\mu^{-}$) is
captured by an atomic nucleus and forms a hydrogen-like system. For this
capture process to take place, muons need to be decelerated to a few eV, where
they are then captured in a high Rydberg state (main quantum number $n\geq$
14). From this state they subsequently cascade down to the $1s$ state via
collisions and Auger and radiative transitions, during the latter step
emitting so-called muonic x rays. The details of this capture and cascade
depend on the target material and density STANISLAUS1987642 ; Markushin:1998ve
, the entire process transpires however within at most a few nanoseconds for
low-$Z$ materials111Except for transitions passing through the $2s$ state,
which can be metastable. (much less for higher-$Z$ elements), which is less
than the time resolution of a typical x-ray detector. Once a muonic atom is
formed, the muon is unlikely to transfer to another atom, with the exception
of muonic hydrogen which is a neutral system and thus can easily penetrate the
electron cloud of nearby atoms. In the ground state, the muon will either
decay in orbit, or be captured by the atomic nucleus Suzuki:1987jf . This
process becomes dominant for $Z$$\geq$14 due to a strong $Z^{4}$ dependence,
thereby considerably reducing the lifetime of the exotic atom.

Due to the relatively large muon mass of 105.7 MeV/c2, the muonic Bohr radius
is $\sim$207 times smaller compared to regular atoms leading to a large
overlap between the muonic and nuclear wave function at lower $n$ orbits. This
makes muonic atoms a very sensitive system to probe the nuclear charge
distribution, first and foremost the nuclear charge radius $\sqrt{<r^{2}>}$
which is typically derived from the $2p-1s$ transition energy Fri95 ;
Knecht:2020npz . In this way muonic atoms provide complementary information to
electron scattering and laser spectroscopy data, respectively ANGELI201369 ;
agmb2009 ; Frois:1987hk ; Cheal12 ; Neugart:2017oqg .

To measure the characteristic muonic x rays from a particular isotope, one
needs to stop a secondary muon beam available at a proton accelerator facility
such as HIPA at the Paul Scherrer Institute (PSI), MUSE at J-PARC, the muon
sources at TRIUMF and RAL, and MuSIC at Osaka. These muons have a typical
momentum of 10-100 MeV/c. To efficiently stop such muons, one usually needs a
solid target thickness of $\mathcal{O}$(1 mm) GROOM2001183 , which limits the
available muonic x-ray data to stable isotopes from which one can produce such
a macroscopic target. However, the charge radii of several long lived
radioactive isotopes are of interest to benchmark isotope shift measurements
from laser spectroscopy Cocolios:2717871 , or to serve as input for atomic
parity violation experiments with radium PhysRevA.78.050501 , which was the
motivation to develop the method presented in this paper. In order to perform
laser spectroscopy on the lightest muonic atoms, the Lamb Shift experiment at
PSI has developed a dedicated low-energy muon beamline to slow down muons to a
few keV in kinetic energy Antognini:2021icf . In this paper we present an
alternative approach, where a standard negative muon beam is stopped in a
high-pressure gaseous hydrogen target, and the negative muon is then
transported to a few $\mathrm{\SIUnitSymbolMicro g}$/cm2 target by means of
diffusion and a series of transfer reactions. A total efficiency per incoming
muon of order one percent is achieved in this way.

The main focus of the paper is the detailed description of the developed
method (Section 2) and apparatus (Section 4) allowing us to perform muonic
atom spectroscopy with such microgram targets. Additionally, we have developed
a Monte Carlo program to simulate all relevant processes taking place in the
high-pressure gas cell. (Section 3). The paper concludes in Section 5 with a
description of the first results that show the performance of the apparatus in
comparison to simulation and a first proof-of-principle measurement with a 5
$\mathrm{\SIUnitSymbolMicro g}$ gold target.

Figure 1: Sketch of the method employed to measure muonic x rays emitted from
a target with only microgram mass. The negative muon enters a 100 bar hydrogen
gas cell with a small admixture of deuterium. Upon stopping, the muon forms
muonic hydrogen $\mu p$ and transfers to a deuteron upon collision with a
deuterium molecule forming muonic deuterium $\mu d$. Due to its low scattering
cross section, $\mu d$ can travel over a large distance potentially reaching
the microgram target at the back of the cell. Here, the muon transfers again
to the target nucleus thereby emitting the characteristic muonic x rays.

##  2 Method

The method presented in this paper for performing muonic atom spectroscopy
with microgram target material relies on transfer reactions taking place
inside a high-pressure hydrogen gas cell operating at 100 bar, corresponding
to about 10% of liquid hydrogen density, with a small admixture of deuterium.
The method is inspired by the work described in Refs. kra89 ; str05 ; Str09
and the wealth of knowledge gained in the pursuit of muon catalyzed fusion
(see, e.g., Refs. Pon90 ; pet01 ) on the behavior and interaction of the
muonic hydrogen isotopes inside gas cells. A sketch of the method can be found
in Fig. 1 with the details given in the following paragraphs.

Once the negative muon comes to rest inside the hydrogen gas it is quickly
captured by a hydrogen molecule forming the excited $\mu$-molecular complex
$(pp\mu e)^{*}$ with the proton $p$ and electron $e$. This complex splits
apart kor96  by either direct dissociation

$(pp\mu e)^{*}\rightarrow(\mu p)^{*}+(pe)$ (1)

or by electron emission and subsequent dissociation

$(pp\mu e)^{*}\rightarrow(pp\mu)^{*}+e\rightarrow(\mu p)^{*}+p+e\,.$ (2)

The formed muonic hydrogen atom $(\mu p)$ quickly de-excites to the ground
state and is thermalized by multiple scatterings on the surrounding hydrogen
molecules. Due to the high-pressure the diffusion radius of the muonic
hydrogen atom is limited to only about 0.5 mm at the given conditions. The key
to increasing the mobility and distance that the muons can travel lies in the
small admixture of deuterium that typically amounts to 0.25% at the chosen
pressure of 100 bar. Once the muonic hydrogen atom scatters with deuterium the
muon is quickly transferred – due to the increased binding energy – forming
muonic deuterium $(\mu d)$ in the $1s$ ground state, where $d$ is the
deuteron:

$(\mu p)+d\rightarrow(\mu d)+p$ (3)

The rate of this transfer at the present conditions is $\sim 3.5\times
10^{6}$/s and07 . Because of the increased binding energy the formed muonic
deuterium atom is accelerated resulting in a kinetic energy of 45 eV Mul06 .
This energy is quickly reduced in subsequent scatterings. However, due to the
Ramsauer-Townsend effect ram21  the transport scattering cross section222For a
definition of the transport cross section see, e.g., Ref. ada07 . of $\mu d$
on H2 features a deep minimum at around 4 eV as shown in Fig. 2 Ada96 ; Mul06
; ada07 . This effect increases the range of the $\mu d$ atom in the gas cell
by about a factor 10 over the one of the thermalized $\mu p$ atom allowing
some of the muons to reach the microgram target material mounted at the back
of the gas cell.

Figure 2: Calculated total cross sections for the elastic scattering of $\mu
d$ on H2 and D2 and transport cross sections (indicated with
$\sigma_{\text{mt}}$) for the elastic scattering of $\mu d$ on H2 Ada96 . The
gain in kinetic energy by the transfer from $\mu p$ to $\mu d$ is indicated by
the vertical green line. The cross sections for the scattering of $\mu d$ on
D2 depend on the hyperfine state $F$ of the $\mu d$ atom and are included in
brown.

Once the $\mu d$ atom reaches the target material, the tightly-bound and
neutral atom easily penetrates the electron cloud and the muon quickly
transfers yet again to the target nucleus $Z$ ger62 ; pon67 :

$(\mu d)+Z\rightarrow(\mu Z)^{*}+d$ (4)

The transfer inside the solid target material is not well understood
theoretically and has not been studied in great detail experimentally.
Estimates for the transfer rate can be obtained from the calculations of Refs.
fio76 ; bra78 . For gold the estimated transfer rate is $\sim 1.6\times
10^{12}$/s as given by the equation in the conclusions of Ref. fio76 .
Neglecting any scattering effects and assuming an energy of the $\mu d$ atom
of 4 eV, this translates to a characteristic length of 12 nm at which the
fraction of remaining $\mu d$ atoms has dropped in gold to $1/e$.

##  3 Simulation

An efficient measurement requires as many muons as possible to reach the
target nuclei after having stopped in the target gas and formed muonic
hydrogen atoms. We have conducted Monte Carlo simulations of the transport
process to optimize the target geometry and the gas conditions. In order to
simulate the drift of $\mu p$ and $\mu d$ atoms through the gas volume, it is
necessary to quantify the collisions between the atoms in the $1s$ state and
the molecules of the target gas. In addition to the transfer reaction
discussed in Equation (3), the following scattering processes take place:

$\mu a\left(F\right)+M_{2}\rightarrow\mu a\left(F^{\prime}\right)+M_{2}$ (5)

Here, $a$ denotes a proton or deuteron bound to a negative muon and $M_{2}$
denotes either one of the molecules H2, D2 or HD. During the scattering
process, the hyperfine state $F$ of the muonic atom can be changed when the
$\mu a$ atom undergoes a spin-flip reaction in which the atomic nucleus $a$ is
exchanged with one of the nuclei of the molecule. At collision energies of eV
and more, the differential cross sections for the molecular scattering
processes are approximately equivalent to those for the corresponding nuclear
processes, which consider three-body Coulomb interactions between the muonic
atom and the molecular nuclei and have been addressed in numerous efforts
Ponomarev1979 ; Bracci1989 ; Bracci1990 ; Bubak1987 . In the experiment,
however, the muonic hydrogen atoms can have kinetic energies well below
$0.1~{}$eV for a significant amount of time. In this regime, molecular binding
effects, electron screening and spin correlations of specific rotational
states can not be neglected any more. Furthermore, excitations and de-
excitations of rotational and rotational-vibrational molecular states
introduce additional inelastic components to the collisions. For this reason,
molecular cross sections need to be used for a realistic simulation of the
diffusion process. The method of calculating such molecular cross sections has
been described in detail in Adamczak2006 .

