\section{Introduction}
\label{sec:intro}

Multimodal image fusion aims to integrate complementary information from different imaging modalities to produce a more informative and comprehensive representation than any single modality alone. In remote sensing, a widely studied task involves fusing a high-resolution (HR) multispectral image with a low-resolution (LR) hyperspectral image to generate a high-resolution hyperspectral image that simultaneously preserves fine spatial details and rich spectral information. The fused HR hyperspectral images find extensive applications in precision agriculture \cite{10850648}, environmental monitoring \cite{HIMEUR202244}, urban planning \cite{9877906}, and defense \cite{3636155}, enabling more accurate crop health assessment \cite{prasad2024design}, land cover classification \cite{roy2023multimodal}, pollution detection, and target recognition \cite{ma2021stdfusionnet}. However, this task presents significant challenges due to the intrinsic differences between multispectral and hyperspectral data, such as disparities in spatial resolution, sensor characteristics, and statistical distributions. Bridging the spatial-spectral gap effectively is essential for improving the quality of remote sensing imagery, driving scientific discovery, and supporting critical decision-making across various application domains. \par


% Multimodal image fusion combines complementary information from different modalities to produce richer representations. In remote sensing, a common task fuses high-resolution multispectral and low-resolution hyperspectral images to generate high-resolution hyperspectral imagery with fine spatial details and rich spectral content. Such fused images enable applications in precision agriculture \cite{10850648}, environmental monitoring \cite{HIMEUR202244}, and urban planning \cite{9877906}. However, significant challenges arise from differences in spatial resolution, sensor properties, and statistical distributions. Bridging this spatial-spectral gap is crucial for improving image quality and supporting decision-making across domains. \par


The fusion of HR multispectral and LR hyperspectral images aims to reconstruct an HR hyperspectral image that preserves both fine spatial structures and rich spectral details. In this process, the HR multispectral image provides detailed spatial information but limited spectral resolution, while the LR hyperspectral image offers abundant spectral signatures but lacks spatial precision \cite{10443302}. These two modalities can be regarded as complementary, degraded observations of the latent HR hyperspectral image—one in the spectral domain and the other in the spatial domain. Reconstructing the HR hyperspectral image from these sources constitutes a highly ill-posed inverse problem, further complicated by differences in resolution, sensor characteristics, and noise distributions \cite{jia2023multiscale}. The core challenges include accurately aligning spatial structures with corresponding spectral features, suppressing spectral distortions, and ensuring consistency across all spectral bands. Overcoming these challenges has garnered increasing attention in recent years, establishing HR multispectral and LR hyperspectral fusion as a key research focus in both remote sensing and multimodal image processing. \par

% The fusion of high-resolution (HR) multispectral and low-resolution (LR) hyperspectral images aims to reconstruct HR hyperspectral imagery that retains both fine spatial structures and rich spectral details. The HR multispectral image offers detailed spatial information but limited spectral resolution, while the LR hyperspectral image provides abundant spectral signatures with poor spatial precision \cite{10443302}. These modalities can be viewed as complementary degraded observations of the latent HR hyperspectral image. Reconstructing it constitutes a highly ill-posed inverse problem, further complicated by resolution disparities, sensor differences, and noise \cite{jia2023multiscale}. Core challenges include aligning spatial and spectral features, suppressing distortions, and maintaining consistency across bands. Addressing these issues has attracted growing interest, making this fusion task a central topic in remote sensing and multimodal image processing.\par 

