[a]Chris Allton

# Recent Results from the FASTSUM Collaboration

Gert Aarts  Ryan Bignell  Tim Burns  Sergio Chaves  Simon Hands  Benjamin
Jäger  Seyong Kim  Maria Paola Lombardo  Ben Page  Sinéad M. Ryan  Jon-Ivar
Skullerud  Thomas Spriggs

###### Abstract

The Fastsum Collaboration has developed a comprehensive research programme in
thermal QCD using $2+1$ flavour, anisotropic ensembles. In this talk, we
summarise some of our recent results including thermal hadron spectrum
calculations using our “Generation 2L” ensembles which have pion masses of
239(1) MeV. These include open charm mesons and charm baryons. We also
summarise our work using the Backus Gilbert approach to determining the
spectral function of the NRQCD bottomonium system. Finally, we review our
determination of the interquark potential in the same system, but using our
“Generation 2” ensembles which have heavier pion masses of 384(4) MeV.

##  1 Introduction

The thermal spectrum of QCD is of great interest for intrinsic reasons in
order to understand how confinement is manifest in hadronic systems. It is
also crucial to aid the analysis of heavy-ion collision experiments. Here we
present an update of the Fastsum Collaboration’s thermal hadronic spectrum
research, focussing on open charm mesons, charm baryons and bottomonium. We
also present an update of our studies of the interquark potential in
bottomonium.

We use 2+1 flavour dynamical simulations with anisotropic latices where the
temporal lattice spacing, $a_{\tau}$, is smaller than the spatial one, $a_{s}$
[1, 2]. Our anisotropy is designed to maximise information from thermal
temporal correlators, noting that they are constrained in temporal extent,
$L_{\tau}$, since the temperature, $T=1/L_{\tau}$. We use stout-linked,
clover-improved Wilson fermions and Symanzik-improved gauge fields.

The lattice ensembles used in this work are our Generation 2 and 2L ensembles
which have parameters listed in Tables 1 and 2. These ensembles have a pion
mass of 384(4) and 239(1) MeV respectively and span temperatures both below
and above the pseudocritical temperature $T_{\text{pc}}$.

$N_{\tau}$ | 128 | 40 | 36 | 32 | 28 | 24 | 20 | 16
---|---|---|---|---|---|---|---|---
$T\,\,\text{[MeV]}$ | 44 | 141 | 156 | 176 | 201 | 235 | 281 | 352
Table 1: Parameters for the Fastsum Generation 2 ensembles used in this work. The lattice sizes are $24^{3}\times N_{\tau}$, with lattice spacings $a_{s}=0.1205(8)$ fm and $a_{\tau}=35.1(2)\,$am, and pion mass $m_{\pi}=384(4)$ MeV. The vertical line indicates the position of $T_{\text{pc}}\approx 181$ MeV. Full details in [1, 2]. $N_{\tau}$ | 128 | 64 | 56 | 48 | 40 | 36 | 32 | 28 | 24 | 20 | 16
---|---|---|---|---|---|---|---|---|---|---|---
$T\,\,\text{[MeV]}$ | 47 | 95 | 109 | 127 | 152 | 169 | 190 | 217 | 253 | 304 | 380
Table 2: Parameters for the Fastsum Generation 2L ensembles used in this work.
The lattice sizes are $32^{3}\times N_{\tau}$, with lattice spacings
$a_{s}=0.1121(3)$ fm and $a_{\tau}=32.46(7)\,$am, and pion mass
$m_{\pi}=239(1)$ MeV [3]. The vertical line indicates the position of
$T_{\text{pc}}\approx 167$ MeV. Full details in [2, 4].

##  2 Charm hadron spectrum

Here we summarise results from both open-charm mesons and charmed baryons
using our Generation 2L ensembles.

Unlike hiddened-charmed mesons at non-zero temperature, which have been
extensively studied on the lattice [5], open-charmed mesons have received less
attention [8, 6, 7]. In [4], we present results for the open charm meson
spectrum for $T\lesssim T_{pc}$. Because the states are confined, we proceed
with conventional analysis techniques and assess up to which temperatures
these are applicable. We extend these techniques to determine the variation of
the masses as a function of temperature.

The results are shown in Fig.1. These show a small temperature variation where
both the pseudoscalar, $D_{\rm{(s)}}$, and vector, $D^{\ast}_{\rm{(s)}}$,
mesons’ masses decrease as the temperature approches $T_{pc}$. However, note
that this temperature shift is at the percent level. In contrast, the
analogous thermal effects for the axial vector and scalar channel are very
strong, see [4] for details.

Figure 1: Temperature dependence of the groundstate masses in the hadronic
phase, for $D$ and $D^{*}$ (left) and $D_{s}$ and $D_{s}^{*}$ (right) mesons.
The vertical band indicates $T_{pc}$ and the horizontal stubs at $T=0$
represent the PDG values [9].

In another analysis [10, 11], we study the charm baryon spectrum paying
particular attention to both parity states. We extract the masses in the
confined phase and use a method based on a direct analysis of the correlation
function to determine whether the parity states approach degeneracy for $T\geq
T_{pc}$.

In Fig.2 we plot the masses for baryons with a variety of charm content as a
function of $T$ up to just beyond $T_{pc}$ where the simple pole fits become
unreliable.

Figure 2: Mass spectrum of of the $J^{1/2}{}^{+}$ baryons with positive (left)
and negative parity (right) as a function of temperature. Dashed lines are
zero-temperature experimental results [9] to guide the eye. The inner (outer)
shaded regions represents the statistical (systematic) errors. See [10] for
the corresponding plots for the $J^{3/2}{}^{+}$ states.

For $T>T_{pc}$ we cannot assume that the charm baryon states are bound and so
a conventional pole fitting ansatze cannot be applied. To gain information
about the mass of the two parity states, we therefore define the ratio $R$,

$R(n_{0})=\frac{\sum_{n=n_{0}}^{\frac{1}{2}N_{\tau}-1}\,\mathcal{R}(\tau_{n})/\sigma_{\mathcal{R}}^{2}(\tau_{n})}{\sum_{n=n_{0}}^{\frac{1}{2}N_{\tau}-1}\,1/\sigma_{\mathcal{R}}^{2}(\tau_{n})},\;\;\;\;\;\;\;\;\;{\rm
where}\;\;\;\;\;\mathcal{R}(\tau)=\frac{G^{+}(\tau)-G^{-}(\tau)}{G^{+}(\tau)+G^{-}(\tau)}.$
(1)

Typically we use $n_{0}=4$, and our results are not qualitatively sensitive to
this choice. $R$ will be unity in the limit of $M^{+}\ll M^{-}$ and zero for
the degenerate case. In Fig.3 we plot $R$ for a number of channels. We can see
an approach to degeneracy above $T_{pc}$ which is most pronounced for baryons
with the least charm content. By fitting the data to cubic splines, we can
determine estimates of the transition temperatures from the inflection points
and these are indicated as vertical lines in the figure. We note that the
location of the inflection points coincide, within a few MeV, with the
pseudocritical temperature, $T_{\text{pc}}=167(3)$ MeV, determined via the
chiral condensate [4] and hence are a manifestion of chiral symmetry
restoration in the charmed baryon sector. Further details about these points
are elucidated in [10, 11].

Figure 3:  R value from Eq.(1) for $J=\frac{1}{2}$ baryons with the lines from
cubic splines. The transition temperature estimates obtained from the
inflection point of the splines are shown as vertical lines.

##  3 Bottomonium (NRQCD) spectrum

Bottomonium states are important probes of the quark gluon plasma phase in
heavy-ion collision experiments because they are created very early and do not
reach chemical equilibrium. The lattice approach to analysing bottomonium
spectra invariably relies on extracting the spectral function at temperature
$T$, $\rho(\omega,T)$, defined from the correlation function,

$G(\tau;T)=\int_{\omega_{\text{min}}}^{\infty}\frac{d\omega}{2\pi}K(\tau,\omega)\rho(\omega;T),$
(2)

where the kernel for NRQCD quarks is defined as,

$K(\tau,\omega)=e^{-\omega\tau}.$ (3)

Note that since NRQCD introduces an additive energy shift, the lower limit of
the integral in Eq.(2), $\omega_{\text{min}}$, is not necessarily zero. The
spectral function gives complete information about the spectrum of a
particular channel, including the widths of the states. The Fastsum
Collaboration has studied the bottomonium spectrum using NRQCD bottom quarks
in a number of publications, e.g. [12], using a variety of methods to extract
the spectral function.

In [13] we extend this work by using the Backus Gilbert [14] method to obtain
the spectral function with our Generation 2L ensembles, and we report on this
work here. We note that we can introduce two “hyper-parameters” in our
analysis. The first is the “whitening” factor, $\alpha$, in the Tikhonov-like
method [15], which governs how much of the identity (white noise) is added to
the kernel. To remove the $\alpha$ dependency in the final result, the
$\alpha\rightarrow 0$ limit is taken. The second parameter is an energy shift,
$\Delta$. Since Eq.(2) is a Laplace transform, we can trivially shift
$\rho(\omega)\rightarrow\rho(\omega+\Delta)$ by multiplying $G(\tau)$ by
$e^{\Delta\tau}$. Increasing the energy shift, $\Delta$, moves the ground
state feature closer to $\omega_{\text{min}}$ which is advantageous because
that is where the Backus Gilbert method has the greatest resolution. Note
however that the value of $\Delta$ needs to be limited to ensure that no
spectral feature is pushed into the $\omega<\omega_{\text{min}}$ region. Hence
we remove the dependency on the $\Delta$ hyper-parameter via this requirement.

In Fig.4 we plot the $\chi_{b1}$ spectral function obtained from local
correlators with various $\Delta$ values to illustrate that the ground state
feature becomes better resolved as $\Delta$ increases. Full results and
predictions of masses and widths obtained using this method are in [13].

Figure 4: The $\chi_{b1}$ spectral function obtained via the Backus Gilbert
method for a variety of $\Delta$ energy shifts at $T=47$ MeV.

We can also perform an interesting statistical analysis of the correlation
functions. As pointed out by Parisi and Lepage [16, 17], the statistical error
of the hadronic correlation function, $\langle{\cal O}(t){\cal O}(O)\rangle$
at large time is determined by the lightest states that can be composed from
${\cal O}^{2}$. Typically this will be the pseudoscalar state.

We have analysed the statistical error in the bottomonium correlation
functions by measuring their covariance matrices’ singularity as the energy
shift, $e^{\Delta\tau}$, is applied. The value of $\Delta$ corresponding to
the most singular covariance matrix, $\Delta_{\rm{sing}}$, is a prediction of
(half) the ground state mass in the ${\cal O}^{2}$ channel. We used the
condition number of the covariance matrix to determine $\Delta_{\rm{sing}}$.
Following the Parisi and Lepage analysis, we expect to find
$\Delta_{\rm{sing}}=M_{\eta_{b}}$ i.e. the mass of the pseudoscalar in the
bottomonium sector.

In Fig.5 we plot results for the channels considered
($\eta_{b},\Upsilon,h_{b}$ and $\chi_{\rm{b0},\rm{b1},\rm{b2}}$) as a function
of $\tau_{2}$ where the covariance matrices are analysed over the time
interval $[0,\tau_{2}/a_{\tau}]$. Smeared operators at both the source and
sink were used. Figure 5 shows a convergence, as $\tau_{2}$ increases, to a
mass value compatible with the pseudoscalar, $\eta_{b}$, independent of the
channel, as expected from an analysis following [16, 17]. Further details of
this work are discussed in [13].

Figure 5: The value of the energy shift, $\Delta_{\text{sing}}$, (i.e. the
predicted mass) which gives the most singular covariance matrix for a variety
of bottomonium channels as a function of $1/\tau_{2}$. The covariance matrices
are defined over the time interval $[0,\tau_{2}/a_{\tau}]$ and therefore the
best results are obtained as $\tau_{2}\rightarrow\infty$. Experimental mass
values are also shown as horizontal lines [9]. This indicates the method
correctly predicts the $\eta_{b}$ (i.e. the pseudoscalar) mass as $\tau_{2}$
increases.

