
\section{Background and Methods}
\label{sec:sec2}


\subsection{Accelerated Dynamic MRI Acquisition and Reconstruction}
\label{sec:sec2.1}
Given a series of fully-sampled dynamic multi-coil $k$-space data $\vec{y} \in\mathbb{C}^{n \times n_c \times n_f}$, the underlying dynamic image $\vec{x}^{*}\in\mathbb{C}^{n \times n_f}$, can be obtained by applying the inverse Fast Fourier transform $\mathcal{F}^{-1}$: $\vec{x}^{*} = \mathcal{F}^{-1}(\vec{y})$, where $n = n_1 \times n_2$, $n_c$, and $n_f$ represent the spatial dimensions, the number of coils, and the number of frames (time-steps) of the dynamic acquisition, respectively.  To accelerate the acquisition, the $k$-space is subsampled. The acquired subsampled dynamic multi-coil $k$-space $\tilde{\vec{y}} \in\mathbb{C}^{n \times n_c \times n_f}$ can be defined by the forward problem for $k=1,..,n_c, \, \, \, t=1,..,n_f$:
% 
\begingroup
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\setlength{\belowdisplayskip}{6pt}  
\begin{equation}
    \tilde{\vec{y}}_t^k \, = \,  \mathcal{A}_{\Lambda^{t}, \mat{S}^{k}} \left(\vec{x}_{t}^{*}\right) \, + \,\tilde{\vec{e}}_{t}^{k},\quad \mathcal{A}_{\Lambda^{t}, \mat{S}^{k}}^{k}\,:=\, \mat{U}_{\Lambda^{t}} \mathcal{F} \mat{S}^{k}, 
    % \quad k=1,..,n_c, \, \, \, t=1,..,n_f,
    \label{eq:multicoil_subsampled}
\end{equation}
\endgroup
% 
\noindent
where $\mat{U}_{M}$ denotes a subsampling operator acting on a set $M \subseteq \Omega$ as follows:
% 
\begingroup
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\begin{equation}
        \vec{z}_{M}\,: (\vec{z}_{M})_{i} \,= \,(\mat{U}_{M}\vec{z})_{i}\,=\, \vec{z}_{i} \cdot \mathbb{1}_{M}(i),
   \quad i \in \Omega  := \{1,2,\cdots\} , \quad \vec{z} \in \mathbb{C}^{n \times n_c}.
   \label{eq:mask_definition}
\end{equation}
\endgroup
% 
\noindent
Here, $\Omega$ denotes the sample space comprising all possible sampling options, where $|\Omega|$ equals $n_2$ or $n$ for 1D (column) sampling and 2D (point) sampling, respectively. $\mathcal{F}$ denotes the forward FFT,  $\mat{S}^k \,\in \, \mathbb{C}^{n\times n}$ the sensitivity map of the $k$-th coil, and $\vec{e}_t^k$ measurement noise. The acceleration factor (AF)  of the acquisition of $\tilde{\vec{y}}$ is determined by the dynamic acquisition set $\Lambda = \{\Lambda^{t}\}_{t=1}^{n_f} \subset \Omega^{n_f}$: $\texttt{AF}(\Lambda) = \frac{n_f |\Omega|}{\sum_{t=1}^{n_f} |\Lambda^{t}|}$, where $|\cdot|$ denotes the cardinality. 

The goal of dynamic MRI reconstruction involves obtaining an approximation of $\vec{x}^{*}$ using $\tilde{\vec{y}}$, formulated as a regularized least-squares optimization problem \cite{bertero2006regularization}:
% 
\begingroup
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{\small
\begin{equation}
\hat{\vec{x}}
= \mathop{\arg\min}_{\vec{x}\in\mathbb{C}^{n\times n_f}}
\frac{1}{2} \sum_{t=1}^{n_f}\sum_{k=1}^{n_c}
\bigl\|\mathcal{A}_{\Lambda^{t}, \mat{S}^{k}}^{k}(\vec{x}_{t})
- \tilde{\vec{y}}_{t}^{k}\bigr\|_2^2
+ \mathcal{G}(\vec{x})
\tag{\theequation}
\stepcounter{equation}
\label{eq:variational_problem_dynamic}
\end{equation}
}
\endgroup
% 
\noindent
where $\mathcal{G}: \mathbb{C}^{n \times n_f} \to \mathbb R$ represents an arbitrary regularization functional imposing prior reconstruction information. 