% Currently, extensive research has focused on enhancing image detail through multimodal fusion, and numerous methods for fusing HR multispectral and LR hyperspectral images are proposed. These approaches can be broadly categorized into three groups: decomposition-based methods, deep learning-based methods, and pan-sharpening-based methods \cite{dian2021recent}. Decomposition-based methods aim to reconstruct the latent HR hyperspectral image by factorizing it into low-dimensional components that capture its underlying spatial and spectral structures. Typically, these approaches model the HR hyperspectral image as a combination of a spectral basis and corresponding spatial coefficients, with different modalities providing complementary cues for estimating these components. Matrix factorization-based methods decompose the HR hyperspectral image into spectral basis and coefficient matrices. Some estimate the spectral basis from LR hyperspectral images and derive spatial coefficients from HR multispectral images, while others jointly optimize both components using coupled factorization frameworks \cite{5995457, 5982386}. To better preserve intrinsic multidimensional correlations, tensor factorization methods treat the data as high-order tensors and apply Tucker, CP, or tensor ring decompositions. Representative methods such as CSTF \cite{8359412}, NLSTF \cite{8917657}, and CTRF \cite{he2022hyperspectral} have shown superior performance by effectively capturing spatial–spectral dependencies and incorporating non-local structural priors. \par

Extensive research has focused on enhancing image detail through multimodal fusion, leading to numerous methods for fusing HR multispectral and LR hyperspectral images. These approaches can be broadly categorized into decomposition-based, deep learning-based, and pan-sharpening-based methods \cite{dian2021recent}. Decomposition-based methods reconstruct latent HR hyperspectral images by factorizing them into low-dimensional components capturing spatial and spectral structures. Typically, the image is modeled as a combination of a spectral basis and spatial coefficients, with each modality providing complementary cues. Matrix factorization methods decompose the image into spectral bases and coefficient matrices, estimating them either separately from LR and HR inputs or jointly via coupled optimization \cite{5995457,5982386}. To better preserve multidimensional correlations, tensor factorization methods treat the data as high-order tensors and apply Tucker, CP, or tensor ring decompositions. Representative methods such as CSTF \cite{8359412}, NLSTF \cite{8917657}, and CTRF \cite{he2022hyperspectral} have demonstrated strong performance by effectively capturing spatial-spectral dependencies and leveraging non-local priors. Yang et al. proposes a feature-driven variational method to improve the quality of images \cite{10751756}. A residual sparse method based on morphological reconstruction and wavelet framework is proposed to model the detailed information of images \cite{9216162}.\par 

% Extensive research has aimed to improve detailed reconstruction in multimodal fusion of HR multispectral and LR hyperspectral images. Existing methods can be broadly categorized into decomposition-based, deep learning-based, and pan-sharpening approaches \cite{dian2021recent}. Decomposition-based methods reconstruct latent HR hyperspectral images by factorizing them into low-dimensional spatial and spectral components. Matrix factorization techniques estimate spectral bases and spatial coefficients separately or jointly \cite{5995457,5982386}. To better capture multidimensional correlations, tensor factorization treats the data as high-order tensors and applies Tucker, CP, or tensor ring decompositions. Methods such as CSTF \cite{8359412} and CTRF \cite{he2022hyperspectral} have shown strong performance by modeling spatial-spectral dependencies and exploiting non-local priors.\par 

Although decomposition-based methods have demonstrated promising performance, their reliance on hand-crafted priors and sensitivity to hyperparameter settings remain major limitations. With the advent of deep learning, fusion mappings are increasingly learned directly from data, reducing the dependence on explicit priors \cite{xie2019multispectral}. Spatial–spectral dependencies are commonly modeled using 3D convolutional neural networks with principal component analysis applied to reduce dimensionality and computational overhead \cite{7869277}. To further enhance structural fidelity, edge-conditioned feature extraction mechanisms have been introduced \cite{9530284}. Given the scarcity of labeled hyperspectral data, various unsupervised and semi-supervised frameworks have been developed. These include encoder–decoder architectures that extract spatial and spectral features without supervision \cite{qu2018unsupervised}, hybrid supervised–unsupervised adaptation schemes \cite{zhang2020unsupervised}, and model-driven designs that jointly estimate both degradation parameters and fusion mappings \cite{liu2022model}. In addition, pan-sharpening techniques—originally developed for enhancing multispectral image resolution—have been extended to hyperspectral fusion. In these approaches, hyperspectral bands are enhanced using multispectral channels, either individually or through adaptive multi-band combination strategies \cite{picone2017band,7134741}. \par