The Monte Carlo simulations are based on the implementation of custom
molecular scattering processes in the Geant4 framework AGOSTINELLI2003250
with use of the G4beamline program Roberts2007G4beamline . The implemented
scattering processes work with double-differential scattering rates in the
laboratory reference frame which have been obtained from the differential
cross sections in the center-of-mass provided by Ada96 . When transferring to
the lab frame, the thermal motion of the target molecules is taken into
account by averaging over the Maxwell-Boltzmann distribution of the momenta of
the gas molecules.

In addition to the custom molecular scattering processes, the simulations
include the physics processes provided by Geant4 for the interaction of the
negative muons with materials in the target region. The simulations start with
muons from the beam passing through the entrance detector in front of the
target cell. We assume the muons in the beam to be normally distributed around
the beam center with a standard deviation of 10 mm. The momentum is
distributed normally around the mean momentum with a momentum bite of 4% (full
width at half maximum). Muons that enter the gas volume through the window and
stop in the gas are substituted with $\mu p$ atoms in the $1s$ state or
directly with $\mu d$ atoms (with low probability corresponding to the
deuterium concentration). The muonic atoms undergo multiple collisions with
molecules including the transfer reaction while drifting through the gas
volume. Eventually, the tracks are killed when reaching the target surface or
another surface inside the gas cell, corresponding to the transfer towards
heavy nuclei. All simulation results presented in this text correspond to a
target disk of 15 mm diameter which is placed 15 mm behind the window of the
gas cell.

Figure 3: Time distributions of $\mu d$ and $\mu p$ atoms reaching the surface
of the target disk for various deuterium concentrations $c_{D}$. The time $t$
corresponds to the time after the incoming muons have triggered the entrance
detector. The distributions are normalized by the number of entrance counts.

In reality, the formation process of muonic atoms with the subsequent fast de-
excitation cascade is highly complex. Collisions of the large, excited muonic
atoms with surrounding molecules influence the kinetic energy distribution of
the atoms after reaching the $1s$ state. Based on the standard cascade model
Leon1962 , different theoretical models of the cascade process have been
developed Borie1980 ; Markushin1984 ; Jensen2002 ; Faifman2008 ; Popov2012 ;
Popov:2017PRA ; Popov:2021PRA . Their results for the kinetic energy of muonic
atoms after the cascade, however, show strong variations. In accordance with
experimental findings Abb97 ; poh10 , the most recent calculations
Popov:2017PRA ; Popov:2021PRA  find that a part of the atoms ends up
thermalized after the cascade while other atoms cover the whole energy range
up to several tens of eV, with few atoms even reaching energies of 100 to 200
eV. For the results of the transport simulations described here, no
significant effect of the initial energy distribution could be observed. Due
to the high density of around 10% of liquid hydrogen, the $\mu p$ atoms at a
kinetic energy of 100 eV have mean free paths of only a few
$\mathrm{\SIUnitSymbolMicro m}$, which makes them slow down after minimal
distances. For the simulation results presented in this text, the initial
kinetic energy of the muonic atoms in the $1s$ state was assumed to be
distributed according to a Maxwell-Boltzmann distribution with a mean energy
of 1 eV.

Figure 4: Energy distributions of $\mu d$ and $\mu p$ atoms reaching the
surface of the target disk for two different deuterium concentrations $c_{D}$.
Only atoms that reach within the first $1~{}\mu$s after the entrance detector
has been triggered are considered here. The distributions are normalized by
the number of counts in the entrance detector.

Time distributions of muonic hydrogen atoms reaching the target disk are
presented in Fig. 3 for various deuterium concentrations in the hydrogen gas.
The distributions were obtained for a muon beam with a mean momentum of
$28.0~{}$MeV/c and have been normalized to the number of entrance counts. The
distribution for pure hydrogen ($c_{\mathrm{D}}=0\%$) shows a prompt peak
stemming from $\mu p$ atoms that reach the target disk directly after
formation. The other distributions show significant peaks at later time which
are associated with $\mu d$ atoms reaching the cell wall after the isotopic
exchange of the muon. Increasing deuterium concentration raises the rate of
the isotopic muon exchange from $\mu p$ to $\mu d$, which enables the $\mu d$
atoms to reach the target disk faster. On the other hand, a higher
concentration of deuterium in the mixture increases the probability for $\mu
d$ atoms to scatter with deuterium molecules. The scattering cross sections
for this process exhibit no Ramsauer-Townsend minima (see Fig. 2) and can
therefore limit the free path of the $\mu d$ atoms. The optimal deuterium
concentration which balances these concurrent effects typically lies between
0.1 and 0.25% depending on the exact target and beam conditions. Energy
distributions of the muonic atoms at the moment they reach the target disk are
shown in Fig. 4 for two different deuterium concentrations. As can be seen
from the distributions, most $\mu d$ atoms have lost the energy gained in the
isotopic exchange reaction when reaching the target. A peak of the energy
distribution around the Ramsauer-Townsend minimum at $\sim 4~{}$eV is visible
for the lower deuterium concentration ($c_{\mathrm{D}}=0.1\%$). For the higher
concentration, this peak has been washed out due to the increased scattering
on deuterium molecules.

Figure 5: Parameter scan for the momentum of the muon beam $p_{\mu}$ and the
deuterium concentration $c_{D}$. The figure of merit is the probability for
muons which are detected in the entrance detector to eventually be transferred
to the target disk within maximally 1 $\mathrm{\SIUnitSymbolMicro s}$. In the
simulations, a gaussian muon beam momentum distribution with 4% full width at
half maximum was assumed.

Figure 5 shows a parameter scan performed with the transport simulations.
Here, the efficiency of the transport process is defined as the number of
muonic atoms reaching the target disk normalized by the number of muons
passing through the entrance detector. For this, only muonic atoms are
considered that reach the target disk within 1000 ns after entering the target
cell. Note, that the transfer to the heavy nuclei in the target is not
considered in the simulations. In reality and depending on the thickness of
the target, only a fraction of the muonic hydrogen atoms reaching the target
surface transfer the muon to the actual target nuclei, while the rest of the
muons are transferred to the target backing material. For the scan, the
efficiency has been determined for various beam momenta and deuterium
concentrations. For this geometry, a maximal efficiency is reached with an
average momentum of 27.8 MeV/c – at which point a significant fraction of the
muons stop directly in the target backing material – and a deuterium
concentration of 0.15%. Comparisons of simulation results with experimental
data will be presented in Sec. 5 of this text.

##  4 Apparatus

Figure 6: Sketch of the muX apparatus with its different components labelled.
For more details see the dedicated sections.

Figure 6 shows an overview of the apparatus and its different components,
which has been discussed in parts also in Refs. Ada18 ; ska19 ; Knecht:2020npz
; Wauters:2021cze . The different components and the layout were developed in
order to enable the method described in Section 2 with the central gas cell
featuring a geometry and dimensions optimized through the simulation described
in Section 3. The following subsections describe the different elements in
detail.

###  4.1 Muon beamline

The measurements were performed at the $\pi$E1 beamline in the experimental
hall of PSI’s high-intensity proton accelerator HIPA Grillenberger:2021kyv . A
static $\mathbf{E}\times\mathbf{B}$ separator installed at the beginning of
the experimental area allowed to reduce the electron contamination in the beam
to negligible levels thereby providing a very pure negative muon beam at the
chosen momentum of around 28 MeV/c. Using a slit system mounted in the
beamline at a point of large dispersion allowed to adjust the momentum
acceptance of the channel to around 3-4% full width at half maximum.

The apparatus was installed about 1.3 m downstream of the last quadrupole
triplet of the $\pi$E1 beamline. At this position the beamspot had a size of
about 10 mm (gaussian sigma) in horizontal and vertical direction. Typical
rates on the muon entrance scintillator (see Sec. 4.2) located behind the beam
veto scintillator (see Sec. 4.2) with an opening of 18 mm were around 20 kHz
at a proton beam intensity of 2 mA.

###  4.2 Beam veto, muon entrance and decay-electron veto scintillators

|  |
---|---|---
(a) | (b) | (c)
Figure 7: Plastic scintillator detectors used in the muX apparatus: a) beam
veto scintillator, b) muon entrance scintillator, and c) decay-electron veto
scintillators. For more details see text.

As shown in Fig. 6, the muX apparatus employs three different plastic
scintillator detectors. They are presented in Fig. 7. The first detector in
beam direction is the beam veto scintillator (Fig. 7 a). This is based on a
$80\times 80$ mm2 BC-404 scintillator plate with a thickness of 4 mm and a
central hole with a diameter of 18 mm. It is read out on two sides by a total
of 12 $3\times 3$ mm2 silicon photomultipliers (model ASD-NUV3S-P by AdvanSiD
advansid ) mounted on a PCB. This detector serves as an active collimator for
the muon beam and vetoes against background coming down the beamline outside
of the acceptance of the entrance detector.

The beam veto detector is followed by the muon entrance detector (Fig. 7 b).
The detector is based on a $20\times 20$ mm2 BC-400 scintillator foil with a
thickness of 200 $\mathrm{\SIUnitSymbolMicro m}$. The scintillator is glued
with optical cement on a UV-transparent plexiglas frame, which in turn is
coupled to 12 $3\times 3$ mm2 silicon photomultipliers as for the beam veto
detector. This detector serves to provide a precise timing for the arrival of
a muon in the high-pressure gas cell and to distinguish a muon from some
remaining electron background in the beam through the amount of deposited
energy.

The last scintillator detectors are the decay-electron veto detectors shown in
Fig. 7c that surround the gas cell. These are large $180\times 180$ mm2 EJ204
scintillator plates with a thickness of 5 mm. The light from the scintillator
is collected through wavelength shifting fibers that are embedded into grooves
in the scintillator plate. They guide the light to a single $3\times 3$ mm2
silicon photomultiplier (model S13360-3050CS by Hamamatsu hamamatsu ). One of
the detectors features a 35 mm diameter central hole such that it can be
mounted around the tube through which the muons enter the gas cell (see Fig.
6). These decay-electron veto detectors serve to efficiently reduce the
background generated in the germanium detectors (directly or indirectly)
coming from muon Michel decays generating electrons with energies up to 53
MeV.

###  4.3 High-pressure gas cell

Figure 8: Rendered CAD image of the high-pressure gas cell with a cutout in
order to show the placement of the target and the arrangement of the entrance
window and supporting grids. For reference the volume between the target and
entrance window has a length of 14 mm and a diameter of 30 mm. See text for
more information on the different parts.

