EMORe: Motion-Robust 5D MRI Reconstruction via Expectation-Maximization-Guided Binning Correction and Outlier Rejection

Abstract

We propose EMORe, an adaptive reconstruction method designed to enhance motion robustness in free-running, free-breathing self-gated 5D cardiac magnetic resonance imaging (MRI). Traditional self-gating-based motion binning for 5D MRI often results in residual motion artifacts due to inaccuracies in cardiac and respiratory signal extraction and sporadic bulk motion, compromising clinical utility. EMORe addresses these issues by integrating adaptive inter-bin correction and explicit outlier rejection within an expectation-maximization (EM) framework, whereby the E-step and M-step are executed alternately until convergence. In the E-step, probabilistic (soft) bin assignments are refined by correcting misassignment of valid data and rejecting motion-corrupted data to a dedicated outlier bin. In the M-step, the image estimate is improved using the refined soft bin assignments. Validation in a simulated 5D MRXCAT phantom demonstrated EMORe’s superior performance compared to standard compressed sensing reconstruction, showing significant improvements in peak signal-to-noise ratio, structural similarity index, edge sharpness, and bin assignment accuracy across varying levels of simulated bulk motion. In vivo validation in 13 volunteers further confirmed EMORe’s robustness, significantly enhancing blood-myocardium edge sharpness and reducing motion artifacts compared to compressed sensing, particularly in scenarios with controlled coughing-induced motion. Although EMORe incurs a modest increase in computational complexity, its adaptability and robust handling of bulk motion artifacts improves image quality, supporting its potential for improved clinical applicability and diagnostic confidence of 5D cardiac MRI.

I. Introduction

Volumetric cardiovascular magnetic resonance imaging (CMR) allows a comprehensive assessment of the whole heart, capturing the 3D volume in different motion phases. Volumetric imaging circumvents the major limitations of standard 2D CMR, such as breath hold requirement, through-plane motion, misregistration between slices, and the need for precise pre-planning [1]. Most volumetric CMR sequences on commercial scanners acquire the data for several minutes under electrocardiogram (ECG) guidance and prospective respiratory gating using navigator echoes to resolve cardiac and respiratory motion [2]. However, depending on the breathing pattern and the extent of arrhythmia, this approach may lead to unpredictably long acquisition times, sometimes exceeding ten minutes [3]. In addition, navigator echoes disrupt the steady-state of magnetization and thus are not compatible with several common CMR pulse sequences [4]. As an alternative to navigator echoes, respiratory bellows [5] can be used for prospective or retrospective gating, but their use has been limited due to inconsistent performance [6]. More recently, there has been increased interest in free-running self-gated (SG) volumetric imaging (FRV-CMR) [7]–[11], in which data are collected continuously for several minutes without the guidance of navigator echoes, respiratory bellows, or even ECG. Furthermore, FRV-CMR offers flexibility in retrospectively controlling temporal resolution by selecting the number of cardiac and respiratory bins after the scan [12].

In SG acquisition [13], a readout passing through the center of the k-space is periodically sampled. The temporal changes in this readout are attributed to physiological motion. A common approach for extracting respiratory and cardiac signals from these repeatedly sampled readouts relies on blind source separation. These surrogate motion signals are then employed to assign the rest of the k-space data into motion bins, leading to motion-resolved volumetric CMR [14]. There are various approaches for retrospectively binning the cardiorespiratory data, according to clinical application and computational constraints. The first approach is respiratory-motion-compensated and cardiac-motion-resolved imaging [15]–[17]; this is achieved by keeping and processing the k-space data from only one respiratory phase while suppressing data from other phases using either hard gating (data rejection) [15] or soft gating (amplitude-based down-weighting) [16]. The limitation of this approach is that data efficiency is 50% or lower because the data from only one respiratory phase are utilized. The second approach is respiratory-motion-corrected and cardiac-motion-resolved imaging [18]–[20]; this is achieved by registering or deforming all other respiratory phases to one reference phase during the reconstruction. Although this approach provides 100% data efficiency, it requires estimation of high-quality deformation fields, which is a difficult task. Moreover, this approach averages the effect of respiratory phases on cardiac function. The third and more recent approach is cardiorespiratory-motion-resolved imaging, commonly known as 5D MRI [14], [21], [22]. There has been increased interest in 5D MRI among researchers and clinicians due to its 100% imaging efficiency, ability to resolve respiratory effects on cardiac function [23], and introduction of a respiratory dimension for enhanced temporal regularization [9], [24]. Nevertheless, the use of SG in all three approaches assumes the availability of reliable cardiac and respiratory motion signals, which are often extracted using blind source separation techniques. While these methods do not require perfectly periodic motion, irregularities such as patient movement, beat-to-beat variability, or inconsistent breathing can affect the accuracy of signal extraction and bin assignment [25]. Consequently, in all three binning approaches, motion suppression may be degraded by inaccuracies in the extracted signals and by motion outliers from involuntary bulk motion such as deep breaths, coughing, sneezing, or twitches. These inaccuracies can lead to incorrect assignment of k-space data to motion bins, resulting in residual motion artifacts and image blurring. Although Pilot Tone (PT) [26] provides an alternative to SG, the extraction of physiological motion from PT faces similar challenges. Likewise, conventional pre-rejection approaches such as beat rejection based on RR-interval irregularity [15] primarily target arrhythmias rather than sporadic bulk patient motion.

