\section{Introduction}

\file{elsarticle.cls} is a thoroughly re-written document class
for formatting \LaTeX{} submissions to Elsevier journals.
The class uses the environments and commands defined in \LaTeX{} kernel
without any change in the signature so that clashes with other
contributed \LaTeX{} packages such as \file{hyperref.sty},
\file{preview-latex.sty}, etc., will be minimal.
\file{elsarticle.cls} is primarily built upon the default
\file{article.cls}.  This class depends on the following packages
for its proper functioning:

\begin{enumerate}
\item \file{natbib.sty} for citation processing;
\item \file{geometry.sty} for margin settings;
\item \file{fleqn.clo} for left aligned equations;
\item \file{graphicx.sty} for graphics inclusion;
\item \file{txfonts.sty} optional font package, if the document is to
  be formatted with Times and compatible math fonts;
\item \file{hyperref.sty} optional packages if hyperlinking is
  required in the document;
\item \file{endfloat.sty} optional packages if floats to be placed at
 end of the PDF.
\end{enumerate}

All the above packages (except some optional packages) are part of any
standard \LaTeX{} installation. Therefore, the users need not be
bothered about downloading any extra packages.  Furthermore, users are
free to make use of \textsc{ams} math packages such as
\file{amsmath.sty}, \file{amsthm.sty}, \file{amssymb.sty},
\file{amsfonts.sty}, etc., if they want to.  All these packages work in
tandem with \file{elsarticle.cls} without any problems.

\section{Major Differences}

Following are the major differences between \file{elsarticle.cls}
and its predecessor package, \file{elsart.cls}:

\begin{enumerate}[\textbullet]
\item \file{elsarticle.cls} is built upon \file{article.cls}
while \file{elsart.cls} is not. \file{elsart.cls} redefines
many of the commands in the \LaTeX{} classes/kernel, which can
possibly cause surprising clashes with other contributed
\LaTeX{} packages;

\item provides preprint document formatting by default, and
optionally formats the document as per the final
style of models $1+$, $3+$ and $5+$ of Elsevier journals;

\item some easier ways for formatting \verb+list+ and
\verb+theorem+ environments are provided while people can still
use \file{amsthm.sty} package;

\item \file{natbib.sty} is the main citation processing package
  which can comprehensively handle all kinds of citations and
works perfectly with \file{hyperref.sty} in combination with
\file{hypernat.sty};

\item long title pages are processed correctly in preprint and
  final formats.

\end{enumerate}

\section{Installation}

The package is available at author resources page at Elsevier
(\url{http://www.elsevier.com/locate/latex}).
It can also be found in any of the nodes of the Comprehensive
\TeX{} Archive Network (\textsc{ctan}), one of the primary nodes
being
\url{http://tug.ctan.org/tex-archive/macros/latex/contrib/elsarticle/}.
Please download the \file{elsarticle.dtx} which is a composite
class with documentation and \file{elsarticle.ins} which is the
\LaTeX{} installer file. When we compile the
\file{elsarticle.ins} with \LaTeX{} it provides the class file,
\file{elsarticle.cls} by
stripping off all the documentation from the \verb+*.dtx+ file.
The class may be moved or copied to a place, usually,
\verb+$TEXMF/tex/latex/elsevier/+,
or a folder which will be read                   
by \LaTeX{} during document compilation.  The \TeX{} file
database needs updation after moving/copying class file.  Usually,
we use commands like \verb+mktexlsr+ or \verb+texhash+ depending
upon the distribution and operating system.


\section{Usage}\label{sec:usage}
The class should be loaded with the command:

\begin{vquote}
 \documentclass[<options>]{elsarticle}
\end{vquote}

\noindent where the \verb+options+ can be the following:


\begin{description}

\item [{\tt\color{verbcolor} preprint}]  default option which format the
  document for submission to Elsevier journals.

\item [{\tt\color{verbcolor} review}]  similar to the \verb+preprint+
option, but increases the baselineskip to facilitate easier review
process.

\item [{\tt\color{verbcolor} 1p}]  formats the article to the look and
feel of the final format of model 1+ journals. This is always single
column style.

\item [{\tt\color{verbcolor} 3p}] formats the article to the look and
feel of the final format of model 3+ journals. If the journal is a two
column model, use \verb+twocolumn+ option in combination.

\item [{\tt\color{verbcolor} 5p}] formats for model 5+ journals. This
is always of two column style.

\item [{\tt\color{verbcolor} authoryear}] author-year citation style of
\file{natbib.sty}. If you want to add extra options of
\file{natbib.sty}, you may use the options as comma delimited strings
as arguments to \verb+\biboptions+ command. An example would be:
\end{description}

\begin{vquote}
 \biboptions{longnamesfirst,angle,semicolon}
\end{vquote}

\begin{description}
\item [{\tt\color{verbcolor} number}] numbered citation style. Extra options
  can be loaded with\linebreak \verb+\biboptions+ command.

\item [{\tt\color{verbcolor} sort\&compress}] sorts and compresses the
numbered citations. For example, citation [1,2,3] will become [1--3].

\item [{\tt\color{verbcolor} longtitle}] if front matter is unusually long, use
  this option to split the title page across pages with the correct
placement of title and author footnotes in the first page.

\item [{\tt\color{verbcolor} times}] loads \file{txfonts.sty}, if
available in the system to use Times and compatible math fonts.

\item [{\tt\color{verbcolor} reversenotenum}] Use alphabets as
author--affiliation linking labels and use numbers for author
footnotes. By default, numbers will be used as author--affiliation
linking labels and alphabets for author footnotes. 

\item [{\tt\color{verbcolor} lefttitle}] To move title and
author/affiliation block to flushleft. \verb+centertitle+ is the
default option which produces center alignment.

\item [{\tt\color{verbcolor} endfloat}] To place all floats at the end
of the document.

\item [{\tt\color{verbcolor} nonatbib}] To unload natbib.sty.

\item [{\tt\color{verbcolor} doubleblind}] To hide author name, 
affiliation, email address etc. for double blind refereeing purpose.

\item[] All options of \file{article.cls} can be used with this
  document class.

\item[] The default options loaded are \verb+a4paper+, \verb+10pt+,
  \verb+oneside+, \verb+onecolumn+ and \verb+preprint+.

\end{description}

\section{Frontmatter}

There are two types of frontmatter coding:
\begin{enumerate}[(1)]
\item each author is
connected to an affiliation with a footnote marker; hence all
authors are grouped together and affiliations follow;
\pagebreak
\item authors of same affiliations are grouped together and the
relevant affiliation follows this group. 
\end{enumerate}

An example of coding the first type is provided below.

\begin{vquote}
 \title{This is a specimen title\tnoteref{t1,t2}}
 \tnotetext[t1]{This document is the results of the research
    project funded by the National Science Foundation.}
 \tnotetext[t2]{The second title footnote which is a longer 
    text matter to fill through the whole text width and 
    overflow into another line in the footnotes area of the 
    first page.}
\end{vquote}

\begin{vquote}
\author[1]{Jos Migchielsen\corref{cor1}%
  \fnref{fn1}}
\ead{J.Migchielsen@elsevier.com}

\author[2]{CV Radhakrishnan\fnref{fn2}}
\ead{cvr@sayahna.org}

\author[3]{CV Rajagopal\fnref{fn1,fn3}}
\ead[url]{www.stmdocs.in}

 \cortext[cor1]{Corresponding author}
 \fntext[fn1]{This is the first author footnote.}
 \fntext[fn2]{Another author footnote, this is a very long 
   footnote and it should be a really long footnote. But this 
   footnote is not yet sufficiently long enough to make two 
   lines of footnote text.}
 \fntext[fn3]{Yet another author footnote.}

 \affiliation[1]{organization={Elsevier B.V.}, 
                 addressline={Radarweg 29},
                 postcode={1043 NX}, 
                 city={Amsterdam}, 
                 country={The Netherlands}}

 \affiliation[2]{organization={Sayahna Foundation},
                 addressline={JWRA 34, Jagathy}, 
                 city={Trivandrum}
                 postcode={695014}, 
                 country={India}}
\end{vquote}

\begin{vquote}

 \affiliation[3]{organization={STM Document Engineering 
                               Pvt Ltd.},
                 addressline={Mepukada, Malayinkil}, 
                 city={Trivandrum}
                 postcode={695571}, 
                 country={India}}
\end{vquote}

The output of the above \TeX{} source is given in Clips~\ref{clip1} and
\ref{clip2}. The header portion or title area is given in
Clip~\ref{clip1} and the footer area is given in Clip~\ref{clip2}.