Figure 8 shows an overview of the high-pressure gas cell. The main body of the
cell is made of aluminum. An aluminum gas line screws into the main body from
the back (feeding in the gas just behind the target) and is sealed using glue
(Model Stycast 2850 FT). The cell is closed towards the muon beamline by a 600
$\mathrm{\SIUnitSymbolMicro m}$ thick carbon fiber-reinforced polymer window
that allows the muons to pass through. In order to withstand the high
pressure, the window is supported by a combination of a first 4 mm thick
titanium grid and a second 2 mm carbon fiber-reinforced polymer grid mounted
on top. The 2 mm carbon fiber-reinforced polymer grid faces the muon beamline
and greatly reduces the background from muons stopping in titanium. The grids
have an open area of 74%. During measurements a transmission through this grid
and window arrangement of around 60% was observed (with a variation of around
$\pm$5% over the various experimental campaigns depending on beam setup) and
thus smaller than the open area due to the divergence of the beam and stops in
the window. The grids and window are pressed down by a 3 mm thick stainless
steel ring with an open diameter of 20 mm. The overall gas seal is achieved by
an O-ring. The cell was pressure tested up to 350 bar at which point the
screws securing the stainless steel ring and titanium and carbon fiber grids
are ripped out of the aluminum body.

A polystyrene tube (not shown in Fig. 8) extends from the stainless steel ring
to the entrance detector ensuring that muons can only stop in low-$Z$ material
on their passage from the entrance detector to the gas cell. Also the inside
of the gas cell is lined with a polystyrene tube removing any potential
background from muonic hydrogen or deuterium atoms reaching the aluminum walls
of the pressure cell.

The inner space of the gas cell has a diameter of 30 mm and a length of 20 mm.
At the end of this space the target in its target holder is mounted reducing
the distance between the entrance window and target surface to 14 mm, which is
the available length for the muons to stop in the gas.

###  4.4 Hydrogen/deuterium gas system

A dedicated gas handling system was built that allows the controlled operation
and mixing of gases from three different sources and the filling of the gas
cell described above through an approximately 3 m long stainless steel tube
with 4 mm inner diameter. It uses various absolute pressure gauges that also
monitor the pressure inside the gas cell during normal operation. Before
filling the gas cell with the hydrogen/deuterium mixture, the cell is flushed
several times with hydrogen in order to reduce residual air contamination and
impurities in the gas as much as possible.

In 2017/2018, the gas mixture was prepared by first filling the tube and gas
cell with the desired pressure of deuterium and in a next step filling
hydrogen up to a pressure of 100 bar. By comparing the measurements of the
transfer reactions with the simulation (and later also by performing Raman
spectroscopy on the deuterium/hydrogen gas mixtures), it was realized that
this procedure leads to much higher and somewhat unreproducible deuterium
concentrations as the filling with hydrogen compresses all the deuterium left
in the filling tube into the gas cell. Due to the high pressures the diffusion
times leading to an equal mixture are very high and thus no equilibrium is
reached before the start of the measurements nub18 . From 2019 onward, we thus
switched to the use of pre-mixed gases obtained commercially.

###  4.5 Target

Figure 9: Typical gold target used during the measurements described in this
paper mounted in its polystyrene holder.

The backing disks for the target materials are made of glassy carbon (Sigradur
from HTW htw ) with a diameter of 16 mm and a thickness of 1 mm. The gold
targets employed for the measurements described here were manufactured by
chemical vapor deposition using the in-house facilities available at PSI.

The glassy carbon disks are mounted in a holder made of polystyrene, which as
described above is fastened at the back of the gas cell using PEEK screws. The
holder reduces the open area of the target to a diameter of 15 mm. In order to
avoid any potential contamination of the gas in the cell the screws were
vented through small holes and generally only procedures applicable for high-
vacuum environments were employed during the design, construction and handling
of the high-pressure gas cell.

###  4.6 High-purity germanium detector array

Figure 10: Germanium detector array used during the 2017/2018 experimental
campaigns. Figure 11: Miniball germanium detector array mounted in its custom,
highly adjustable frame in the $\pi$E1 area. The photograph shows the array
moved backwards on a dedicated rail system in order to gain easy access to the
muon entrance detector and gas cell.

The germanium detector setups evolved over the course of the different
experimental campaigns. In 2017 and 2018 (see Fig. 10), the array was mounted
in a custom frame on rails and consisted of seven single-crystal coaxial
detectors with $\sim$60% efficiency from the IN2P3/STFC French/UK Ge loan pool
loan_pool , two single-crystal coaxial Ge detector with 75% and 70% efficiency
(an additional 15% detector was mounted in 2018), a Miniball cluster
consisting of three individual crystals of around 60% each war13  and a planar
detector optimized for low-energy gamma rays. The frame allowed a modest
flexibility in positioning the detectors at distances of around 10 to 15 cm
away from the target.

In 2019 due to the long shutdown at CERN, the full Miniball detector array
war13  was brought from ISOLDE to PSI and mounted for an extended period in
the $\pi$E1 area. The full array consisted of 8 clusters of three crystals
with efficiencies of around 60% each. The array mounted in its original frame
at PSI is shown in Fig. 11. The array was supplemented by the 70% coaxial and
planar detector already used in 2017 and 2018. Due to the larger number, the
detectors could only be mounted at a distance of around 15 cm to the target.

With these configurations, we reached about 1.8% and 2.2% full energy
detection efficiency at 1332 keV with the full arrays for 2017/2018 and 2019,
respectively.

###  4.7 Calibration

While during dedicated calibration runs a large variety of standard gamma
calibration sources were used, a scheme was implemented to provide continuous
calibration lines during normal muon beam operation. As shown in Fig. 6, a 100
$\mathrm{\SIUnitSymbolMicro m}$ thick layer of 208Pb was mounted behind a 600
$\mathrm{\SIUnitSymbolMicro m}$ thick carbon fiber-reinforced polymer sheet
(the same as the entrance window of the gas cell) – both with a central hole
of 18 mm – and installed in front of the beam veto detector with the carbon
fiber-reinforced polymer sheet facing the beam. This allowed a fraction of the
muon beam (those outside the 18 mm acceptance of the beam veto scintillator)
to stop in 208Pb and produce a continuous source of well-known muonic x rays
reaching up to high energies Ber88  for calibration. In addition, a weak 2.6
kBq 60Co source was placed close to the target that provided additional
continuous calibration lines and could also be used for an automated analysis
and check of the full detector array by monitoring for each detector the
properties of the measured 60Co lines.

##  5 Results

###  5.1 Optimization of transfer process

The transfer process inside the gas cell is typically optimized through a
series of measurements employing a 50 nm thick gold target prepared as
described in Sec. 4.5. The 50 nm thickness was chosen under the assumption
that such a layer would lead to the full transfer of all muons from the muonic
hydrogen and deuterium atoms to gold as they penetrate into the layer. The
total target mass amounts to a bit over 200 $\mathrm{\SIUnitSymbolMicro g}$,
which allows for swift measurements while at the same time only having a
negligible amount of direct muon stops inside the gold layer. The optimization
procedure basically follows the parameter space outlined in Fig. 5 by scanning
the deuterium concentration and muon beam momentum and counting the amount of
emitted muonic $2p-1s$ gold x rays at 5591.7 and 5760.8 keV (the higher energy
transition is distributed across two lines due to hyperfine splitting) Ack66 ;
Fri95 .

The observed total counts of $2p-1s$ gold x rays $N_{\gamma}$ is used to
estimate the efficiency $\epsilon_{T}$ of the full transfer process by
relating them to the number of detected muons by the entrance detector
$N_{\mu}$:

$\epsilon_{T}=\frac{N_{\gamma}}{N_{\mu}}\frac{1}{I\epsilon_{det}\zeta},$ (6)

with $I$ the absolute yield per muon of the $2p-1s$ line, $\epsilon_{det}$ the
detection efficiency of the full germanium detector array at this energy and
$\zeta$ the transmission through the grids and the entrance window of the gas
cell (see Sec. 4.3).

The yields of the $2p-1s$ transition after transfer is distinctly different
and about a factor 2 lower compared to direct capture as was, e.g, measured
for several low-$Z$ elements Ber78 ; Jac88 ; Mul93 ; Jac95 . This can be
understood by the fact that the initial distribution across the angular
momentum states is shifted to much lower values Haf77 , which boosts the
yields of higher $np-1s$ transitions compared to the direct capture. As no
trustworthy measurements of the $2p-1s$ yield after transfer for elements
higher than argon exist, it was estimated for gold after transfer to be
$65.7\pm 7.1$%, and thus significantly higher than seen in the measurements at
lower atomic numbers, by extrapolating a series of transfer measurements in a
hydrogen gas cell with small admixtures of argon, krypton and xenon. A
dedicated publication on these measurements is in preparation Ska21 .

---
(a)
(b)
Figure 12: (a) Transfer efficiency as a function of deuterium concentration $c_{D}$ at a muon beam momentum of 27.25 MeV/c. (b) Transfer efficiency as a function of muon beam momentum $p$ at a deuterium concentration of 0.25%. The two time windows correspond to the time after a muon entered the gas cell over which the recorded muonic gold x rays are counted to calculate the transfer efficiency. Also shown are the simulation results from Fig. 5 that demonstrate excellent agreement. In the comparisons here, the momentum of the simulated results was shifted by $-0.65$ MeV/c in order to compensate for the not precisely known material budget traversed by the muon beam and to obtain the maximum efficiency at the same momentum as in the measurement. |
---|---
(a) | (b)
|
(c) | (d)
Figure 13: Evolution of the normalized muonic gold $2p-1s$ x rays as a
function of time after a muon entered the gas cell for different deuterium
concentrations: (a) $c_{D}=0.1\%$, (b) $c_{D}=0.25\%$, (c) $c_{D}=0.75\%$, and
(d) $c_{D}=1.5\%$. The results from the simulation (in red) nicely agree with
the measured data (in black).

