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\usepackage{graphicx} 
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\title{Interpretable Matching of Optical-SAR Image via Dynamically Conditioned Diffusion Models}

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% \author{Ben Trovato}
% \authornote{Both authors contributed equally to this research.}
% \email{trovato@corporation.com}
% \orcid{1234-5678-9012}
% \author{G.K.M. Tobin}
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%   \streetaddress{P.O. Box 1212}
%   \city{Dublin}
%   \state{Ohio}
%   \country{USA}
%   \postcode{43017-6221}
% }

\author{Shuiping Gou}
\email{shpgou@mail.xidian.edu.cn}
\affiliation{%
	\institution{Xidian University}
	\city{Xi'an}
	\country{China}
}


\author{Xin Wang}
\email{23171214441@stu.xidian.edu.cn}
\affiliation{%
	\institution{Xidian University}
	\city{Xi'an}
	\country{China}
}

\author{Xinlin Wang}
\authornote{Corresponding author.}
\email{wangxinlin@xidian.edu.cn}
%%\orcid{1234-5678-9012}
\affiliation{%
	\institution{Xidian University}
	\city{Xi'an}
	\country{China}
}

\author{Yunzhi Chen}
\email{2016010026@hzvtc.edu.cn}
\affiliation{%
	\institution{Hangzhou Vocational and Technical College}
	\city{Hangzhou}
	\country{China}
}


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%%
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%% the work being presented. Separate the keywords with commas.
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%   Your, Paper}

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%   \caption{Seattle Mariners at Spring Training, 2010.}
%   \Description{Enjoying the baseball game from the third-base
%   seats. Ichiro Suzuki preparing to bat.}
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% \received{20 February 2007}
% \received[revised]{12 March 2009}
% \received[accepted]{5 June 2009}

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\maketitle

\begin{figure*}[t]
	\centering
	\includegraphics[width=\textwidth]{./Figures/supp_sen.pdf}
	\caption{Visualisation of translated SAR template images (Template $\hat{S}$), the translated features (Template $O'$), and the similarity maps from different areas of the OSdataset and SEN1-2 datasets. The auto-encoder trained on the OSdataset possesses a certain degree of universality, which is a one-time investment.}
	\label{supp_sen}
\end{figure*}

\section{PRELIMINARIES}
Most diffusion models are based on the framework of DDPMs \cite{ddpm}, which consist of a forward diffusion process and a reverse generation process, both of which are modelled as Markov chains. During forward diffusion, Gaussian noise is slowly added to the original data ${x}_{0}$ according to a fixed variance schedule $\left\{\beta_{1}, \beta_{2} \ldots \beta_{T}\right\}$ in $T$ steps. Finally, a sequence of noisy samples \{${x}_{0}, ..., {x}_{t}, ..., {x}_{T}$\} is produced, where $x_T \sim N(0, \mathbf{I})$. Each diffusion step is formulated as: 
\begin{equation}
	q\left(x_t \mid x_{t-1}\right):=N\left(x_t ; \sqrt{1-\beta_t} x_{t-1}, \beta_t \mathbf{I}\right)
\end{equation}

By setting the $\alpha_t=1-\beta_t$ and the $\overline{\alpha_t}=\Pi_{t=1}^T \alpha_i$, $x_T$ can be obtained at any moment $t$ by the following equation:
\begin{equation}
	q\left(x_t \mid x_0\right):=N\left(x_t ; \sqrt{\bar{\alpha}_t} x_0,\left(1-\bar{\alpha}_t\right) \mathbf{I}\right)
\end{equation}

During the reverse generation process $p\left(x_{t-1} \mid x_t\right)$, the noise is gradually removed and the original data is reconstructed. To solve the problem, the DDPM learns the parametric Gaussian transform $p_\theta\left(x_{t-1} \mid x_t\right)$. Essentially, it predicts the mean of the Gaussian distribution $\mu_\theta\left(x_t, t\right)$. The reverse process is expressed as follows:
\begin{equation}
	p_\theta\left(x_{t-1} \mid x_t\right)=N\left(x_{t-1} ; \mu_\theta\left(x_t, t\right), \sigma_t^2 \mathbf{I}\right)
\end{equation}

During training, the denoising neural network $f_\theta\left(x_t, t\right)$ is trained to minimize the training objective via L2 loss, that is, predicting $x_0$ from the $x_t$ \cite{ddpm} :
\begin{equation}
	L=\left\|f_\theta\left(x_t, t\right)-x_0\right\|^2
\end{equation}
In the inference phase, data samples $x_0$ are reconstructed from the noise $x_T$ in an iterative manner using the model $f_\theta$ and updating rules \cite{ddpm}.