\deforange{blue!70}
\src{Header of the title page.}
\includeclip{1}{130 612 477 707}{1psingleauthorgroup.pdf
\deforange{orange}

\deforange{blue!70}
\src{Footer of the title page.}
\includeclip{1}{93 135 499 255}{1pseperateaug.pdf
\deforange{orange}

Most of the commands such as \verb+\title+, \verb+\author+,
\verb+\affiliation+ are self explanatory.  Various components are
linked to each other by a label--reference mechanism; for
instance, title footnote is linked to the title with a footnote
mark generated by referring to the \verb+\label+ string of
the \verb=\tnotetext=.  We have used similar commands
such as \verb=\tnoteref= (to link title note to title);
\verb=\corref= (to link corresponding author text to
corresponding author); \verb=\fnref= (to link footnote text to
the relevant author names).  \TeX{} needs two compilations to
resolve the footnote marks in the preamble part.  
Given below are the syntax of various note marks and note texts.


\begin{vquote}
  \tnoteref{<label(s)>}
  \corref{<label(s)>}
  \fnref{<label(s)>}
  \tnotetext[<label>]{<title note text>}
  \cortext[<label>]{<corresponding author note text>}
  \fntext[<label>]{<author footnote text>}
\end{vquote}

\noindent where \verb=<label(s)>= can be either one or more comma
delimited label strings. The optional arguments to the
\verb=\author= command holds the ref label(s) of the address(es)
to which the author is affiliated while each \verb=\affiliation=
command can have an optional argument of a label. In the same
manner, \verb=\tnotetext=, \verb=\fntext=, \verb=\cortext= will
have optional arguments as their respective labels and note text
as their mandatory argument.

The following example code provides the markup of the second type
of author-affiliation.

\begin{vquote}
\author{Jos Migchielsen\corref{cor1}%
  \fnref{fn1}}
\ead{J.Migchielsen@elsevier.com}
 \affiliation[1]{organization={Elsevier B.V.}, 
                 addressline={Radarweg 29},
                 postcode={1043 NX}, 
                 city={Amsterdam}, 
                 country={The Netherlands}}

\author{CV Radhakrishnan\fnref{fn2}}
\ead{cvr@sayahna.org}
 \affiliation[2]{organization={Sayahna Foundation},
                 addressline={JWRA 34, Jagathy}, 
                 city={Trivandrum}
                 postcode={695014}, 
                 country={India}}

\author{CV Rajagopal\fnref{fn1,fn3}}
\ead[url]{www.stmdocs.in}
 \affiliation[3]{organization={STM Document Engineering 
                               Pvt Ltd.},
                 addressline={Mepukada, Malayinkil}, 
                 city={Trivandrum}
                 postcode={695571}, 
                 country={India}}
\end{vquote}

\vspace*{-.5pc}

\begin{vquote}
\cortext[cor1]{Corresponding author}
\fntext[fn1]{This is the first author footnote.}
\fntext[fn2]{Another author footnote, this is a very long 
  footnote and it should be a really long footnote. But this 
  footnote is not yet sufficiently long enough to make two lines 
  of footnote text.}
\end{vquote}

The output of the above \TeX{} source is given in Clip~\ref{clip3}.

\deforange{blue!70}
\src{Header of the title page..}
\includeclip{1}{119 563 468 709}{1pseperateaug.pdf
\deforange{orange}

Clip~\ref{clip4} shows the output after giving \verb+doubleblind+ class option. 

\deforange{blue!70}
\src{Double blind article}
\includeclip{1}{124 567 477 670}{elstest-1pdoubleblind.pdf
\deforange{orange}

\vspace*{-.5pc}
The frontmatter part has further environments such as abstracts and
keywords.  These can be marked up in the following manner:

\begin{vquote}
 \begin{abstract}
  In this work we demonstrate the formation of a new type of 
  polariton on the interface between a ....
 \end{abstract}
\end{vquote} 

\vspace*{-.5pc}
\begin{vquote}
 \begin{keyword}
  quadruple exiton \sep polariton \sep WGM
 \end{keyword}
\end{vquote}

\noindent Each keyword shall be separated by a \verb+\sep+ command.
\textsc{msc} classifications shall be provided in 
the keyword environment with the commands
\verb+\MSC+. \verb+\MSC+ accepts an optional
argument to accommodate future revisions.
eg., \verb=\MSC[2008]=. The default is 2000.\looseness=-1

\subsection{New page}
Sometimes you may need to give a page-break and start a new page after
title, author or abstract. Following commands can be used for this
purpose.

\begin{vquote}
  \newpageafter{title}
  \newpageafter{author}
  \newpageafter{abstract}
\end{vquote}


\begin{itemize}
\leftskip-2pc
\item [] {\tt\color{verbcolor} \verb+\newpageafter{title}+} typeset the title alone on one page.

\item [] {\tt\color{verbcolor} \verb+\newpageafter{author}+}  typeset the title
and author details on one page.

\item [] {\tt\color{verbcolor} \verb+\newpageafter{abstract}+}
typeset the title,
author details and abstract \& keywords one one page.

\end{itemize}

\section{Floats}
{Figures} may be included using the command, \verb+\includegraphics+ in
combination with or without its several options to further control
graphic. \verb+\includegraphics+ is provided by \file{graphic[s,x].sty}
which is part of any standard \LaTeX{} distribution.
\file{graphicx.sty} is loaded by default. \LaTeX{} accepts figures in
the postscript format while pdf\LaTeX{} accepts \file{*.pdf},
\file{*.mps} (metapost), \file{*.jpg} and \file{*.png} formats. 
pdf\LaTeX{} does not accept graphic files in the postscript format. 

The \verb+table+ environment is handy for marking up tabular
material. If users want to use \file{multirow.sty},
\file{array.sty}, etc., to fine control/enhance the tables, they
are welcome to load any package of their choice and
\file{elsarticle.cls} will work in combination with all loaded
packages.

\section[Theorem and ...]{Theorem and theorem like environments}

\file{elsarticle.cls} provides a few shortcuts to format theorems and
theorem-like environments with ease. In all commands the options that
are used with the \verb+\newtheorem+ command will work exactly in the same
manner. \file{elsarticle.cls} provides three commands to format theorem or
theorem-like environments: 

\begin{vquote}
 \newtheorem{thm}{Theorem}
 \newtheorem{lem}[thm]{Lemma}
 \newdefinition{rmk}{Remark}
 \newproof{pf}{Proof}
 \newproof{pot}{Proof of Theorem \ref{thm2}}
\end{vquote}

The \verb+\newtheorem+ command formats a
theorem in \LaTeX's default style with italicized font, bold font
for theorem heading and theorem number at the right hand side of the
theorem heading.  It also optionally accepts an argument which
will be printed as an extra heading in parentheses. 

\begin{vquote}
  \begin{thm} 
   For system (8), consensus can be achieved with 
   $\|T_{\omega z}$
   ...
     \begin{eqnarray}\label{10}
     ....
     \end{eqnarray}
  \end{thm}
\end{vquote}  

Clip~\ref{clip5} will show you how some text enclosed between the
above code\goodbreak \noindent looks like:

\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newtheorem}}
\includeclip{2}{1 1 453 120}{jfigs.pdf}
\deforange{orange}

The \verb+\newdefinition+ command is the same in
all respects as its\linebreak \verb+\newtheorem+ counterpart except that
the font shape is roman instead of italic.  Both
\verb+\newdefinition+ and \verb+\newtheorem+ commands
automatically define counters for the environments defined.

\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newdefinition}}
\includeclip{1}{1 1 453 105}{jfigs.pdf}
\deforange{orange}

The \verb+\newproof+ command defines proof environments with
upright font shape.  No counters are defined. 

\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newproof}}
\includeclip{3}{1 1 453 65}{jfigs.pdf}
\deforange{orange}

Users can also make use of \verb+amsthm.sty+ which will override
all the default definitions described above.