In order to measure the detection efficiency of the germanium array at the
energies of the muonic gold $2p-1s$ x rays, a 300 $\mathrm{\SIUnitSymbolMicro
m}$ thick natural lead target with a diameter of 30 mm was mounted in vacuum
and without the entrance window and support grids at the target position
inside the gas cell. This arrangement guaranteed that any muon registered by
the entrance detector hits the lead target and stops within. By using the
measured yield of the $2p-1s$ transition $\mathrm{BR}(2p-1s)=0.899$ (from gold
Har82  instead of lead as the corresponding values found in literature for
lead But76 ; And69  of $\mathrm{BR}(2p-1s)\sim 0.6$ are clearly not correct
and as only a negligible change of the $2p-1s$ yields are expected for these
two close, high-$Z$ elements), the efficiency can be calculated by the ratio
of detected number of x rays to the number of emitted x rays given by the
number of muons multiplied by the yield:

$\epsilon_{det}=\frac{N_{\gamma}}{N_{\mu}\times\mathrm{BR}(2p-1s)}$ (7)

The number of detected x rays $N_{\gamma}$ was obtained by fitting the
corresponding $2p-1s$ lines in the measured spectrum. Additionally and by
taking full advantage of the large germanium detector array, a $\gamma-\gamma$
coincidence analysis was performed by triggering one of the germanium
detectors on the $3d-2p$ transition and counting the number of observed
$2p-1s$ x rays in the rest of detector array. The two approaches resulted in
perfectly consistent values for the detector efficiency of
$\epsilon_{det}=(0.46\pm 0.01)\%$ at $\sim$5.6 MeV energy.

The window transmission $\zeta$ was measured just following the detector
efficiency measurements by still employing the natural lead target, but
closing the cell with the carbon fiber window and support grids and comparing
the amount of emitted lead x rays per incoming muon with and without the
entrance window. The extracted value amounts to $\zeta=54.5\%$ and is
consistent across several of the muonic lead lines.

With these parameters at hand the transfer efficiency can be calculated and
plotted for the 2019 measurements as a function of deuterium concentration and
beam momentum in Fig. 12, which reaches a maximum of approximately 5% at a
deuterium concentration of 0.25% and a momentum of 27.25 MeV/c. The plots show
the results for two different time windows of 500 and 1000 ns after the muon
entrance highlighting the effect of the deuterium concentration and momentum
on the speed of the transfer process to the gold target. Shown in addition to
the measurements are also the results of the simulation from Fig. 5. Excellent
agreement is found between the measurements and simulation suggesting good
understanding of the processes occurring in the gas cell and corroborating the
assumption that a 50 nm gold layer is thick enough such that all muons
transfer to the target material. Good agreement between simulation and
measurements is also found by plotting the time evolution of the muonic gold
$2p-1s$ x rays as a function of time after a muon entered the gas cell – see
Fig. 13.

###  5.2 Measurements with a few microgram target

---
(a)
(b)
Figure 14: Two dimensional histogram of the measured x-ray energy versus the
time after a muon entered the gas cell (a) and its projection on energy (b).
Visible are the $2p_{1/2}-1s_{1/2}$ and the single escape (S.E.)
$2p_{3/2}-1s_{1/2}$ lines from the continuous muonic 208Pb calibration source,
a gamma line from $p\mu d$ fusion and the muonic $2p_{1/2}-1s_{1/2}$ and
$2p_{3/2}-1s_{1/2}$ x rays from the 5 $\mathrm{\SIUnitSymbolMicro g}$ gold
target. The $2p_{3/2}-1s_{1/2}$ in muonic gold shows additionally a hyperfine
structure. The black line in (b) is a simple Gaussian fit with linear
background to the $2p_{1/2}-1s_{1/2}$ line of muonic gold. For details, see
text.

Already in 2017 in a first proof-of-principle measurement, we have aimed at a
measurement with only 5 $\mathrm{\SIUnitSymbolMicro g}$ of gold as motivated
by the maximum allowed quantity for a potential 226Ra measurement. We did this
in a series of measurements where the target mass was reduced from about 470
$\mathrm{\SIUnitSymbolMicro g}$ down to 5 $\mathrm{\SIUnitSymbolMicro g}$. As
we only discovered the issues with the hydrogen/deuterium mixture (as
described in Sec. 4.4) later, we cannot give precise values for these series
of measurements as they are plagued with uncertainty due to changes in the
deuterium concentration for the different measurements, but will only give
some rough estimates in the next paragraph.

In the measurement sequence we started with a 50 nm gold coating on a diameter
of 25 mm (the target mount was somewhat different in these first measurements
compared to the one described in Sec. 4.5) and gradually reduced it to a 3 nm
coating on a $10\times 10$ mm2 area. Each of the reductions – 50 nm to 3 nm
and 491 mm2 to 100 mm2 – were accompanied by a loss in gold x rays of about a
factor 3. For the final gold measurement with the 5
$\mathrm{\SIUnitSymbolMicro g}$ target material (3 nm coating on a $10\times
10$ mm2), the estimated total transfer efficiency was slightly above 1%. The
result of a 18.5 h long measurement with this target is shown in Fig. 14. The
figure shows the measured energy spectrum versus the time after a muon entered
the gas cell (Fig. 14a) and its projection on energy (Fig. 14b). Clearly
visible are the peaks from the muonic 208Pb x rays used for the continuous
calibration (see Sec. 4.7) that appear constant in time and the gold x rays
from the target that only start to appear with the muon entering the gas cell.
Additionally, the 5.5 MeV gamma ray emitted in the $p\mu d$ fusion process can
be seen Fri91 .

Figure 14 also shows a simple Gaussian fit with linear background to the
$2p_{1/2}-1s_{1/2}$ line of muonic gold. The resulting mean of the Gaussian is
fitted to 5591.65(26) keV perfectly in agreement and with only slightly worse
precision than the value found in literature of 5591.71(15) keV Fri04
demonstrating the excellent performance of the apparatus and method for such
microgram target quantities. Based on this measured energy of the
$2p_{1/2}-1s_{1/2}$, one could proceed to follow the standard procedure in
muonic atom spectroscopy to extract a nuclear charge radius through the
matching of the measured energy with calculations that have the nuclear charge
radius as a free parameter Fri04 .

With the dominant uncertainty on any extracted, high-$Z$ nuclear charge radius
being of theoretical nature, a measurement as shown in Fig. 14 that results in
transition energies of the muonic atom with $\sim$200 eV accuracy is
sufficient to extract the charge radius of such high-$Z$ elements with a
relative uncertainty of a few times $10^{-4}$ Fri04 .

##  6 Conclusion

In conclusion, we have developed a method and built an apparatus to perform
muonic atom spectroscopy with targets that are only available in microgram
quantities such as highly radioactive or scarce isotopically pure elements.
The method relies on repeated transfer reactions taking place inside a 100 bar
hydrogen gas cell with an admixture of 0.25% deuterium at room temperature.
The apparatus consists of a muon entrance detector, beam and decay electron
veto detectors, a high-pressure gas cell containing the target and a high-
purity germanium detector array to measure the energies of the x rays emitted
in the muonic cascade.

In order to optimize and understand the apparatus, a detailed simulation was
developed that models all of the processes taking place inside the gas cell. A
comparison of the results of the simulation with the measured data shows
excellent agreement and underlines the good understanding of the transfer
processes taking place in such environments.

As a proof of principle, the muonic $2p-1s$ x rays from a 5
$\mathrm{\SIUnitSymbolMicro g}$ gold target were measured showing good signal
to background levels and demonstrating the ability to perform high sensitivity
measurements with microgram target materials. Employing this method, we
succeeded to measure 248Cm, the heaviest element ever probed with muonic atom
spectroscopy sps22 . A measurement of 226Ra is also within reach. Additional
measurements of elements that were so far inaccessible are being planned.

Acknowledgments We gratefully acknowledge fruitful discussions with P. Kammel
that laid the basis to the development of the method presented in this paper.
The experiments were performed at the $\pi$E1 beam line of PSI. We would like
to thank the accelerator and support groups for the excellent conditions.
Additional technical support by F. Barchetti, F. Burri, M. Hildebrandt, M.
Meier, L. Noorda, A. Stoykov, and A. Weber from PSI is gratefully
acknowledged.

Funding was provided by FWO-Vlaanderen (Belgium), the Paul Scherrer Institute
through the Career Return Program, the Cluster of Excellence “Precision
Physics, Fundamental Interactions, and Structure of Matter” (PRISMA EXC 1098
and PRISMA+ EXC 2118/1) funded by the German Research Foundation (DFG) within
the German Excellence Strategy (Project ID 39083149), the German Research
Foundation (DFG) under Project WA 4157/1 and 407008443, the German BMBF under
contracts 05P18PKCIA and ‘Verbundprojekt’ 05P21PKCI1, and the Swiss National
Science Foundation through the Marie Heim-Vögtlin Grant no. 164515 and project
Grant no. 200021_165569. P.I. is a member of the Allianz Program of the
Helmholtz Association, contract no EMMI HA-216 “Extremes of Density and
Temperature: Cosmic Matter in the Laboratory”.