\begin{figure*}[t]  
	\centering  
	\includegraphics[width=\textwidth]{./Figures/atten_map.pdf}  
	\caption{Visualisation of ground truth, pre-calibrated, and post-calibrated attention maps on 4 samples. The two samples (a) and (b) are from urban built-up areas, and the samples (s) and (d) are from Rural farmland areas. Samples (a) and (c) are from the OSdataset and samples (b) and (d) are from the SEN1-2 datasets. These results show that the proposed attention calibration block has an effective local calibration capability and some global calibration capability.}
	\label{atten_map}
\end{figure*}

\section{Additional Experimental Results}
\subsection{The universality of interpretable latent space}

The interpretable latent space proposed is derived from an auto-encoder pre-trained on the OSdataset. However, SAR images from different remote sensors exhibit distinct characteristics. To demonstrate the matching generality of this interpretable latent space, as depicted in Figure \ref{supp_sen}, inference was conducted on images from both the OSdataset and the SEN1-2 datasets. The translated images (Template $\hat{S}$), translated features (Template $O'$), and similarity maps were visualized. For samples from the OSdataset, the translated template images and features closely resemble the actual template images (Template $S$) and features (Template $S'$), yielding correct matching results. Conversely, on the diverse SEN1-2 datasets, the translated SAR templates exhibit reduced overall brightness compared to the ground truth SAR templates, owing to the higher overall brightness in the OSdataset. Nevertheless, this disparity does not affect the matching. Moreover, the SAR images in the SEN1-2 datasets display more noise than those in the OSdataset. Despite this, the translated images and features exhibit less noise while preserving the structural connectivity features. Furthermore, the resulting similarity maps continue to exhibit smooth single peaks, precisely located within the correct matching regions. These observations collectively underscore the generality of the auto-encoder trained on the OSdataset, representing a one-time investment.

\subsection{Visualisation of the impact of the ACB}
The proposed Attention Calibration Block (ACB) operates on 2D attention maps following cross-attention. To assess the smoothness and calibration efficacy of our proposed ACB on attention maps, as depicted in Figure \ref{atten_map}, we visualize the ground truth, pre-correction, and post-correction attention maps of four samples during the inference process on two datasets. Specifically, we selected an urban built-up area and a rural cropland area for each dataset, with the rural farmland area posing greater matching difficulty due to its numerous highly similar areas. Samples (a) and (c) are from the OSdataset, while samples (b) and (d) are from SEN1-2. The sampling steps were set to 5, and at the final sampling iteration, we visualize the attention map before and after ACB with a feature stride of 8.

Observing the attention maps, it is evident that they exhibit diagonal sparsity. For samples (a) and (b) from the urban built-up area, cross-attention accurately perceives rich texture details; however, the attention maps display numerous points with inconsistent values and a few with incorrect locations. Following ACB, the corrected attention maps no longer contain tokens with incorrect locations and are smoothed. It is noteworthy that while ACB effectively calibrates the attention map, a slight inconsistency with the ground truth is observed. Nonetheless, our translation results still yield correct matches. This is because although the attention map contains information about the match location, refining the attention region step by step is necessary, and direct matching using the attention map is suboptimal compared to matching using the translated image.

Rural farmland areas comprise numerous similar rectangular regions, often resulting in erroneous perceptions by cross-attention. In samples (c) and (d), it is evident that the values and locations of most tokens in the uncalibrated attention map are confusing and inconsistent. However, surprisingly, the ACB module can still consolidate all the information from the attentional graph to produce a more consistent attentional graph. Calibration in rural farmland areas is slightly inferior compared to urban built-up areas due to the challenges associated with matching in rural farmland areas. These findings demonstrate that the proposed attention calibration block possesses effective local calibration capabilities and some global calibration capabilities, leading to more accurate attention perception and matching effects.

\subsection{Analysis of Interpretability}
Existing methods extract common features of multimodal images for matching, which is abstract, and hard to understand its similarity.
%The common features is difficulty to define, which is hard to determine the reasons for matching failures. 
However, our approach is built on cross-modal translation, which translates one modality into the other modality in the latent space, and completes matching in the translated feature space. We adopt structural similarity index measure (SSIM) to constrain the translation process. Figure \ref{fig_inter} (a) visually shows the matching feature similarities of SAR-optical images. It shows that our model can capture similar structures (red circle) via modality translation, whereas common cross-modal features extracted by MARU-Net are inconsistent. Figure \ref{fig_inter} (b) quantitatively shows that the more similar the features, the better the matching. 