\section[Enumerated ...]{Enumerated and Itemized Lists}
\file{elsarticle.cls} provides an extended list processing macros
which makes the usage a bit more user friendly than the default
\LaTeX{} list macros.   With an optional argument to the
\verb+\begin{enumerate}+ command, you can change the list counter
type and its attributes.

\begin{vquote}
 \begin{enumerate}[1.]
 \item The enumerate environment starts with an optional
   argument `1.', so that the item counter will be suffixed
   by a period.
 \item You can use `a)' for alphabetical counter and '(i)' for
   roman counter.
  \begin{enumerate}[a)]
    \item Another level of list with alphabetical counter.
    \item One more item before we start another.
\end{vquote}

\deforange{blue!70}
\src{List -- Enumerate}
\includeclip{4}{1 1 453 185}{jfigs.pdf}
\deforange{orange}

Further, the enhanced list environment allows one to prefix a
string like `step' to all the item numbers.  

\begin{vquote}
 \begin{enumerate}[Step 1.]
  \item This is the first step of the example list.
  \item Obviously this is the second step.
  \item The final step to wind up this example.
 \end{enumerate}
\end{vquote}

\deforange{blue!70}
\src{List -- enhanced}
\includeclip{5}{1 1 313 83}{jfigs.pdf}
\deforange{orange}


\section{Cross-references}
In electronic publications, articles may be internally
hyperlinked. Hyperlinks are generated from proper
cross-references in the article.  For example, the words
\textcolor{black!80}{Fig.~1} will never be more than simple text,
whereas the proper cross-reference \verb+\ref{tiger}+ may be
turned into a hyperlink to the figure itself:
\textcolor{blue}{Fig.~1}.  In the same way,
the words \textcolor{blue}{Ref.~[1]} will fail to turn into a
hyperlink; the proper cross-reference is \verb+\cite{Knuth96}+.
Cross-referencing is possible in \LaTeX{} for sections,
subsections, formulae, figures, tables, and literature
references.

\section[Mathematical ...]{Mathematical symbols and formulae}

Many physical/mathematical sciences authors require more
mathematical symbols than the few that are provided in standard
\LaTeX. A useful package for additional symbols is the
\file{amssymb} package, developed by the American Mathematical
Society. This package includes such oft-used symbols as
$\lesssim$ (\verb+\lesssim+), $\gtrsim$ (\verb+\gtrsim+)  or 
$\hbar$ (\verb+\hbar+). Note that your \TeX{}
system should have the \file{msam} and \file{msbm} fonts installed. If
you need only a few symbols, such as $\Box$ (\verb+\Box+), you might try the
package \file{latexsym}.

Another point which would require authors' attention is the
breaking up of long equations.  When you use
\file{elsarticle.cls} for formatting your submissions in the 
\verb+preprint+ mode, the document is formatted in single column
style with a text width of 384pt or 5.3in.  When this document is
formatted for final print and if the journal happens to be a double column
journal, the text width will be reduced to 224pt at for 3+
double column and 5+ journals respectively. All the nifty 
fine-tuning in equation breaking done by the author goes to waste in
such cases.  Therefore, authors are requested to check this
problem by typesetting their submissions in final format as well
just to see if their equations are broken at appropriate places,
by changing appropriate options in the document class loading
command, which is explained in section~\ref{sec:usage},
\nameref{sec:usage}. This allows authors to fix any equation breaking
problem before submission for publication.
\file{elsarticle.cls} supports formatting the author submission
in different types of final format.  This is further discussed in
section \ref{sec:final}, \nameref{sec:final}.

\enlargethispage*{\baselineskip}

\subsection*{Displayed equations and double column journals}

Many Elsevier journals print their text in two columns. Since
the preprint layout uses a larger line width than such columns,
the formulae are too wide for the line width in print. Here is an
example of an equation  (see equation 6) which is perfect in a
single column preprint format:

In normal course, articles are prepared and submitted in single column
format even if the final printed article will come in a double column
format journal. Here the problem is that when the article is typeset by
the typesetters for paginating and fit within the single column width,
they have to break the lengthy equations and align them properly. Even
if most of the tasks in preparing your proof is automated, the equation
breaking and aligning requires manual judgement, hence this task is manual.
When there comes a manual operation that area is error prone. Author
needs to check that equation pretty well. 

However if authors themselves break the equation to the single column
width typesetters need not want to touch these area and the proof authors
get will be without any errors.

\setlength\Sep{6pt}
\src{See equation (6)}
\deforange{blue!70}
\includeclip{4}{105 500 500 700}{1psingleauthorgroup.pdf}
\deforange{orange}
                 	
\noindent When this document is typeset for publication in a
model 3+ journal with double columns, the equation will overlap
the second column text matter if the equation is not broken at
the appropriate location.

\vspace*{6pt}
\deforange{blue!70}
\src{See equation (6) overprints into second column}
\includeclip{3}{59 421 532 635}{elstest-3pd.pdf}
\deforange{orange}
\vspace*{6pt}

\noindent The typesetter will try to break the equation which
need not necessarily be to the liking of the author or as it
happens, typesetter's break point may be semantically incorrect.
Therefore, authors may check their submissions for the incidence
of such long equations and break the equations at the correct
places so that the final typeset copy will be as they wish.

\section{Bibliography}

Three bibliographic style files (\verb+*.bst+) are provided ---
\file{elsarticle-num.bst}, \file{elsarticle-num-names.bst} and
\file{elsarticle-harv.bst} --- the first one can be used for the
numbered scheme, second one for numbered with new options of 
\file{natbib.sty}. The third one is for the author year
scheme.

In \LaTeX{} literature, references are listed in the
\verb+thebibliography+ environment.  Each reference is a
\verb+\bibitem+ and each \verb+\bibitem+ is identified by a label,
by which it can be cited in the text:

\verb+\bibitem[Elson et al.(1996)]{ESG96}+ is cited as
\verb+\citet{ESG96}+. 

\noindent In connection with cross-referencing and
possible future hyperlinking it is not a good idea to collect
more that one literature item in one \verb+\bibitem+.  The
so-called Harvard or author-year style of referencing is enabled
by the \LaTeX{} package \file{natbib}. With this package the
literature can be cited as follows:

\begin{enumerate}[\textbullet]
\item Parenthetical: \verb+\citep{WB96}+ produces (Wettig \& Brown, 1996).
\item Textual: \verb+\citet{ESG96}+ produces Elson et al. (1996).
\item An affix and part of a reference:
\verb+\citep[e.g.][Ch. 2]{Gea97}+ produces (e.g. Governato et
al., 1997, Ch. 2).
\end{enumerate}

In the numbered scheme of citation, \verb+\cite{<label>}+ is used,
since \verb+\citep+ or \verb+\citet+ has no relevance in the numbered
scheme.  \file{natbib} package is loaded by \file{elsarticle} with
\verb+numbers+ as default option.  You can change this to author-year
or harvard scheme by adding option \verb+authoryear+ in the class
loading command.  If you want to use more options of the \file{natbib}
package, you can do so with the \verb+\biboptions+ command, which is
described in the section \ref{sec:usage}, \nameref{sec:usage}.  For
details of various options of the \file{natbib} package, please take a
look at the \file{natbib} documentation, which is part of any standard
\LaTeX{} installation.

In addition to the above standard \verb+.bst+ files, there are 10
journal-specific \verb+.bst+ files also available.
Instruction for using these \verb+.bst+ files can be found at 
\href{http://support.stmdocs.in/wiki/index.php?title=Model-wise_bibliographic_style_files}
{http://support.stmdocs.in}

\section[Graphical ...]{Graphical abstract and highlights}
A template for adding graphical abstract and highlights are available
now. This will appear as the first two pages of the PDF before the
article content begins.

\pagebreak
Please refer below to see how to code them.

\begin{vquote}
....
....

\end{abstract}

\begin{graphicalabstract}
\end{graphicalabstract}

\begin{highlights}
\item Research highlight 1
\item Research highlight 2
\end{highlights}

\begin{keyword}
....
....
\end{vquote}

\section{Final print}\label{sec:final}

The authors can format their submission to the page size and margins
of their preferred journal.  \file{elsarticle} provides four
class options for the same. But it does not mean that using these
options you can emulate the exact page layout of the final print copy. 


\lmrgn=3em
\begin{description}
\item [\texttt{1p}:] $1+$ journals with a text area of
384pt $\times$ 562pt or 13.5cm $\times$ 19.75cm or 5.3in $\times$
7.78in, single column style only.