## References

  *   * (1) Stanislaus, S., Entezami, F., Measday, D.F.: Atomic capture of muons and density. Nuclear Physics A 475(4), 642–656 (1987). https://doi.org/10.1016/0375-9474(87)90230-2
  * (2) Markushin, V.E.: Cascade in muonic and pionic atoms with Z = 1. Hyperfine Interact. 119, 11 (1999). https://doi.org/10.1023/A:1012642215730
  * (3) Suzuki, T., Measday, D.F., Roalsvig, J.P.: Total Nuclear Capture Rates for Negative Muons. Phys. Rev. C 35, 2212 (1987). https://doi.org/10.1103/PhysRevC.35.2212
  * (4) Fricke, G., Bernhardt, C., Heilig, K., Schaller, L.A., Schellenberg, L., Shera, E.B., Dejager, C.W.: Nuclear Ground State Charge Radii from Electromagnetic Interactions. Atomic Data and Nuclear Data Tables 60, 177 (1995). https://doi.org/10.1006/adnd.1995.1007
  * (5) Knecht, A., Skawran, A., Vogiatzi, S.M.: Study of nuclear properties with muonic atoms. Eur. Phys. J. Plus 135(10), 777 (2020). https://doi.org/10.1140/epjp/s13360-020-00777-y
  * (6) Angeli, I., Marinova, K.P.: Table of experimental nuclear ground state charge radii: An update. Atomic Data and Nuclear Data Tables 99(1), 69–95 (2013). https://doi.org/10.1016/j.adt.2011.12.006
  * (7) Angeli, I., Gangrsky, Y.P., Marinova, K.P., Boboshin, I.N., Komarov, S.Y., Ishkhanov, B.S., Varlamov, V.V.: N and Z Dependence of Nuclear Charge Radii. J. Phys. G 36(8), 085102 (2009). https://doi.org/10.1088/0954-3899/36/8/085102
  * (8) Frois, B., Papanicolas, C.N.: Electron Scattering and Nuclear Structure. Ann. Rev. Nucl. Part. Sci. 37, 133–176 (1987). https://doi.org/10.1146/annurev.nucl.37.1.133
  * (9) Cheal, B., Cocolios, T.E., Fritzsche, S.: Laser spectroscopy of radioactive isotopes: Role and limitations of accurate isotope-shift calculations. Phys. Rev. A 86, 042501 (2012). https://doi.org/10.1103/PhysRevA.86.042501
  * (10) Neugart, R., Billowes, J., Bissell, M.L., Blaum, K., Cheal, B., Flanagan, K.T., Neyens, G., Nörtershäuser, W., Yordanov, D.T.: Collinear laser spectroscopy at ISOLDE: new methods and highlights. J. Phys. G 44(6), 064002 (2017). https://doi.org/10.1088/1361-6471/aa6642
  * (11) Groom, D.E., Mokhov, N.V., Striganov, S.I.: Muon Stopping Power and Range Tables 10 MeV?100 TeV. Atomic Data and Nuclear Data Tables 78(2), 183–356 (2001). https://doi.org/10.1006/adnd.2001.0861
  * (12) Cocolios, T.E.: Absolute charge radii of radioactive isotopes measured by muonic x-ray spectroscopy at PSI. Technical report, CERN, Geneva (May 2020). https://cds.cern.ch/record/2717871
  * (13) Wansbeek, L.W., Sahoo, B.K., Timmermans, R.G.E., Jungmann, K., Das, B.P., Mukherjee, D.: Atomic parity nonconservation in Ra+. Phys. Rev. A 78, 050501 (2008). https://doi.org/10.1103/PhysRevA.78.050501
  * (14) Antognini, A., Kottmann, F., Pohl, R.: Laser spectroscopy of light muonic atoms and the nuclear charge radii. SciPost Phys. Proc. 5, 021 (2021). https://doi.org/10.21468/SciPostPhysProc.5.021
  * (15) Kraiman, J.B., et al.: Diffusion of Muonic Deuterium in $D_{2}$ Gas. Phys. Rev. Lett. 63, 1942–1945 (1989). https://doi.org/10.1103/PhysRevLett.63.1942
  * (16) Strasser, P., Nagamine, K., Matsuzaki, T., Ishida, K., Matsuda, Y., Iwasaki, M.: Muonic Atoms of Unstable Nuclei. AIP Conf. Proc. 793(1), 242–257 (2005). https://doi.org/10.1063/1.2121987
  * (17) Strasser, P., et al.: Muon spectroscopy with trace alkaline-earth and rare-earth isotopes implanted in solid D2. Hyperfine Interact. 193, 121 (2009). https://doi.org/10.1007/s10751-009-0055-0
  * (18) Ponomarev, L.I.: Muon catalysed fusion. Contemporary Physics 31, 219 (1990). https://doi.org/10.1080/00107519008222019
  * (19) Petitjean, C.: The $\mu$CF Experiments at PSI –A Conclusive Review. Hyperfine Interactions 138(1), 191 (2001). https://doi.org/10.1023/A:1020883708302
  * (20) Korenman, G.Y.: Mechanisms and kinetics of exotic atom formation in hydrogen and helium. Hyperfine Interactions 101(1), 81–89 (1996). https://doi.org/%****␣muX_method.tex␣Line␣800␣****10.1007/BF02227608
  * (21) Andreev, V.A., et al.: Measurement of the muon capture rate in hydrogen gas and determination of the proton’s pseudoscalar coupling ${g}_{P}$. Phys. Rev. Lett. 99, 032002 (2007). https://doi.org/10.1103/PhysRevLett.99.032002
  * (22) Mulhauser, F., et al.: Ramsauer-Townsend effect in muonic atom scattering. Phys. Rev. A 73, 034501 (2006). https://doi.org/10.1103/PhysRevA.73.034501
  * (23) Ramsauer, C.: Über den Wirkungsquerschnitt der Gasmoleküle gegenüber langsamen Elektronen. Annalen der Physik 369(6), 513–540 (1921). https://doi.org/10.1002/andp.19213690603
  * (24) Adamczak, A., Gronowski, J.: Diffusion radius of muonic hydrogen atoms in h-d gas. The European Physical Journal D 41(3), 493 (2007). https://doi.org/10.1140/epjd/e2006-00252-6
  * (25) Adamczak, A., et al.: Atlas of Cross Sections for Scattering of Muonic Hydrogen Atoms on Hydrogen Isotope Molecules. Atom. Data and Nucl. Data Tables 62, 255 (1996). https://doi.org/10.1006/adnd.1996.0006
  * (26) Gershtein, S.S. Sov. Phys. JETP 16, 501 (1962)
  * (27) Ponomarev, L.I., Puzynina, T.P.: The Two-center Problem in Quantum Mechanics. Sov. Phys. JETP 25, 846 (1967)
  * (28) Fiorentini, G., Torelli, G.: Muon Transfer to Heavy Atoms: A Probe for the Study of the $(\mu p)_{2s}$ System. Nuovo Cim. A 36, 317–330 (1976). https://doi.org/10.1007/BF02724519
  * (29) Bracci, L., Fiorentini, G.: MESON TRANSFER TO ATOMS AND MOLECULES. Nuovo Cim. A 50, 373 (1979). https://doi.org/10.1007/BF02813242
  * (30) Ponomarev, L.I., Somov, L.N., Faifman, M.P.: A simple approach to the description of the three-body system in the physics of mesic atoms. Sov. J. Nucl. Phys 29, 67 (1979)
  * (31) Bracci, L., Chiccoli, C., Fiorentini, G., Melezhik, V.S., Pasini, P., Ponomarev, L.I., Wozniak, J.: The atlas of the cross sections of mesic atomic process I. The processes $p\mu+p$,$d\mu+d$ and $t\mu+t$. Muon Catalyzed Fusion 4, 247 (1989)
  * (32) Bracci, L., Chiccoli, C., Fiorentini, G., Melezhik, V.S., Pasini, P., Wozniak, J.: Calculation of the elastic scattering cross sections for $p\mu+p$ and $t\mu+t$ collisions in adiabatic representation. Phys. Lett. A 149, 463 (1990)
  * (33) Bubak, M., Faifman, M.P.: Cross sections for hydrogen muonic atomic processes in two-level approximation of the adiabatic framework. JINR Preprint E4-87-464 (1987)
  * (34) Adamczak, A.: Differential cross sections for muonic atom scattering from hydrogenic molecules. Phys. Rev. A 74 (2006). https://doi.org/10.1103/PhysRevA.74.042718
  * (35) Agostinelli, S., et al.: Geant4?a simulation toolkit. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 506(3), 250–303 (2003). https://doi.org/10.1016/S0168-9002(03)01368-8
  * (36) Roberts, T.J., Kaplan, D.M.: G4beamline simulation program for matter-dominated beamlines. 2007 IEEE Particle Accelerator Conference (PAC), 3468–3470 (2007)
  * (37) Leon, M., Bethe, H.A.: Negative meson absorption in liquid hydrogen. Phys. Rev. 127, 636 (1962). https://doi.org/10.1103/PhysRev.127.636
  * (38) Borie, E., Leon, M.: X-ray yields in protonium and mesic hydrogen. Phys. Rev. A 21, 1460 (1980). https://doi.org/10.1103/PhysRevA.21.1460
  * (39) Markushin, V.E.: Atomic cascade in muonic hydrogen and the problem of kinetic-energy distribution in the ground state. Phys. Rev. A 50, 1137 (1984). https://doi.org/10.1103/PhysRevA.50.1137
  * (40) Jensen, T.S., Markushin, V.E.: Collisional deexcitation of exotic hydrogen atoms in highly excited states. Eur. Phys. J. D. 21, 271 (2002). https://doi.org/10.1140/epjd/e2002-00208-x
  * (41) Faifman, M.P., Men’shikov, L.I.: Study of cascade processes in $p\mu$ and $p{K}^{-}$ atoms based on a new approach. In: Proceedings of International Conference on Muon Catalyzed Fusion and Related Topics ($\mu$CF-07), JINR, Dubna, p. 233 (2008)
  * (42) Popov, V.P., Pomerantsev, V.N.: Kinetic energy distributions of muonic and pionic hydrogen atoms. Hyperfine Interact. 209, 75 (2012). https://doi.org/10.1007/s10751-011-0514-2
  * (43) Popov, V.P., Pomerantsev, V.N.: Isotopic effects in scattering and kinetics of the atomic cascade of excited ${\mu}^{-}p$ and ${\mu}^{-}d$ atoms. Phys. Rev. A 95, 022506 (2017). https://doi.org/10.1103/PhysRevA.95.022506
  * (44) Popov, V.P., Pomerantsev, V.N.: Collision-induced radiative quenching and other disintegration modes of the $2s$ state of muonic hydrogen and deuterium atoms. Phys. Rev. A 105, 042804 (2022). https://doi.org/10.1103/PhysRevA.105.042804
  * (45) Abbott, D.J., et al.: Diffusion of muonic deuterium and hydrogen atoms. Phys. Rev. A 55, 214 (1997). https://doi.org/10.1103/PhysRevA.55.214
  * (46) Pohl, R., et al.: The size of the proton. Nature 466, 213 (2010). https://doi.org/10.1038/nature09250
  * (47) Adamczak, A., Antognini, A., Berger, N., Cocolios, T.E., Dressler, R., Eggenberger, A., Eichler, R., Indelicato, P., Jungmann, K., Kirch, K., Knecht, A., Papa, A., Pohl, R., Pospelov, M., Rapisarda, E., Reiter, P., Ritjoho, N., Roccia, S., Severijns, N., Skawran, A., Wauters, F., Willmann, L.: Nuclear structure with radioactive muonic atoms. EPJ Web Conf. 193, 04014 (2018). https://doi.org/10.1051/epjconf/201819304014
  * (48) Skawran, A., et al.: Towards nuclear structure with radioactive muonic atoms. Nuovo Cim. 42, 125 (2019). https://doi.org/10.1393/ncc/i2019-19125-7
  * (49) Wauters, F., Knecht, A.: The muX project. SciPost Phys. Proc. 5, 022 (2021). https://doi.org/10.21468/SciPostPhysProc.5.022
  * (50) Grillenberger, J., Baumgarten, C., Seidel, M.: The High Intensity Proton Accelerator Facility. SciPost Phys. Proc. 5, 002 (2021). https://doi.org/10.21468/SciPostPhysProc.5.002
  * (51) AdvanSiD. https://advansid.com/. Accessed: 2022-09-20
  * (52) Hamamatsu. https://www.hamamatsu.com. Accessed: 2022-09-20
  * (53) Nuber, J.: Simulation of the diffusion process of muonic atoms in hydrogen targets. Master’s thesis, Karlsruhe Intitute of Technology (2018)
  * (54) HTW Hochtemperatur-Werkstoffe GmbH. http://htw-germany.com. Accessed: 2022-09-20
  * (55) IN2P3/STFC French/UK Ge Pool. https://gepool.in2p3.fr. Accessed: 2022-09-20
  * (56) Warr, N., et al.: The Miniball spectrometer. The European Physical Journal A 49(3), 40 (2013). https://doi.org/10.1140/epja/i2013-13040-9
  * (57) Bergem, P., et al.: Nuclear polarization and charge moments of 208Pb from muonic x rays. Phys. Rev. C 37, 2821 (1988). https://doi.org/10.1103/PhysRevC.37.2821
  * (58) Acker, H.L., Backenstoss, G., Daum, C., Sens, J.C., De Wit, S.A.: Measurement and analysis of muonic x-ray spectra in spherical nuclei. Nuclear Physics 87, 1 (1966). https://doi.org/10.1016/0029-5582(66)90355-5
  * (59) Bertin, A., Ferrari, F., Massa, I., Piccinini, M., Vannini, G., Vitale, A.: Muon transfer effect on the Lyman series of $\mu$Al muonic atoms. Physics Letters A 68, 201 (1978). https://doi.org/10.1016/0375-9601(78)90803-4
  * (60) Jacot-Guillarmod, R., Bienz, F., Boschung, M., Piller, C., Schaller, L.A., Schellenberg, L., Schneuwly, H., Reichart, W., Torelli, G.: Muon transfer from hydrogen to argon and helium at 10–15 bars. Phys. Rev. A 38, 6151 (1988). https://doi.org/10.1103/PhysRevA.38.6151
  * (61) Mulhauser, F., Schneuwly, H.: Muon transfer to sulphur dioxide 26, 4307 (1993). https://doi.org/10.1088/0953-4075/26/22/032
  * (62) Jacot-Guillarmod, R.: Muon transfer from hydrogen and deuterium atoms to neon. Phys. Rev. A 51, 2179 (1995). https://doi.org/10.1103/PhysRevA.51.2179
  * (63) Haff, P.K., Rodrigo, E., Tombrello, T.A.: Muon transfer in gas targets. Annals of Physics 104, 363 (1977). https://doi.org/10.1016/0003-4916(77)90336-0
  * (64) Skawran, A.: Development of a new method to perform muonic atom spectroscopy with microgram targets. PhD thesis, ETH Zürich (2021). https://doi.org/10.3929/ethz-b-000489410
  * (65) Hartmann, F.J., Bergmann, R., Daniel, H., et al.: Measurement of the muonic x-ray cascade in Mg, AI, In, Ho, and Au. Z Physik A 305, 189 (1982). https://doi.org/10.1007/BF01417434
  * (66) Butsev, V.S., et al.: Intensities of muonic X rays in lead, thorium and uranium. JETP Letters 23, 491 (1976)
  * (67) Anderson, H.L., Hargrove, C.K., Hincks, E.P., McAndrew, J.D., McKee, R.J., Barton, R.D., Kessler, D.: Precise Measurement of the Muonic X Rays in the Lead Isotopes. Phys. Rev. 187, 1565 (1969). https://doi.org/10.1103/PhysRev.187.1565
  * (68) Friar, J.L., Gibson, B.F., Jean, H.C., Payne, G.L.: Nuclear transition rates in $\mu$-catalyzed p-d fusion. Phys. Rev. Lett. 66, 1827 (1991). https://doi.org/10.1103/PhysRevLett.66.1827
  * (69) Fricke, G., Heilig, K.: Nuclear Charge Radii. Springer-Verlag, Berlin Heidelberg (2004). https://doi.org/10.1007/b87879
  * (70) Vogiatzi, S.M.: Muonic atom spectroscopy with radioactive targets. Talk at the 2022 Swiss Physical Society meeting: https://indi.to/jCk85 (2022)