Recent works on FRV-CMR reconstruction based on compressive recovery [11], [24], [27] or deep learning models [28] do not account for the discussed imperfections in motion estimation. We posit that residual motion from imperfect retrospective binning can arise from two categories: (i) valid data assigned to a wrong motion bin due to an inaccurate self-gating signal and (ii) inclusion of outliers from, for example, bulk motion. For outlier rejection, static imaging protocols often utilize quantitative motion assessment techniques, such as correlation with reference images or temporal consistency, to identify and exclude motion-corrupted data using a fixed threshold [29]–[31]. However, these approaches lack adaptability to the dynamic spatiotemporal correlations inherent in CMR. Recent advances in outlier rejection in dynamic imaging leverage physics-guided group sparsity [25] or use of expectation-maximization (EM) [32] to determine inliers and outliers using a mixture model of the two classes [33]. However, these outlier rejection techniques suffer from the major limitation of discarding incorrectly binned, yet valid, data as outliers and effectively increasing the acceleration rate, as they lack the flexibility to correct inaccurate bin assignments. There is recent work on intra-bin motion correction [27] proposing rigid motion correction within a respiratory bin to reduce respiratory blurring in 5D MRI. While their method improves sharpness within respiratory bins, it also assumes perfect signal extraction from self-gating and accurate initial bin assignments.

Our proposed reconstruction framework, expectation-maximization (EM)-guided binning correction and outlier rejection (EMORe), integrates adaptive outlier rejection and binning refinement within a single iterative reconstruction framework using EM. Unlike previous works, EMORe does not assume self-gating-based binning as final, but rather leverages intra-bin inconsistencies to iteratively reassign data readouts to their correct motion bins or to reject a readout to an outlier bin, with corresponding improvement in image estimates. This study extends the preliminary results presented as abstracts at the SCMR 2025 and 2026 Annual Scientific Sessions [34], [35]. In this work, we present the detailed implementation of the EMORe algorithm for free-breathing volumetric CMR. We evaluate its performance against standard compressed sensing using a 5D MRXCAT phantom. We further assess its effectiveness in 5D MRI reconstruction and validate its utility for cardiac functional assessment using data acquired from healthy human subjects.

II. Methods

A. EM–Guided Binning Correction and Outlier Rejection (EMORe)

In FRV-CMR, the acquired $N$ data readouts are retrospectively assigned to $K$ target cardiorespiratory motion bins using SG. However, as discussed in Section I, the initial SG-based bin assignments of $N$ acquired readouts to $K$ motion states, or bins, is inherently imperfect. Let $g_n\in \{1,\dots ,K\}$ denote the SG-based bin assignment for the $n^{\text{th}}$ readout. To mitigate inaccurate data binning, the proposed EMORe framework leverages the EM algorithm by treating the true motion bin assignments as latent random variables, represented by $r_n\in \{1,\dots ,K,K+1\}$, where the additional $(K+1{)}^{\text{th}}$ bin corresponds to an outlier bin for discarding readouts corrupted by bulk motion that do not belong to any valid motion bin $\{1,\dots ,K\}$. The algorithm learns the probabilities that a readout belongs to any bin, given the k-space data and the current image estimates; a readout then participates in image formation for any motion state bin weighted by these probabilities.

The EMORe algorithm iteratively performs two steps until convergence: the E-step and the M-step, as shown in Fig. 1. In the E-step, EMORe refines the probabilistic (soft) participation of acquired readouts across the valid motion bins and outlier bin based on the current estimates of the images for all motion states. Subsequently, the M-step updates the image estimates utilizing the refined bin participation obtained from the E-step. By integrating explicit inter-bin correction and outlier rejection within this EM framework, EMORe enhances robustness against incorrect binning, bulk motion and other types of data corruption, thereby significantly improving the quality of reconstructed images. Detailed implementations of the E-step and M-step in the EMORe framework are described below.

Fig. 1.

Schematic overview of the proposed EMORe framework. (Left) In the E-step (1), we refine bin participation of readouts to valid motion bins and an outlier bin, given the prior bin participation and current image estimate. (Right) In the M-step (2), we improve the image estimate using the refined bin participation. Both steps are repeated until convergence, resulting in motion-compensated images.

1). E-step:

In the E-step of the EMORe framework, we use the current estimate of the images and the acquired data to update the probabilistic (soft) participation weights for each readout across all $K+1$ motion bins. The participation weight of the $n^{\text{th}}$ readout to $k^{\text{th}}$ bin at iteration $t$, denoted $w_{(n,k)}^{\left(t\right)}$, represents the posterior probability of the readout belonging to the bin. Via Bayes’ theorem, these weights are given by:

$$\begin{gathered} w_{(n,k)}^{\left(t\right)}=p\left(r_n=k\mid {\mathit{y}}_n,{\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)}\right), \\ =\frac{p\left({\mathit{y}}_n\mid {\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)},r_n=k\right)\cdot p\left(r_n=k\mid {\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)}\right)}{\sum _{j=1}^{K+1}p\left({\mathit{y}}_n\mid {\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)},r_n=j\right)\cdot p\left(r_n=j\mid {\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)}\right)}. \end{gathered}$$

Assuming that the prior probability of the $n^{\text{th}}$ readout belonging to $k^{\text{th}}$ bin, $p\left(r_n=k\right)={\theta }_{(n,k)}$, is independent of the image estimate,

$$w_{(n,k)}^{\left(t\right)}=\frac{p\left({\mathit{y}}_n\mid {\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)},r_n=k\right)\cdot {\theta }_{(n,k)}}{\sum _{j=1}^{K+1}p\left({\mathit{y}}_n\mid {\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)},r_n=j\right)\cdot {\theta }_{(n,j)}}.$$ (1a)

