\section{Introduction}
\label{sec:sec1}


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\begin{figure*}[t]
    \centering
    \includegraphics[width=0.9\textwidth]{figs/fig1.pdf}
    \vspace{-7pt}
    \caption{Overview of the Adaptive Dynamic Sampler for frame-specific $k$-space sampling. Initial multi-coil dynamic $k$-space data $\tilde{\vec{y}}_{0}$ and sensitivity maps $\mat{S}$ are used to perform a SENSE reconstruction, which serves as the input to the encoder $\mathcal{E}_{\boldsymbol{\theta_1}}$ in the ADS module. The encoder output is flattened and passed to a multi-layer perceptron $\mathcal{M}_{\boldsymbol{\psi_1}}$, which generates probability vectors for each potential sampling location. These probabilities are softplused, rescaled to meet the acceleration factor $R$, and binarized using a straight-through estimator layer to produce the sampling pattern $\vec{U}_{\Lambda}$, ensuring \texttt{AF}($\Lambda$) = $R$. For brevity here we assume that $\tilde{\vec{y}}_{0}$ comprises only of ACS data and we also assume only a single ADS cascade ($N=1$).}
    \label{fig:ads}
    \vspace{-15pt}
\end{figure*}
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Magnetic Resonance Imaging (MRI) represents an important clinical imaging modality. However, its inherently slow acquisition and susceptibility to motion-stemming from respiratory, bowel, or cardiac movements-pose significant challenges, especially for dynamic imaging. These factors hinder the collection of fully-sampled dynamic frequency-domain data, known as $k$-space, leading to degraded image quality. Consequently, patients are often required to perform actions like voluntary breath-holding to minimize motion effects. To address these issues, accelerated dynamic acquisition techniques, which involve subsampling of the $k$-space below the Nyquist ratio \cite{1697831}, have been employed.

Recent advancements have seen Deep Learning (DL)-based MRI reconstruction methods significantly outperforming traditional approaches \cite{lustig2008compressed,gamper2008compressed}. These DL methods excel in reconstructing MRI images from highly-accelerated scans \cite{guo2023reconformer,Yiasemis_2022_CVPR}. Despite the predominant focus on static MRI reconstruction due to limited reference dynamic acquisitions, recent developments have extended to dynamic MRI reconstruction \cite{LONNING2024102348,Qin2019,Yiasemis2024,Yoo2021,zhang2023radial}.

Nonetheless, a critical limitation persists in DL-based dynamic MRI reconstruction: current methodologies typically rely on predetermined, often arbitrary, subsampling patterns, uniformly applied across all temporal frames. This approach overlooks the benefits of exploiting temporal correlations present in adjacent frames and the potential superiority of DL-based learned adaptive sampling schemes, which constitutes our primary motivation.


Previous studies have demonstrated the feasibility of learning optimized on training data sampling schemes at specific acceleration factors, trained jointly with a reconstruction network in single \cite{Bahadir2019} or multi-coil \cite{zhang2020extending} settings. In the context of adaptive sampling, various DL-based sampling approaches have been proposed in both single \cite{pineda2020,pmlr-v139-van-gorp21a,yin2021end,bakker2020experimental} and multi-coil \cite{bakker2022on,gautam2024patientadaptive} scenarios.

Among the existing literature, most approaches remain limited to static imaging. One study \cite{shor2023multi} extended this direction to dynamic acquisitions by learning a single, non-adaptive optimized non-Cartesian trajectory jointly with a reconstruction model. To our knowledge, this remains the only example explicitly targeting dynamic sampling from training data. Yet, it does not explore \textit{adaptiveness}, nor does it support varying acceleration rates, leaving the field of adaptive dynamic Cartesian subsampling unexplored.  Our work aims to fill this gap. Our contributions can be summarized as follows:

\begin{itemize}[label=$\bullet$,leftmargin=*]
    \item We introduce E2E-ADS-Recon, a novel end-to-end \textit{learned adaptive dynamic subsampling and reconstruction method} for dynamic, multi-coil, Cartesian MRI data. This method integrates an adaptive dynamic sampling model with a sensitivity map prediction module and a dynamic MRI reconstruction method, all trained \textit{end-to-end}. Our approach is designed for simultaneous training across \textit{varying acceleration factors}.
    \item We propose two dynamic sampling methodologies: one that learns distinct adaptive trajectories for each frame (\textit{frame-specific}) and another that learns a uniform adaptive trajectory for all frames (\textit{unified}), with both methods applied to 1D and 2D sampling.
    \item We evaluate our pipeline on a multi-coil, dynamic cardiac MRI dataset \cite{cmrxrecon2023}. Our evaluations include comparisons with pipelines where dynamic subsampling schemes are dataset-optimized or use fixed or random schemes. 
\end{itemize}