# BASED: Benchmarking, Analysis, and Structural Estimation of Deblurring
††thanks: This study was supported by Russian Science Foundation under grant
22-21-00478, https://rscf.ru/en/project/22-21-00478/

Nikita Alutis1 Egor Chistov1 Mikhail Dremin1 Dmitriy Vatolin12  Lomonosov
Moscow State University1
MSU Institute for Artificial Intelligence2
Moscow, Russia
{nikita.alutis, egor.chistov, mikhail.dremin<EMAIL_ADDRESS>
###### Abstract

This paper discusses the challenges of evaluating deblurring-methods quality
and proposes a reduced-reference metric based on machine learning. Traditional
quality-assessment metrics such as PSNR and SSIM are common for this task, but
not only do they correlate poorly with subjective assessments, they also
require ground-truth (GT) frames, which can be difficult to obtain in the case
of deblurring. To develop and evaluate our metric, we created a new motion-
blur dataset using a beam splitter. The setup captured various motion types
using a static camera, as most scenes in existing datasets include blur due to
camera motion. We also conducted two large subjective comparisons to aid in
metric development. Our resulting metric requires no GT frames, and it
correlates well with subjective human perception of blur.

###### Index Terms:

deblurring, video quality assessment, deblurring dataset

##  I Introduction

Numerous learning-based approaches to both single-image and video deblurring
have recently been proposed. Today, however, there is no comprehensive way to
assess deblurring quality. Instead, common methods for this task include the
traditional PSNR and SSIM[1] image quality metrics, as well as the deep-
learning-based LPIPS metric [2] and super-resolution quality ERQA metric [3].
However as we show later, they all correlate poorly with subjective human
perception of blur, and they require GT-frames, which can be hard to obtain
when creating a deblurring dataset.

The best way to assess methods performance is to conduct a subjective
comparisons, but they are too costly to employ every time a method needs
evaluation. Therefore, we propose a new machine-learning-based reduced-
reference metric to specifically evaluate deblurring quality.

Training the metric and evaluating its performance requires datasets with real
blur, because as [6] proves, real blur differs considerably from synthetic
blur. Three recent datasets created using a beam-splitter have emerged [4, 5,
6]; among them, RSBlur [6] is the most diverse.

Figure 1: Example of BASED and LPIPS metrics on crop from RSBlur dataset.
Right image is significantly less blurred, however LPIPS shows that right
image is more blurred than left, while BASED metric accurately reflects
increase in sharpness

Even this dataset has a drawback, however: it mostly contains scenes in which
blur results from camera motion, so the blurring is nearly even over the
entire frame. But in the real world, blur often occurs due to the movement of
one or more fast-moving objects in the frame. These videos are much more
difficult for neural networks to correctly process (as we show later), because
only a small part of the image actually suffers from blurring; other regions
provide little or no information to aid in restoration.

Therefore, to ensure maximum generalization ability of the resulting metric,
we created the Benchmarking, Analysis, and Structural Estimation of Deblurring
(BASED) dataset using a beam-splitter setup that employs a static camera to
capture a variety of motion types.

Finally, to train the metric to correlate with human perception, we needed to
get subjective values for each crop of the dataset. Thus, we performed two
large independent subjective comparisons on both the BASED and RSBlur
datasets, each using a slightly different sets of methods.

Our contributions are the following:

  * •

We created a real-motion-blur dataset using a beam-splitter setup and tested
the performance of state-of-the-art deblurring methods.

  * •

We developed a comprehensive deblurring benchmark to quantify the ability of
deblurring models to restore blurred videos. The benchmark is publicly
available at https://videoprocessing.ai/benchmarks/deblurring.html.

  * •

We conducted two large subjective comparisons to analyze existing quality
metrics and collected reference values to aid in developing our metric.

  * •

We produced a machine-learning-based metric that evaluates deblurring quality
without requiring GT frames.

The code for the resulting metric can be found at
https://github.com/illaitar/based.

##  II Related Works

Many neural-network-based deblurring methods are in use today [7, 8, 9, 10,
11, 12, 13, 15, 14]. Some are image based, including MFFDNet[7] and NAFNet[8],
and some are video based, including VRT[10]. Most research articles that
propose such methods employ traditional full-reference image quality
assessment through, for example, PSNR or SSIM. The ERQA metric for super-
resolution-quality assessment has also served to assess deblurring quality.
And recently, several neural-network-based metrics have emerged, the most
popular of which is LPIPS, but they rarely appear in such articles.
Researchers have also created specialized no-reference blur metrics, such as
[18] and [21]. Our aim is to evaluate how well all these metrics handle image
and video deblurring.

Datasets for deblurring typically fall into two categories: synthetic blur,
such as GoPro [16] and Reds [17], and real-world blur, such as BSD[4],
RealBlur[5], and RSBlur[6]. Synthetic blur, as [6] shows, differs too greatly
from real-world blur to judge a method’s performance on arbitrary videos. At
the same time, real-blur datasets all suffer from problems related to image
and data processing:

  * •

BSD, RealBlur, and RSBlur contain only images, not videos.

  * •

BSD has low resolution.

  * •

RealBlur mostly includes low-light scenes, but real-world blur often occurs in
well-lit scenes.

  * •

All three datasets are mostly based on camera motion, so the frames exhibit
even blurring. In the real world, however, blur is often due to the motion of
one or more fast-moving objects; therefore, only some areas in these videos
contain this artifact.

##  III Proposed Dataset

We propose a new dataset of videos containing real-world motion blur. We
collected it using a beam-splitter optical device.

###  III-A Scene Selection

We filmed various objects and motion types to diversify the scenes. Motion
types in our dataset include swinging, falling, rolling, linear movement, and
turning. We set up a white background and a static light source to focus on
the blurred objects. Out of 135 videos we handpicked 23 to represent movement
type and speed, texture complexity, and temporal and spatial complexity.