##  4 Interquark potentials in Bottomonium

For temperatures below $T_{pc}$, the interquark potential is known to be
confining, i.e. with a non-zero string tension, whereas above $T_{pc}$ it is
expected to flatten allowing unbound quark states. Lattice studies of the
interquark potential have been obtained from Wilson loops, which correspond to
infinitely heavy quarks [19, 18], the NRQCD bottomonium system [20], and from
charmonium using the HAL QCD method [21, 22, 23].

In this work, we study the interquark potential in the bottomonium system also
using the HAL QCD approach, see [24]. The bottom quarks were simulated using
the NRQCD approximation and our Generation 2 ensembles were used, see Table 1.

The HAL QCD method [25] uses Bethe Salpeter wavefunctions, $\psi(t,\vec{r})$,
obtained from temporal correlators of non-local heavy quark–antiquark meson
operators, $\overline{Q}(\tau,\vec{x})\Gamma Q(\tau,\vec{x}+\vec{r})$. It
represents the mesonic system with a Schrödinger equation,

$\left[\frac{p^{2}}{2\mu}+V(\vec{r})\right]\psi(\tau,\vec{r})=E\psi(\tau,\vec{r}),$
(4)

where $\mu$ is the reduced quark mass in the centre of mass frame. The
residual, non-physical $\tau-$dependency has to be carefully handled and this
is discussed in [24].

Figure 6 shows the preliminary results for the interquark bottomonium
potential for a variety of temperatures. In each pane, the same time window
was considered, thus nullifying any systematic effects from this fitting
artefact. These plots show evidence of the expected flattening of the
potential as the temperature increases.

Figure 6: The interquark potential in (NRQCD) bottomonium via the HAL QCD
procedure. In each pane, all the temperatures used the same time window:
$[13,14]$ (left) and $[17,18]$ (right). This is to isolate thermal effects
from $\tau$ systematics. The expected flattening of the potential with
temperature can be seen.

##  5 Conclusions

Recent results from Fastsum Collaboration’s thermal spectrum research [4, 10,
11, 13] and interquark potentials [24] have been presented.

##  6 Acknowledgements

This work is supported by STFC grant ST/T000813/1. SK is supported by the
National Research Foundation of Korea under grant NRF-2021R1A2C1092701 and
Grant NRF-2021K1A3A1A16096820, funded by the Korean government (MEST). BP has
been supported by a Swansea University Research Excellence Scholarship
(SURES). This work used the DiRAC Extreme Scaling service at the University of
Edinburgh, operated by the Edinburgh Parallel Computing Centre and the DiRAC
Data Intensive service operated by the University of Leicester IT Services on
behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was
funded by BEIS capital funding via STFC capital grants ST/R00238X/1,
ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grants ST/R001006/1
and ST/R001014/1. DiRAC is part of the UK National e-Infrastructure. This work
was performed using PRACE resources at Cineca (Italy), CEA (France) and
Stuttgart (Germany) via grants 2015133079, 2018194714, 2019214714 and
2020214714. We acknowledge the support of the Swansea Academy for Advanced
Computing, the Supercomputing Wales project, which is part-funded by the
European Regional Development Fund (ERDF) via Welsh Government, and the
University of Southern Denmark and ICHEC, Ireland for use of computing
facilities. We are grateful to the Hadron Spectrum Collaboration for the use
of their zero temperature ensemble in our Generation 2 work.

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# Compactness in Lorentz sequence spaces

Paweł Sawicki  Doctoral School of Quantum Information Theory, University of
Gdańsk, 80-308 Gdańsk, Poland<EMAIL_ADDRESS>
###### Abstract.

In this paper we are going to discuss compactness in Lorentz sequence spaces.
Firstly, it will be shown how to define such a space, check whether a sequence
belongs to it and calculate its norm. Equipped with this knowledge, we will
proceed to propose usable compactness criteria for Lorentz sequence spaces,
employing the concept of seminorms.

###### Key words and phrases:

Lorentz sequence spaces, compactness criteria, equinormed sets

###### 2020 Mathematics Subject Classification:

46B50, 46E30

##  1\. Introduction

Lorentz spaces were defined in year 1950 by George Gunter Lorentz in the paper
[1] as functional spaces consisting of measurable functions on a bounded
interval of $\mathbb{R}$. This definition can be naturally generalised to all
possible spaces with measure, as in [2]. This source expands significantly on
the known uses of Lorentz spaces and possibilities of further research.

As a special case, the space with measure can be chosen as $\mathbb{N}$ with
the counting measure, thus defining a $l_{p,q}$ space. It is discussed in the
article [3]. A somewhat more general version, with a weight sequence $w$, is
given by [4]. Thus defined _Lorentz sequence spaces_ will be of main interest
in our further considerations.

They can also be defined in a different but equivalent manner, through
permutations, as in [5]. It is mentioned there that the concept of
_nonincreasing rearrangement_ can be used to conduct the definition more
smoothly. In the later sections here, it will be described precisely by the
means of injective maps, serving the purpose of transforming a fully abstract
definition into a constructive one.

To the best of our knowledge, there are no known compactness criteria in
Lorentz spaces. In this paper, we are stating (and proving the correctness of)
one for Lorentz sequence spaces.

Let us give a short reminder about compactness: a set in a metric space is
called _compact_ if every its open cover has a finite subcover. A set is
called _relatively compact_ when its closure is compact. A set is called
_precompact_ if, for every $\varepsilon>0$, it has a finite cover by sets of
radius at most $\varepsilon$. These two notions are equivalent when applied to
subsets of a complete space. In metric spaces, a set is compact exactly when
it is relatively compact and complete.

Being able to determine whether a set is (relatively) compact can be useful in
a number of ways. For example, it enables methods efficient in solving
nonlinear equations (Schauder fixed point theorem, Leray-Schauder degree), as
well as investigation of compact operators, notable for their spectral
properties.

##  2\. Basic definitions

######  Definition 2.1 (Space $l_{p}$).

For $a=(a_{1},a_{2},a_{3}...)\in\mathbb{R}^{\mathbb{N}}$ and $p\in[1,\infty)$
let $||a||_{p}=(\sum_{i=1}^{\infty}|a_{i}|^{p})^{\frac{1}{p}}$. $||a||_{p}$ is
known as a $p$-norm of a sequence. The space of all sequences with finite
$p$-norm is known as $l_{p}$. It is a Banach space.

######  Definition 2.2 (Permutations of $\mathbb{N}$).

Let us denote the set of all permutations of $\mathbb{N}$ as
$\Pi\subset\mathbb{N}^{\mathbb{N}}$. Thus, for any
$\sigma=(\sigma_{1},\sigma_{2},\sigma_{3},...)\in\mathbb{N}^{\mathbb{N}}$,
$\sigma\in\Pi$ if and only if the mapping defined by $\sigma$ (index to value)
is bijective. Additionally, we can define a set $\Psi$ of all injective maps,
so that $\Pi\subset\Psi\subset\mathbb{N}^{\mathbb{N}}$, and $\sigma\in\Psi$ if
and only if the mapping defined by $\sigma$ is injective (one-to-one).

######  Definition 2.3 (Weight sequence).

A sequence $w$ of positive real numbers will be called a _weight sequence_
when it satisfies $w_{1}=1\geq w_{2}\geq w_{3}...\xrightarrow[\infty]{}0$ and
$\sum_{i=1}^{\infty}w_{i}=+\infty$.

######  Definition 2.4 (Lorentz space).

For any $a=(a_{1},a_{2},a_{3},...)\in\mathbb{R}^{\mathbb{N}}$,
$p\in[1,\infty)$, weight sequence $w$ and permutation $\sigma$, let us define
the sequence $L(a,p,w,\sigma)$: $L_{i}=a_{\sigma_{i}}\cdot\sqrt[p]{w}_{i}$,
$i\in\mathbb{N}$. The Lorentz norm $||\cdot||_{p,w}$ is given as

$||a||_{p,w}=\sup_{\sigma\in\Pi}:||L(a,p,w,\sigma)||_{p}.$

All sequences with finite Lorentz $p,w$-norm form a Lorentz space $L_{p,w}$.

######  Theorem 2.1.

For any $a$, $p$, $w$ defined as above,

$||a||_{p,w}=\sup_{\sigma\in\Psi}:||L(a,p,w,\sigma)||_{p}.$

###### Proof.

It is obvious that
$\sup_{\sigma\in\Pi}:||L(a,p,w,\sigma)||_{p}\leq\sup_{\sigma\in\Psi}:||L(a,p,w,\sigma)||_{p}$.
We can concentrate on proving the reverse inequality, which can be rewritten
as $\sup_{\sigma\in\Psi}:\sum_{i=1}^{\infty}|a_{\sigma_{i}}|^{p}\cdot
w_{i}\leq\sup_{\sigma\in\Pi}:\sum_{i=1}^{\infty}|a_{\sigma_{i}}|^{p}\cdot
w_{i}$. By definition, the left-hand side equals
$\sup_{\sigma\in\Psi}:\lim_{n\to\infty}\sum_{i=1}^{n}|a_{\sigma_{i}}|^{p}\cdot
w_{i}$. However, for any $\psi\in\Psi$ and $n\in\mathbb{N}$, there exists a
permutation $\psi_{n}\in\Pi$ such that $\psi_{i}=\psi_{n,i}$ for any $i\leq
n$. Thus,

$\forall\varepsilon>0\;\exists\psi\in\Psi,m\in\mathbb{N},\psi_{m}\in\Pi:\;\sup_{\sigma\in\Psi}:\lim_{n\to\infty}\sum_{i=1}^{n}|a_{\sigma_{i}}|^{p}\cdot
w_{i}\leq$
$\leq\varepsilon+\lim_{n\to\infty}\sum_{i=1}^{n}|a_{\psi_{i}}|^{p}\cdot
w_{i}\leq 2\varepsilon+\sum_{i=1}^{m}|a_{\psi_{i}}|^{p}\cdot w_{i}\leq$ $\leq
2\varepsilon+\lim_{n\to\infty}\sum_{i=1}^{n}|a_{\psi_{m,i}}|^{p}\cdot
w_{i}\leq
2\varepsilon+\sup_{\sigma\in\Pi}:\sum_{i=1}^{\infty}|a_{\sigma_{i}}|^{p}\cdot
w_{i}.$

Hence indeed
$\sup_{\sigma\in\Psi}:\sum_{i=1}^{\infty}|a_{\sigma_{i}}|^{p}\cdot
w_{i}\leq\sup_{\sigma\in\Pi}:\sum_{i=1}^{\infty}|a_{\sigma_{i}}|^{p}\cdot
w_{i}$ and this completes the proof. For divergent series ($a\notin L_{p,w}$),
the reasoning is fully analogous. ∎

######  Theorem 2.2 (Lorentz norm supremum is achieved).

For any values of $a$, $p$, $w$ such that $a\in L_{p,w}$, there exists
$\sigma\in\Psi$ such that $||a||_{p,w}=||L(a,p,w,\sigma)||_{p}$.

###### Proof.

Initial observation: we can assume without loss of generality that the
sequence $a$ has infinitely many nonzero entries, otherwise $\sigma$
permutates these nonzero entries and the problem is trivial.