% Although decomposition-based methods have shown strong performance, their reliance on hand-crafted priors and sensitivity to hyperparameters remain key limitations. With deep learning, fusion mappings are increasingly learned directly from data, reducing dependence on explicit priors \cite{xie2019multispectral}. Spatial–spectral dependencies are commonly modeled with 3D CNNs and principal component analysis to lower computational cost \cite{7869277}. To improve structural fidelity, edge-conditioned features have been proposed \cite{9530284}. Given limited labeled data, unsupervised and semi-supervised frameworks have emerged, including encoder–decoder architectures \cite{qu2018unsupervised}, hybrid adaptation schemes \cite{zhang2020unsupervised}, and model-driven designs that jointly estimate degradation and fusion \cite{liu2022model}. Additionally, pan-sharpening techniques developed for multispectral enhancement have been adapted to hyperspectral fusion via individual or adaptive multi-band refinement \cite{picone2017band,7134741}. \par

Despite significant progress in multimodal image fusion, most existing methods primarily focus on learning mappings in the spatial domain, aiming to directly reconstruct high-resolution hyperspectral images by modeling spatial–spectral correlations. However, the role of frequency-domain information—closely tied to the image degradation process—remains largely underexplored. In detail enhancement tasks within multimodal fusion, such as fused image super-resolution, degradation typically results in the suppression or loss of high-frequency components during downsampling, making reconstruction essentially a problem of recovering missing frequency content \cite{Fuoli_2021_ICCV}. Overlooking this frequency-domain perspective may limit the ability of spatial-domain approaches to fully recover fine-grained details and maintain structural consistency. \par 

To address the aforementioned limitations, we propose a Dual-Domain Learning Network (DDLN) for multimodal image fusion. Specifically, we design a Dual-Domain Fusion (DDF) module to integrate spatial and frequency domain information from multimodal inputs. The DDF module consists of two parallel branches: a spatial domain branch and a frequency domain branch. In the spatial domain branch, spatial attention is employed to refine spatial features and facilitate interactions between modalities, thereby enabling effective fusion in the spatial domain. In the frequency domain branch, the discrete Fourier transform (DFT) is applied to extract amplitude and phase components from different modalities, allowing frequency-domain fusion of multimodal information. This branch effectively expands the receptive field to cover the entire image, enabling the modeling of richer global contextual dependencies and enhancing the model’s reasoning capability. Extensive experiments on several publicly available datasets demonstrate the superior performance of the proposed DDLN, which consistently outperforms existing representative methods.\par 
Our contributions are summarized as follows.
\begin{itemize}
    \item We propose a Dual-Domain Learning Network (DDLN) to enhance multimodal fusion reasoning by jointly exploring spatial and frequency domain information.
    \item We design a Dual-Domain Fusion (DDF) module, consisting of a frequency-domain branch and a spatial-domain branch, to effectively learn complementary information from both domains and enhance the fine-grained reasoning ability of the multimodal fusion model.
    \item Experimental results on multiple publicly available datasets demonstrate that the proposed DDLN framework consistently outperforms existing state-of-the-art methods across both qualitative and quantitative metrics.
\end{itemize}

The remainder of this paper is organized as follows. Section 2 details the proposed DDLN framework. Section 3 presents experimental results and analysis. Finally, Section 4 concludes the paper.\par




% Please follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press.
% This style guide now has several important modifications (for example, you are no longer warned against the use of sticky tape to attach your artwork to the paper), so all authors should read this new version.

% %-------------------------------------------------------------------------
% \subsection{Language}

% All manuscripts must be in English.

% \subsection{Dual submission}

% Please refer to the author guidelines on the \confName\ \confYear\ web page for a
% discussion of the policy on dual submissions.

% \subsection{Paper length}
% Papers, excluding the references section, must be no longer than eight pages in length.
% The references section will not be included in the page count, and there is no limit on the length of the references section.
% For example, a paper of eight pages with two pages of references would have a total length of 10 pages.
% {\bf There will be no extra page charges for \confName\ \confYear.}

% Overlength papers will simply not be reviewed.
% This includes papers where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide.
% Note that this \LaTeX\ guide already sets figure captions and references in a smaller font.
% The reason such papers will not be reviewed is that there is no provision for supervised revisions of manuscripts.
% The reviewing process cannot determine the suitability of the paper for presentation in eight pages if it is reviewed in eleven.