\begin{figure*}[t] 
	\centering 
	\includegraphics[width=\linewidth]{./Figures/fig_inter.pdf}
	\caption{The interpretability explanation.}
	\label{fig_inter}
\end{figure*}

\subsection{Analysis of Robustness} 
Most methods extract unknown common features of multimodal images, which is vulnerable to the interference. 
Our translation-based method focuses on consistency structural features, which is insensitive to noises.
Figure \ref{fig_noise} visually compares the correct matching rate (CMR) \cite{LoFTR,IF} of SOTA methods on noisy optical template images in the OSdataset without further training. 
It shows that the CMR of all methods decreases with the addition of more noises.
However, the CMR of our model only has a very slight drop when the noise variance is less than 5\%, and consistently remains higher than that of others on different noisy data. 

\subsection{Analysis of Generalization} 
In the submission, the experimental datasets, OSdataset and SEN1-2, consist of optical-SAR image pairs.
To evaluate the generalization, we conduct experiments on an extra NIR (Near Infrared)-SAR dataset containing 100 NIR-SAR image pairs with a size of 5556 $\times$ 3704 pixels and 5m spatial resolution. These large-scene images are crop into 5521 image pairs with the size of 512 $\times$ 512. The NIR images are treated as templates, with the training, validation and testing datasets split in a 7:2:1 ratio. 
Firstly, we trained SOTA models on the NIR-SAR dataset (w/ Training) to evaluate each approach. 
In addition, we tested the NIR-SAR testing dataset directly using SOTA models trained on the OSdataset (w/o Training) to further assess the generalization.
Table \ref{tab2} quantitatively shows the generalization performance from different views.  

\begin{table}[!h]
	\caption{The comparison of SOTAs on the NIR-SAR dataset. }
	\label{tab2}	
	\setlength{\tabcolsep}{1.3mm}
	\renewcommand{\arraystretch}{1.0} 	
	\scalebox{0.64}{
		\begin{tabular}{c|c c c ||c c c }
			\hline
			\multirow{2}{*}{Methods} & \multicolumn{3}{c||}{w/ Training} & \multicolumn{3}{c}{w/o Training} \\ 
			\cline{2-7}
			& {CMR (T=5)} & {RMSE (T=5)} & {RMSE (all)} & {CMR (T=5)} & {RMSE (T=5)} & {RMSE (all)}\\ 
			\hline
			OSMnet   & 0.7852 & 2.7871 & 13.2734 & 0.5917 & 3.1245 & 16.1871 \\ 
			MARU-Net & 0.8129 & 2.4707 & 10.1147 & 0.6226 & 2.6082 & 15.9857 \\ 
			RSOMNet\cite{IF}  & 0.8160 & \textbf{2.1015} & 8.9358  & 0.6832 & \textbf{2.3044} & 14.0608\\
			Ours     & \textbf{0.8341} & 2.2312 & \textbf{8.7958}  & \textbf{0.7188} & 2.3874 & \textbf{11.2455} \\ 
			\hline
	\end{tabular}}
\end{table}

\begin{table}[h]
	\caption{The performance of SOTA methods on two datasets. }
	\label{tab1}	
	\setlength{\tabcolsep}{0.75mm}
	\renewcommand{\arraystretch}{1.0} 	
	\begin{threeparttable} %
		\scalebox{0.64}{
			\begin{tabular}{c|c c c c||c c c c}
				\hline
				\multirow{2}{*}{Methods} & \multicolumn{4}{c||}{OSdataset} & \multicolumn{4}{c}{SEN1-2} \\ 
				\cline{2-9}
				& {CMR(T=5)} & {RMSE(T=5)} & {RMSE(all)} & {Time(s)} & {CMR(T=5)} & {RMSE(T=5)} & {RMSE(all)} & {Time(s)}\\ 
				\hline
				RSOMNet & 0.8296 & 2.0834 & 7.9752 & 0.1847 &0.9108  &1.4793 &5.0752 & 0.1021  \\
				\hline \hline  
				& {Pars (M)} & {Com (G)} & {Mem (GB)} & {Time (s)} & {Pars (M)} & {Com (G)} & {Mem (GB)} & {Time (s)}\\ 
				\hline
				VSMatch & 3.06 & 2.2e5 & 0.86 & 451.2569 &3.06  &5.5e4 &0.52 & 87.155  \\ 
				%		\hline
				OSMnet & 5.53 & 42.92 & 1.29 & 0.0913 &5.53 & 11.2& 0.67& 0.0568 \\ 
				%		\hline
				MARU-Net & 18.82 & 39.6 & 1.14 & 0.0895 & 18.82 & 10.3 & 0.64 &0.0346 \\ 
				RSOMNet & 13.57 & 140.9 & 3.24 & 0.1847 & 13.57 & 21.6 & 1.20 &0.1152 \\ 
				%		\hline
				Ours (w/o)& 20.7+5.6 & 61.7 & 2.09 & 0.1572 & 20.7+5.6 & 13.4 &1.01 &0.1021 \\ 
				%		\hline
				Ours (w/)& 21.1+5.6 & 51.1\scalebox{0.8}{$\downarrow$} & 1.45\scalebox{0.8}{$\downarrow$} & 0.1093 & 21.1+5.6 & 12.6\scalebox{0.8}{$\downarrow$} &0.74\scalebox{0.8}{$\downarrow$} &0.0621 \\ 
				\hline
		\end{tabular}}
		\begin{tablenotes}\scriptsize %Ìí¼Ó´Ë´¦
			\item 			
			Pars: Parameters  $\quad$ 
			Com: Computational complexity $\quad$
			Mem: Memory usage $\quad$
		\end{tablenotes} %Ìí¼Ó´Ë´¦
	\end{threeparttable} %
\end{table}