First part: we will show that when there exists $\sigma\in\Psi$ for which
$|a_{\sigma}|$ is nonincreasing and covers all nonzero elements of the
sequence, for this $\sigma$ the function $L$ attains its maximum. For this
purpose, let us choose any other injective map $\psi$. We need to show that

$||L(a,p,w,\sigma)||_{p}\geq||L(a,p,w,\psi)||_{p},$

or equivalently

$\sum_{i=1}^{\infty}|a_{\sigma_{i}}|^{p}\cdot
w_{i}\geq\sum_{i=1}^{\infty}|a_{\psi_{i}}|^{p}\cdot w_{i}.$

For clarity of notation, we can pick a (nonincreasing positive) sequence $c$
by $c_{i}=|a_{\sigma_{i}}|^{p}$, with a corresponding map
$\xi:\mathbb{N}\to\mathbb{N}$ such that $c_{\xi_{i}}=|a_{\psi_{i}}|^{p}$. Now
we wish to prove that $\sum_{i=1}^{\infty}c_{i}\cdot
w_{i}\geq\sum_{i=1}^{\infty}c_{\xi_{i}}\cdot w_{i}$. Because the sum of a
series is defined as the limit of its partial sums, it is enough to show for
any $n\in\mathbb{N}$ that

$\sum_{i=1}^{n}c_{i}\cdot w_{i}\geq\sum_{i=1}^{n}c_{\xi_{i}}\cdot w_{i}.$

We can assume without loss of generality that $\xi$ maps $\\{1,2,...,n\\}$
onto $\\{1,2,...,n\\}$, as otherwise the right-hand side of our inequality
could be trivially increased. Now the simplest proof is by the principle of
mathematical induction. The first step (for $n=1$) is obvious, so let us
assume that the lemma is true for $n-1$. Moreover, let $\xi$ map $l$ to $n$
and $n$ to $k$. We can define a new map $\xi^{\prime}$ which differs from
$\xi$ only by mapping $l$ to $k$ and $n$ to $n$. By the inductive assumption,
$\sum_{i=1}^{n}c_{i}\cdot w_{i}\geq\sum_{i=1}^{n}c_{\xi^{\prime}_{i}}\cdot
w_{i}$. There also occurs

$\sum_{i=1}^{n}c_{\xi^{\prime}_{i}}\cdot w_{i}-\sum_{i=1}^{n}c_{\xi_{i}}\cdot
w_{i}=c_{k}w_{l}+c_{n}w_{n}-c_{k}w_{n}-c_{n}w_{l}=(c_{k}-c_{n})(w_{l}-w_{n})\geq
0.$

Therefore $\sum_{i=1}^{n}c_{i}\cdot w_{i}\geq\sum_{i=1}^{n}c_{\xi_{i}}\cdot
w_{i}$. This completes the first part of the proof.

Second part: we will show that when there is no $\sigma\in\Psi$ for which
$|a_{\sigma}|$ would be nonincreasing and would contain all nonzero elements
of the sequence $a$, such a sequence $a$ does not belong to any $L_{p,w}$. For
this purpose, let us define the sequence $\sigma$ by the following conditions:
$\mathbb{N}_{i}=\mathbb{N}\backslash\\{\sigma_{1},\sigma_{2},...,\sigma_{i-1}\\}$,
$\sigma_{i}\in\mathbb{N}_{i}$, $\forall
j\in\mathbb{N}_{i}:\;|a_{\sigma_{i}}|\geq|a_{j}|\land(|a_{\sigma_{i}}|=|a_{j}|\implies\sigma_{i}\leq
j)$. Now try to assume that these conditions do not define a unique
$\sigma_{i}$ (and does define unique $\sigma_{h}$ for all natural numbers
$h<i$). Then, by contraposition,

$\forall j\in\mathbb{N}_{i}\;\exists
m\in\mathbb{N}_{i}:\;|a_{m}|\geq|a_{j}|\land m>j.$

Therefore it would be possible to construct an infinite sequence in
$\mathbb{N}_{i}$ $j_{1},j_{2},j_{3}...$ such that $j_{k}$ would be strictly
increasing and $|a_{j_{k}}|$ would be increasing. However, by the initial
observation, it is possible to choose such $j_{1}\in\mathbb{N}_{i}$ that
$|a_{j_{1}}|>0$. Thus we could define $\psi\in\Psi=(j_{1},j_{2},j_{3},...)$,
and for any weight sequence $w$ and $p\in[1,\infty)$ the series
$|a_{\psi_{i}}|^{p}\cdot w_{i}$ would be divergent (because the left-hand
factors are greater or equal $|a_{j_{1}}|^{p}>0$ and the series of the right-
hand factors is divergent), showing that in such a case $a\notin L_{p,w}$.
Hence we have proven that, for $a\in L_{p,w}$, our definition of the injective
map $\sigma\in\Psi$ is correct and renders a nonincreasing $|a_{\sigma}|$,
realising the Lorentz norm supremum. ∎

######  Remark 2.1 (Behaviour of $|a_{\sigma}|$).

When $\lim_{n\to\infty}|a_{\sigma_{n}}|^{p}>0$, $a\notin L_{p,w}$. When the
limit equals $0$ and the series is convergent, $a\in L_{p,w}$. When the limit
equals $0$ but the series is divergent, $a$ might or might not belong to
$L_{p,w}$.

######  Case 2.1.

For $p=1$, $|a_{\sigma_{i}}|=w_{i}=\frac{1}{i}$ there occurs $a\in L_{p,w}$,
$||a||_{p,w}=\frac{\pi^{2}}{6}$.

######  Case 2.2.

For $p=1$, $|a_{\sigma_{i}}|=w_{i}=\frac{1}{\sqrt{i}}$ there occurs $a\notin
L_{p,w}$.

##  3\. Compactness

For determining compactness criteria in Lorentz sequence spaces, we will use
the seminorm approach, as laid out in [6].

######  Definition 3.1 (Seminorm).

For any real vector space $V$, a _seminorm_ is a function
$s:V\rightarrow[0,\infty)$ such that $\forall x\in\mathbb{R},v\in V\;s(x\cdot
v)=|x|\cdot s(v)$, $\forall v,w\in V\;s(v)+s(w)\geq s(v+w)$. A seminorm for
which $s(v)=0\iff v=0$ is called a _norm_ , as used in the previous section.

The following two definitions are repeated from the aforementioned article.

######  Definition 3.2 (Regular family of seminorms).

When on a normed space $E,||\cdot||$ there is a family of seminorms
$\\{||\cdot||_{i}\\}_{i\in I}$, it is known as _regular_ if it satisfies three
conditions:

1) $\forall x\in E\;||x||=\sup_{i\in I}\;||x||_{i}$;

2) the family $\\{||\cdot||_{i}\\}_{i\in I}$ forms a directed set, i.e.

$\forall i,j\in I\;\exists k\in I\;\forall x\in
E\;||x||_{k}\geq||x||_{i}\land||x||_{k}\geq||x||_{j};$

3) every bounded sequence in $E$ has a subsequence which is Cauchy with
respect to all seminorms in the family.

######  Definition 3.3 (Equinormed set).

A set $A\subset E$ is _equinormed_ (with respect to a given family of
seminorms indexed by set $I$) if

$\forall\varepsilon>0\;\exists i\in I\;\forall x\in
A\;||x||\leq||x||_{i}+\varepsilon.$

######  Remark 3.1.

For any positive $p$, a bounded set $A\in E$ is equinormed if and only if

$\forall\varepsilon>0\;\exists i\in I\;\forall x\in
A\;||x||^{p}\leq||x||_{i}^{p}+\varepsilon.$

This follows from the fact that the power function $x\mapsto x^{p}$ is
uniformly continuous on bounded sets.

The following theorem is given in [6] as Theorem 3.19.

######  Theorem 3.1.

For any regular family of seminorms, a non-empty set $A\subset E$ is
precompact when and only when $A-A$ is equinormed and $A$ is bounded.

Now we need to define a regular family of seminorms for a Lorentz space
$L_{p,w}$. This family will be indexed by $i\in\mathbb{N}$ and defined as
$||a||_{p,w,i}=||r_{i}(a)||_{p,w}$, where $r_{i}:L_{p,w}\to L_{p,w}$ is a
linear projection given by

$r_{i}(a)=(a_{1},a_{2},...,a_{i},0,0,...).$

######  Remark 3.2.

$r_{i}(L_{p,w})\subset L_{p,w}$ is a subset closed under addition and scalar
multiplication, and there occurs $a\in
r_{i}(L_{p,w})\implies||a||_{p,w}=||a||_{p,w,i}$. Herefrom it follows
instantly that $||\cdot||_{p,w,i}$ is a seminorm. Its value is achieved on
some permutation of the set $\\{1,...,i\\}$.

At this point it is necessary to show that this family is regular.

######  Lemma 3.2 (Second condition of regularity).

For any appropriate $a$, $p$, $w$, $n$ there occurs
$||a||_{p,w,n}\leq||a||_{p,w,n+1}$ and hence the family
$\\{||\cdot||_{p,w,i}\\}_{i\in\mathbb{N}}$ fulfils the second condition of
regularity.

###### Proof.

Let $\sigma$ be a permutation which achieves the supremum for $r_{n}(a)$ and
$\psi$ for $r_{n+1}(a)$. $\sigma$ and $\psi$ can be chosen so that
$\sigma_{n+1}=n+1$, $\sigma(\\{1,...,n\\})=\\{1,...,n\\}$ and
$\psi(\\{1,...,n+1\\})=\\{1,...,n+1\\}$. Then it suffices to show that

$\sum_{i=1}^{\infty}|r_{n}(a)_{\sigma_{i}}|^{p}\cdot
w_{i}\leq\sum_{i=1}^{\infty}|r_{n+1}(a)_{\psi_{i}}|^{p}\cdot w_{i}.$

And indeed,

$\sum_{i=1}^{\infty}|r_{n}(a)_{\sigma_{i}}|^{p}\cdot
w_{i}=\sum_{i=1}^{n}|a_{\sigma_{i}}|^{p}\cdot
w_{i}\leq\sum_{i=1}^{n+1}|a_{\sigma_{i}}|^{p}\cdot
w_{i}\leq\sum_{i=1}^{n+1}|a_{\psi_{i}}|^{p}\cdot
w_{i}=\sum_{i=1}^{\infty}|r_{n+1}(a)_{\psi_{i}}|^{p}\cdot w_{i}.$

∎

######  Lemma 3.3 (First condition of regularity).

For any appropriate $a$, $p$, $w$, $n$ there occurs
$||a||_{p,w}=\lim_{n\to\infty}||a||_{p,w,n}$.

###### Proof.

Let $\sigma$ be an injective map which achieves the supremum for $a$ and
$\sigma^{(n)}$ for $r_{n}(a)$. We need to prove that

$\lim_{n\to\infty}\sum_{i=1}^{n}|a_{\sigma_{i}^{(n)}}|^{p}\cdot
w_{i}=\sum_{i=1}^{\infty}|a_{\sigma_{i}}|^{p}\cdot w_{i}.$

Let $\chi$ denote the characteristic function (equal $1$ if the expression in
brackets is true and $0$ otherwise). Obviously,

$\lim_{n\to\infty}\sum_{i=1}^{n}|a_{\sigma_{i}^{(n)}}|^{p}\cdot
w_{i}\geq\lim_{n\to\infty}\sum_{i=1}^{n}|a_{\sigma_{i}}|^{p}\cdot
w_{i}\cdot\chi(\sigma_{i}\leq n)=\sum_{i=1}^{\infty}|a_{\sigma_{i}}|^{p}\cdot
w_{i}.$

On the other hand,

$\forall n\in\mathbb{N}\;\sum_{i=1}^{n}|a_{\sigma_{i}^{(n)}}|^{p}\cdot
w_{i}\leq\sum_{i=1}^{\infty}|a_{\sigma_{i}^{(n)}}|^{p}\cdot
w_{i}\leq\sum_{i=1}^{\infty}|a_{\sigma_{i}}|^{p}\cdot w_{i}.$

Since inequality is preserved in the limit, this completes the proof. ∎

######  Lemma 3.4 (Third condition of regularity).

Every bounded sequence in $L_{p,w}$ has a subsequence that is Cauchy for all
seminorms.