% %-------------------------------------------------------------------------
% \subsection{The ruler}
% The \LaTeX\ style defines a printed ruler which should be present in the version submitted for review.
% The ruler is provided in order that reviewers may comment on particular lines in the paper without circumlocution.
% If you are preparing a document using a non-\LaTeX\ document preparation system, please arrange for an equivalent ruler to appear on the final output pages.
% The presence or absence of the ruler should not change the appearance of any other content on the page.
% The camera-ready copy should not contain a ruler.
% (\LaTeX\ users may use options of \texttt{iccv.sty} to switch between different versions.)

% Reviewers:
% note that the ruler measurements do not align well with lines in the paper --- this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly.
% Just use fractional references (\eg, this line is $087.5$), although in most cases one would expect that the approximate location will be adequate.


% \subsection{Paper ID}
% Make sure that the Paper ID from the submission system is visible in the version submitted for review (replacing the ``*****'' you see in this document).
% If you are using the \LaTeX\ template, \textbf{make sure to update paper ID in the appropriate place in the tex file}.


% \subsection{Mathematics}

% Please number all of your sections and displayed equations as in these examples:
% \begin{equation}
%   E = m\cdot c^2
%   \label{eq:important}
% \end{equation}
% and
% \begin{equation}
%   v = a\cdot t.
%   \label{eq:also-important}
% \end{equation}
% It is important for readers to be able to refer to any particular equation.
% Just because you did not refer to it in the text does not mean some future reader might not need to refer to it.
% It is cumbersome to have to use circumlocutions like ``the equation second from the top of page 3 column 1''.
% (Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers).
% All authors will benefit from reading Mermin's description of how to write mathematics:
% \url{http://www.pamitc.org/documents/mermin.pdf}.

% \subsection{Blind review}

% Many authors misunderstand the concept of anonymizing for blind review.
% Blind review does not mean that one must remove citations to one's own work---in fact it is often impossible to review a paper unless the previous citations are known and available.

% Blind review means that you do not use the words ``my'' or ``our'' when citing previous work.
% That is all.
% (But see below for tech reports.)

% Saying ``this builds on the work of Lucy Smith [1]'' does not say that you are Lucy Smith;
% it says that you are building on her work.
% If you are Smith and Jones, do not say ``as we show in [7]'', say ``as Smith and Jones show in [7]'' and at the end of the paper, include reference 7 as you would any other cited work.

% An example of a bad paper just asking to be rejected:
% \begin{quote}
% \begin{center}
%     An analysis of the frobnicatable foo filter.
% \end{center}

%    In this paper we present a performance analysis of our previous paper [1], and show it to be inferior to all previously known methods.
%    Why the previous paper was accepted without this analysis is beyond me.

%    [1] Removed for blind review
% \end{quote}


% An example of an acceptable paper:
% \begin{quote}
% \begin{center}
%      An analysis of the frobnicatable foo filter.
% \end{center}

%    In this paper we present a performance analysis of the  paper of Smith \etal [1], and show it to be inferior to all previously known methods.
%    Why the previous paper was accepted without this analysis is beyond me.

%    [1] Smith, L and Jones, C. ``The frobnicatable foo filter, a fundamental contribution to human knowledge''. Nature 381(12), 1-213.
% \end{quote}

% If you are making a submission to another conference at the same time, which covers similar or overlapping material, you may need to refer to that submission in order to explain the differences, just as you would if you had previously published related work.
% In such cases, include the anonymized parallel submission~\cite{Authors14} as supplemental material and cite it as
% \begin{quote}
% [1] Authors. ``The frobnicatable foo filter'', F\&G 2014 Submission ID 324, Supplied as supplemental material {\tt fg324.pdf}.
% \end{quote}

% Finally, you may feel you need to tell the reader that more details can be found elsewhere, and refer them to a technical report.
% For conference submissions, the paper must stand on its own, and not {\em require} the reviewer to go to a tech report for further details.
% Thus, you may say in the body of the paper ``further details may be found in~\cite{Authors14b}''.
% Then submit the tech report as supplemental material.
% Again, you may not assume the reviewers will read this material.