\item [\texttt{3p}:] $3+$ journals with a text area of 468pt
$\times$ 622pt or 16.45cm $\times$ 21.9cm or 6.5in $\times$
8.6in, single column style.

\item [\texttt{twocolumn}:] should be used along with 3p option if the
journal is $3+$ with the same text area as above, but double column
style. 

\item [\texttt{5p}:] $5+$ with text area of 522pt $\times$
682pt or 18.35cm $\times$ 24cm or 7.22in $\times$ 9.45in,
double column style only.
\end{description}

Following pages have the clippings of different parts of
the title page of different journal models typeset in final
format.

Model $1+$ and $3+$  will have the same look and
feel in the typeset copy when presented in this document. That is
also the case with the double column $3+$ and $5+$ journal article
pages. The only difference will be wider text width of
higher models.  Here are the specimen single and double column journal 
pages.



\begin{comment}
\begin{center}
\hypertarget{bsc}{}
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{\bf [Specimen single column article -- Click here]}
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{\bf [Specimen double column article -- Click here]}
}
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\end{document}


\section{}
\label{}






\section{}
\label{}







\section{Introduction}
\label{}
Distortion or missing data problems often occur in obtained signals and images owing to the limitations of image sensing, compression artifacts, and transmission errors [1]. This paper examines the problem associated with image restoration [2][3], that is, the reconstruction or estimation of missing data in images, which are immeasurable or corrupted by noise. Fundamentally, image restoration is an uncertain inverse problem-solving process, and appropriate signal priors are necessary to regulate this underdetermined inverse problem. The use of prior information about images can be model-based, which explicitly represents priors, such as total variation (TV) [4][5] and sparsity priors [6][7], or neural network-based, which explores priors by learning, such as deep learning (DL) [8]. Generally, DL-based methods provide superior performance compared to model-based methods in image restoration. However, pure DL schemes require large training data with annotations that are time-consuming and difficult to obtain. In contrast, model-based image restoration methods are relatively straightforward.

If the positions of the pixels in an image are considered as nodes and the neighborhood relations between pixel points as edges, then the geometry of the image can be considered as a graph, and the color or brightness information of the image can be considered as a signal defined on the graph. Thus, graph signal processing (GSP) technology [9][10][11] can be used for image restoration. The GSP technology performs the common signal operations, such as filtering, smoothing, and Fourier transform, on the signals defined on a graph. It then combines the signal information defined on the graph with the common graph operations, such as the graph Laplacian matrix [12], to form the prior information of the graph. Owing to these merits, GSP has become a state-of-the-art method for analyzing irregular structures [13][14]. 

Model-based image restoration methods generally include fidelity and regularization terms. Prior information is typically represented by a regularization term. A commonly used fidelity term is convolutional sparse representation [15][16]. Similar to the Fourier transform or wavelet transform, a signal can be decomposed and represented by a linear combination of a small number of dictionary elements. "Sparse" denotes that there are many dictionary elements present; however, only a few are necessary for a given signal. When the linear combination is substituted by a convolutional operation, the sparse representation is extended to convolutional sparse representation or convolutional sparse coding (CSC), which allows a sparsity prior to be applied globally rather than locally [17]. Common CSC models used for image restoration include the convolutional basis pursuit denoising (CBPDN) model [18] and online convolutional sparse coding [19].

Liu and Wohlberg first combined the CSC model with graph signal information to address the image inpainting problem [20]. In this method, an image was partitioned into overlapping patches, each of which corresponded to a vertex of the graph, and the similarity information between the image patches was represented by the edge weights of the graph. The graph signals defined on the vertices were the color or brightness vectors corresponding to the image patches. The Dirichlet energy of the graph signal was then used as a regularization term owing to its non-local smoothing properties.

Another method of constructing a graph signal through an image is the dual graph method [21][22]. The dual graph includes a row graph and a column graph. A vertex of the row graph corresponds to a row of the image, and a vertex of the column graph corresponds to a column of the image. The definitions of edge weights and signal information are similar to those of the single graph defined by [20]. The structure of the dual graph is simple compared to that of the single graph, and it relaxes the conditions, which is beneficial in image restoration when using graph signal priors.

In this paper, the dual graph signal prior was combined with the CBPDN model for image restoration. The contribution of this paper is twofold.
\begin{enumerate}
\item Introducing the dual graph prior information into the CSC model can solve the shortcomings of the $l^{1}$ regularizer in CSC and also compensate for the insufficiency of the single graph prior in [20]. To the best of our knowledge, this is the first paper to apply dual graph prior information to CSC.
\item A CSC model with prior information of the dual graph is proposed in this paper. The image denoising results of the proposed model are significantly better than those of the CSC model with single graph prior information, in terms of objective evaluation and visual quality.
\end{enumerate}


The outline of this paper is as follows. Section 2 introduces the related work and motivation of this article. Section 3 introduces the CSC model with a dual graph smoothing prior. Section 4 discusses the solution process of the model used in this peper. The experimental results and analysis are presented in Section 5. Finally, Section 6 provides a summary of this paper. 
\section{Related Work}
CSC is a classic image restoration model. It is a new variant of sparse coding, in which the entire signal or image is decomposed into a set of filters and the convolutional sum of the corresponding coefficient maps. The coefficient maps and signals or images are of the same size and have corresponding dictionary filters. A prominent formula for this problem is the CBPDN:
\begin{equation}\mathop {\min }\limits_{{{\mathbf{x}}_k}} \frac{1}{2}\left\| {\sum\limits_k {{{\mathbf{d}}_k}*{{\mathbf{x}}_k} - {\mathbf{s}}} } \right\|_2^2 + \lambda \sum\limits_k {{{\left\| {{{\mathbf{x}}_k}} \right\|}_1}} \label{eq}\end{equation}
where $\{ {{\mathbf{d}}_k}\} $ is a set of dictionary filters, $ * $ denotes convolution, and $\{ {{\mathbf{x}}_k}\} $ is a set of coefficient maps.

The $l^{1}$ regularizer in the standard CSC model can lead to high spatial sparsity of coefficient mapping, which consequently reduces the number of training image patches that play a role in the formation of the dictionary. Wohlberg et al. proposed an improved form of CBPDN that includes an additional regularization term based on the Laplacian of the non-local graph of the image. The graph is constructed by introducing each image patch corresponding to a vertex of the graph into the graph Laplacian regularizer $\sum\limits_k {\left\langle {{{\mathbf{x}}_k},{\mathbf{L}}{{\mathbf{x}}_k}} \right\rangle } $, as shown in Equation (2). Thus, the regularizer is an explicit penalty to enforce similar image patches with similar sparse representations.
\begin{equation}\mathop {\min }\limits_{{{\mathbf{x}}_k}} \frac{1}{2}\left\| {\sum\limits_k {{{\mathbf{d}}_k}*{{\mathbf{x}}_k} - {\mathbf{s}}} } \right\|_2^2 + \lambda \sum\limits_k {{{\left\| {{{\mathbf{x}}_k}} \right\|}_1}\rm{ + }}\frac{\mu }{2}\sum\limits_k {\left\langle {{{\mathbf{x}}_k},{\mathbf{L}}{{\mathbf{x}}_k}} \right\rangle} 
 \end{equation}