For valid motion bins $(k=1,\dots ,K)$,

$$p\left({\mathit{y}}_n\mid {\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)},r_n=k\right)=\frac{\text{exp}\left(-\frac{1}{L{\sigma }^2}{‖{\mathit{A}}_{(n,k)}{\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)}-{\mathit{y}}_n‖}_2^2\right)}{\pi {\sigma }^2};$$ (1b)

and, for the outlier motion bin $(k=K+1)$,

$$p\left({\mathit{y}}_n\mid {\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)},r_n=K+1\right)=\frac{\text{exp}\left(-\frac{{\tau }^2}{{\sigma }^2}\right)}{\pi {\sigma }^2}.$$ (1c)

Here, ${\mathit{A}}_{(n,k)}\in {\mathbb{C}}^{L\times M}$ is the multi-coil forward operator that incorporates pixel-wise multiplication with sensitivity maps, Fourier transform, and selection of the $n^{\text{th}}$ readout assigned to the $k^{\text{th}}$ motion state bin. Additionally, $\sigma$ is the standard deviation of the circularly symmetric Gaussian noise in the measured data, $L$ is the collective length of a readout from all coils, and $M$ is the number of voxels in the volumetric image for each motion state. Assuming that the duration of a single readout is negligible relative to physiological motion, we assume motion affects the entire readout vector ${\mathit{y}}_n$ and normalize the squared residual norm by $L$ in (1b) to estimate probabilities per readout. The parameter $\tau$ determines the degree of outlier data rejection: a smaller value of $\tau$ corresponds to more aggressive outlier rejection at the potential cost of discarding valid data. The choice of bin prior probabilities, ${\theta }_{(n,k)}$, is detailed in Section II-A.3.

2). M-step:

In the M-step of the EMORe framework, we use the updated readout participation weights $w_{(n,k)}^{\left(t\right)}$ and user-defined regularization parameters to re-estimate the images ${\stackrel{\mathit{ˆ}}{\mathit{x}}}^{\left(t\right)}$. This is achieved by maximizing their posterior probability given data and the probabilistic binning assignments. For additive white Gaussian measurement noise and a regularizing prior proportional to $e^{-{ℛ}_x}$, the corresponding optimization problem is formulated as:

$$\begin{gathered} {\stackrel{\mathit{ˆ}}{\mathit{x}}}^{\left(t\right)}=\text{arg}\underset{\mathit{x}}{\text{min}}\sum _{n=1}^N\sum _{k=1}^K\left[-w_{(n,k)}^{\left(t\right)}\text{log}p\left({\mathit{y}}_n\mid r_n=k,\mathit{x}\right)\right]+{ℛ}_{\mathit{x}} \\ =\text{arg}\underset{\mathit{x}}{\text{min}}\sum _{n=1}^N\sum _{k=1}^K\left[\frac{w_{\left(n,k\right)}^{\left(t\right)}}{L{\sigma }^2}{\Vert {\mathit{A}}_{\left(n,k\right)}\mathit{x}-{\mathit{y}}_n\Vert }_2^2\right]+{\lambda }_s{‖{\nabla }_s\mathit{x}‖}_1+{\lambda }_c{‖{\nabla }_c\mathit{x}‖}_1+{\lambda }_r{‖{\nabla }_r\mathit{x}‖}_1. \end{gathered}$$ (2)

In (2), the regularization term ${ℛ}_{\mathit{x}}$ is the sum of three components: (i) the anisotropic total variation (TV) regularization term, ${\lambda }_s{‖{\nabla }_s\mathit{x}‖}_1$, enforcing smoothness along three spatial dimensions; (ii) the cardiac temporal regularization term, ${\lambda }_c{‖{\nabla }_c\mathit{x}‖}_1$; and, (iii) the respiratory temporal regularization term, ${\lambda }_r{‖{\nabla }_r\mathit{x}‖}_1$. The hyperparameters ${\lambda }_s,{\lambda }_c$, and ${\lambda }_r$ control the strengths of regularization in their respective dimensions and are chosen to reflect the distinct spatial and temporal characteristics of 5D cardiac MRI, with weaker regularization along spatial dimensions to preserve anatomical edges, stronger regularization along the cardiac dimension due to smoother inter-phase transitions, and comparatively weaker regularization along the respiratory dimension to accommodate larger motion-induced variability. The optimization problem in (2) is solved using the alternating direction method of multipliers (ADMM) [36]. In (2), the data fidelity term is normalized by $L$ to maintain consistency with (1b).

3). Algorithm parameters:

The complete EMORe algorithm for FRV-CMR reconstruction is presented in Algorithm 1. The EM algorithm is known to be sensitive to initialization [32], so we initialize ${\stackrel{\mathit{ˆ}}{\mathit{x}}}^{\left(0\right)}$ by partially solving (2) for $I_1$ ADMM iterations, using initial readout participation weights derived from SG-based bin assignment as follows,

$$w_{(n,k)}^{\left(0\right)}=\left\{\begin{array}{ll}1,& \text{if}k=g_n,\\ 0,& \text{otherwise.}\end{array}\forall n,k.\right.$$ (3)

While this SG-based assignment is imperfect, it provides adequate initialization for convergence of the algorithm to a reasonable local minimum. $I_1$ was set to 10, which was the minimum number of iterations required to achieve a stable initialization for the algorithm, balancing convergence reliability and computational efficiency.