\subsection{Model Performance}
Figure \ref{fig_vis} visually compares SOTA models on SEN1-2 dataset, which displays the superiority of our method. 
Table \ref{tab1} provides a quantitative comparison of an advanced matching method published in 2024, RSOMNet\cite{IF}. It is observed that our method performs better than RSOMNet on all metrics on the SEN1-2 dataset. For the OSdataset, our approach (RMSE(T=5)=2.29) is inferior to RSOMNet (RMSE(T=5)=2.0834) on the RMSE(T=5). The OSdataset is a high-resolution dataset, which contains richer details. However, our translation-based method focuses more on the structural features. Therefore, our method is not state-of-the-art on the pixel-level metric RMSE, but shows promising match accuracy.  
`
\begin{figure}[H] 
	\centering 
	\includegraphics[width=0.9\linewidth]{./FIgures/noise.pdf}
	\caption{CMR of SOTAs on noising OSdataset.}
	\label{fig_noise}
\end{figure}

\begin{figure*}[t] 
	\centering 
	\includegraphics{./Figures/vis2.pdf}
	\caption{The visualization of SOTAs on SEN1-2 dataset.}
	\label{fig_vis}
\end{figure*}

\subsection{Model Parameters} 
Table \ref{tab1} further details parameters, computation, and memory usage of several SOTA methods during inference.   
For the large-scale OSdataset, the sparse attention mechanism (Ours(w/)), compared to the method without sparse attention (Ours(w/o)), significantly reduces memory and computation resource usage while only increasing parameters by 1.5\%.
Our model consists of both trainable and frozen parameters (the encoder), where trainable parameters are comparable to MARU-Net's. RSOMNet requires very large computation resources.