where ${\mathbf{L}}$ is a graph Laplacian matrix for a non-local graph.
However, this method has its shortcomings. In the image, the rows and columns share similar information. The above method ignores the adjacent information between the image rows and columns. Specifically, it uses the local information of the image, without fully considering the information and similarity between the rows and columns of the image. Therefore, in this paper, this article propose another variant of CBPDN that uses dual graph smoothing priors to regularize feature maps. The dual graph Laplacian regularizer [21] is introduced by using the rows and columns of the image as the vertices to form the row and column graphs, respectively. This variant effectively utilizes the adjacent information of image rows and columns. This paper adopted ideas from manifold learning [23] to constrain the solution for this article to be smooth on the graphs to implicitly enforce row and column proximities.
\section{Proposed Model}
In this section, this paper discuss the processes for constructing row  and column graphs, and introduce the CSC model with a dual graph smoothing prior.
\subsection{Column Graph}
This article uses $\mathbf{s} \in \mathbb{R}^{M \times N}$ to represent the original image matrix and consider each column of ${\mathbf{s}}$ as a node to form an undirected weighted column graph ${{\cal G}_c} = \{ {{\cal V}_c},{{\cal E}_c},{{\cal W}_c}\} $, its nodes ${{\cal V}_c} \in \{ 1,2, \cdots ,N\} $, edges ${{\cal E}_c} \in {{\cal V}^c} \times {{\cal V}^c}$, and weight matrix $\mathbf{W}_{c} \in \mathbb{R}^{N \times N}$. The weight matrix $\mathbf{W}_{c}$ describes the pairwise similarity between nodes in $\mathcal{G}_{c}$. $\mathbf{W}_{c}$ is defined as: 
\begin{equation}{{\mathbf{W}}_c}(i,j) = \left\{ {\begin{array}{*{20}{c}}
{\exp ( - \frac{{{{\rm{d}}_{ij}}^2}}{{{\delta ^2}}}),j \in {{\cal N}_i}}\\
{0,o.w.}
\end{array}} \right. \end{equation}
where $\mathcal{N}_{i}$ is the neighborhood defined by $i$ as the center pixel, $\delta$ is a factor that controls the shape of the function for weight calculation, and ${{\rm{d}}_{ij}}$  is a measure of the cosine distance [24] between vertex $i$ and vertex $j$, with a value range of [-1,1]. The more similar the two vertices, the closer they are to one. Thus, the graph $\mathcal{G}_{c}$ is completely and uniquely determined by the weight matrix $\mathbf{W}_{c}$.

Next, this article define the graph Laplacian matrix corresponding to column graph $\mathcal{G}_{c}$ as $\mathbf{L}_{c}=\mathbf{D}_{c}-\mathbf{W}_{c} \in \mathbb{R}^{N \times N}$, where the degree matrix $\mathbf{D}_{c}$ is a diagonal matrix with entries $
\mathbf{D}_{c}(i, i)=\sum_{j \neq i} \mathbf{W}_{\mathbf{c}}(i, i)
$. Given that the edge weights are non-negative in (3), it can be proved that the matrix $\mathbf{L}_{c}$ is positive semi-definite (PSD). This paper consider ${{\mathbf{x}}_k}$ as a collection of M-dimensional column vectors denoted by subscripts ${{\mathbf{x}}_k} = ({{\mathbf{x}}_{k,1}},{{\mathbf{x}}_{k,2}}, \cdots ,{{\mathbf{x}}_{k,N}})$. Regarding the columns ${{\mathbf{x}}_{k,1}},{{\mathbf{x}}_{k,2}}, \cdots ,{{\mathbf{x}}_{k,N}}$ as a vector-valued function defined on the vertices ${{\cal V}_c}$, the assumption of the smoothness of the implies that ${{\mathbf{x}}_{k,i}} \approx {{\mathbf{x}}_{k,j}}$, if $(i,j) \in {{\cal E}_c}$. Specifically, \begin{equation}\sum\limits_{i,j \in {{\cal V}_c}} {{{\mathbf{W}}_c}} (i,j){({{\mathbf{x}}_{k,i}} - {{\mathbf{x}}_{k,j}})^2} = {\mathop{\rm tr}\nolimits} ({{\mathbf{x}}_k}{{\mathbf{L}}_c}{\mathbf{x}}_k^{\rm T}) \end{equation}
needs to be small. Equation (4) is referred to as the column graph Laplacian regularizer (CGLR) of the feature maps ${{\mathbf{x}}_k}$. When the nodes $i$ and $j$ are more similar, ${{\mathbf{W}}_c}(i,j)$ is smaller; thus, when similar vertices present similar values, Equation (4) becomes small.
\subsection{Row Graph}
Row graphs can be defined similarly to the column graphs. Each row of ${\mathbf{s}}$ is considered as a node to form an undirected weighted row graph ${{\cal G}_r} = \{ {{\cal V}_r},{{\cal E}_r},{{\cal W}_r}\} $, with nodes ${{\cal V}_r} = \{ 1,2, \cdots ,M\} $, edges ${{\cal E}_r} \in {{\cal V}^r} \times {{\cal V}^r}$, and weight matrix $\mathbf{W}_{r} \in \mathbb{R}^{M \times M}$. The graph Laplacian matrix of ${{\cal G}_r}$ is $\mathbf{L}_{r}=\mathbf{D}_{r}-\mathbf{W}_{r} \in \mathbb{R}^{M \times M}$. This paper consider ${{\mathbf{x}}_k}$ as a collection of N-dimensional row vectors denoted by subscripts ${{\mathbf{x}}_k} = {(\left(\mathbf{x}_{k}^{1}\right)^{\mathrm{T}},\left(\mathbf{x}_{k}^{2}\right)^{\mathrm{T}}, \cdots ,\left(\mathbf{x}_{k}^{M}\right))^{\mathrm{T}}}$. Regarding the rows $\left(\mathbf{x}_{k}^{1}\right)^{\mathrm{T}},\left(\mathbf{x}_{k}^{2}\right)^{\mathrm{T}}, \cdots,\left(\mathbf{x}_{k}^{M}\right)^{\mathrm{T}}$   
 as a vector-valued function defined on the vertices 
 ${{\cal V}_r}$, the row graph Laplacian regularizer (RGLR) of the feature maps ${{\mathbf{x}}_k}$ is ${\mathop{\rm tr}\nolimits}({\mathbf{x}}_k^{\rm T}{{\mathbf{L}}_r}{{\mathbf{x}}_k})$.
 \subsection{CSC Model with Dual Graph Smoothing Prior}
By combining CGLR and RGLR, the dual graph Laplacian regularizer (DGLR) of the feature maps ${{\mathbf{x}}_k}$ can be obtained.
\begin{equation}{\mathop{\rm G}\nolimits} ({{\mathbf{x}}_k}) = {\mathop{\rm tr}\nolimits} ({{\mathbf{x}}_k}{{\mathbf{L}}_c}{\mathbf{x}}_k^{\rm T}) + {\mathop{\rm tr}\nolimits} ({\mathbf{x}}_k^{\rm T}{{\mathbf{L}}_r}{{\mathbf{x}}_k})\end{equation}
The CSC model with DGLR is as follows:
\[\mathop {\min }\limits_{\{ {{\mathbf{x}}_k}\} } \frac{1}{2}\left\| {\sum\limits_k {{{\mathbf{d}}_k}*{{\mathbf{x}}_k} - {\mathbf{s}}} } \right\|_2^2 + \lambda \sum\limits_k {{{\left\| {{{\mathbf{x}}_k}} \right\|}_1}}  + \frac{\alpha }{2}\sum\limits_k {{\mathop{\rm tr}\nolimits} ({{\mathbf{x}}_k}{{\mathbf{L}}_c}{\mathbf{x}}_k^{\rm T})}  \]
\begin{equation}+ \frac{\beta }{2}\sum\limits_k {{\mathop{\rm tr}\nolimits} ({\mathbf{x}}_k^{\rm T}{{\mathbf{L}}_r}{{\mathbf{x}}_k})} \end{equation}
$\lambda$, $\alpha$, and $\beta $ are trade-off parameters between the three regularization terms.

\section{Energy Optimization}
This paper employed the alternating direction method of multiplication (ADMM) [25] framework to solve Equation (6). Based on the ADMM single split method, Equation (6) is rewritten by separating the variable $\{ {{\mathbf{y}}_k}\} $,
\[\mathop {\min }\limits_{\{ {{\mathbf{x}}_k}\} } \frac{1}{2}\left\| {\sum\limits_k {{{\mathbf{d}}_k}*{{\mathbf{x}}_k} - {\mathbf{s}}} } \right\|_2^2 + \lambda \sum\limits_k {{{\left\| {{{\mathbf{y}}_k}} \right\|}_1}}  + \frac{\alpha }{2}\sum\limits_k {{\mathop{\rm tr}\nolimits} ({{\mathbf{y}}_k}{{\mathbf{L}}_c}{\mathbf{y}}_k^{\rm T})} \]
\begin{equation} + \frac{\beta }{2}\sum\limits_k {{\mathop{\rm tr}\nolimits} ({\mathbf{y}}_k^{\rm T}{{\mathbf{L}}_r}{{\mathbf{y}}_k})} 	\qquad {\mathop{\rm s}\nolimits} .t.\;{{\mathbf{x}}_k} = {{\mathbf{y}}_k}\end{equation}
Subsequently, equation (7) can be decomposed into the following three sub-problems:

\noindent(a) First, we fix the auxiliary variable$\{ {{\mathbf{y}}_k}\} $ and the multiplier $\{ {{\mathbf{u}}_k}\} $, and update the coefficient maps $\{ {{\mathbf{x}}_k}\} $.
\[{\{ {{\mathbf{x}}_k}\} ^{(t + 1)}} = \mathop {\min }\limits_{\{ {{\mathbf{x}}_k}\} } \frac{1}{2}\left\| {\sum\limits_k {{{\mathbf{d}}_k}*{{\mathbf{x}}_k} - {\mathbf{s}}} } \right\|_{\rm{2}}^{\rm{2}}\]
\begin{equation} + \frac{\rho }{2}\sum\limits_k {\left\| {{{\mathbf{x}}_k} - {\mathbf{y}}_k^{(t)} + {\mathbf{u}}_k^{(t)}} \right\|_{\rm{2}}^{\rm{2}}} \end{equation}
(b) We then fix the coefficient maps $\left\{ {{{\mathbf{x}}_k}} \right\}$ and the multiplier $\left\{ {{{\mathbf{u}}_k}} \right\}$, and update the auxiliary variable $\left\{ {{{\mathbf{y}}_k}} \right\}$.
\[{\{ {{\mathbf{y}}_k}\} ^{(t + 1)}} = \mathop {\min }\limits_{\{ {{\mathbf{y}}_k}\} } \lambda \sum\limits_k {{{\left\| {{{\mathbf{y}}_k}} \right\|}_1}}  + \frac{\alpha }{2}\sum\limits_k {{\mathop{\rm tr}\nolimits} ({{\mathbf{y}}_k}{{\mathbf{L}}_c}{\mathbf{y}}_k^{\rm T}) } \]
\begin{equation}+\frac{\beta }{2}\sum\limits_k {{\mathop{\rm tr}\nolimits} ({\mathbf{y}}_k^{\rm T}{{\mathbf{L}}_r}{{\mathbf{y}}_k})}  + \frac{\rho }{2}\sum\limits_k {\left\| {{\mathbf{x}}_k^{(t + 1)} - {{\mathbf{y}}_k} + {\mathbf{u}}_k^{(t)}} \right\|_2^2} \end{equation}
(c) Finally, the fixed coefficient maps $\left\{ {{{\mathbf{x}}_k}} \right\}$, the auxiliary variable $\{ {{\mathbf{y}}_k}\} $, and the multiplier $\left\{ {{{\mathbf{u}}_k}} \right\}$ are updated.
\begin{equation}{{\rm{\{ }}{{\mathbf{u}}_k}\} ^{(t + 1)}} = {{\mathbf{u}}_k} + {\mathbf{x}}_k^{(t + 1)} - {\mathbf{y}}_k^{(t + 1)}\end{equation}
The update of $\left\{ {{{\mathbf{x}}_k}} \right\}$ is similar to that in the standard convolutional learning case, and it can be solved effectively in the Fourier domain [26]. $\{ {{\mathbf{y}}_k}\} $ can also be solved using the ADMM algorithm. We separate the variable $\{ {{\mathbf{z}}_k}\} $ and rewrite Equation (9).
\[{\{ {{\mathbf{y}}_k}\} ^{(t + 1)}} = \mathop {\min }\limits_{\{ {{\mathbf{y}}_k}\} } \lambda \sum\limits_k {{{\left\| {{{\mathbf{y}}_k}} \right\|}_1}}  + \frac{\alpha }{2}\sum\limits_k {{\mathop{\rm tr}\nolimits} ({{\mathbf{z}}_k}{{\mathbf{L}}_c}{\mathbf{z}}_k^{\rm T})  } \]
\[+\frac{\beta }{2}\sum\limits_k {{\mathop{\rm tr}\nolimits} ({\mathbf{z}}_k^{\rm T}{{\mathbf{L}}_r}{{\mathbf{z}}_k})}  + \frac{\rho }{2}\sum\limits_k {\left\| {{\mathbf{x}}_k^{(t + 1)} - {{\mathbf{y}}_k} + {\mathbf{u}}_k^{(t)}} \right\|_2^2} \]
\begin{equation}{\mathop{\rm s}\nolimits} .t.\;{{\mathbf{y}}_k} = {{\mathbf{z}}_k}\end{equation}
Subsequently, we introduce the Lagrange multiplier $\left\{ {{{\mathbf{v}}_k}} \right\}$, and Equation (11) can be decomposed into the following three sub-problems:
\[{\{ {{\mathbf{y}}_k}\} ^{(t + 1)}} = \mathop {\min }\limits_{\{ {{\mathbf{y}}_k}\} } \lambda \sum\limits_k {{{\left\| {{{\mathbf{y}}_k}} \right\|}_1}}  + \frac{\rho }{2}\sum\limits_k {\left\| {{\mathbf{x}}_k^{(t + 1)} - {{\mathbf{y}}_k} + {\mathbf{u}}_k^{(t)}} \right\|_2^2} \]
\begin{equation} + \frac{\eta }{2}\sum\limits_k {\left\| {{{\mathbf{y}}_k}-{\mathbf{  z}}_k^{(t)} + {\mathbf{v}}_k^{(t)}} \right\|_2^2} \end{equation}
\[{\{ {{\mathbf{z}}_k}\} ^{(t + 1)}} = \mathop {\min }\limits_{\{ {{\mathbf{z}}_k}\} } \frac{\alpha }{2}\sum\limits_k {{\mathop{\rm tr}\nolimits} ({{\mathbf{z}}_k}{{\mathbf{L}}_c}{\mathbf{z}}_k^{\rm T}) + } \frac{\beta }{2} \sum\limits_k {{\mathop{\rm tr}\nolimits}({\mathbf{z}}_k^{\rm T}{{\mathbf{L}}_r}{{\mathbf{z}}_k})} \]
\begin{equation} + \frac{\eta }{2}\sum\limits_k {\left\| {{\mathbf{y}}_k^{(t + 1)} - {{\mathbf{z}}_k} + {\mathbf{v}}_k^{(t)}} \right\|_2^2} \end{equation}
\begin{equation}{{\rm{\{ }}{{\mathbf{v}}_k}\} ^{(t + 1)}} = {{\mathbf{v}}_k} + {\mathbf{y}}_k^{(t + 1)} - {\mathbf{z}}_k^{(t + 1)}\end{equation}
$\left\{ {{{\mathbf{y}}_k}} \right\}$ can be efficiently solved using the soft-threshold method. The update of $\left\{ {{{\mathbf{z}}_k}} \right\}$ is a convex optimization problem; hence, a closed solution can be directly obtained. To solve the unconstrained QP problem (13), this article refer to [21] to derive the objective function for ${{\mathbf{z}}_k}$, then set it to zero, and solve for ${{\mathbf{z}}_k}$ to obtain the linear equation system of unknown ${\mathop{\rm vec}\nolimits} ({\mathbf{z}}_k^*)$:
\[(\alpha {{\mathbf{L}}_c} \otimes {{\mathbf{I}}_M} + \beta {{\mathbf{I}}_N} \otimes {{\mathbf{L}}_r} + \eta {\mathbf{I}}){\mathop{\rm vec}\nolimits} ({\mathbf{z}}_k^*)\]
\begin{equation} = \eta {\mathop{\rm vec}\nolimits} ({{\mathbf{y}}_k} + {{\mathbf{v}}_k}),k = 1,2, \cdots \end{equation}
Here, ${\mathop{\rm vec}\nolimits} ( \cdot )$ refers to the vector form of a matrix superimposed by columns, and $ \otimes $ is a defined operator. The calculation result of $\mathbf{Q}=\alpha \mathbf{L}_{c} \otimes \mathbf{I}_{M}+\beta \mathbf{I}_{N} \otimes \mathbf{L}_{r}, \mathbf{Q} \in \mathbb{R}^{M N \times M N}$ is represented by Equation (16). Because the coefficient matrix ${\mathbf{Q}}$ is generally symmetric, sparse, and positive definite, a plethora of mature numerical linear algebra methods such as conjugate gradient (CG) [27] can be used to solve equation (15) effectively. A unique advantage of using the dual graph signal smoothness prior in Equation (1) is that it requires only solving a system of linear equations when computing its solution. For the relative residual stopping rule, this paper refer to [28].
\[
\mathbf{Q}=\alpha\left[\begin{array}{cccc}
l_{11} \mathbf{I}_{M} & l_{12} \mathbf{I}_{M} & \cdots & l_{1 N} \mathbf{I}_{M} \\
l_{21} \mathbf{I}_{M} & l_{22} \mathbf{I}_{M} & \cdots & l_{2 N} \mathbf{I}_{M} \\
\vdots & \vdots & \ddots & \vdots \\
l_{N 1} \mathbf{I}_{M} & l_{N 2} \mathbf{I}_{M} & \cdots & l_{N N} \mathbf{I}_{M}
\end{array}\right]\]
\begin{equation}+\beta\left[\begin{array}{cccc}
\mathbf{L}_{r} & 0 & \cdots & 0 \\
0 & \mathbf{L}_{r} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \mathbf{L}_{r}
\end{array}\right]
\end{equation}
Here, $l_{i j}, i, j=1,2, \cdots, N$ is the element in the row and column of the matrix ${{\mathbf{L}}_c}$.
\section{Experiment}
This paper used random Gaussian white noise for the denoising experiments to verify the performance of the proposed model in image restoration. In this paper, two types of datasets, namely, "kodim" dataset and "drone" image dataset, were used to perform Gaussian white noise denoising experiments. The experiments were run on a computer configured with IntelCorei7-6700U and a CPU of 3.41 GHz, using Matlab 2018b version.