###  III-B Recording Rig

Our beam-splitter rig consisted of the beam-splitting glass, which has a 0.5
reflection coefficient, and two cameras with zero visual stereo base between
them. We used a setup similar to that used for the RealBlur dataset.

A box protected our cameras and the beam splitter from external light; its
inside was covered with matte light-absorbing fabric. This arrangement
eliminated the camera-lens reflections that previous datasets exhibited. We
disabled optical stabilization and manually fixed all settings, such as ISO,
shutter speed, and color temperature.

To create the dataset, we had to simultaneously capture blurred (target) and
nonblurred (reference) videos. To do so, our two cameras employed different
exposure times. We set the one filming the blurred view to a 0.1-second
exposure and the one filming ground truth to 0.005 seconds. The other settings
were the same for both cameras, so all splitter-glass distortions were
insignificant. The color-correction step compensated for brightness variation
due to different shutter speeds.

###  III-C Parallax Minimization

To obtain accurate and reliable ground-truth data for a deblurring dataset,
filming scenes with minimal parallax is essential. Parallax refers to the
apparent shift in an object’s position when viewed from different lines of
sight. Although post-processing techniques can effectively correct for affine
mismatches between cameras, such as scale, rotation, and translation,
correcting for parallax is more difficult. Eliminating parallax at this stage
requires the use of complex optical flow and inpainting algorithms, which can
potentially compromise the quality of the ground-truth data.

Parallax correction is crucial for several reasons. First, it helps maintain
the dataset’s integrity by ensuring that the captured images are consistent
and reliable. This, in turn, enables the development of more accurate and
robust deblurring algorithms. Second, minimizing parallax simplifies the data
processing pipeline by reducing the need for complex and computationally
expensive optical-flow algorithms. This approach streamlines the development
process and saves valuable time and resources.

Parallax minimization can employ a two-step process. First, the lenses of the
left distorted and left ground-truth cameras must be visually aligned to
create a visual zero stereo base. This alignment ensures that the cameras
share a common point of view, reducing the potential for parallax. Next, the
cameras must aim at the same scene to maximize the overlapping region, using a
photo captured by one camera and a video stream captured by the other. Doing
so enables direct comparison of the images to ensure no parallax is present.

Once the absence or near absence of parallax is confirmed, all scenes can be
filmed in a single uninterrupted session. The ground-truth data is therefore
consistent, reliable, and free of parallax-induced distortions — vital
features for the development of effective deblurring algorithms.

###  III-D Post-processing

Because the resulting views have little or no parallax present, we were able
to implement a simple post-processing pipeline that avoids any significant
frame distortion. We first flipped the target camera view horizontally, then
we matched it to the ground truth view using a homography transformation. For
the estimation of transformation parameters, we utilized SIFT[25] extractor,
Brute-Force matcher, and the MAGSAC++ algorithm[24]. Global brightness
difference and small color mismatches between two views were corrected using
the method [22]. Temporal alignment employed audio streams captured by the
cameras. Fig 2 presents a scheme of our pipeline.

Figure 2: Dataset-creation pipeline. Incoming light enters a beam splitter,
and two cameras receive the same image. After initial filming we applied
temporal, spatial, and color alignment.

##  IV Benchmark

We tested 7 methods and in total 11 their variations (i.e. trained on
different datasets) on BASED: NAFNet (REDS), MAXIM (REDS), Restormer (GoPro),
VRT (REDS), MPR local (GoPro) , VRT (GoPro), DeepRFT (GoPro), DeepRFT (REDS),
MAXIM (GoPro), DeblurGAN Inception (GoPro), Restormer local (GoPro). Our
examination employed PSNR, SSIM, LPIPS, and ERQA to test the results. We
selected these metrics because they are the most common in articles that cover
deblurring. The comparison results are shown on Table I. Examples of methods
work are present on Fig 4.

It is obvious that all metrics show a different picture in terms of methods
performance. It can also be seen that clearly, some metrics provide
misinformation in terms of methods performance.

##  V Subjective Comparison

We measured the performance of various groups of deblurring methods on the
BASED and RSBlur datasets to obtain the most diverse dataset for metric
training. We cropped the original dataset frames to three resolutions to test
method performance at different scales: 512x512, 256x256, and 128x128. Our
comparison used 253 crops from BASED and 1,200 from RSBlur. Crops from RSBlur
and BASED are shown on Fig 5 and Fig 4.

We conducted a side-by-side pairwise evaluation using the Subjectify.us
service, which enables crowdsourced subjective comparisons. To estimate
information fidelity, we asked participants to avoid choosing the most
beautiful video and instead choose the one that exhibits better detail
restoration. Because participants are not experts in this field, they lack
professional biases. Every participant was shown 25 video pairs and in each
case was asked to choose the best video (“indistinguishable” was also an
option). Each pair was shown to 10–15 participants until the confidence
interval stopped changing. Of the 25 pairs for each participant, 3 are for
verification, so the final results exclude those responses. Verification
questions are necessary because we want to ensure that participants understand
the task and avoid making random choices — a critical facet of effective
large-scale crowdsourced comparisons. All other responses from 1,904
successful RSBlur participants and 648 successful BASED participants went into
calculating subjective scores using the Bradley-Terry model[19].

Table I and Fig 3 present the results of our subjective comparison. NAFNet was
the best method across two datasets.

TABLE I: Results using BASED. (G) indicates methods trained on the GoPro dataset, (R) indicates methods trained on the REDS dataset, and L indicates the local version of the method[12]. The best result appears in bold. Method Metric | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | ERQA$\uparrow$ | Subj.$\uparrow$
---|---|---|---|---|---
NAFNet (R) | 30.5480 | 0.9504 | 0.0856 | 0.7451 | 2.84
VRT (G) | 31.4295 | 0.9450 | 0.0817 | 0.7487 | 2.39
VRT (R) | 30.9788 | 0.9460 | 0.0824 | 0.7506 | 1.57
DeblurGAN (G) | 31.1717 | 0.9430 | 0.0887 | 0.7430 | 1.04
MAXIM (R) | 30.6573 | 0.9496 | 0.0784 | 0.7428 | 1.01
DeepRFT (G) | 31.5761 | 0.9448 | 0.0832 | 0.7432 | 0.54
DeepRFT (R) | 31.3234 | 0.9448 | 0.0814 | 0.7434 | 0.46
MPR {L} (G) | 31.6504 | 0.9454 | 0.0832 | 0.7452 | 0.44
MAXIM (G) | 31.3634 | 0.9439 | 0.0919 | 0.7444 | 0.21
Restormer (G) | 31.7611 | 0.9463 | 0.0824 | 0.7478 | 0.12
Restormer {L} (G) | 31.1234 | 0.9421 | 0.0825 | 0.7387 | 0.12

Figure 3: Subjective-comparison results for RSBlur. The study comprises input
from 1,904 participants. NAFNet performed much better than other methods.

##  VI Proposed Metric

###  VI-A Concept

We set out to create a reduced-reference metrict hat operates without GT
frames. It takes two pictures — an original blurred one and a resulting
deblurred one, and provides a quality score.

We trained Random Forest Regressor [20] on several metrics traditionally used
for texture analysis and sharpness estimation tasks. We tested the model with
different hyper-parameters and found out that 220 estimators (Table IV) and a
squared error criterion are the best combination. Our approach used the same
crops as the subjective comparison, allowing us to employ the subjective
values when training and testing the metric. We have also tried gradient
boosting approach[26], but it tended to overfit on both datasets.

We wanted to completely avoid overfitting on our data, so we trained the
metric only on a smaller BASED dataset. Later, we tested it on RSBlur to prove
its generalizability. In addition, our analysis did not use RSBlur
correlations to tweak individual features; it only used BASED values as a
reference.

###  VI-B Features

We studied and tweaked many features and these showed the best overall results
in terms of individual correlation and correlation of whole metric on BASED:

####  VI-B1 Laplacian

The Laplacian is a second-order-derivative operator that enables detection of
edges and intensity changes in an image. Assessing blur using this operator
involves computing the Laplacian of an image and examining its properties to
determine the degree of blur.

####  VI-B2 FFT

The FFT is a mathematical technique that converts a signal from the time
domain into the frequency domain, allowing us to analyze its spectral content.
Assessing blur using the Fast Fourier Transform (FFT) involves analyzing the
frequency content of an image in the frequency domain.

For our task, we filtered all low frequencies with the standard threshold of
30, and then compared two resulting functions.

####  VI-B3 Gabore Filtering

Gabor filters are a type of linear filter used in image processing and
computer vision to extract features from images. They are constructed by
multiplying a Gaussian kernel and a complex sinusoidal wave, yielding a filter
that is sensitive to both spatial location and frequency. The response of a
Gabor filter to an image is obtainable by convolving the filter with the
image. Doing so produces a filtered image that highlights areas of the
original that match the filter’s frequency and orientation selectivity.

We used a set of three Gabore kernels to estimate their response and then
calculated the difference between the resulting vectors.

####  VI-B4 Hough

The basic idea behind the Hough transform is to represent the image in a
parameter space such that each point in the image corresponds to a curve in
that space.

####  VI-B5 HOG

Histogram of oriented gradients (HOG) is a feature-extraction technique by
which computer-vision and image-processing algorithms detect object shapes and
their boundaries. It divides an image into small cells, computes the gradient
orientations and magnitudes for every pixel in each cell, and then aggregates
the gradients into a histogram of gradient orientations for each cell.

To obtain the result of this metric, we calculated the L2 norm between the HOG
for the original image and the HOG for the deblurred one.

####  VI-B6 SSIM-M

SSIM is a traditional metric used to measure the similarity between two images
based on their structural content. We modified SSIM to work with YUV images
and calculated the result as $6\times SSIM(Y)+SSIM(U)+SSIM(V)$, which has been
proven to correlate better with human perception[23].

####  VI-B7 Sobel

The Sobel operator works by convolving an image with two filters to calculate
the image gradient. One filter detects horizontal edges, while the other
detects vertical edges. After applying the filters, the resulting gradient
images are combined to produce an edge map that highlights the edges in the
original image.

We used Sobel kernels with the size of 13 and then calculated the L2 norm of
the residual of two resulting gradient images.

####  VI-B8 LBP

Local binary patterns (LBP) is a histogram-based approach that represents an
image’s local structure by computing a binary code for each pixel on the basis
of its relationship with neighboring pixels.