%Although our model contains some frozen parameters, 

%ACM's consolidated article template, introduced in 2017, provides a
%consistent \LaTeX\ style for use across ACM publications, and
%incorporates accessibility and metadata-extraction functionality
%necessary for future Digital Library endeavors. Numerous ACM and
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%of publication, from review to final ``camera-ready'' copy, to the
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%\begin{itemize}
%\item {\verb|acmconf|}: The default proceedings template style.
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%\item{\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.
%\item{\verb|sigplan|}: Used for SIGPLAN conference articles.
%\end{itemize}
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%In addition to specifying the {\itshape template style} to be used in
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%which modify some part of the applied template style. A complete list
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%\item{\verb|authorversion|}: Produces a version of the work suitable
%  for posting by the author.
%\item{\verb|screen|}: Produces colored hyperlinks.
%\end{itemize}
%
%This document uses the following string as the first command in the
%source file:
%\begin{verbatim}
%\documentclass[sigconf,authordraft]{acmart}
%\end{verbatim}
%
%\section{Modifications}
%
%Modifying the template --- including but not limited to: adjusting
%margins, typeface sizes, line spacing, paragraph and list definitions,
%and the use of the \verb|\vspace| command to manually adjust the
%vertical spacing between elements of your work --- is not allowed.
%
%{\bfseries Your document will be returned to you for revision if
%  modifications are discovered.}
%
%\section{Typefaces}
%
%The ``\verb|acmart|'' document class requires the use of the
%``Libertine'' typeface family. Your \TeX\ installation should include
%this set of packages. Please do not substitute other typefaces. The
%``\verb|lmodern|'' and ``\verb|ltimes|'' packages should not be used,
%as they will override the built-in typeface families.
%
%\section{Title Information}
%The title of your work should use capital letters appropriately -
%\url{https://capitalizemytitle.com/} has useful rules for
%capitalization. Use the {\verb|title|} command to define the title of
%your work. If your work has a subtitle, define it with the
%{\verb|subtitle|} command.  Do not insert line breaks in your title.
%
%If your title is lengthy, you must define a short version to be used
%in the page headers, to prevent overlapping text. The \verb|title|
%command has a ``short title'' parameter:
%\begin{verbatim}
%  \title[short title]{full title}
%\end{verbatim}
%
%\section{Authors and Affiliations}
%
%Each author must be defined separately for accurate metadata
%identification. Multiple authors may share one affiliation. Authors'
%names should not be abbreviated; use full first names wherever
%possible. Include authors' e-mail addresses whenever possible.
%
%Grouping authors' names or e-mail addresses, or providing an ``e-mail
%alias,'' as shown below, is not acceptable:
%\begin{verbatim}
%  \author{Brooke Aster, David Mehldau}
%  \email{dave,judy,steve@university.edu}
%  \email{firstname.lastname@phillips.org}
%\end{verbatim}
%
%The \verb|authornote| and \verb|authornotemark| commands allow a note
%to apply to multiple authors --- for example, if the first two authors
%of an article contributed equally to the work.
%
%If your author list is lengthy, you must define a shortened version of
%the list of authors to be used in the page headers, to prevent
%overlapping text. The following command should be placed just after
%the last \verb|\author{}| definition:
%\begin{verbatim}
%  \renewcommand{\shortauthors}{McCartney, et al.}
%\end{verbatim}
%Omitting this command will force the use of a concatenated list of all
%of the authors' names, which may result in overlapping text in the
%page headers.
%
%The article template's documentation, available at
%\url{https://www.acm.org/publications/proceedings-template}, has a
%complete explanation of these commands and tips for their effective
%use.
%
%Note that authors' addresses are mandatory for journal articles.
%
%\section{Rights Information}
%
%Authors of any work published by ACM will need to complete a rights
%form. Depending on the kind of work, and the rights management choice
%made by the author, this may be copyright transfer, permission,
%license, or an OA (open access) agreement.
%
%Regardless of the rights management choice, the author will receive a
%copy of the completed rights form once it has been submitted. This
%form contains \LaTeX\ commands that must be copied into the source
%document. When the document source is compiled, these commands and
%their parameters add formatted text to several areas of the final
%document:
%\begin{itemize}
%\item the ``ACM Reference Format'' text on the first page.
%\item the ``rights management'' text on the first page.
%\item the conference information in the page header(s).
%\end{itemize}
%
%Rights information is unique to the work; if you are preparing several
%works for an event, make sure to use the correct set of commands with
%each of the works.
%
%The ACM Reference Format text is required for all articles over one
%page in length, and is optional for one-page articles (abstracts).
%
%\section{CCS Concepts and User-Defined Keywords}
%
%Two elements of the ``acmart'' document class provide powerful
%taxonomic tools for you to help readers find your work in an online
%search.
%
%The ACM Computing Classification System ---
%\url{https://www.acm.org/publications/class-2012} --- is a set of
%classifiers and concepts that describe the computing
%discipline. Authors can select entries from this classification
%system, via \url{https://dl.acm.org/ccs/ccs.cfm}, and generate the
%commands to be included in the \LaTeX\ source.
%
%User-defined keywords are a comma-separated list of words and phrases
%of the authors' choosing, providing a more flexible way of describing
%the research being presented.
%
%CCS concepts and user-defined keywords are required for for all
%articles over two pages in length, and are optional for one- and
%two-page articles (or abstracts).
%
%\section{Sectioning Commands}
%
%Your work should use standard \LaTeX\ sectioning commands:
%\verb|section|, \verb|subsection|, \verb|subsubsection|, and