\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0cm}

 \begin{center}
\includegraphics[scale=0.4]{fig.png}
\end{center}

\caption{The average denoising results of our model and SCSC under different noise levels.}

\label{fig1}

\end{figure}
 
 To evaluate the robustness of this model to different noise levels, noise levels of 5$\%$, 10$\%$, 15$\%$, 20$\%$, 25$\%$, 30$\%$, and 35$\%$, were used to distort the clean image. Separate denoising experiments were performed for each level of noise corruption. Furthermore, this article compared the denoising performance of this model with three other models: 1) CSC; 2) CSC based on a single graph (SCSC); and 3) three-dimensional block matching method (BM3D), which is a popular denoising technology [29]. The proposed CSC model based on the dual graph is denoted as DCSC herein. SCSC and DCSC are extensions of the CSC model, and were compared based on innovation. BM3D is the most popular denoising method among current methods. The addition of BM3D as a comparison method was based on its effective denoising effect.
This paper used the PSNR value of the image and visualization as a metric to compare the denoising performance of the models. 

Each model had several parameters to be selected during the verification phase of the experiment. Both SCSC and DCSC added different regular terms to the CSC model. To perform a fair comparison of the three models, this paper first adjusted the parameters of the CSC to the optimal values, and subsequently adjusted the remaining parameters in the SCSC and DCSC models. After debugging, this article set the parameters of (2) and (6) to $\lambda=0.28$, $\quad \mu=0.1$ and$\quad \alpha=\beta=0.2$. In the convolutional sparse coding model, the dictionary needs to be trained through the training set. In order to avoid the amount of calculation and to ensure the fairness of the experiment, this paper uses the same independently trained 12×12×36 dictionary for all models to solve the model.

\begin{table*}
\setlength{\abovecaptionskip}{-1.5cm}
\setlength{\belowcaptionskip}{0.1cm}
\renewcommand\arraystretch{1.2}
\caption{Comparison of PSNR values of denoising results of different methods under different noise levels}
\label{table1}
\centering
\begin{tabular}{|m{1.5cm}<{\centering}|m{1.5cm}<{\centering}|m{1.5cm}<{\centering}| m{1.5cm}<{\centering}|m{1.5cm}<{\centering}|m{1.5cm}<{\centering}|m{1.5cm}<{\centering}|m{1.5cm}<{\centering}|m{1.5cm}<{\centering}|}
  \hline
 \multirow{2}{*}{Image} &  \multicolumn{4} {|c|} { $\delta {\rm{ = }}15$ }&   \multicolumn{4} {|c|} { $\delta {\rm{ = }}20$}   \\    
  \cline{2-9}
           
                        & CSC   & SCSC  & BM3D  & Ours  & CSC   & SCSC  & BM3D  & Ours   \\
   \hline
1                       & 31.32 & 31.29 & \bf34.19 & 31.49 & 31.17 & 31.14 & \bf33.00    & 31.30   \\
\hdashline
2                       & 27.40  & 27.35 & \bf30.68 & 27.57 & 27.36 & 27.33 & \bf29.24 & 27.53  \\
\hdashline
3                       & 26.80  & 26.75 & \bf29.79 & 26.97 & 26.73 & 26.69 & \bf28.36 & 26.90   \\
\hdashline
4                       & 24.89 & 24.85 & \bf28.19 & 25.06 & 24.89 & 24.85 & \bf26.68 & 25.06  \\
\hdashline
5                       & 30.20  & 30.17 & \bf32.95 & 30.35 & 30.14 & 30.11 & 30.25 & \bf30.27  \\
\hdashline
6                       & 30.97 & 30.97 & \bf32.52 & 31.14 & 30.75 & 30.73 & \bf32.19 & 30.91  \\
\hdashline
7                       & 25.15 & 25.09 & \bf28.73 & 25.34 & 25.07 & 25.02 & \bf27.07 & 25.24  \\
\hdashline
8                       & 30.13 & 30.08 & \bf33.70  & 30.32 & 29.87 & 29.84 & \bf32.25 & 30.05  \\
\hdashline
9                       & 28.36 & 28.31 & \bf31.53 & 28.53 & 28.25 & 28.20  & \bf30.03 & 28.40   \\
\hdashline
10                      & 31.59 & 31.54 & \bf33.36 & 31.75 & 31.29 & 31.25 & \bf32.92 & 31.42  \\
\hdashline
average                 & 28.69 & 28.64 & \bf31.56 & 28.85 & 28.55 & 28.52 & \bf30.19 & 28.71  \\
 \hline
 \multirow{2}{*}{Image} &  \multicolumn{4} {|c|} { $\delta {\rm{ = }}25$ }&   \multicolumn{4} {|c|} { $\delta {\rm{ = }}30$}   \\    
    \cline{2-9}     
                       & CSC   & SCSC  & BM3D  & Ours  & CSC   & SCSC  & BM3D  & Ours   \\
     \hline
1                       & 30.47 & 30.46 & \bf31.97 & 30.48 & 29.20  & 29.02 & 29.18 & \bf29.29  \\
\hdashline
2                       & 27.17 & 27.16 & \bf28.34 & 27.29 & 26.42 & 26.43 & 26.44 & \bf26.46  \\
\hdashline
3                       & 26.62 & 26.59 & 26.70 & \bf26.72 & 25.97 & 26.00    & 25.98 & \bf26.00     \\
\hdashline
4                       & 24.81 & 24.78 & 24.83 & \bf24.94 & 24.42 & 24.43 & 24.45 & \bf24.49  \\
\hdashline
5                       & 29.73 & 29.73 & 29.74 & \bf29.75 & 28.55 & 28.41 & 28.59 & \bf28.63  \\
\hdashline
6                       & 30.17 & 30.18 & \bf30.69 & 30.19 & 29.00    & 29.09 & \bf29.15 & 28.84  \\
\hdashline
7                       & 24.91 & 24.89 & \bf25.23 & 25.05 & 24.49 & 24.51 & 24.53 & \bf24.57  \\
\hdashline
8                       & 29.48 & 29.47 & 29.52 & \bf29.53 & 28.32 & 28.21 & 28.34 & \bf28.39  \\
\hdashline
9                       & 27.90  & 27.89 & 27.95 & \bf27.98 & 27.18 & 27.14 & 27.20  & \bf27.23  \\
\hdashline
10                      & 30.64 & 30.62 & 30.54 & \bf30.65 & 29.30  & 29.10  & 29.36 & \bf29.38  \\
\hdashline
average                 & 28.19 & 28.18 & \bf28.55 & 28.26 & 27.29 & 27.23 & 27.32 & \bf27.33 \\
\hline
\end{tabular}
\label{table1}
\end{table*}

\begin{figure*}[htbp]
\setlength{\abovecaptionskip}{0cm}

 \begin{center}

  \subfigure[Clean image]{\includegraphics[width=0.3\textwidth]{fig2-a.png}}
  \subfigure[Noisy image ($\delta {\rm{ = }}30$)]
 {\includegraphics[width=0.3\textwidth]{fig2-b.png}}
  \subfigure[CSC (PSNR:29.30)]{\includegraphics[width=0.3\textwidth]{fig2-c.png}}
  \subfigure[SCSC (PSNR:29.10)]{\includegraphics[width=0.3\textwidth]{fig2-d.png}}
  \subfigure[BM3D (PSNR:29.36)]{\includegraphics[width=0.3\textwidth]{fig2-e.png}}
  \subfigure[Ours (PSNR:\bf29.38)]{\includegraphics[width=0.3\textwidth]{fig2-f.png}}