In the high-dimensional inverse problem of FRV-CMR, the EM algorithm can become unstable [37], especially when the total number of data bins $K$ is large (e.g., $K=80$, for 20 cardiac × 4 respiratory bins). To mitigate this instability, we leverage a bin-assignment prior ${\theta }_{(n,k)}$, where $\sum _{k=1}^{K+1}{\theta }_{(n,k)}=1$ for every $n$. We define an informative prior guided by SG-based initial assignments as follows:

$${\theta }_{(n,k)}=\left\{\begin{array}{ll}{\alpha }_g,& \text{if}k=g_n,\\ {\alpha }_o,& \text{if}k=K+1,\\ \left(1-{\alpha }_g-{\alpha }_o\right)/(K-1),& \text{otherwise.}\end{array}\right.$$ (4)

The hyperparameters ${\alpha }_g$ and ${\alpha }_o$ are probabilities, chosen to balance the effectiveness of residual motion compensation against algorithm convergence stability. Specifically, a higher value of ${\alpha }_g$ increases reliance on the initial SG-based binning, enhancing stability but limiting motion correction. Whereas, a higher value of ${\alpha }_o$ promotes aggressive rejection of outliers, at the potential cost of discarding valid data. The hyperparameter values $\left\{{\alpha }_g,{\alpha }_o,{\lambda }_s,{\lambda }_c,{\lambda }_r\right\}$ reported in Table I were optimized using SSIM on a single digital phantom dataset across a range of simulated bulk motion levels (0 – 20%). The parameter $\tau$ was selected as $3\sigma$, following the classical three-sigma rule for identifying outliers [38]. Additionally, we performed a hyperparameter sensitivity analysis to study the impact of parameters on reconstruction quality over a range of simulated bulk motion outliers. We compared reconstruction SSIM for various parameter values, while keeping all other reconstruction settings fixed. We evaluated performance for $\tau \in \{1\sigma ,3\sigma ,5\sigma ,\infty \}$, where $\tau =\infty$ corresponds to disabling outlier rejection. In parallel, we examined three prior configurations: (i) the proposed prior, $\left\{{\alpha }_g,{\alpha }_o\right\}=\{0.85,0.05\}$, combining SG-guidance with active outlier rejection; (ii) a SG–only prior, $\left\{{\alpha }_g,{\alpha }_o\right\}=\{0.9,0\}$, without any outlier rejection; and (iii) a uniform prior, $\left\{{\alpha }_g,{\alpha }_o\right\}=\left\{\frac{1}{K+1},\frac{1}{K+1}\right\}$, disregarding SG-guidance with active outlier rejection. For regularization, we compared the proposed values for $\left\{{\lambda }_s,{\lambda }_c,{\lambda }_r\right\}$ with scaled variations of 0.5× and 2× these values. The results of this analysis are presented in Section III-A.

TABLE I.

Imaging and Reconstruction Parameters for Phantom and In Vivo 5D MRI Studies

Parameter	Phantom Study	In Vivo Study
Imaging parameters
Number of Datasets	50	13
Spatial Resolution (mm)	2 × 2 × 2	1.5–2.1 × 1.5–2.1 × 2.0–2.8
Matrix Size	90 × 82 × 76	96 × 144 × 80
Scan Time (min)	5	5
Acceleration Rate $R$	7.3	9.3–9.8
Sex Distribution, Male / Female	20 / 30	7 / 6
Age Range (Mean), years	–	20–68 (36)
BMI Range (Mean), kg/m2	–	22–37 (28)
TE / TR (ms)	1.2 / 4.0	1.2 / 3.0–3.2
Sampling Pattern	3D Cartesian	3D Cartesian
Flip Angle (°)	–	10–14
Receiver Bandwidth (Hz/pixel)	–	789
Reconstruction parameters
Regularization ${\lambda }_s,{\lambda }_c,{\lambda }_r$	2 × 10−2, 10 × 10−2, 6 × 10−2
Convergence threshold $\eta$	10−4
Outlier threshold $\tau$	$3\sigma$
Prior ${\alpha }_g,{\alpha }_o$	0.85, 0.05
Inner iterations $I_1,I_2$	10, 4
Max outer iterations $J$	60

Moreover, the M-step (2) presents a large-scale optimization problem due to the large size of each volumetric image. Therefore, obtaining a closed-form solution or fully solving the M-step problem through iterative procedures is computationally expensive and impractical. To address this, we employ a generalized EM [32] approach, where the M-step is approximately solved via $I_2$ ADMM iterations. $I_2=4$ was selected as the minimum number of inner ADMM iterations required to achieve monotonic improvement in image update in each M-step without incurring prohibitive computational overhead. The stopping criterion of the EMORe algorithm is either the maximum allowed number of EM iterations, set to $J=60$, or satisfying a convergence threshold $\eta ={10}^{-4}$ on the normalized squared image difference between consecutive EM iterations – whichever is achieved first.

All algorithm hyperparameters are summarized in Table I. Once selected, the same configuration was applied unchanged across all phantom and in vivo experiments to ensure consistency and avoid dataset-specific tuning.

Algorithm 1.