%\verb|paragraph|. They should be numbered; do not remove the numbering
%from the commands.
%
%Simulating a sectioning command by setting the first word or words of
%a paragraph in boldface or italicized text is {\bfseries not allowed.}
%
%\section{Tables}
%
%The ``\verb|acmart|'' document class includes the ``\verb|booktabs|''
%package --- \url{https://ctan.org/pkg/booktabs} --- for preparing
%high-quality tables.
%
%Table captions are placed {\itshape above} the table.
%
%Because tables cannot be split across pages, the best placement for
%them is typically the top of the page nearest their initial cite.  To
%ensure this proper ``floating'' placement of tables, use the
%environment \textbf{table} to enclose the table's contents and the
%table caption.  The contents of the table itself must go in the
%\textbf{tabular} environment, to be aligned properly in rows and
%columns, with the desired horizontal and vertical rules.  Again,
%detailed instructions on \textbf{tabular} material are found in the
%\textit{\LaTeX\ User's Guide}.
%
%Immediately following this sentence is the point at which
%Table~\ref{tab:freq} is included in the input file; compare the
%placement of the table here with the table in the printed output of
%this document.
%
%\begin{table}
%  \caption{Frequency of Special Characters}
%  \label{tab:freq}
%  \begin{tabular}{ccl}
%    \toprule
%    Non-English or Math&Frequency&Comments\\
%    \midrule
%    \O & 1 in 1,000& For Swedish names\\
%    $\pi$ & 1 in 5& Common in math\\
%    \$ & 4 in 5 & Used in business\\
%    $\Psi^2_1$ & 1 in 40,000& Unexplained usage\\
%  \bottomrule
%\end{tabular}
%\end{table}
%
%To set a wider table, which takes up the whole width of the page's
%live area, use the environment \textbf{table*} to enclose the table's
%contents and the table caption.  As with a single-column table, this
%wide table will ``float'' to a location deemed more
%desirable. Immediately following this sentence is the point at which
%Table~\ref{tab:commands} is included in the input file; again, it is
%instructive to compare the placement of the table here with the table
%in the printed output of this document.
%
%\begin{table*}
%  \caption{Some Typical Commands}
%  \label{tab:commands}
%  \begin{tabular}{ccl}
%    \toprule
%    Command &A Number & Comments\\
%    \midrule
%    \texttt{{\char'134}author} & 100& Author \\
%    \texttt{{\char'134}table}& 300 & For tables\\
%    \texttt{{\char'134}table*}& 400& For wider tables\\
%    \bottomrule
%  \end{tabular}
%\end{table*}
%
%Always use midrule to separate table header rows from data rows, and
%use it only for this purpose. This enables assistive technologies to
%recognise table headers and support their users in navigating tables
%more easily.
%
%\section{Math Equations}
%You may want to display math equations in three distinct styles:
%inline, numbered or non-numbered display.  Each of the three are
%discussed in the next sections.
%
%\subsection{Inline (In-text) Equations}
%A formula that appears in the running text is called an inline or
%in-text formula.  It is produced by the \textbf{math} environment,
%which can be invoked with the usual
%\texttt{{\char'134}begin\,\ldots{\char'134}end} construction or with
%the short form \texttt{\$\,\ldots\$}. You can use any of the symbols
%and structures, from $\alpha$ to $\omega$, available in
%\LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few
%examples of in-text equations in context. Notice how this equation:
%\begin{math}
%  \lim_{n\rightarrow \infty}x=0
%\end{math},
%set here in in-line math style, looks slightly different when
%set in display style.  (See next section).
%
%\subsection{Display Equations}
%A numbered display equation---one set off by vertical space from the
%text and centered horizontally---is produced by the \textbf{equation}
%environment. An unnumbered display equation is produced by the
%\textbf{displaymath} environment.
%
%Again, in either environment, you can use any of the symbols and
%structures available in \LaTeX\@; this section will just give a couple
%of examples of display equations in context.  First, consider the
%equation, shown as an inline equation above:
%\begin{equation}
%  \lim_{n\rightarrow \infty}x=0
%\end{equation}
%Notice how it is formatted somewhat differently in
%the \textbf{displaymath}
%environment.  Now, we'll enter an unnumbered equation:
%\begin{displaymath}
%  \sum_{i=0}^{\infty} x + 1
%\end{displaymath}
%and follow it with another numbered equation:
%\begin{equation}
%  \sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f
%\end{equation}
%just to demonstrate \LaTeX's able handling of numbering.
%
%\section{Figures}
%
%The ``\verb|figure|'' environment should be used for figures. One or
%more images can be placed within a figure. If your figure contains
%third-party material, you must clearly identify it as such, as shown
%in the example below.
%
%
%\begin{figure}[h]
%  \centering
%    \fbox{\rule{0pt}{2.5in} \rule{0.9\linewidth}{0pt}}
%  % \includegraphics[width=\linewidth]{sample-franklin}
%  \caption{Example of caption}
%\end{figure}
%
%
%Your figures should contain a caption which describes the figure to
%the reader.
%
%Figure captions are placed {\itshape below} the figure.
%
%Every figure should also have a figure description unless it is purely
%decorative. These descriptions convey what’s in the image to someone
%who cannot see it. They are also used by search engine crawlers for
%indexing images, and when images cannot be loaded.
%
%A figure description must be unformatted plain text less than 2000
%characters long (including spaces).  {\bfseries Figure descriptions
%  should not repeat the figure caption – their purpose is to capture
%  important information that is not already provided in the caption or
%  the main text of the paper.} For figures that convey important and
%complex new information, a short text description may not be
%adequate. More complex alternative descriptions can be placed in an
%appendix and referenced in a short figure description. For example,
%provide a data table capturing the information in a bar chart, or a
%structured list representing a graph.  For additional information
%regarding how best to write figure descriptions and why doing this is
%so important, please see
%\url{https://www.acm.org/publications/taps/describing-figures/}.
%
%\subsection{The ``Teaser Figure''}
%
%A ``teaser figure'' is an image, or set of images in one figure, that