###### Proof.

As a minor lemma, let us prove the thesis for a particular seminorm
$||\cdot||_{p,w,n}$, for a fixed $n\in\mathbb{N}$. We consider a sequence
$(a_{1},a_{2},a_{3},...)$ where each element
$a_{i}=(a_{i,1},a_{i,2},a_{i,3},...)\in L_{p,w}$. Let $\sigma^{(i)}$ denote
the permutation with which the supremum for $r_{n}(a_{i})$ is attained. As we
can see, $\sigma^{(i)}(\\{1,...,n\\})=\\{1,...,n\\}$ and we may associate
$a_{i}$ with a permutation of the finite set $\\{1,...,n\\}$. Since there are
finitely many (namely $n!$) such permutations, one of them must be chosen for
infinitely many elements of the sequence $(a_{i})$. This subsequence will be
denoted as $(a_{n_{1}},a_{n_{2}},a_{n_{3}},...)$ and we can write a universal
formula $||a_{n_{j}}||_{p,w,n}=(\sum_{i=1}^{n}|a_{n_{j},\sigma_{i}}|^{p}\cdot
w_{i})^{\frac{1}{p}}$. Now this can be interpreted as a norm in
$\mathbb{R}^{n}$, and because all norms in $\mathbb{R}^{n}$ are equivalent, it
has the desired property that a bounded sequence has a Cauchy subsequence.
Being bounded in $||\cdot||_{p,w}$ implies being bounded in all
$||\cdot||_{p,w,n}$, so the proof of the lemma is finished.

Now it is possible to use the diagonal argument to extend the minor lemma to
all seminorms. From any bounded sequence $(a_{1},a_{2},a_{3},...)$ in
$L_{p,w}$ we can choose a subsequence $(a_{b_{1}},a_{b_{2}},a_{b_{3}},...)$
which is Cauchy with respect to $||\cdot||_{p,w,1}$, from it a subsubsequence
$(a_{b_{c_{1}}},a_{b_{c_{2}}},a_{b_{c_{3}}},...)$ which is Cauchy also with
respect to $||\cdot||_{p,w,2}$, and proceed so _ad infinitum_. The cross-
subsequence $(a_{1},a_{b_{2}},a_{b_{c_{3}}},...)$ fulfils the condition. ∎

######  Remark 3.3 (Equinormedness).

With our family of seminorms, Definition 3.3 can be trivially interpreted as
stating that a set $A\in L_{p,w}$ is equinormed when
$\lim_{i\to\infty}||a||_{p,w,i}=||a||_{p,w}$ uniformly on $A$. And
equivalently, by Remark 3.1, when
$\lim_{i\to\infty}||a||_{p,w,i}^{p}=||a||_{p,w}^{p}$ uniformly on $A$.

######  Corollary 3.1.

A set $A\subset L_{p,w}$ is precompact if and only if it is bounded in
$||\cdot||_{p,w}$ and the set $A-A$ is equinormed with respect to
$||\cdot||_{p,w,i}$. This is a direct consequence of Theorem 3.1.

##  4\. Main result

Now we have the necessary tools to begin proving the main theorem of this
article, namely that a set $A\subset L_{p,w}$ is precompact if and only if it
is bounded in $||\cdot||_{p,w}$ and equinormed with respect to
$||\cdot||_{p,w,i}$. Therefore, the core of the proof is to show that the
equinormedness of $A$ implies the equinormedness of $A-A$ (the reverse is much
easier). This will be done through a series of lemmas, showing the uniform
properties of $A$ and acquiring other interesting observations in the
meantime.

######  Lemma 4.1 (Sequences $a\in A$ have a bounded number of element
repetitions).

If $A$ is bounded, then for any $d>0$ there is such a number $\lambda(d)$ that
no sequence in $A$ has more than $\lambda(d)$ elements greater or equal $d$ in
their absolute values, or in other words

$\forall d>0\;\exists\lambda(d)\in\mathbb{N}\;\forall a\in
A\;|a_{\sigma_{\lambda(d)+1}}|<d,$

where, by our previous notation, $\sigma$ is a nonincreasing natural injective
map.

###### Proof.

Since $A$ is bounded, let $M$ be its upper bound in $||\cdot||_{p,w}$. Trying
to assume that for some $d$ there is no such upper bound $\lambda(d)$ on the
amount of repetitions, we get

$\forall n\in\mathbb{N}\;\exists a\in A\;M\geq||a||_{p,w}\geq
d^{p}\cdot\sum_{i=1}^{n}w_{i}.$

Hence for any $n$ there would occur $\frac{M}{d^{p}}\geq\sum_{i=1}^{n}w_{i}$,
and yet we know that the series of the weight sequence $w$ is divergent. This
proves the lemma by contradiction. ∎

For clarity of notation, with a pre-determined weight sequence $w$ and power
$p$, let us denote $||a||_{p,w,i}^{p}$ as $S_{i}(a)$. When $A$ is equinormed
with respect to $||\cdot||_{p,w,i}$, it means that

$\forall\varepsilon>0\;\exists n(\varepsilon)\in\mathbb{N}\;\forall a\in
A\;||a||_{p,w}^{p}-S_{n(\varepsilon)}(a)<\varepsilon.$

Hereforth, by convention, we will denote by $n(\varepsilon)$ the lowest
natural number having said property.

In the following lemma, we will split our hypothesis into two possible cases
and prove for one of them that $A-A$ is equinormed, so that we will be able to
use the other case as an assumption to continue the proof.

######  Lemma 4.2.

If $\exists m\in\mathbb{N}\;\forall\varepsilon>0\;n(\varepsilon)\leq m$, then
$A-A$ is equinormed.

###### Proof.

By the definition of $n(\varepsilon)$, this condition means that

$\exists n\in\mathbb{N}\;\forall a\in A\;||a||_{p,w}^{p}=S_{n}(a).$

This occurs if and only if $a_{i}=0$ for any $i>n$, because any nonzero values
would increase the norm without impacting the seminorm. However, in this case
the set $A-A$ holds the same property, so it is equinormed. ∎

Starting from now on, we will be able to use the assumption that $\forall
m\in\mathbb{N}\;\exists\varepsilon>0\;n(\varepsilon)>m$ (the index of
seminorms increases as $\varepsilon$ approaches $0$).

######  Lemma 4.3 (Sequences $a\in A$ approach $0$ uniformly).

If $A$ is bounded and equinormed, then

$\forall d>0\;\exists N\in\mathbb{N}\;\forall a\in A\;\forall n\geq
N\;|a_{n}|<d.$

###### Proof.

As usual, we will conduct an indirect proof. Let us assume that

$\exists d>0\;\forall N\in\mathbb{N}\;\exists a\in A\;\exists n\geq
N\;|a_{n}|\geq d$

in order to show that $A$ would not be equinormed. For this purpose, we want
to estimate $||a||_{p,w}^{p}-S_{k}(a)$ for all values of $k\in\mathbb{N}$ and
show that it cannot be arbitralily small. By Lemmas 4.1, 4.2, on assumption of
the boundedness and equinormedness of $A$, it is possible to check only the
values $k>\lambda(\frac{d}{2})$ whenever the expected difference
$\varepsilon>0$ is small enough that $n(\varepsilon)>\lambda(\frac{d}{2})$. By
the indirect assumption, we could choose a sequence $a\in A$ which has at
least one element $|a_{n}|\geq d$ with $n\geq k$. Denoting by
$k^{\prime}<\lambda(d)$ the number of elements greater or equal $d$ among
$\\{|a_{1}|,|a_{2}|,...,|a_{k}|\\}$, and using the previous notation of
$\sigma^{(k)}$ for a permutation which achieves the supremum for $r_{k}(a)$,
there occurs

$||a||_{p,w}^{p}\geq|a_{\sigma_{1}^{(k)}}|^{p}w_{1}+|a_{\sigma_{2}^{(k)}}|^{p}w_{2}+...+|a_{\sigma_{k^{\prime}}^{(k)}}|^{p}w_{k^{\prime}}+d^{p}w_{k^{\prime}+1}+|a_{\sigma_{k^{\prime}+1}^{(k)}}|^{p}w_{k^{\prime}+2}+...+|a_{\sigma_{k-1}^{(k)}}|^{p}w_{k}.$

Thus, it is possible to lay out the estimation

$||a||_{p,w}^{p}-S_{k}(a)\geq(d^{p}-|a_{\sigma_{k^{\prime}+1}^{(k)}}|^{p})\cdot
w_{k^{\prime}+1}+(|a_{\sigma_{k^{\prime}+1}^{(k)}}|^{p}-|a_{\sigma_{k^{\prime}+2}^{(k)}}|^{p})\cdot
w_{k^{\prime}+2}+...$
$+(|a_{\sigma_{\lambda(\frac{d}{2})}^{(k)}}|^{p}-|a_{\sigma_{\lambda(\frac{d}{2})+1}^{(k)}}|^{p})\cdot
w_{\lambda(\frac{d}{2})+1}\geq(d-\frac{d}{2^{p}})\cdot
w_{\lambda(\frac{d}{2})+1}.$

It suffices to choose such $\varepsilon<(d-\frac{d}{2^{p}})\cdot
w_{\lambda(\frac{d}{2})+1}$ that $n(\varepsilon)>\lambda(\frac{d}{2})$ to see
that for any appropriate $k$ and some $a\in A$ there is
$||a||_{p,w}^{p}-S_{k}(a)>\varepsilon$. For this reason, $A$ cannot be
equinormed after all, and it proves the lemma by contradiction. ∎

######  Remark 4.1.

Lemma 4.3 is a rather interesting result on its own. To see it, let us define
a function $\gamma:\mathbb{R}\rightarrow\mathbb{N}$ where $\gamma(d)=N$ (the
minimum value satisfying the condition from Lemma 4.3). As it turns out, its
naturally defined reverse sequence
$\gamma^{-1}(n)=\sup\\{x\in\mathbb{R}:\gamma(x)\geq n\\}$ is a majorizing
sequence for all $a\in A$. However, there is no guarantee that $\gamma^{-1}\in
L_{p,w}$.

In order to proceed with the subsequent lemmas, it will be reasonable to
introduce, for given values of $p$ and $w$, a few more convenient symbols
similar to $S_{i}(a)$ for seminorms. For a given $i$, the sequence $a\in
A\subset L_{p,w}$ can be naturally divided into its head $a_{(1...i)}$ and its
tail $a_{(i+1...\infty)}=(a_{n+i})_{n\in\mathbb{N}}$. Now we can define the
following concepts:

######  Definition 4.1 (Tail complement of the seminorm).

$\tilde{S}_{i}(a)=||a_{(i+1...\infty)}||_{p,w_{(i+1...\infty)}}^{p}$

This formula can be naturally described as _tail complement of the seminorm_
because $S_{i}(a)+\tilde{S}_{i}(a)$ is a certain reconfiguration of the
sequence $a$ multiplied piecewise by the weight sequence $w$.

######  Definition 4.2 (Inverse function of $\sigma$).

For $\sigma\in\Psi$, we will use $\sigma^{-1}$ in its natural sense, as the
inverse function of the map $\sigma$. By slight abuse of notation, for any
indices $i\in\mathbb{N}\setminus\sigma(\mathbb{N})$ we assign
$w_{\sigma_{i}^{-1}}=0$. Thereupon, any sum over $\sigma^{-1}(\mathbb{N})$ can
be written without additional constraints, as any element outside the domain
would be equal $0$ and thus would not influence the sum.

The definitions given below, just like Definition 4.1, are stated for a fixed
$a\in L_{p,w}$. Whenever they refer to an injective map $\sigma$, it means the
optimal map existing for $a$ by Theorem 2.2, and ”norm formula” means the
formula defining the $p$-th power of the Lorentz norm.

######  Definition 4.3 (Head of the norm).

$H_{i}(a)=\sum_{j=1}^{i}|a_{j}|^{p}\cdot w_{\sigma_{j}^{-1}}$

This is the part of the norm formula for sequence $a$ counted over its first
$i$ elements.