Our objectives in this project involve learning an adaptive dynamic sampling  pattern $\mat{U}_{\Lambda} := (\mat{U}_{\Lambda^1},\cdots,\mat{U}_{\Lambda^{n_f}})$ that maximizes the information content of the acquired subsampled data $\tilde{\vec{y}}$ which is adapted based on some initial dynamic $k$-space data $\tilde{\vec{y}}_{\Lambda_0}$, and within the same framework train a DL-based dynamic reconstruction technique. The idea is that both sampling and reconstruction are improved and co-trained by exploiting cross-frame information found across the dynamic data.


\subsection{Sensitivity Map Prediction}
\label{sec:sec2.3}

Coil sensitivities are estimated using the fully-sampled central $k$-space region, or autocalibration signal (ACS). This is further refined by a 2D U-Net-based \cite{10.1007/978-3-319-24574-4_28} Sensitivity Map Predictor (SMP), $\mathcal{S}_{\boldsymbol{\omega}}$, an approach proven effective in enhancing sensitivity map prediction \cite{sriram2020end, peng2022deepsense, Yiasemis_2022_CVPR}.


\subsection{Adaptive Dynamic Sampler}
\label{sec:sec2.4}


To adapt the subsampling pattern to each case, we introduce the Adaptive Dynamic Sampler (ADS), extending a previous approach for static adaptive sampling \cite{bakker2022on} trained jointly with a variational network reconstruction model \cite{sriram2020end}. Given initial measurements $\tilde{\vec{y}}_{0} \in \mathbb{C}^{n \times n_c \times n_f}$ acquired from an initial set $\Lambda_0 \subset \Omega^{n_f}$ (e.g., $\Lambda_0$ = $\Lambda_{\text{acs}}$), ADS allocates a sampling budget online: $\vec{n_b}  = (n_b^1, \cdots, n_b^{n_f}) = (\frac{n_a}{R} - |\Lambda_0^1|, \cdots, \frac{n_a}{R} - |\Lambda_0^{n_f}|)$. Here, $n_a = |\Omega|$ denotes the total number of potential samples, and $R$ an arbitrary acceleration factor. We focus on frame-specific sampling (for unified set $n_f = 1$).

ADS operates through a number of $N$ cascades, dividing the sampling budget evenly as $\frac{\vec{n_b}}{N}$ across cascades. Each cascade comprises an encoder module $\mathcal{E}_{\boldsymbol{\theta_m}}$ and a multi-layer perceptron (MLPs) $\mathcal{M}_{\boldsymbol{\psi_m}}$. The encoders follow a U-Net \cite{10.1007/978-3-319-24574-4_28} encoder structure, alternating between $l_\text{enc}$ 3D convolutional layers ($3^3$ kernels) with instance normalization\cite{8099920} and ReLU \cite{xu2015empirical} activations, and max-pooling layers ($2^3$ kernels), except for the first layer. MLPs consist of $l_\text{mlp}$ linear layers, with leaky ReLU \cite{xu2015empirical} (with negative slope 0.01) activation, except for the final layer.

Each $\mathcal{E}_{\boldsymbol{\theta_m}}$ receives (subsampled) multi-coil $k$-space measurements  as input, which are reconstructed into a single image via complex conjugate sensitivity map sum (SENSE reconstruction). The resulting image is subsequently flattened and introduced into $\mathcal{M}_{\boldsymbol{\psi_m}}$, generating probability vectors $\vec{p}_m = (\vec{p}_m^1,\cdots, \vec{p}_m^{n_f}) \in \mathbb{R}^{n_f \times n_a}$ such that $\vec{p}_m^t \in \mathbb{R}^{n_a}$. Probabilities in $\vec{p}_m^t$ corresponding to previously acquired measurements on $\bigcup_{j=0}^{m-1}\Lambda_j^t$ are set to zero. A softplus activation function is applied to each $\vec{p}_m^t$, followed by rescaling (see \Algorithm{rescale}) to ensure \(\mathbb{E}(\vec{p}_m^t) = \frac{n_b^t}{N \times n_a}\).