 \end{center}

\caption{Denoising results on image parrot from the "kodim" dataset by different methods $\delta {\rm{ = }}30$.}

\label{fig2}

\end{figure*}

\begin{figure*}[htbp]
\setlength{\abovecaptionskip}{0cm}

 \begin{center}

  \subfigure[Clean image]{\includegraphics[width=0.15\textwidth]{fig3-a.png}}\hspace{0.1mm}
  \subfigure[Noisy image]{\includegraphics[width=0.15\textwidth]{fig3-b.png}}\hspace{0.1mm}
  \subfigure[CSC(25.56 dB)]{\includegraphics[width=0.15\textwidth]{fig3-c.png}}\hspace{0.1mm}
  \subfigure[SCSC(25.51 dB)]{\includegraphics[width=0.15\textwidth]{fig3-d.png}}\hspace{0.1mm}
  \subfigure[GCSC(25.40 dB)]{\includegraphics[width=0.15\textwidth]{fig3-e.png}}\hspace{0.1mm}
  \subfigure[Ours(\bf25.66 dB)]{\includegraphics[width=0.15\textwidth]{fig3-f.png}}\hspace{0.1mm}
   \subfigure[Clean image]{\includegraphics[width=0.15\textwidth]{fig4-a.png}}\hspace{0.1mm}
  \subfigure[Noisy image]{\includegraphics[width=0.15\textwidth]{fig4-b.png}}\hspace{0.1mm}
  \subfigure[CSC(27.84 dB)]{\includegraphics[width=0.15\textwidth]{fig4-c.png}}\hspace{0.1mm}
  \subfigure[SCSC(27.81 dB)]{\includegraphics[width=0.15\textwidth]{fig4-d.png}}\hspace{0.1mm}
  \subfigure[GCSC(27.74 dB)]{\includegraphics[width=0.15\textwidth]{fig4-e.png}}\hspace{0.1mm}
  \subfigure[Ours(\bf27.91 dB)]{\includegraphics[width=0.15\textwidth]{fig4-f.png}}\hspace{0.1mm}

 \end{center}

\caption{Denoising results on images "highway" and "beach" from the UVA dataset by different methods $\delta {\rm{ = }}20$.}

\label{fig3}

\end{figure*}




Table \ref{table1} shows the PSNR value comparison of the denoising results of 10 images in the "kodim" dataset with different models. The most outstanding results (highest PSNR value) at each noise level are highlighted in bold. It can be observed that the denoising performance of the model proposed in this article is comparable to that of the BM3D method. When the noise level was lower than 25$\%$, the denoising performance of BM3D was relatively high; however, when the noise level was higher than 25$\%$, the performance of the DCSC approximated or exceeded that of BM3D. This is because the BM3D algorithm mostly led to inaccurate block matching for images with high complexity or different noise intensities. Fig.\ref{fig1} shows the average denoising results of DCSC and SCSC under different noise levels. It can be observed that the PSNR value of the denoising result of the proposed model is higher than that of the SCSC. Fig.\ref{fig2} provides a visual display of the denoising results of a 768×512 image with 30$\%$ Gaussian noise in the "kodim" dataset with different denoising models. Based on the visualization results, it can be observed that the model proposed in this article achieved better denoising results than SCSC, and contained fewer noise points. Although the BM3D result appears clearer and smoother than that of the proposed model, it is excessively smooth, resulting in the loss of image details.

To further verify the performance of the model, a substantial number of images in the "drone" dataset were randomly selected for experiments, and the experimental results were compared with those of the recently proposed GCSC model that adds a gradient regular term [30] to the CSC model. Table \ref{table2} shows the average PSNR value of the denoising results of 10 images under different models, and the noise level was between 15$\%$ to 35$\%$. The best result (the highest PSNR value) under each noise level presented the most coarseness. It can be seen from Table \ref{table2} that the model proposed in this article maintained the highest PSNR value under different noise levels. Finally, a visual comparison of denoising results of "highway" and "beach" images captured by a UAV under different models is shown in Fig.\ref{fig3}, and the noise level is 20$\%$. The noise points of the model proposed in this article were less than those of the other models; therefore, the clarity of the image was improved.

\begin{table}
\setlength{\abovecaptionskip}{0cm}
\setlength{\belowcaptionskip}{0.1cm}
\renewcommand\arraystretch{1.5}
\caption{Display of the average PSNR value of the de-noising results of 10 UAV images in different models}
\label{table2}
\centering
\begin{tabular}{|m{1.2cm}<{\centering}|m{1.2cm}<{\centering}|m{1.2cm}<{\centering}| m{1.2cm}<{\centering}|m{1.2cm}<{\centering}|}  

\hline
 $\delta $ & CSC   & SCSC  & GCSC  & Ours   \\ 
\hline
15 & 26.74 & 26.57 & 26.54 & \bf26.85  \\ 
\hdashline
20 & 26.58 & 26.54 & 26.44 & \bf26.70  \\ 
\hdashline
25 & 26.36 & 26.34 & 26.26 & \bf26.43  \\ 
\hdashline
30 & 25.75 & 25.77 & 25.79 & \bf25.79  \\ 
\hdashline
35 & 24.64 & 24.75 & \bf24.92 & 24.76  \\
\hline
\end{tabular}
\label{table2}
\end{table}


The maximum number of iterations of the experiment in this article is 50. In order to make the experimental results more convincing, this article shows in Table \ref{table3} the average PSNR value corresponding to the denoising results of the 10 UAV images in Table \ref{table2} under different models when the experimental iterations are 25 times. It can be seen from Table \ref{table3} that when the experiment is iterated for 25 times, the changing trend of the denoising results of each model under different noise levels is consistent with that of Table \ref{table2}, and the model proposed in this paper still maintains the highest PSNR value.



\begin{table}
\setlength{\abovecaptionskip}{0cm}
\setlength{\belowcaptionskip}{0.1cm}
\renewcommand\arraystretch{1.5}
\caption{The average PSNR value of the denoising results of 10 UAV images in 25 iterations of different models}
\label{table3}
\centering
\begin{tabular}{|m{1.2cm}<{\centering}|m{1.2cm}<{\centering}|m{1.2cm}<{\centering}| m{1.2cm}<{\centering}|m{1.2cm}<{\centering}|}  

\hline
 $\delta $ & CSC   & SCSC  & GCSC  & Ours   \\ 
\hline
15 & 26.72 & 26.67 & 26.52 & \bf26.83  \\ 
\hdashline
20 & 26.60 & 26.54 & 26.43 & \bf26.68  \\ 
\hdashline
25 & 26.35 & 26.32 & 26.25 & \bf26.42  \\ 
\hdashline
30 & 25.64 & 25.67 & 25.70 & \bf25.71  \\ 
\hdashline
35 & 24.58 & 24.58 & \bf24.86 & 24.67  \\
\hline
\end{tabular}
\label{table3}
\end{table}


\section{Summary}
In this paper, the CSC model was integrated with the dual graph Laplacian regularizer to effectively utilize the similarity between image rows and columns. Compared with a single graph, the dual graph has a simple structure, relaxes the conditions, and is more conducive to image restoration using image signal priors. The corresponding optimization problem was formulated, and the ADMM algorithm was employed in the DFT domain. Finally, denoising experiments were performed using Gaussian white noise. The results proved that the model proposed in this article is superior to other models integrated with a single graph Laplacian regularization term. However, this article also has shortcomings, which are also areas that need to be improved in the future. Although the introduction of the dual graph improves the denoising performance of the original model to a certain extent, the dual graph is constructed from the original input image. Its structure is not updated with the update of the feature map, so there is no guarantee that each feature map has the structure of the map defined on the input image. From a theoretical analysis, if the dual graph is constructed on the feature graph, then it is constantly updated according to the iteration of the algorithm, which can further limit the smoothness of the feature graph on the graph and make the model obtain better performance. This perspective can be used as a research direction of the CSC model in the future. 