######  Definition 4.4 (Tail of the norm).

$\tilde{H}_{i}(a)=\sum_{j=i+1}^{\infty}|a_{j}|^{p}\cdot w_{\sigma_{j}^{-1}}$

This is the other part of the norm formula. There occurs
$H_{i}(a)+\tilde{H}_{i}(a)=||a||_{p,w}^{p}$.

######  Definition 4.5 (Head by weight of the norm).

$W_{i}(a)=\sum_{j=1}^{i}|a_{\sigma_{j}}|^{p}\cdot w_{j}$

This is the part of the norm formula for sequence $a$ counted over its first
$i$ elements, ordered by weight.

######  Definition 4.6 (Tail by weight of the norm).

$\tilde{W}_{i}(a)=\sum_{j=i+1}^{\infty}|a_{\sigma_{j}}|^{p}\cdot w_{j}$

This is, once again, the other part of the norm formula. There occurs
$W_{i}(a)+\tilde{W}_{i}(a)=||a||_{p,w}^{p}$.

######  Definition 4.7 (Norm of the tail).

$T_{i}(a)=||a_{(i+1...\infty)}||_{p,w}^{p}$

This differs from the tail complement of the seminorm in that all values from
$w$ can be used again.

######  Lemma 4.4 (Relationships between the defined terms).

$S_{i}(a)+\tilde{S}_{i}(a)\leq||a||_{p,w}^{p}=H_{i}(a)+\tilde{H}_{i}(a)=W_{i}(a)+\tilde{W}_{i}(a)\leq
S_{i}(a)+T_{i}(a)$

###### Proof.

The first inequality follows directly from the fact that
$S_{i}(a)+\tilde{S}_{i}(a)$ cannot be greater than the $p$-th power of the
Lorentz norm, which is realised by $H_{i}(a)+\tilde{H}_{i}(a)$. In the second
inequality, analogously, there is $H_{i}(a)\leq S_{i}(a)$ by definition of the
seminorm, so we only need to show that $\tilde{H}_{i}(a)\leq T_{i}(a)$. For
this purpose, let us concentrate on any particular summand associated with
$a_{j}$ ($j>i$). In $\tilde{H}_{i}(a)$, it takes the form of $|a_{j}|^{p}\cdot
w_{\sigma_{j}^{-1}}$. In $T_{i}(a)$, it can be similarly described as
$|a_{j}|^{p}\cdot w_{\psi_{j}^{-1}}$, with the difference that while the map
$\sigma_{j}^{-1}$ is equal to the number of of all elements of $a$ greater
than $|a_{j}|$, the map $\psi_{j}^{-1}$ counts only over the elements that
have indices above $i$. Hence $\sigma_{j}^{-1}\geq\psi_{j}^{-1}$,
$w_{\sigma_{j}^{-1}}\leq w_{\psi_{j}^{-1}}$ and finally $\tilde{H}_{i}(a)\leq
T_{i}(a)$, completing the proof of the lemma. ∎

######  Remark 4.2.

Recombining our inequality yields $||a||_{p,w}^{p}-S_{i}(a)\leq T_{i}(a)$.
Therefore, when in any set $A$ the norms of the tails approach $0$ uniformly
for $i\rightarrow\infty$, said set is equinormed.

######  Lemma 4.5 (Uniform behaviour of norms of the tails).

When norms of the tails in a set $A\subset L_{p,w}$ approach $0$ uniformly for
$i\rightarrow\infty$, the same is true for $A-A$.

###### Proof.

We can use the norm properties of $||\cdot||_{p,w}$, applying the triangle
inequality to claim instantly that

$\forall\varepsilon>0\;\exists k\in\mathbb{N}\;\forall a,b\in
A\;||a_{(k+1...\infty)}||_{p,w}<\varepsilon,\;||b_{(k+1...\infty)}||_{p,w}<\varepsilon\implies$
$\implies||(a-b)_{(k+1...\infty)}||_{p,w}<2\cdot\varepsilon.$

Therefore

$\forall\varepsilon>0\;\exists k\in\mathbb{N}\;\forall a,b\in
A\;||a_{(k+1...\infty)}||_{p,w}^{p}<\varepsilon,\;||b_{(k+1...\infty)}||_{p,w}^{p}<\varepsilon\implies$
$\implies||(a-b)_{(k+1...\infty)}||_{p,w}<2\cdot\sqrt[p]{\varepsilon}\iff||(a-b)_{(k+1...\infty)}||_{p,w}^{p}<2^{p}\cdot\varepsilon.$

∎

These results give us a clear path towards proving the entire theorem. Now it
is enough to show that equinormedness of $A$ implies that norms of the tails
in $A$ approach $0$ uniformly for $i\rightarrow\infty$. There is an intuitive
and fairly trivial step of showing the same for tails by weight of the norms
in $A$, and making these two steps will be the content of our final two
lemmas.

######  Lemma 4.6 (Tails by weight of the norms approach $0$ uniformly).

When $A$ is bounded and equinormed,

$\forall\varepsilon>0\;\exists N\in\mathbb{N}\;\forall a\in A\;\forall i\geq
N\;\tilde{W}_{i}(a)=\sum_{j=i+1}^{\infty}|a_{\sigma_{j}}|^{p}\cdot
w_{j}<\varepsilon.$

(Reading this notation, it is important to remember that the map $\sigma$
depends on the choice of $a$.)

###### Proof.

Because $W_{i}(a)+\tilde{W}_{i}(a)=||a||_{p,w}^{p}$, and the $p$-th powers of
seminorms approach $||a||_{p,w}$ uniformly on $A$, it is enough to show that
$\forall a\in A\;\forall i\in\mathbb{N}\;S_{i}(a)\leq W_{i}(a)$. And we have
already proven this, given that $W_{i}(a)$ is constructed according to the
procedure described in Theorem 2.2. ∎

######  Lemma 4.7 (Norms of the tails approach $0$ uniformly).

When $A$ is bounded and equinormed,

$\forall\varepsilon>0\;\exists N\in\mathbb{N}\;\forall a\in A\;\forall i\geq
N\;T_{i}(a)=||a_{(i+1...\infty)}||_{p,w}^{p}<\varepsilon.$

###### Proof.

By the previous lemma, there exists such $k\in\mathbb{N}$ that $\forall a\in
A\;\tilde{W}_{k}(a)<\frac{\varepsilon}{2}$. Now let $b$ be the $i$-th tail of
some $a\in A$, that is $b=a_{(i+1...\infty)}$. It is easy to notice that also
$\tilde{W}_{k}(b)<\frac{\varepsilon}{2}$. Indeed,

$\tilde{W}_{k}(a)-\tilde{W}_{k}(b)=\sum_{j=k+1}^{\infty}w_{j}(|a_{\sigma_{j}}|^{p}-|b_{\psi_{j}}|^{p}),$

where $\psi$ is a map such that $|b_{\psi}|$ is nonincreasing (hence it is the
optimal injective map for the sequence $b$). Yet, $b$ is a subsequence of $a$,
so $b_{\psi}$ is a subsequence of the nonincreasing $|a_{\sigma}|$. Therefore,
for all natural $j$, $|a_{\sigma_{j}}|^{p}-|b_{\psi_{j}}|^{p}>0$ and every
summand in the expression is nonnegative.

At the same time, $T_{i}(a)=W_{k}(b)+\tilde{W}_{k}(b)$. Now we only need to
find such $i$ that $W_{k}(b)<\frac{\varepsilon}{2}$. Since
$b=a_{(i+1...\infty)}$, applying Lemma 4.3 shows that for any $d>0$ choosing
$N=\gamma(d^{\frac{1}{p}})$ returns

$\forall j\in\mathbb{N},\;i\geq N,\;b\in
A_{(i+1...\infty)}\;|b_{\psi_{j}}|^{p}<d.$

Because $W_{k}(b)=\sum_{j=1}^{k}w_{j}\cdot|b_{\psi_{j}}|^{p}$,

$N=\gamma((\frac{\varepsilon}{2\sum_{j=1}^{k}w_{j}})^{\frac{1}{p}})$

gives us the desired result. ∎

We have carried out all the preparations necessary to prove the main theorem
in a few short steps.

######  Theorem 4.8.

A set $A\subset L_{p,w}$ is precompact if and only if it is bounded and
equinormed.

###### Proof.

When $A$ is bounded and equinormed, by Lemma 4.7 norms of the tails in $A$
approach $0$ uniformly, so by Lemma 4.5 the same is true for the tails in
$A-A$. Hence, Remark 4.2 shows that $A-A$ is equinormed, and we can claim that
$A$ is precompact thanks to Theorem 3.1. On the other hand, it is well known
that a precompact set is bounded. The fact that it is equinormed, with respect
to a regular family of seminorms (in its normed space as per Definition 3.2),
is given in [6], as Proposition 3.13 states directly that every precompact
subset is equinormed. ∎

On a final note, we will provide an alternative formulation of our compactness
condition, which is ostensibly stronger but might be easier to use in
particular cases. The proof will reuse many techniques from the previous two
lemmas.

######  Theorem 4.9 (Tails of the norms approach $0$ uniformly).

When $A$ is bounded, it is equinormed if and only if

$\forall\varepsilon>0\;\exists N\in\mathbb{N}\;\forall a\in A\;\forall i\geq
N\;\tilde{H}_{i}(a)=\sum_{j=i+1}^{\infty}|a_{j}|^{p}\cdot
w_{\sigma_{j}^{-1}}<\varepsilon.$

###### Proof.

The implication from the stated fact to the equinormedness is rather simple.
Since $H_{i}(a)+\tilde{H}_{i}(a)=||a||_{p,w}^{p}$, the values of $H_{i}(a)$
approach $||a||_{p,w}^{p}$ uniformly on $A$, and as we have mentioned in the
proof of Lemma 4.4, there is $H_{i}(a)\leq S_{i}(a)$ by definition of the
seminorm, so (using Remark 3.1) the seminorms also exhibit the same behaviour
on $A$. We can concentrate on proving the implication in the opposite
direction.

Similarly, because $H_{i}(a)+\tilde{H}_{i}(a)=||a||_{p,w}^{p}$, and seminorms
approach $||a||_{p,w}^{p}$ uniformly on $A$, it is enough to show that

$\forall\varepsilon>0\;\forall k\in\mathbb{N}\;\exists N\in\mathbb{N}\;\forall
a\in A\;\forall i\geq N\;S_{k}(a)-H_{i}(a)<\varepsilon.$

Now, let us try to estimate the expression $S_{k}(a)-H_{i}(a)$, once again
breaking it down by elements. For any summand of $S_{k}(a)$ denoted as
$w_{l}\cdot|a_{\sigma_{l}^{(k)}}|^{p}$ ($1\leq l\leq k$), there are two
possibilities: its unique counterpart $w_{l}\cdot|a_{\sigma_{l}}|^{p}$ occurs
as a summand in $H_{i}(a)$ or it does not, the second type happening exactly
when $\sigma_{l}>i$. Let us recall that $|a_{\sigma_{l}^{(k)}}|$ can be
interpreted as the $l$-th greatest absolute value taken from the first $k$
elements of sequence $a$, while $|a_{\sigma_{l}}|$ is the $l$-th greatest
absolute value from the whole sequence, so obviously
$|a_{\sigma_{l}^{(k)}}|\leq|a_{\sigma_{l}}|$.