To enable differentiability of the binarization process and allow end-to-end training, we apply a straight-through estimator \cite{yin2019understanding} (STE) for gradient approximation, following prior work \cite{bakker2022on,yin2021end,zhang2020extending}. This stochastically generates $\Lambda_m^t$ by binarizing $\vec{p}_m^t$ through rejection sampling to meet the sampling budget $\frac{\vec{n_b}}{N}$, with gradients approximated using a sigmoid function (slope = 10). The STE's forward and backward passes are detailed in Algorithms \ref{alg:ste_forward} and \ref{alg:ste_backward}, with further details in \Appendix{appendix2-ste}.

The first ADS cascade processes the initial data $\tilde{\vec{y}}_{0}$, and each subsequent cascade $m$ takes as input $k$-space data $\tilde{\vec{y}}_{{m-1}}$ produced from the previous cascade aiming to produce a new acquisition set $\Lambda_{m} \subset \Omega^{n_f}$. The new $k$-space data is then acquired from the predicted set $\Lambda_{m}$ and is subsequently aggregated with previous data, equivalent to an acquisition on $\bigcup_{j=0}^{m} \Lambda_j$. This sequential approach yields the final set $\Lambda := \bigcup_{m=0}^{N} \Lambda_m  \subset \Omega^{n_f}$ satisfying \texttt{AF}($\Lambda$) = $R$.  See \Fig{ads} for a depiction of the ADS module, with further details in  Algorithms \ref{alg:ads} (frame-specific sampling) and \ref{alg:ads_unified} (unified sampling).

% The first ADS cascade processes the initial data $\tilde{\vec{y}}_{0}$, and each subsequent cascade $m$ takes as input $k$-space data $\tilde{\vec{y}}_{{m-1}}$ produced from the previous cascade aiming to produce a new acquisition set $\Lambda_{m} = ( \Lambda_{m}^1,\cdots,\Lambda_{m}^{n_f}) \subset \Omega^{n_f}$, ensuring $\Lambda_{m-1}^t \bigcap \Lambda_{m}^t = \emptyset$ and $\ |\Lambda_m^t| = \frac{n_b^t}{N}  + |\Lambda_0^t|$. The new $k$-space data is then acquired from the predicted set $\Lambda_{m}$ as $\vec{U}_{\Lambda_m} \vec{y} = (\mat{U}_{\Lambda_m^1}\vec{y}^1, \cdots, \mat{U}_{\Lambda_m^{n_f}}\vec{y}^{n_f})$, which is subsequently aggregated with previous data, equivalent to an acquisition on $\bigcup_{j=0}^{m} \Lambda_j$:
% % 
% \begingroup
% \setlength{\abovedisplayskip}{6pt}    % space above
% \setlength{\belowdisplayskip}{6pt}   
% \begin{equation}
%     \tilde{\vec{y}}_{m} := \tilde{\vec{y}}_{m-1} + \vec{U}_{\Lambda_m} {\vec{y}} \quad \Longleftrightarrow \quad \tilde{\vec{y}}_{m} = \vec{U}_{\cup_{j=0}^{m}\Lambda_{j}}  {\vec{y}}.
% \end{equation}
% \endgroup
% % 
% This sequential approach yields the final set $\Lambda := \bigcup_{m=0}^{N} \Lambda_m  \subset \Omega^{n_f}$ satisfying $|\Lambda| = \sum_{m=1}^{N}\sum_{t=1}^{n_f}|\Lambda_N^t| = n_f \times \frac{n_a}{R}$, ensuring \texttt{AF}($\Lambda$) = $R$.  See \Fig{ads} for a depiction of the ADS module, with further details in  Algorithms \ref{alg:ads} (frame-specific sampling) and \ref{alg:ads_unified} (unified sampling).

\subsection{Dynamic MRI Reconstruction}
\label{sec:sec2.5}

Our proposed pipeline (\Section{sec2.6}) is designed to be independent of the underlying reconstruction algorithm. Throughout this work, we primarily employ vSHARP \cite{vsharp2023,Yiasemis2024}, a state-of-the-art (SOTA) model that achieved leading performance in the CMRxRecon Challenges \cite{lyu2024state,wang2025towards}. vSHARP efficiently solves \eqref{eq:variational_problem_dynamic} via half-quadratic variable splitting and ADMM-unrolled optimization over $T$ iterations.

To verify that our pipeline generalizes beyond a single reconstruction backbone, we additionally integrate MEDL-Net \cite{qiao2023medl}, another SOTA method that addresses the same variational problem through an iterative architecture with dense inter-cascade connections, enabling fewer cascades without loss of fidelity.