% Sometimes your paper is about a problem which you tested using a tool that is widely known to be restricted to a single institution.
% For example, let's say it's 1969, you have solved a key problem on the Apollo lander, and you believe that the 1970 audience would like to hear about your
% solution.
% The work is a development of your celebrated 1968 paper entitled ``Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties'', by Zeus \etal.

% You can handle this paper like any other.
% Do not write ``We show how to improve our previous work [Anonymous, 1968].
% This time we tested the algorithm on a lunar lander [name of lander removed for blind review]''.
% That would be silly, and would immediately identify the authors.
% Instead write the following:
% \begin{quotation}
% \noindent
%    We describe a system for zero-g frobnication.
%    This system is new because it handles the following cases:
%    A, B.  Previous systems [Zeus et al. 1968] did not  handle case B properly.
%    Ours handles it by including a foo term in the bar integral.

%    ...

%    The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don't you know.
%    It displayed the following behaviours, which show how well we solved cases A and B: ...
% \end{quotation}
% As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors.
% A reviewer might think it likely that the new paper was written by Zeus \etal, but cannot make any decision based on that guess.
% He or she would have to be sure that no other authors could have been contracted to solve problem B.
% \medskip

% \noindent
% FAQ\medskip\\
% {\bf Q:} Are acknowledgements OK?\\
% {\bf A:} No.  Leave them for the final copy.\medskip\\
% {\bf Q:} How do I cite my results reported in open challenges?
% {\bf A:} To conform with the double-blind review policy, you can report results of other challenge participants together with your results in your paper.
% For your results, however, you should not identify yourself and should not mention your participation in the challenge.
% Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\medskip\\

% \begin{figure}[t]
%   \centering
%   \fbox{\rule{0pt}{2in} \rule{0.9\linewidth}{0pt}}
%    %\includegraphics[width=0.8\linewidth]{egfigure.eps}

%    \caption{Example of caption.
%    It is set in Roman so that mathematics (always set in Roman: $B \sin A = A \sin B$) may be included without an ugly clash.}
%    \label{fig:onecol}
% \end{figure}

% \subsection{Miscellaneous}

% \noindent
% Compare the following:\\
% \begin{tabular}{ll}
%  \verb'$conf_a$' &  $conf_a$ \\
%  \verb'$\mathit{conf}_a$' & $\mathit{conf}_a$
% \end{tabular}\\
% See The \TeX book, p165.

% The space after \eg, meaning ``for example'', should not be a sentence-ending space.
% So \eg is correct, {\em e.g.} is not.
% The provided \verb'\eg' macro takes care of this.

% When citing a multi-author paper, you may save space by using ``et alia'', shortened to ``\etal'' (not ``{\em et.\ al.}'' as ``{\em et}'' is a complete word).
% If you use the \verb'\etal' macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher \etal.
% However, use it only when there are three or more authors.
% Thus, the following is correct:
%    ``Frobnication has been trendy lately.
%    It was introduced by Alpher~\cite{Alpher02}, and subsequently developed by
%    Alpher and Fotheringham-Smythe~\cite{Alpher03}, and Alpher \etal~\cite{Alpher04}.''

% This is incorrect: ``... subsequently developed by Alpher \etal~\cite{Alpher03} ...'' because reference~\cite{Alpher03} has just two authors.

% \begin{figure*}
%   \centering
%   \begin{subfigure}{0.68\linewidth}
%     \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}}
%     \caption{An example of a subfigure.}
%     \label{fig:short-a}
%   \end{subfigure}
%   \hfill
%   \begin{subfigure}{0.28\linewidth}
%     \fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}}
%     \caption{Another example of a subfigure.}
%     \label{fig:short-b}
%   \end{subfigure}
%   \caption{Example of a short caption, which should be centered.}
%   \label{fig:short}
% \end{figure*}