For this reason, the summand differences of the first type can be estimated
from above by
$w_{l}\cdot|a_{\sigma_{l}^{(k)}}|^{p}-w_{l}\cdot|a_{\sigma_{l}}|^{p}\leq 0$,
and the summands of the second type by
$w_{l}\cdot|a_{\sigma_{l}^{(k)}}|^{p}\leq w_{l}\cdot|a_{\sigma_{l}}|^{p}$.
This breakdown of the expression may ignore some summands from $H_{i}(a)$, but
they are nonnegative and, in subtraction, can also be estimated from above by
$0$. Finally, we are able to write the inequality

$S_{k}(a)-H_{i}(a)\leq\sum_{j=1}^{k}w_{j}\cdot|a_{\sigma_{j}}|^{p}\cdot\chi(\sigma_{j}>i),$

with $\chi$, like earlier, denoting the characteristic function (equal $1$ if
the expression in brackets is true and $0$ otherwise). Now it is enough to
apply Lemma 4.3, showing that for any $d>0$ it is possible to choose
$N=\gamma(d^{\frac{1}{p}})$ and guarantee that

$\forall j\in\mathbb{N},\;i\geq N,\;a\in
A\;|a_{\sigma_{j}}|^{p}\cdot\chi(\sigma_{j}>i)<d.$

This gives us $S_{k}(a)-H_{i}(a)<\sum_{j=1}^{k}w_{j}\cdot d$, yielding the
thesis of this theorem for

$N=\gamma((\frac{\varepsilon}{\sum_{j=1}^{k}w_{j}})^{\frac{1}{p}}).$

∎

######  Corollary 4.1.

A set $A\subset L_{p,w}$ is precompact if and only if it is bounded in
$||\cdot||_{p,w}$ and

$\forall\varepsilon>0\;\exists N\in\mathbb{N}\;\forall a\in A\;\forall i\geq
N\;\sum_{j=i+1}^{\infty}|a_{j}|^{p}\cdot w_{\sigma_{j}^{-1}}<\varepsilon.$

This is a direct consequence of Theorem 4.9.

Acknowledgements. The author would like to thank Professor Jacek Gulgowski for
his invaluable support through all the research and editorial process,
particularly concerning the idea of investigating compactness with use of
seminorms.

## References

  * [1] G. G. Lorentz, _Some New Functional Spaces_ , in: Annals of Mathematics, 1950, volume 51
  * [2] Loukas Grafakos, _Classical Fourier Analysis_ , Springer 2014
  * [3] Chander, Subhash and Datt, Gopal and Verma, Satish, _Operators on Lorentz sequence spaces_ , in: Mathematica Bohemica, 2009, volume 134
  * [4] M. Ciesielski, G. Lewicki, _Sequence Lorentz Spaces and Their Geometric Structure_ , in: Journal of Geometric Analysis, 2019, volume 29, pp. 1929–1952
  * [5] J. Lindenstrauss, L. Tzafriri, _Classical Banach Spaces I: Sequence Spaces_ , Springer, Berlin - Heidenberg 2013
  * [6] J. Gulgowski, P. Kasprzak, P. Maćkowiak, _Compactness in normed spaces: a unified approach through semi-norms_ , in: Topological Methods in Nonlinear Analysis, 2023, volume 62, pp. 105–134


# Image Restoration with Point Spread Function Regularization and Active
Learning

Peng Jia,1,2,5 Jiameng Lv,1 Runyu Ning,1 Yu Song,1 Nan Li,3 Kaifan Ji,4
Chenzhou Cui, 3 Shanshan Li 3
1College of Electronic Information and Optical Engineering, Taiyuan, 030024,
China
2Peng Cheng Lab, Shenzhen, 518066, China
3National Astronomical Observatories, Beijing, 100101,China
4Yunnan Observatories, Kunming, Yunnan, ,China
5Department of Physics, Durham University, Durham, DH1 3LE, UK
robinmartin20@gmail.comlvjiameng1224@link.tyut.edu.cnnan.li@nao.cas.cnjkf<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Large-scale astronomical surveys can capture numerous images of celestial
objects, including galaxies and nebulae. Analysing and processing these images
can reveal intricate internal structures of these objects, allowing
researchers to conduct comprehensive studies on their morphology, evolution,
and physical properties. However, varying noise levels and point spread
functions can hamper the accuracy and efficiency of information extraction
from these images. To mitigate these effects, we propose a novel image
restoration algorithm that connects a deep learning-based restoration
algorithm with a high-fidelity telescope simulator. During the training stage,
the simulator generates images with different levels of blur and noise to
train the neural network based on the quality of restored images. After
training, the neural network can directly restore images obtained by the
telescope, as represented by the simulator. We have tested the algorithm using
real and simulated observation data and have found that it effectively
enhances fine structures in blurry images and increases the quality of
observation images. This algorithm can be applied to large-scale sky survey
data, such as data obtained by LSST, Euclid, and CSST, to further improve the
accuracy and efficiency of information extraction, promoting advances in the
field of astronomical research.

###### keywords:

techniques: image processing – methods: numerical – software: data analysis

††pubyear: 2015††pagerange: Image Restoration with Point Spread Function
Regularization and Active Learning–Image Restoration with Point Spread
Function Regularization and Active Learning

##  1 Introduction

Sky survey projects obtain numerous observational images containing celestial
objects with extended structures, including nebulae and galaxies, which are of
utmost interest and importance for astronomical research. Scientists select
these targets for in-depth analysis. However, aberrations and noises in
astronomical observations can degrade image quality, resulting in blurred and
distorted images of celestial objects. These factors significantly affect the
precision of scientific information extracted from such images. Thus, it is
crucial to develop and utilize image restoration algorithms to improve image
quality and enable further scientific exploration. Recent developments in deep
neural networks have led to the emergence of several new image restoration
algorithms, which take advantage of the powerful representation capabilities
of deep neural networks and have demonstrated impressive results. Two primary
types of image restoration algorithms have been proposed based on different
representation strategies: algorithms based on the properties of astronomical
images and those based on the properties of the degradation process, such as
the point spread function (PSF) of telescopes and the characteristics of
noise. These algorithms can enhance the quality of images obtained from
astronomical observations.

Image restoration algorithms based on image properties are often developed
with generative neural networks, such as the VAE (Kingma & Welling, 2013), the
U-Net (Ronneberger et al., 2015), or the GAN (Goodfellow et al., 2020). These
neural networks are trained on a large number of high signal-to-noise ratio
(SNR) and high spatial resolution real observation images, from which they
learn features. The learned features are then used to restore low SNR or low
spatial resolution images. Properly selected training data can lead to
effective results for galaxy images (Schawinski et al., 2017; Arcelin et al.,
2021; Jia et al., 2021; Gan et al., 2021; Li et al., 2022) or solar images
(Jia et al., 2019). However, manual interventions are often required to obtain
appropriate training data, and overfitting or training on improper data can
result in the generation of fake structures by the neural network (Jia et al.,
2021).

Image restoration algorithms based on the properties of PSFs aim to learn the
generalized inverse function of PSFs. The method first constructs a PSF model
using wavefront decomposition (Fétick et al., 2019; Beltramo-Martin et al.,
2019; Fusco et al., 2020; Jia et al., 2020) or PSF basis (Jia et al., 2017;
Sun et al., 2020), and then uses this model as prior knowledge to restore
images through supervised or unsupervised learning. In unsupervised learning
algorithms, the PSFs are decomposed into parameters, and the image restoration
algorithm fits these parameters through training (Qi et al., 2014; Gao et al.,
2017; Sureau et al., 2020). However, these algorithms also face the parameter-
tuning problem encountered by classical image restoration algorithms. In
supervised learning algorithms, PSFs are used to generate blurred images as
the training set (Jia et al., 2020; Li & Alexander, 2022; Wang et al., 2022).
However, two issues limit the application of supervised algorithms. First,
PSFs are necessary as prior information for supervised learning algorithms.
Although some methods have been proposed to extract star images as PSF
templates (Terry et al., 2022), obtaining appropriate PSFs is complicated due
to the presence of sky background noise or read-out noise. Furthermore,
obtaining star images from extended target images (such as galaxies or
nebulae) is challenging (Long et al., 2019). Second, the neural network’s
generalization ability is dependent on the training set. Because PSFs can
change significantly, it is difficult to obtain PSFs and develop datasets that
can reflect different states of real observation data. Improper PSF datasets
may introduce epistemic uncertainty, which can limit the performance of
trained image restoration algorithms (Hüllermeier & Waegeman, 2021).

Therefore, we present a novel framework for processing images from large-scale
astronomical sky surveys. Our framework integrates a high-fidelity simulator
of a specific telescope and a deep neural network-based image restoration
algorithm with the active learning strategy. The simulator is customized for
the telescope used in the sky survey and can generate simulated images with
various PSFs and noise levels typically encountered in real observations. The
image restoration algorithm restores these images, and we evaluate the quality
of the restored images by computing the mean square error (MSE) between the
original and restored images. The active learning strategy governs the
simulator to generate more images with PSFs or noise levels that the image
restoration neural network could not restore well, thereby further training
the neural network. This approach enables us to obtain an effective neural
network for image restoration. We discuss our framework in detail in Section 2
and demonstrate its ability to restore simulated and real observation images
with varying levels of blur in Section 3. Finally, we draw our conclusions and
outline future work in Section 4.

##  2 The Framework

Large-scale astronomical surveys produce a vast number of blurred images with
varying quality, making it impractical to manually process them one by one
using human intervention-based algorithms. Therefore, an image restoration
framework capable of producing stable results for images captured by the sky
survey project is necessary. This paper proposes a framework that addresses
this need and is illustrated in Figure 1. The framework consists of three
parts: the Monte Carlo simulation part, the image restoration part, and the
parameter selection part. To expedite the training process of the framework,
we propose using parallel computing technology to run the data generation and
image restoration parts on different computers or processors. We will discuss
details of our framework in the following subsections.

Figure 1: The image restoration framework is illustrated in the diagram, which
is composed of three main parts: the Monte Carlo simulation part, the image
restoration part, and the parameter selection part. In the Monte Carlo
simulation part, PSFs are generated under different observation conditions,
and sample images are convolved to generate blurred images as training data.
The image restoration part restores these blurred images. Meanwhile, the
parameter selection part evaluates the quality of the restored images and
generates parameter dictionaries for the data generation part. To speed up the
training process, we employ the MPI technology to run the data generation and
image restoration parts in separate processors or computers.

###  2.1 The Monte Carlo Simulation Part

The Monte Carlo simulation part generates point spread functions (PSFs) of the
telescope for various observation conditions and convolves sample images to
create blurred images as the training data. The imaging process of an optical
telescope can be modeled with the equation 2:

$\displaystyle Img(x,y)=[Obj(x,y)*PSF(x,y)]_{pixel(x,y)}+Noise(x,y)$ (1)

where $Obj(x,y)$ and $Img(x,y)$ are original and observed images. $PSF(x,y)$
is the point-spread function (PSF) of the telescope. $[]_{pixel(x,y)}$ stands
for the pixel response function of the detector and $Noise(x,y)$ stands for
the noise from the background and the detector.

This paper primarily focuses on studying long-exposure images captured by
ground-based optical telescopes, but our framework can be applied to images
collected by any telescope, provided that we have an appropriate simulator. In
the case of ground-based long exposure observations, atmospheric turbulence
has a significant impact on the PSFs, which can be accurately modeled by using
the Moffat model with $\beta$ equal to 4.765 (Moffat, 1969). We set the Full-
Width Half Maximum (FWHM) of the PSF as the first free parameter in the
simulator. Additionally, we use the Gaussian random number to model various
levels of noise caused by the detector and background, and we set the standard
deviation as the second free parameter in the simulator. By using these
parameters, we can generate images with realistic PSFs and noise, which better
represent actual observation conditions for image restoration algorithms.
Finally, we convolve several high-resolution template images with PSFs and add
noise to generate the training data.

###  2.2 The Image Restoration Part

The image restoration neural network we use in this study is the PSF-NET,
which was proposed by Jia et al. (2020). The PSF-NET consists of two neural
networks, namely the PSF network (PSF) and the RESTORE network (RESTORE), as
shown in the Figure 2. Here is a detailed description of their roles and
functionalities: Firstly, the primary task of the PSF network is to model the
image’s blurring process. It achieves this by learning and representing the
characteristics of the PSF and noise, transforming a high-resolution image
into a blurred version. The objective of this step is to capture the
fundamental information regarding image blurring, thereby enhancing the
accuracy and efficiency of the subsequent restoration process. Secondly, the
RESTORE network plays the role of a deconvolution algorithm, tasked with
generating high-resolution, clear images from the blurred images. After being
trained, the RESTORE network is capable of effectively restoring blurred
images to high-quality images, recovering the fine details and information in
the image. It is noteworthy that both the PSF and RESTORE networks comprise 6
residual blocks, along with several convolutional and transposed convolutional
blocks. The structures of these blocks are depicted in Figure 3. During the
training phase, we adopt a joint training approach to train both the PSF
network and the RESTORE network, thereby improving training efficiency and
mitigating overfitting risks. After training, the RESTORE network can be
deployed for the task of restoring blurred images.