EMORe reconstruction

1:	Initialize ${\stackrel{\mathit{ˆ}}{\mathit{x}}}^{\left(0\right)}$ using $w_{(n,k)}^{\left(0\right)}$ in (2) and performing $I_1$ ADMM iterations.
2:	Set $t\leftarrow 1$ (Initialize iteration counter)
3:	repeat
4:	E-step: Update $w_{(n,k)}^{\left(t\right)}$ using ${\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)}$ in (1).
5:	M-step: Estimate ${\stackrel{\mathit{ˆ}}{\mathit{x}}}^{\left(t\right)}$ by using $w_{(n,k)}^{\left(t\right)}$ in (2) and performing $I_2$ ADMM iterations.
6:	$t\leftarrow t+1$ (Increment iteration counter)
7:	until ${‖{\stackrel{\mathit{ˆ}}{\mathit{x}}}^{\left(t\right)}-{\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)}‖}_2^2/{‖{\stackrel{\mathit{ˆ}}{\mathit{x}}}^{(t-1)}‖}_2^2<\eta \wedge t\ge J$
8:	return ${\stackrel{\mathit{ˆ}}{\mathit{x}}}^{\left(t\right)}$

B. Image comparison & evaluation

Compressed sensing (CS) [39], the predominant FRV-CMR reconstruction method and widely used in both clinical and research settings, was employed as the standard for comparison with EMORe. The CS image estimate, ${\stackrel{\mathit{ˆ}}{\mathit{x}}}_{CS}$, was obtained by solving a regularized weighted least-squares problem in (2) using binary participation weights derived from the initial SG-based bin assignments $w_{(n,k)}^{SG}=w_{(n,k)}^{\left(0\right)}$ (3) for all iterations, $t$.

$${\stackrel{\mathit{ˆ}}{\mathit{x}}}_{cs}=\text{arg}\underset{\mathit{x}}{\text{min}}\sum _{n=1}^N\sum _{k=1}^K\left[\frac{w_{(n,k)}^{SG}}{L{\sigma }^2}{‖{\mathit{A}}_{(n,k)}\mathit{x}-{\mathit{y}}_n‖}_2^2\right]+{\lambda }_s{‖{\nabla }_s\mathit{x}‖}_1+{\lambda }_c{‖{\nabla }_c\mathit{x}‖}_1+{\lambda }_r{‖{\nabla }_r\mathit{x}‖}_1.$$ (5)

Similar to the EMORe, the ADMM algorithm was employed to solve the optimization problem in the CS reconstruction, with iterations terminating upon reaching a maximum count $J$ or when the normalized squared difference between consecutive image iterates fell below a threshold $\eta$.

In phantom studies, where a reference image was available, we compared the quality of EMORe and CS images using the structural similarity index (SSIM↑) [40] and peak signal-to-noise ratio (PSNR↑), defined as $20{\text{log}}_{10}\left(\frac{\text{max}\left(\mathit{x}\right)}{\Vert \stackrel{\mathit{ˆ}}{\mathit{x}}-\mathit{x}{\Vert }_2/\sqrt{N}}\right)(\text{dB})$. Additionally, we measured blood-myocardium edge sharpness of all reconstructed images using edge sharpness assessment [41] presented for MRI. The edge sharpness (↑) was quantified as the mean slope of sigmoid functions fitted to pixel intensity profiles crossing the blood-myocardium boundaries, where residual motion artifacts typically introduce blurring. Higher slope values indicate improved edge sharpness and clearer delineation of anatomical structures. In our 5D MRI data, a single blood–myocardium edge sharpness value was obtained by averaging estimates across all four respiratory states, measured on a fixed sagittal cardiac slice where the blood pool was fully enclosed by the myocardium. Furthermore, in studies where the true bin assignments were available, we assessed the improvement in bin participation of data using EMORe by comparing initial and final Brier score [42], which evaluates the mean squared error of probabilistic bin assignments. The Brier score (↓) at EMORe iteration $t$ is defined as $\frac{1}{N}{\sum }_{n=1}^N{\sum }_{k=1}^K{\left({\tilde{w}}_{(n,k)}-w_{(n,k)}^{\left(t\right)}\right)}^2$, where ${\tilde{w}}_{(n,k)}$ represents the true bin participation of readout $n$ for bin $k$, defined as 1 for the bin corresponding to the true motion state and 0 for all other bins.

In the absence of reference images for the in vivo study, two expert readers independently and blindly evaluated EMORe–CS cine slice pairs for perceived noise and artifacts quality using a 5-point Likert scale (5: excellent, 4: good, 3: fair, 2: poor, 1: non-diagnostic).

For quantitative validation, in vivo 5D MRI was compared against a reference multi-slice 2D cine MRI short-axis stack for the quantification of left-ventricular end-diastolic volume (LV EDV), left-ventricular end-systolic volume (LV ESV), and left-ventricular ejection fraction (LV EF). Agreement was assessed using linear regression and Bland-Altman analysis.

C. Experimental studies

To evaluate the robustness of the proposed EMORe reconstruction framework against uncompensated motion in 5D MRI, we performed phantom and in vivo studies comparing EMORe to conventional CS reconstruction.