The idea behind LBP is to compare a pixel’s intensity with those of
surrounding pixels and generate a binary code for that pixel. Once it computes
these codes for all pixels in the image, it generates a histogram that counts
the frequency of each code.

To obtain a result for this metric, we calculated the L2 norm of two LBP
histograms: one from the original image and one from the deblurred image.

####  VI-B9 Reblur

We used the approach from [21] to calculate the difference between the
reblurred original image and reblurred blur image using the L2 norm of their
absolute difference. Our version applied a Gaussian kernel of size of 17 to
the reblur images.

TABLE II: Mean Pearson (PLCC), Spearman (SRCC), and Kendall (KRCC) correlation coefficients between metric features and subjective-comparison results. The proposed metric was trained on BASED dataset. For results on BASED, 5 fold cross validation was used as final score. The best result appears in bold. | RSBlur | BASED
---|---|---
Metric | PLCC | SRCC | KRCC | PLCC | SRCC | KRCC
Laplacian | 0.8584 | 0.8343 | 0.7421 | 0.8939 | 0.8716 | 0.7521
FFT | 0.7356 | 0.8411 | 0.7504 | 0.8327 | 0.8613 | 0.7430
Reblur | 0.6950 | 0.6672 | 0.5593 | 0.6945 | 0.6432 | 0.5284
Hough | 0.8077 | 0.7918 | 0.6859 | 0.6718 | 0.7500 | 0.6259
Sobel | 0.7553 | 0.7652 | 0.6714 | 0.8984 | 0.8800 | 0.7476
HOG | 0.7188 | 0.7927 | 0.7007 | 0.8745 | 0.8595 | 0.7323
LBP | 0.6914 | 0.7830 | 0.6879 | 0.7653 | 0.8058 | 0.6713
Gabore | 0.5735 | 0.5965 | 0.5036 | 0.8831 | 0.8445 | 0.7233
SSIM-M | 0.8584 | 0.8188 | 0.7250 | 0.9242 | 0.8777 | 0.7521
Proposed | 0.8349 | 0.8520 | 0.7635 | 0.9531 | 0.9053 | 0.8821
TABLE III: Mean Pearson (PLCC), Spearman (SRCC) and Kendall (KRCC) correlation coefficients between metrics and subjective-comparison results. The best result appears in bold. | RSBlur | BASED
---|---|---
Metric | PLCC | SRCC | KRCC | PLCC | SRCC | KRCC
SSIM | 0.8177 | 0.7623 | 0.6489 | 0.8821 | 0.8555 | 0.7216
PSNR | 0.7836 | 0.7644 | 0.6714 | 0.9044 | 0.8753 | 0.7476
LPIPS | 0.7520 | 0.7558 | 0.6290 | 0.9290 | 0.8833 | 0.7688
ERQA | 0.6980 | 0.7006 | 0.5983 | 0.8052 | 0.7741 | 0.6577
CPBD | 0.1176 | 0.0811 | 0.0618 | 0.0157 | 0.1112 | 0.1015
Reblur | 0.6747 | 0.6223 | 0.5132 | 0.6952 | 0.6408 | 0.5239
Proposed | 0.8349 | 0.8520 | 0.7635 | 0.9531 | 0.9053 | 0.8821
Figure 4: Method results on BASED. NAFNet clearly shows the best result. Figure 5: Methods results on RSBlur. On this challenging example MAXIM, which was trained on real blur dataset, shows a better performance. TABLE IV: Dependence of metric performance quality on the number of trees in the ensemble. The best result appears in bold. | RSBlur
---|---
Estimators | PLCC | SRCC | KRCC
50 | 0.8312 | 0.8387 | 0.7434
100 | 0.8334 | 0.8512 | 0.7595
150 | 0.8323 | 0.8475 | 0.7546
200 | 0.8341 | 0.8502 | 0.7610
220 | 0.8349 | 0.8520 | 0.7635
250 | 0.8349 | 0.8509 | 0.7617
300 | 0.8348 | 0.8487 | 0.7574
350 | 0.8352 | 0.8484 | 0.7581
400 | 0.8353 | 0.8493 | 0.7586

###  VI-C Results

We achieved good results for all three correlations. The results of each
individual component can be observed on Table II. Relative to traditional
metrics, such as SSIM, PNSR, ERQA, and LPIPS, we also achieved state-of-the-
art performance (Table III). But these metrics were not designed for a
reduced-reference analysis (i.e., without the use of GT frames) and thus
exhibit worse results than they do in the full-reference case. We also tested
openly available No-Reference blur metrics: the CPBD metric and the original
version of Reblur[21] which we previously modified to use as one of the
components.

##  VII Conclusion

In this paper we created a real-motion-blur dataset using a beam splitter and
used it to test the performance of state-of-the-art deblurring methods. We
also conducted two large subjective comparisons, one on our dataset and one on
the existing RSBlur dataset, to both show the failure of current quality
metrics and to collect reference values for metric development. Finally, we
developed a machine-learning-based metric to evaluate deblurring quality
without using GT frames or subjective comparisons.

##  VIII Acknowledgments

Dataset preparation and subjective comparison were supported by Russian
Science Foundation under grant 22-21-00478. Autors are also grateful to Daniil
Davydov and Alexander Fominykh for detailed discussions of the study.

## References

  * [1] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE transactions on image processing, vol. 13, no. 4, pp. 600–612, 2004.
  * [2] R. Zhang, P. Isola, A. A. Efros, E. Shechtman, and O. Wang, “The unreasonable effectiveness of deep features as a perceptual metric,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp. 586–595.
  * [3] A. Kirillova, E. Lyapustin, A. Antsiferova, and D. Vatolin, “ERQA: Edge-restoration quality assessment for video super-resolution,” arXiv preprint arXiv:2110.09992, 2021.
  * [4] Z. Zhong, Y. Gao, Y. Zheng, B. Zheng, and I. Sato, “Real-world video deblurring: A benchmark dataset and an efficient recurrent neural network,” International Journal of Computer Vision, vol. 131, no. 1, pp. 284–301, 2023.
  * [5] J. Rim, H. Lee, J. Won, and S. Cho, “Real-world blur dataset for learning and benchmarking deblurring algorithms,” in Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XXV 16, 2020, pp. 184–201.
  * [6] J. Rim, G. Kim, J. Kim, J. Lee, S. Lee, and S. Cho, “Realistic blur synthesis for learning image deblurring,” in Computer Vision–ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part VII, 2022, pp. 487–503.
  * [7] C. Guo, Q. Wang, H.-N. Dai, and P. Li, “Multi-stage feature-fusion dense network for motion deblurring,” Journal of Visual Communication and Image Representation, vol. 90, p. 103717, 2023.
  * [8] J. Liang et al., “Vrt: A video restoration transformer,” arXiv preprint arXiv:2201.12288, 2022.
  * [9] L. Chen, X. Chu, X. Zhang, and J. Sun, “Simple baselines for image restoration,” in Computer Vision–ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part VII, 2022, pp. 17–33.
  * [10] Z. Tu et al., “Maxim: Multi-axis mlp for image processing,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 5769–5780.
  * [11] S. W. Zamir, A. Arora, S. Khan, M. Hayat, F. S. Khan, and M.-H. Yang, “Restormer: Efficient transformer for high-resolution image restoration,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 5728–5739.
  * [12] X. Chu, L. Chen, C. Chen, and X. Lu, “Improving image restoration by revisiting global information aggregation,” in Computer Vision–ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part VII, 2022, pp. 53–71.
  * [13] O. Kupyn, V. Budzan, M. Mykhailych, D. Mishkin, and J. Matas, “Deblurgan: Blind motion deblurring using conditional adversarial networks,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp. 8183–8192.
  * [14] C. Guo, Q. Wang, H.-N. Dai, and P. Li, “Multi-stage feature-fusion dense network for motion deblurring,” Journal of Visual Communication and Image Representation, vol. 90, p. 103717, 2023.
  * [15] X. Mao, Y. Liu, W. Shen, Q. Li, and Y. Wang, “Deep Residual Fourier Transformation for Single Image Deblurring,” CoRR, vol. abs/2111.11745, 2021
  * [16] S. Nah, T. H. Kim, and K. M. Lee, “Deep Multi-Scale Convolutional Neural Network for Dynamic Scene Deblurring,” Jul. 2017.
  * [17] S. Nah et al., “NTIRE 2019 Challenge on Video Deblurring and Super-Resolution: Dataset and Study,” Jun. 2019.
  * [18] N. D. Narvekar and L. J. Karam, “A No-Reference Image Blur Metric Based on the Cumulative Probability of Blur Detection (CPBD),” IEEE Transactions on Image Processing, vol. 20, no. 9, pp. 2678–2683, 2011, doi: 10.1109/TIP.2011.2131660.
  * [19] Bradley, R. A. and Terry, M. E. (1952). Rank analysis of in- complete block designs: I. the method of paired com- parisons. Biometrika, 39(3/4):324–345.
  * [20] L. Breiman, “Random forests,” Machine learning, vol. 45, pp. 5–32, 2001.
  * [21] D. B. Bong and B. E. Khoo, “An efficient and training-free blind image blur assessment in the spatial domain,” IEICE TRANSACTIONS on Information and Systems, vol. 97, no. 7, pp. 1864–1871, 2014.
  * [22] S. Lavrushkin, V. Lyudvichenko, and D. Vatolin, “Local method of color-difference correction between stereoscopic-video views,” in 2018-3DTV-Conference: The True Vision-Capture, Transmission and Display of 3D Video (3DTV-CON), 2018, pp. 1–4.
  * [23] A. Antsiferova, A. Yakovenko, N. Safonov, D. L. Kulikov, A. Gushin, and D. S. Vatolin, “Objective video quality metrics application to video codecs comparisons: choosing the best for subjective quality estimation,” CoRR, vol. abs/2107.10220, 2021
  * [24] D. Barath, J. Noskova, M. Ivashechkin, and J. Matas, “MAGSAC++, a fast, reliable and accurate robust estimator,” CoRR, vol. abs/1912.05909, 2019
  * [25] D. G. Lowe, “Object recognition from local scale-invariant features,” in Proceedings of the seventh IEEE international conference on computer vision, 1999, vol. 2, pp. 1150–1157.
  * [26] J. H. Friedman, “Stochastic gradient boosting,” Computational statistics and data analysis, vol. 38, no. 4, pp. 367–378, 2002.