Figure 2: This figure shows the structure of the PSF-NET. It includes an image
restoration neural network (RESTORE) and a PSF generation neural network
(PSF).  Figure 3: The left panel of the figure displays the architecture of
the PSF neural network, which is similar to that of the RESTORE neural
network. The middle panel illustrates the convolutional block, while the right
panel depicts the residual block.

An important point to consider is that both the PSF network and the RESTORE
network are trained together, with the PSF network acting as a constraint for
the RESTORE network. Unlike a simple deconvolution algorithm, the RESTORE
network handles deconvolution and noise reduction. Therefore the PSF network
learns impacts brought by the PSF and the noise. This structural design not
only infuses the restoration process with enhanced physical realism but also
bolsters the model’s ability to generalize, enabling it to address various
blurring scenarios effectively. The PSF neural network in this study generates
blurred images from high-resolution images, while the Restore neural network
generates high-resolution images from blurred images. To reflect the functions
of these two neural networks, we have designed the loss function as presented
in Equation 2.

$\displaystyle Loss=L_{idt}+L_{rec}+L_{fl},$ (2)

where $L_{idt}$ is the identity loss function, $L_{rec}$ is the cycle loss
function and $L_{fl}$ is the focal frequency loss function. They are defined
in equation 3,

$\displaystyle\begin{aligned} L_{idt}=\|PSF(Img_{org})-Img_{blur}\|_{2}+\\\
\|Restore(Img_{blur})-Img_{org}\|_{2},\\\
L_{rec}=\|Restore(PSF(Img_{org}))-Img_{org}\|_{2}+\\\
\|PSF(Restore(Img_{blur}))-Img_{blur}\|_{2},\\\ L_{ffl}=W\times
L_{fl},\end{aligned}$ (3)

where $PSF$ and $Restore$ stand for operators carried out by the PSF neural
network and the RESTORE neural network, $Img_{blur}$ and $Img_{org}$ stand for
blurred images generated by the simulation method, and original high
resolution images. We use the $L_{idt}$ and the $L_{rec}$ to set the mean
square error (MSE) between the original images and the images restored by the
neural network as small as possible.

Both the PSF neural network and the RESTORE neural network adopt an encoder-
decoder structure. However, since the decoder contains several up-sampling
deconvolution layers that may introduce gaps in restored images, using only
the aforementioned loss function may pose a problem. Our empirical
observations suggest that gaps affect the structure of restored galaxy images
and are more noticeable in the spatial frequency domain. To address this
issue, we propose using the focal frequency loss $L_{ffl}$ to further enhance
the performance of our neural network. The $L_{ffl}$ was originally introduced
by Jiang et al. (2021), and involves the use of regularized weights $W$ to
modulate the power spectral density, with $L_{fl}$ representing the mean
squared error (MSE) between the original and restored images in the spatial
frequency domain, as defined in equation 4,

$\displaystyle\begin{aligned}
W=|FFT(Img_{org})-FFT(Restore(Img_{blur}))|^{\alpha},\\\
L_{fl}=\|FFT(Img_{org})-FFT(Restore(Img_{blur}))\|^{2},\end{aligned}$ (4)

The fast Fourier transform is represented by $FFT$ and the regularized
parameter $\alpha$ is used in the focal frequency loss $L_{ffl}$, which we
defined as 1 in this study.

###  2.3 The Parameter Selection Part

The parameter selection part is the central part of our framework, responsible
for adjusting the proportion of images with various blur or noise levels used
for training the neural network. Additionally, this part oversees the
execution of the simulation and image restoration parts in parallel using the
MPICH (Gropp, 2002). We will discuss the parameter selection component below.

To begin, our parameter selection part involves defining a dictionary to
control the distribution of images with varying blur or noise levels in the
training set. The keys of this dictionary are determined by the FWHM of PSFs
and noise levels, while the values represent the mean loss function of all
images with the respective FWHM or noise level in the test set. These values
are updated at the end of each epoch. Subsequently, we generate a new set of
images based on these updated key values using equation 5,

$\displaystyle\begin{aligned}
test_{count}(r_{i})=(total(r_{i})-10)\times(1-list(r_{i}))+10,\\\
train_{count}(r_{i})=(total(r_{i})-10)\times list(r_{i}),\end{aligned}$ (5)

$test_{count}(r_{i})$ and $train_{count}(r_{i})$ represent the total number of
images with a particular FWHM and noise level ($r_{i}$) in the test set and
the training set, respectively. $total(r_{i})$ represents the total number of
images with the same FWHM or noise level. $list$ represents the parameter
dictionary used to store the normalized mean value of the loss function for
all images with predefined FWHMs of PSFs or noise levels. The values in the
$list$ are used to calculate the percentage of images at a particular FWHM and
noise level. By generating more blurred images that cannot be restored well by
the restoration neural network in the previous epoch, the network is trained
to gain better performance for these images in the next epoch because we have
more images of that particular blur or noise level. With the parameter
selection part, our neural network can effectively sample the space of blur
levels or noise levels for a particular sky survey project and obtain a stable
generalization ability for all images obtained by the project.

To accelerate the computationally expensive simulation and image restoration
parts, we propose dynamically adjusting the computation load on different
computers during the training stage. The parameter selection part controls the
data generation and image restoration parts and is illustrated in Figure 4.
The parameter selection part has two ranks. Rank0 trains and tests the image
restoration part, which requires significant GPU resources on a single
computer. Rank1 generates blur images with the data generation part on several
other computers, which require CPU resources. The training data is set with an
appropriate number based on the data generation and image restoration speeds.
During the training stage, Rank1 generates blur images and sends a "TRUE"
signal to Rank0 when all images in the training set of one epoch are
generated. Rank1 continues to generate blurred images even after sending the
"TRUE" signal. Meanwhile, Rank0 restores these blur images before receiving
the "TRUE" signal from Rank1. Upon receiving the "TRUE" signal, Rank0 stops
training and evaluates the qualities of all restored images. The parameter
selection part calculates the MSE of all restored images in the test set and
updates the parameter dictionary. The framework runs continuously with the
parameter dictionary as input parameters until the set number of iterations is
reached or the loss does not decrease for ten consecutive epochs.

Figure 4: The flow chart of the parameter selection part is shown in this
figure. The parameter selection part comprises two ranks, each with its own
distinct role. Rank0 is responsible for training and testing the image
restoration component, and runs primarily on a computer equipped with an RTX
3090 GPU. On the other hand, Rank1 is responsible for generating blurred
images using the simulation component and runs on several other computers
equipped with CPUs. For this purpose, we utilize older laptops or desktops
that are equipped with i5 or i7 processors to run the simulation code.

##  3 Performance Evaluation with Simulated and Real Observation Images

To assess the effectiveness of our framework, we will test it on both
simulated and real observational data. In Section 3.1, we will introduce
several criteria for evaluating its performance. In Section 3.2, we will use
simulated data to test the framework’s performance. Since we have control over
the blur and noise levels, we can effectively evaluate the neural network’s
generalization ability using simulated data. In Section 3.3, we will evaluate
the framework’s performance using real observational data obtained from the
SDSS project (Abazajian et al., 2009). We send images of galaxies directly to
the neural network and the results demonstrate the effectiveness of our
framework. Further details about our framework will be discussed below.

###  3.1 Performance Evaluation Criteria

We employ the peak signal-to-noise ratio (PSNR) as defined by Xu et al. (2014)
and the structural similarity (SSIM) as defined by Wang et al. (2004) to
quantitatively assess the quality of the images. The equation for PSNR can be
found in equation 6:

$\displaystyle\begin{aligned}
PSNR(img,img_{ref})=20\log_{10}\left(\frac{MAX(img_{ref})}{\sqrt{MSE}}\right),\\\
MSE=\frac{1}{\sqrt{N\times
M}}\sum_{i=1}^{N}\sum_{j=1}^{M}\left(img_{(i,j)}-img_{ref(i,j)}\right)\end{aligned}$
(6)

where $N\times M$ is the size of the image, $img$ and $img_{ref}$ are restored
images and original images. For simulated images, we could obtain the $img$
and $img_{ref}$ and for real observation images, $img_{ref}$ is mean values of
$img$. The PSNR can directly reflect the similarity between restored images
and blurred images. Images with larger PSNR would have better quality.
Meanwhile, the SSIM is defined in the equation 7:

$\displaystyle\begin{aligned} SSIM(img,img_{ref})&=l(img,img_{ref})\\\ &\cdot
c(img,img_{ref})\cdot s(img,img_{ref}),\\\
l(img,img_{ref})&=\frac{2\mu_{img}\mu_{ref}+C_{1}}{\mu^{2}_{img}+\mu^{2}_{ref}+C_{1}},\\\
c(img,img_{ref})&=\frac{2\delta_{img}\delta_{ref}+C_{2}}{\delta^{2}_{img}+\delta^{2}_{ref}+C_{2}},\\\
s(img,img_{ref})&=\frac{\delta_{img,ref}+C_{3}}{\delta_{img}\delta_{ref}+C_{3}},\\\
C_{1}&=(K_{1}L)^{2},\\\ C_{2}&=(K_{2}L)^{2},\\\ C_{3}&=C_{2}/2,\end{aligned}$
(7)

where $\mu$ and $\delta$ are the average and standard deviation of the gray
scale values, respectively, $\delta_{img,ref}$ is the covariance of the gray
scale values, L is the dynamic range of the image, and $K_{1}$ and $K_{2}$ are
small arbitrary values (0.001 in this paper). The SSIM is a perceptual model
that considers the brightness, contrast, and intensity scale of two images
simultaneously. The quality of an image is considered better if its SSIM is
larger.

###  3.2 Performance Evaluation With Simulated Images

In order to evaluate the effectiveness of our framework on simulated images,
we first simulate an observation scenario carried out by a ground-based
telescope with long exposure. We obtain galaxy images in the r band from the
SDSS DR 7 and generate blurred images through simulation. Our simulator
assumes that the PSF follows the Moffat model, with a full-width at half-
maximum (FWHM) distribution ranging from 2.0 to 8.0 pixels (equivalent to
0.792 arcsec to 3.168 arcsec) across 5 different levels, and Gaussian noise
distributed equally across 5 different levels with $\sigma$ ranging from 1.0
to 15.0. This results in a total of 25 different levels of blurred images. We
train our framework using the images generated by the simulator with the
aforementioned parameters. Additionally, we generate blurred images with
higher levels of blur and noise to evaluate the performance of our algorithm.
We divide the FWHM of the PSFs into 13 bins ranging from 2 to 14 pixels
(equivalent to 0.792 arcsec to 5.544 arcsec), with the FWHMs of the PSFs in
each bin considered as random variables. We also divide the noise levels into
24 bins ranging from 2 to 25, with $\sigma$ considered as a random variable in
each bin.

In this section, we apply the evaluation criteria outlined in Section 3.1 to
assess the effectiveness of our framework. We utilize box plots to illustrate
the performance of our algorithm and the Richardson-Lucy method (RL) with the
Gaussian denoising method across varying noise levels and different PSFs. In
Figure 5, results obtained from our algorithm are denoted in green, RL
algorithm results in red, and the original blurred data in blue. Upon
analyzing these box plots, it becomes apparent that our algorithm consistently
outperforms the RL method under diverse conditions, encompassing various noise
levels and distinct PSFs. Notably, our algorithm exhibits a significant
advantage when dealing with datasets characterized by fluctuating noise
levels, demonstrating its robustness and stability in the presence of noisy
input data.