1). 5D MRXCAT phantom study:

We simulated free-breathing self-gated 5D MRI scans using the MRXCAT phantom [43] from five digital subjects. Each subject comprised four respiratory phases (end-inspiratory to end-expiratory) and 20 cardiac phases spanning the full cardiac cycle, resulting in 4 × 20 = 80 cardiorespiratory phases. Respiratory periods ranged from 3.25 to 4.75 s across the subjects, with a random variation of 0 to 1 s introduced between consecutive cycles to mimic irregular breathing. Cardiac activity of the digital subjects was simulated with heart rates ranging between 62 and 83 beats per minute, with additional beat-to-beat variations of 0 to 160 ms in consecutive R–R intervals.

The k-space data were sampled using a pseudo-random Cartesian trajectory with self-gating [44], acquiring 75, 000 readouts over a 5-minute scan with a repetition time (TR) of 4 ms. Coil sensitivity maps for an 8-channel array were generated using the Biot–Savart law, with coils positioned evenly in anterior and posterior planes. Complex circularly symmetric Gaussian noise was added to achieve an SNR of 30 dB. Additional imaging parameters are detailed in Table I.

To simulate sporadic bulk-motion, seven distinct outlier motion states were generated per subject in addition to the reference (resting) state. These included rigid-body translations of ±20 mm along the superior–inferior axis and rotations of ±10° about the anterior–posterior and superior-inferior axes, as well as a rotation of −10° about the left–right axis. A positive rotation about the left–right axis was omitted, reflecting the low likelihood of backward head-tilting motion in typical patient scanning. For each of five digital subjects, ten datasets were generated by introducing varying levels of bulk-motion corruption: 0%, 5%, 10%, 15%, 20%, 30%, 40%, 50%, 60%, and 70%. Within each of these 50 simulated scans, the bulk-motion corruption was distributed into ten different episodes. These bulk-motion episodes ranged in duration from approximately 1.5 s (in the 5% outlier scenario) to 21 s (in the extreme 70% outlier scenario). In each bulk-motion episode, one of the seven outlier motion states was randomly selected to replace the corresponding reference k-space data.

During data acquisition simulation, the SG signal, consisting of a single readout at the k-space center ($k_y=0,k_z=0$), was repeatedly sampled after every ten readouts along the superior-inferior direction. These SG readouts were used exclusively for motion extraction and were excluded from image reconstruction, as in [45], [46]. In the preprocessing of the scanned data, the SG signal was reorganized into a Casorati matrix, and two band-pass filters with passbands of 0.1–0.5 Hz (respiratory) and 0.5–3 Hz (cardiac) were applied along the temporal dimension. Subsequently, principal component analysis (PCA) and independent component analysis (ICA) were performed on the filtered Casorati matrix to estimate respiratory and cardiac surrogate signals [11], [47]. Based on these signals, k-space data were binned into four respiratory bins of equal data efficiency, each further divided into 20 cardiac bins by partitioning the R-R interval into equal-duration phases, resulting in $K=80$ cardiorespiratory bins.

The SG-based binary bin assignments were used to initialize the participation weights $w_{(n,k)}^{\left(0\right)}$ and prior ${\theta }_{(n,k)}$ using (3) and (4), respectively. Due to cardiac and respiratory variability, the binning is imperfect, even in the case of 0% outlier corruption fraction. An initial image estimate ${\stackrel{\mathit{ˆ}}{\mathit{x}}}^{\left(0\right)}$ was obtained by partially solving (2) for $I_1$ ADMM iterations. Subsequently, the EMORe reconstruction iteratively performed the E-step and M-step to refine bin assignments, reject outlier data, and enhance image quality until convergence criteria were satisfied. For comparison, CS reconstructions were performed for all simulated datasets.

Image quality of EMORe and CS reconstructions was quantitatively assessed across all phantom experiments using PSNR, SSIM, and edge sharpness, using the true phantom images as reference. Additionally, improvement in bin assignments achieved by EMORe was evaluated against the initial SG-based binning employed in CS using the Brier score.

2). In vivo 5D MRI study:

We compared EMORe and CS reconstructions using in vivo 5D MRI data acquired from 13 healthy volunteers in a study approved by the institutional review board. Written informed consent was obtained from all participants prior to imaging. Imaging was performed on a 3T clinical scanner (MAGNETOM Vida, Siemens Healthcare, Erlangen, Germany) equipped with a 48-channel receiver coil. Ferumoxytol-enhanced scans were obtained using a free-running, free-breathing acquisition with a fixed duration of approximately 5 minutes. Data were sampled using a pseudo-random Cartesian trajectory with SG as in Section II-C.1. Detailed imaging parameters are summarized in Table I. To evaluate the robustness of EMORe in handling controlled bulk motion, three volunteers were instructed to simulate coughing during the final 30 seconds of their respective scans.

For quantitative validation, each volunteer also underwent a free-breathing real-time 2D cine scan performed with a research sequence [48]. Short-axis and long-axis (4-chamber) stacks were collected with a spatial resolution 2.0–2.1 mm, a temporal resolution 37–39 ms, and a scan time of 6 s per slice. The 2D and 3D acquisitions were completed within 5 min of each other.