Given subsampled measurements $\tilde{\vec{y}}$ and sensitivity maps $\vec{S}$, the reconstruction model predicts the underlying dynamic image: $\hat{\vec{x}} = \mathcal{R}_{\boldsymbol{\phi}}(\tilde{\vec{y}};\vec{S})$. Further algorithmic details can be found in the related literature \cite{vsharp2023,Yiasemis2024,qiao2023medl}. 

% Other dynamic MRI reconstruction approaches, such as \cite{zhang2022compressed,huang2021dynamic,kustner2020cinenet}, have employed methods ranging from compressed sensing to deep learning-based spatiotemporal modeling. 


\subsection{End-to-end Adaptive Dynamic Sampling and Reconstruction}\label{sec:sec2.6}


For our end-to-end approach we integrate the methodologies detailed in Sections \ref{sec:sec2.3}, \ref{sec:sec2.4}, and \ref{sec:sec2.5}. The process is visually summarized in \Fig{diagram} and algorithmically in \Algorithm{end_to_end}. Given ACS data $\tilde{\vec{y}}_{\text{acs}}$, the sensitivities $\vec{S}$ are predicted via $\mathcal{S}_{\boldsymbol{\omega}}$. Subsequently, with initial measurements $\tilde{\vec{y}}_{0}$ and a specified acceleration factor $R$, the adaptive dynamic sampling module $\text{ADS}_{\boldsymbol{\psi}, \boldsymbol{\theta}}$ generates an adaptive dynamic subsampling pattern $\mat{U}_\Lambda$, where $\Lambda$ is as defined in \Section{sec2.4}. Note that for acceleration consistency we choose $ \Lambda_{\text{acs}} \subseteq\Lambda_{0}$. After acquiring the data $\tilde{\vec{y}}_{\Lambda}$ using $\mat{U}_\Lambda$,  $\mathcal{R}_{\boldsymbol{\phi}}$ reconstructs a dynamic image  utilizing $\vec{S}$ and $\tilde{\vec{y}}_{\Lambda}$. 

The training of our framework proceeds in an end-to-end manner, ensuring gradients propagate effectively across all three modules, allowing for simultaneous optimization of sampling and reconstruction. The ADS module, as detailed in \Section{sec2.3}, employs a STE to approximate gradients through the binarization process, ensuring differentiability and enabling reconstruction loss gradients to flow back through the ADS module.  The co-optimization of ADS with SMP and the reconstruction model is achieved through loss computation at the end of the forward pass of the pipeline. In this design, the reconstruction network processes adaptively sampled measurements, and the resulting reconstruction loss (see \Section{sec4.2} for loss calculation) propagates supervision signals throughout the pipeline. This joint optimization ensures that both the sampling patterns and reconstruction quality are systematically improved. Implemented in PyTorch \cite{paszke2019pytorch}, the framework leverages its automatic differentiation capabilities to compute gradients seamlessly across all components, including the probabilistic sampling steps within ADS.


%%%%%%%%%%%%%%------------------FIGURE--------------------------------------------------------%%%%%%%%%%%%%%%

\begin{figure}[!htb]
    \centering
    \vspace{-3pt}
    \includegraphics[width=0.9\textwidth]{figs/fig2.pdf}
    \vspace{-8pt}
    \caption{
   Overview of the E2E-ADS-Recon pipeline for frame-specific $k$-space sampling. Initial multi-coil dynamic $k$-space data $\tilde{\vec{y}}_0$ include ACS data $\tilde{\vec{y}}_{\text{ACS}}$, which are used by the SMP module to generate coil sensitivity maps $\mat{S}$. These sensitivity maps and the initial measurements are input to the ADS module, which generates adaptive sampling patterns $\vec{U}_\Lambda$ based on the desired acceleration factor $R$. These patterns guide subsampled $k$-space acquisitions during dynamic imaging. The subsampled data $\tilde{\vec{y}}$ are processed with the sensitivity maps $\mat{S}$ in the reconstruction module, yielding reconstructions $\hat{\vec{x}}$. The pipeline, including ADS, SMP and reconstruction model, is jointly optimized end-to-end to enhance imaging quality. For simplicity, the illustration assumes a single ADS cascade ($N=1$). }
   \vspace{-10pt}
    \label{fig:diagram}
\end{figure}

%%%%%%%%%%%%%%------------------FIGURE--------------------------------------------------------%%%%%%%%%%%%%%%