%are placed after all author and affiliation information, and before
%the body of the article, spanning the page. If you wish to have such a
%figure in your article, place the command immediately before the
%\verb|\maketitle| command:
%\begin{verbatim}
%  \begin{teaserfigure}
%    \includegraphics[width=\textwidth]{sampleteaser}
%    \caption{figure caption}
%    \Description{figure description}
%  \end{teaserfigure}
%\end{verbatim}
%
%\section{Citations and Bibliographies}
%
%The use of \BibTeX\ for the preparation and formatting of one's
%references is strongly recommended. Authors' names should be complete
%--- use full first names (``Donald E. Knuth'') not initials
%(``D. E. Knuth'') --- and the salient identifying features of a
%reference should be included: title, year, volume, number, pages,
%article DOI, etc.
%
%The bibliography is included in your source document with these two
%commands, placed just before the \verb|\end{document}| command:
%\begin{verbatim}
%  \bibliographystyle{ACM-Reference-Format}
%  \bibliography{bibfile}
%\end{verbatim}
%where ``\verb|bibfile|'' is the name, without the ``\verb|.bib|''
%suffix, of the \BibTeX\ file.
%
%Citations and references are numbered by default. A small number of
%ACM publications have citations and references formatted in the
%``author year'' style; for these exceptions, please include this
%command in the {\bfseries preamble} (before the command
%``\verb|\begin{document}|'') of your \LaTeX\ source:
%\begin{verbatim}
%  \citestyle{acmauthoryear}
%\end{verbatim}
%
%  Some examples.  A paginated journal article \cite{Abril07}, an
%  enumerated journal article \cite{Cohen07}, a reference to an entire
%  issue \cite{JCohen96}, a monograph (whole book) \cite{Kosiur01}, a
%  monograph/whole book in a series (see 2a in spec. document)
%  \cite{Harel79}, a divisible-book such as an anthology or compilation
%  \cite{Editor00} followed by the same example, however we only output
%  the series if the volume number is given \cite{Editor00a} (so
%  Editor00a's series should NOT be present since it has no vol. no.),
%  a chapter in a divisible book \cite{Spector90}, a chapter in a
%  divisible book in a series \cite{Douglass98}, a multi-volume work as
%  book \cite{Knuth97}, a couple of articles in a proceedings (of a
%  conference, symposium, workshop for example) (paginated proceedings
%  article) \cite{Andler79, Hagerup1993}, a proceedings article with
%  all possible elements \cite{Smith10}, an example of an enumerated
%  proceedings article \cite{VanGundy07}, an informally published work
%  \cite{Harel78}, a couple of preprints \cite{Bornmann2019,
%    AnzarootPBM14}, a doctoral dissertation \cite{Clarkson85}, a
%  master's thesis: \cite{anisi03}, an online document / world wide web
%  resource \cite{Thornburg01, Ablamowicz07, Poker06}, a video game
%  (Case 1) \cite{Obama08} and (Case 2) \cite{Novak03} and \cite{Lee05}
%  and (Case 3) a patent \cite{JoeScientist001}, work accepted for
%  publication \cite{rous08}, 'YYYYb'-test for prolific author
%  \cite{SaeediMEJ10} and \cite{SaeediJETC10}. Other cites might
%  contain 'duplicate' DOI and URLs (some SIAM articles)
%  \cite{Kirschmer:2010:AEI:1958016.1958018}. Boris / Barbara Beeton:
%  multi-volume works as books \cite{MR781536} and \cite{MR781537}. A
%  couple of citations with DOIs:
%  \cite{2004:ITE:1009386.1010128,Kirschmer:2010:AEI:1958016.1958018}. Online
%  citations: \cite{TUGInstmem, Thornburg01, CTANacmart}. Artifacts:
%  \cite{R} and \cite{UMassCitations}.
%
%\section{Acknowledgments}
%
%Identification of funding sources and other support, and thanks to
%individuals and groups that assisted in the research and the
%preparation of the work should be included in an acknowledgment
%section, which is placed just before the reference section in your
%document.
%
%This section has a special environment:
%\begin{verbatim}
%  \begin{acks}
%  ...
%  \end{acks}
%\end{verbatim}
%so that the information contained therein can be more easily collected
%during the article metadata extraction phase, and to ensure
%consistency in the spelling of the section heading.
%
%Authors should not prepare this section as a numbered or unnumbered {\verb|\section|}; please use the ``{\verb|acks|}'' environment.
%
%\section{Appendices}
%
%If your work needs an appendix, add it before the
%``\verb|\end{document}|'' command at the conclusion of your source
%document.
%
%Start the appendix with the ``\verb|appendix|'' command:
%\begin{verbatim}
%  \appendix
%\end{verbatim}
%and note that in the appendix, sections are lettered, not
%numbered. This document has two appendices, demonstrating the section
%and subsection identification method.
%
%\section{Multi-language papers}
%
%Papers may be written in languages other than English or include
%titles, subtitles, keywords and abstracts in different languages (as a
%rule, a paper in a language other than English should include an
%English title and an English abstract).  Use \verb|language=...| for
%every language used in the paper.  The last language indicated is the
%main language of the paper.  For example, a French paper with
%additional titles and abstracts in English and German may start with
%the following command
%\begin{verbatim}
%\documentclass[sigconf, language=english, language=german,
%               language=french]{acmart}
%\end{verbatim}
%
%The title, subtitle, keywords and abstract will be typeset in the main
%language of the paper.  The commands \verb|\translatedXXX|, \verb|XXX|
%begin title, subtitle and keywords, can be used to set these elements
%in the other languages.  The environment \verb|translatedabstract| is
%used to set the translation of the abstract.  These commands and
%environment have a mandatory first argument: the language of the
%second argument.  See \verb|sample-sigconf-i13n.tex| file for examples
%of their usage.
%
%\section{SIGCHI Extended Abstracts}
%
%The ``\verb|sigchi-a|'' template style (available only in \LaTeX\ and
%not in Word) produces a landscape-orientation formatted article, with
%a wide left margin. Three environments are available for use with the
%``\verb|sigchi-a|'' template style, and produce formatted output in
%the margin:
%\begin{itemize}
%\item {\verb|sidebar|}:  Place formatted text in the margin.
%\item {\verb|marginfigure|}: Place a figure in the margin.
%\item {\verb|margintable|}: Place a table in the margin.
%\end{itemize}