Acquired k-space data were retrospectively sorted into 80 cardiorespiratory bins using blind-source-separation, following methods outlined in Section II-C.1. This binning strategy achieves approximately equal data efficiency across all motion bins, promoting stability of the data distribution among bins. As a consequence, cardiorespiratory phases that are visited less frequently (e.g., end-inspiration) are represented by wider motion bins, which can lead to increased residual blurring. For computational efficiency, coil compression via singular value decomposition was applied, reducing the original 30 acquisition coils to 8 virtual coils [49]. Coil sensitivity maps were estimated using the approach by Walsh et al. [50]. Subsequently, both EMORe and CS methods were applied to reconstruct 5D MRI images, using frameworks identical to those described for the phantom study (Section II-C.1). Due to the lack of ground truth reference images for in vivo data, quantitative image assessment was limited to blood-myocardium edge sharpness. For qualitative evaluation, blind scoring of artifacts was performed on EMORe and CS sagittal cine slice pairs from end-inspiratory and end-expiratory phases of each volunteer’s reconstruction, yielding a total of 26 scored cine image pairs. For quantitative comparison with 2D cine MRI, the reconstructed 5D volumes from EMORe and CS were reformatted into 2D short-axis cine stacks in DICOM format, consistent with the reference short-axis 2D cine MRI. Left-ventricular (LV) quantification was performed using SuiteHEART (NeoSoft, Pewaukee, Wisconsin) at end expiration for all images.

To enable a controlled within-subject comparison of reconstruction robustness to bulk motion, an additional retrospective experiment was performed for the three volunteers instructed to cough during the final 30 seconds of the 300-second scan. For each of these subjects, two 5D reconstructions were generated using EMORe and CS from the same raw acquisition: (i) a “no instructed motion” reconstruction using only data acquired before the coughing period (first 270 seconds), and (ii) a “with instructed motion” reconstruction using a time window of identical duration that included the coughing interval (last 270 seconds). This ensured that both reconstructions had equal acceleration rates and shared in common a majority of the acquired data. This strategy enables direct isolation of the impact of bulk motion on image quality and assessment of EMORe’s ability to preserve image quality in the presence of bulk motion.

D. Implementation details

EMORe and CS reconstructions were implemented in MATLAB (Mathworks, Natick, MA, USA) and executed on an NVIDIA H100 GPU at the Ohio Supercomputer Center. The average reconstruction times were approximately 23.7 minutes for EMORe and 19.1 minutes for CS in the phantom study, and 58.0 minutes for EMORe and 49.8 minutes for CS in the in vivo study. The source code and one sample dataset are publicly available at github.com/OSU-MR/Motion-Robust-5D-MRI-EMORe.

III. Results

A. 5D MRI phantom Study

The quantitative comparisons between EMORe and CS reconstructions across five MRXCAT digital subjects at ten simulated outlier corruption levels are presented in Fig. 2. Metrics reported include PSNR, SSIM, edge sharpness, and Brier score, averaged across subjects. Statistical significance was assessed using a paired $t$-test across subjects at each outlier level, with significance indicated for $p<0.05$.

Fig. 2.

Quantitative comparison between EMORe (red) and CS (blue) reconstructions for 5D MRXCAT phantom study across varying levels of simulated motion outliers (0–70%). Metrics shown include PSNR (dB), SSIM, edge sharpness, and Brier score, averaged across five digital subjects. Error bars represent standard error of the mean. Asterisks indicate statistical significance ($\mathit{p}<0.05$) using a paired $t$-test across five subjects at each outlier level. The edge sharpness (green) for the true phantom with no outliers is also shown for comparison.

Representative end-inspiratory and end-expiratory short-axis cine slices from the ground truth, CS, and EMORe reconstructions are shown in Fig. 3 for three scenarios: 10%, 20%, and 40% simulated outliers. Corresponding temporal profiles along the $x-t$ and $y-t$ dimensions are also visualized to facilitate qualitative comparison of temporal consistency and boundary definition. A video¹ containing all the cine slices presented in Fig. 3 is available under the “Supplementary Files.”

Fig. 3.

Representative short-axis slices at end-inspiratory and end-expiratory states extracted from 5D MRXCAT reconstructions using CS and EMORe under 10%, 20%, and 40% simulated motion outlier levels. Each group shows the static frame, the corresponding temporal profiles along the $\mathit{x}-\mathit{t}$ and $\mathit{y}-\mathit{t}$ dimensions, extracted at the indicated spatial positions (yellow crosshairs), and corresponding average blood–myocardium edge sharpness values. Arrows highlight motion artifacts and blurring in CS images that are visibly reduced in EMORe reconstructions.

To visualize EMORe’s ability to reject corrupted data while preserving valid readouts, Fig. 4 displays respiratory (blue) and cardiac (teal) surrogate signals overlaid with simulated bulk motion intervals (horizontal black bars) and the assignment percentage to the outlier bin for the corresponding readouts (vertical red bars). Examples are shown for 10%, 20%, and 40% outlier scenarios, corresponding to the reconstructions in Fig. 3. Additionally, Fig. 5a shows EMORe’s convergence behavior for a representative phantom dataset with 10% outliers using the SSIM and Brier scores across iterations.

Fig. 4.

Respiratory (blue) and cardiac (teal) surrogate signals overlaid with simulated bulk motion intervals (horizontal black bars) and the assignment percentage to the outlier bin for the corresponding readout (vertical red bars), shown for three representative cases with 10%, 20%, and 40% motion outlier corruption. The height of each red bar represents the assignment percentage to the outlier bin. Magnified panels (bottom row) show representative 30-second intervals, highlighting the relationship between surrogate signal behavior and outlier-bin assignments under simulated bulk motion.

Fig. 5.

Convergence of EMORe in representative examples across iterations ($\mathit{t}$) for the phantom and in vivo studies. (a) Convergence of SSIM and Brier score for a phantom dataset with 10% outliers. (b) Convergence of normalized squared image (${\mathit{x}}^{\left(\mathit{t}\right)}$) difference and normalized squared probabilistic bin assignments (${\mathit{w}}^{\left(\mathit{t}\right)}$) difference between consecutive EM iterations ($\mathit{t}$).