Furthermore, we conduct a t-test to determine whether a significant
performance difference exists between the RL algorithm and our model. Under
the null hypothesis (H0), we assume there is no performance difference between
the RL algorithm and our model. The alternative hypothesis (H1) posits a
performance difference between the RL algorithm and our model. In this study,
we conduct statistical significance analyses for different scenarios and
calculate p-values, all of which are less than 0.05, as shown in Tables 1, 2,
3, and 4. In all these cases, we reject the null hypothesis, signifying a
significant performance difference between the RL algorithm and our model.
These findings collectively underscore the remarkable image restoration
quality achieved by our algorithm. When evaluated using PSNR or SSIM metrics,
our algorithm consistently outperforms the traditional RL method. In summary,
our test results unequivocally establish the superior performance of our image
restoration framework across various observation conditions.

Figure 5: Comparison of different mage restoration algorithms under different Noise and PSF Conditions. The box plot showing the differences between our restoration algorithm (Green) and the RL (Red) method under different PSF conditions (Top Panel) and different noise levels (Bottom Panel).  Fwhm | 2 | 4 | 6 | 8 | 10 | 12 | 14
---|---|---|---|---|---|---|---
t$\\_$statistic | 17.97 | 20.46 | 17.57 | 20.27 | 20.76 | 18.79 | 24.56
p$\\_$value | 5.96e-38 | 1.23e-43 | 5.08e-37 | 3.40e-43 | 2.72e-44 | 7.41e-40 | 3.21e-52
Table 1: Statistical Analysis of SSIM Performance for RL and Our Algorithm under Different Blur Levels (PSFs). Fwhm | 2 | 4 | 6 | 8 | 10 | 12 | 14
---|---|---|---|---|---|---|---
t$\\_$statistic | 49.69 | 73.94 | 56.31 | 75.21 | 76.49 | 68.05 | 110.96
p$\\_$value | 6.02e-90 | 6.82e-113 | 4.29e-97 | 6.88e-114 | 7.09e-115 | 4.77e-108 | 8.31e-137
Table 2: Statistical Analysis of PSNR Performance for RL and Our Algorithm under Different Blur Levels (PSFs). Sigma | 2 | 9 | 12 | 15 | 18 | 21 | 23 | 25
---|---|---|---|---|---|---|---|---
t$\\_$statistic | 1.70 | 17.81 | 26.46 | 32.40 | 32.12 | 40.43 | 40.60 | 43.98
p$\\_$value | 4.88e-02 | 3.25e-29 | 1.05e-40 | 5.35e-47 | 1.00e-46 | 4.40e-54 | 3.24e-54 | 1.63e-54
Table 3: Statistical Analysis of SSIM Performance for RL and Our Algorithm under Different Noise Levels. Sigma | 2 | 9 | 12 | 15 | 18 | 21 | 23 | 25
---|---|---|---|---|---|---|---|---
t$\\_$statistic | 6.66 | 34.76 | 42.52 | 39.62 | 43.38 | 50.09 | 47.96 | 59.78
p$\\_$value | 3.52e-09 | 3.11e-49 | 1.02e-55 | 1.98e-53 | 2.27e-56 | 4.52e-61 | 1.19e-59 | 6.52e-67
Table 4: Statistical Analysis of PSNR Performance for RL and Our Algorithm
under Different Noise Levels.

To assess the performance of our algorithm in a qualitative manner, we utilize
simulated images to test the effectiveness of our framework. Specifically, we
apply the trained RESTORE neural network to restore simulated images that have
PSFs and noise levels within the range defined by the training set. The
resulting images are presented in Figure 6. The figure illustrates that the
trained RESTORE neural network can significantly enhance the quality of the
images. The fine details, such as spirals and bars of galaxies, can be clearly
observed in the restored images. Additionally, we test the performance of our
framework on simulated images that have PSFs or noise levels that exceed the
range defined in the training set. The results are shown in Figure 7, and it
is evident that the trained RESTORE neural network exhibits a strong
generalization ability, even for images with larger FWHM or higher noise
levels.

Figure 6: The performance of our framework in the restoration of images that
are generated within FWHMs or noise levels defined by the training set. Figure
7: The performance of our framework in the restoration of images that are
generated with larger FWHM or higher noise level defined by the training set.

###  3.3 Performance Evaluation With Real Observation Images

Frist, we demonstrate the efficacy of our framework with real observation data
obtained from the SDSS Data Release 7 (York et al., 2000; Abazajian et al.,
2009). We employ our method to restore images of low surface brightness
galaxies (LSBGs) as detected by Yi et al. (2022). LSBGs are a class of
galaxies with central surface brightness fainter than the sky background,
often exhibiting high gas content and believed to be in the early stages of
galaxy formation or to have recently undergone a burst of star formation
(Impey & Bothun, 1997). Due to their low luminosity, studying LSBGs is
challenging (Du et al., 2015), making image restoration algorithms necessary
to enhance their image quality before any scientific analysis can be
performed. However, it is difficult to use traditional deconvolution-based
image restoration algorithms, which require bright stars as references, and
content-based image restoration methods are ineffective due to very small
number of photons in LSBGs. Thus, we use our framework to restore LSBG images.

These images in the SDSS project are captured by a 2.5-meter telescope and
have a pixel scale of 0.396 arcsec with an exposure time of 53.9 sec. As a
result, the PSF and noise levels used in the simulation model in Section 2.2
can reflect the properties of real observation data. Therefore, we train our
framework with the simulator and the aforementioned parameters to obtain the
weights of the RESTORE neural network. Subsequently, we apply the RESTORE
neural network directly to process the LSBG images captured by the SDSS
project in the r band. Figure 8 exhibits both the original images and the
images restored by our neural network. Additionally, we also employ the RL
deconvolution algorithm (Fish et al., 1995) and select PSF references
according to Infante-Sainz et al. (2020) to restore these images for
comparison. Furthermore, we present the LSGB images acquired by the DESI
Legacy Imaging Surveys (Dey et al., 2019) in this figure to provide a reliable
reference. Since larger telescopes are used to execute the DESI Legacy Imaging
Surveys, the data obtained from it can improve our evaluation of the
effectiveness of our method.

Figure 8: This figure shows original images obtained by the SDSS project in
the r band, images restored by the RL method, with references of PSFs provided
by Infante-Sainz et al. (2020), images restored by our framework and images of
the same LSBG obtained by the DESI Legacy Imaging Surveys. In the upper left
corner of each figure, we show the PSNR of restored images and calculation
speed of different methods in restoration of different images. As shown in
this figure, our framework could effectively restore blurred images.

Firstly, as demonstrated in the upper left corner of each figure, our
framework can efficiently improve the PSNR of LSBGs with less time (around 10
times faster than the RL method with appropriate prior PSFs). Next, we examine
the restored images in detail. Although the RL method could provide effective
results, thanks to the accurate PSF model, images restored by the RL method
are still affected by strong noises. Our framework effectively restores fine
structures of these galaxies, such as spirals, disks, and filaments, and also
reduce effects brought by noises. When comparing the restored images with
those obtained by the DESI Legacy Imaging Surveys, we find that structures
restored by our framework are true. It is worth noting that our framework
builds PSFs and the deconvolution procedures for image restoration, so in
principle it does not generate artificial features. Moreover, our framework
can effectively suppress noise in these images, resulting in some images with
even better quality than those obtained by the DESI Legacy Imaging Surveys.
Overall, our framework can assist scientists in studying the properties and
morphological structures of LSBGs in greater detail.

In our further investigation, our primary focus was on evaluating the
algorithm’s performance concerning the enhancement of photometry accuracy and
detection efficiency. To do this, we randomly select SDSS R band images, each
with dimensions of 1024x1024 pixels. Subsequently, we apply both our image
restoration algorithm and the RL algorithm to restore these images. Following
restoration, we utilize the SExtractor for detection and photometry (Bertin &
Arnouts, 1996). The results of this study are visually presented in Figure 9.
We conduct this analysis for each magnitude, considering a dataset of 2000
images for evaluation. Subsequently, we assess these results through a
rigorous statistical analysis. Our findings indicate that the processed data,
particularly the data treated with our algorithm, show a higher recall rate
and precision rate. This outcome demonstrates that our algorithm significantly
improves the efficiency of celestial object detection, especially for stars
with low signal-to-noise ratio. Moreover, our method has proven effective in
enhancing photometry accuracy, as vividly depicted in the top panel of Figure
9. These results provide strong support for the practical applicability of our
algorithm in the field of astronomy.

Figure 9: Comparison of photometry and detection results for stars with
different magnitudes. MagErr: The average percentage error between photometric
measurements and true values. P-R Curve: The Precision-Recall (P-R) curve is a
graphical representation of the trade-off between precision and recall for a
binary classification model. Precision is the ratio of true positive
predictions to the total number of positive predictions, while recall is the
ratio of true positive predictions to the total number of actual positive
instances.

##  4 Conclusions and future works

We have introduced a novel framework in this paper for restoring blurred
astronomical images by merging deep learning techniques with simulation
algorithms. Our framework actively trains the restoration neural network using
a simulation algorithm that represents a specific telescope. Once trained, the
restoration neural network can produce restored images more efficiently
compared to the traditional RL deconvolution algorithm. We have tested our
method on both simulated and real observational data and have found that it
effectively minimizes the impact of noise and PSFs, making previously unseen
fine structures of galaxies visible.

We have also identified two areas for future improvement. Firstly, we have
shown that the physical parameters and prior model used to represent the image
degradation process are crucial for our framework. To extend our approach to
data from different sky survey projects, we need to develop an adequate
parametric point spread function (PSF) model and an adequate telescope
simulator. Therefore, we will introduce physics-informed machine learning
algorithms as necessary tools to build PSF models. Meanwhile, the digital twin
technology is a promising method for generating simulation data according to
telemetry data and high-fidelity simulators, and we are currently developing a
digital twin as the telescope simulator (Jia et al., 2022; Zhan et al., 2022).
Secondly, we use the $L_{2}$ norm and FFL to train our restoration neural
network, and we could further investigate new regularization methods based on
human attention and big data obtained from previous sky survey projects to
improve the performance of our framework.

Overall, our proposed framework is suitable for restoring images obtained from
future sky survey projects, such as the LSST, the Euclid, and the CSST. Our
framework could help scientists to better recognize the morphology of
galaxies, which could increase outcomes from citizen science platforms.
Besides, our framework could better increase accuracy of shape measurements
for galaxies with low signal to noise ratio. We are now using our framework to
process data obtained by the DECaLS (Dark Energy Survey Collaboration et al.,
2016) for further scientific research. We will deploy our method for data
obtained by the CSST and the Euclid in the future.

## Acknowledgements

This work is supported by National Natural Science Foundation of China (NSFC)
with funding number of 12173027 and 12173062. We acknowledge the science
research grants from the China Manned Space Project with NO. CMS-
CSST-2021-A01. We acknowledge the science research grants from the Square
Kilometer Array (SKA) Project with NO. 2020SKA0110102. Peng Jia acknowledges
support from the Civil Aerospace Technology Research Project (D050105) and the
Major Key Project of PCL. Jiameng Lv acknowledges support from the Shanxi
Graduate Innovation Project (2022Y274).

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science Foundation,
the U.S. Department of Energy, the National Aeronautics and Space
Administration, the Japanese Monbukagakusho, the Max Planck Society, and the
Higher Education Funding Council for England. The SDSS Web Site is
http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the American
Museum of Natural History, Astrophysical Institute Potsdam, University of
Basel, University of Cambridge, Case Western Reserve University, University of
Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, Johns Hopkins University, the Joint Institute for
Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and
Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences
(LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for
Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico
State University, Ohio State University, University of Pittsburgh, University
of Portsmouth, Princeton University, the United States Naval Observatory, and
the University of Washington.

## Data Availability

After acceptance the code will be released in PaperData Repository powered by
China-VO with a DOI number.

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