%%
%% The acknowledgments section is defined using the "acks" environment
%% (and NOT an unnumbered section). This ensures the proper
%% identification of the section in the article metadata, and the
%% consistent spelling of the heading.
% \begin{acks}
% To Robert, for the bagels and explaining CMYK and color spaces.
% \end{acks}

%%
%% The next two lines define the bibliography style to be used, and
%% the bibliography file.
\bibliographystyle{ACM-Reference-Format}
\bibliography{sample-base}

%%
%% If your work has an appendix, this is the place to put it.
% \appendix

% \section{Research Methods}

% \subsection{Part One}

% Lorem ipsum dolor sit amet, consectetur adipiscing elit. Morbi
% malesuada, quam in pulvinar varius, metus nunc fermentum urna, id
% sollicitudin purus odio sit amet enim. Aliquam ullamcorper eu ipsum
% vel mollis. Curabitur quis dictum nisl. Phasellus vel semper risus, et
% lacinia dolor. Integer ultricies commodo sem nec semper.

% \subsection{Part Two}

% Etiam commodo feugiat nisl pulvinar pellentesque. Etiam auctor sodales
% ligula, non varius nibh pulvinar semper. Suspendisse nec lectus non
% ipsum convallis congue hendrerit vitae sapien. Donec at laoreet
% eros. Vivamus non purus placerat, scelerisque diam eu, cursus
% ante. Etiam aliquam tortor auctor efficitur mattis.

% \section{Online Resources}

% Nam id fermentum dui. Suspendisse sagittis tortor a nulla mollis, in
% pulvinar ex pretium. Sed interdum orci quis metus euismod, et sagittis
% enim maximus. Vestibulum gravida massa ut felis suscipit
% congue. Quisque mattis elit a risus ultrices commodo venenatis eget
% dui. Etiam sagittis eleifend elementum.

% Nam interdum magna at lectus dignissim, ac dignissim lorem
% rhoncus. Maecenas eu arcu ac neque placerat aliquam. Nunc pulvinar
% massa et mattis lacinia.

\end{document}
\endinput
%%
%% End of file `sample-authordraft.tex'.