While EMORe successfully assigns a high fraction of readouts to the outlier bin during simulated bulk-motion episodes, not all readouts acquired during bulk-motion intervals are rejected with equal probability. Fig. 6 provides a further insight using an analysis of EMORe’s outlier rejection behavior of a representative phantom experiment with 20% outliers. Fig. 6 shows $k_y-k_z$ (phase-encoding) maps of (i) 2D histogram of simulated outliers (number of bulk-motion–corrupted readouts at each ($k_y,k_z$) location), (ii) true-positive rejection rate (percentage of corrupted readouts correctly rejected), and (iii) false-positive rejection rate (percentage of valid readouts incorrectly rejected). The simulated bulk-motion outliers occur more frequently near the center of k-space due to repeated sampling of low-frequency readouts across motion phases. Correspondingly, EMORe exhibits high true-positive rejection for central k-space while maintaining negligible false-positive rejection across the full $k_y-k_z$ grid. Peripheral k-space readouts show lower rejection probability. We attribute this behavior to motion being primarily encoded in the central parts of the k-space [51], [52] and lower signal energy in the outer parts of the k-space.

Fig. 6.

Analysis of EMORe outlier rejection behavior for a representative phantom experiment with 20% simulated bulk-motion outliers. ${\mathit{k}}_{\mathit{y}}-{\mathit{k}}_{\mathit{z}}$ (phase-encoding) maps show (a) the 2D histogram of simulated bulk-motion–corrupted readouts at each (${\mathit{k}}_{\mathit{y}},{\mathit{k}}_{\mathit{z}}$) location, (b) the true-positive outlier rejection rate, defined as the percentage of corrupted readouts correctly rejected by EMORe, and (c) the false-positive rejection rate, defined as the percentage of valid readouts incorrectly rejected.

As shown in Fig. 7, reconstruction performance degrades in the absence of effective outlier rejection ($\tau =\infty$ or ${\alpha }_o=0$) or when an overly conservative threshold ($(\tau =5\sigma$) is used, particularly under increasing levels of simulated bulk motion. The proposed threshold of $\tau =3\sigma$ yields the overall best performance across varying motion levels (Fig. 7a), indicating a balance between preserving valid data and suppressing motion-corrupted data. Furthermore, Fig. 7b demonstrates that removing the SG–informed prior leads to noticeable performance degradation, potentially due to convergence to suboptimal local minima. In contrast, the proposed prior configuration provides improved results by effectively guiding the bin assignment process. Fig. 7c shows that both under-regularization (0.5×) and over-regularization (2×) of (${\lambda }_s,{\lambda }_c,{\lambda }_r$) result in overall reduced performance compared to the proposed regularization values, across the range of outlier corruption levels.

Fig. 7.

Sensitivity analysis of EMORe with respect to the outlier threshold $\mathit{\tau }$, prior configuration (${\mathit{\alpha }}_{\mathit{g}},{\mathit{\alpha }}_{\mathit{o}}$) and TV regularization strength using SSIM metric. (a) Effect of varying $\mathit{\tau }\in \{1\mathit{\sigma },3\mathit{\sigma },5\mathit{\sigma },\mathbf{\infty }\}$ on reconstruction performance. (b) Effect of different bin-assignment priors $\left({\mathit{\alpha }}_{\mathit{g}},{\mathit{\alpha }}_{\mathit{o}}\right)\in \left\{\right(\mathrm{0.85,0.05}),(\mathrm{0.9,0}),(1/(\mathit{K}+1),1/(\mathit{K}+1)\left)\right\}$, with $\mathit{K}=80,(1/(\mathit{K}+1)\approx 0.0123)$ on reconstruction performance. (c) Effect of regularization strength (${\mathit{\lambda }}_{\mathit{s}},{\mathit{\lambda }}_{\mathit{c}},{\mathit{\lambda }}_{\mathit{r}}$) on reconstruction performance.

B. 5D MRI in vivo study

The blind artifact scores and the edge sharpness comparisons between EMORe and CS for the 13 in vivo scans, including three scans with instructed coughing in the last 30 seconds of a 5-minute scan, are presented in Table II and Fig. 8a. These results are reported for each volunteer, averaged across respiratory phases for the sharpness measure, and averaged across reviewers and respiratory phases for the blind scores. Both image quality assessment metrics show statistically significant improvement ($p<0.01$) by EMORe over CS, using paired $t$-test.

TABLE II.

Comparison of EMORe and CS for edge sharpness and blind artifact scores for in vivo study.

	Blind artifact score (↑)		Edge sharpness (↑)
Volunteer	EMORe	CS	EMORe	CS
1	4.25	3.75	0.759	0.706
2	4.25	3.50	0.831	0.768
3	4.25	4.00	0.716	0.683
4	4.25	3.75	0.728	0.719
5	4.25	3.75	0.609	0.627
6	3.50	3.00	0.591	0.585
7	4.00	4.00	0.672	0.674
8	3.75	3.75	0.877	0.801
9	4.00	3.75	0.642	0.604
10	4.25	4.25	0.760	0.770
11†	4.000	3.25	0.888	0.814
12†	4.000	3.00	0.719	0.713
13†	4.000	3.25	0.595	0.557
Mean	4.06*	3.62	0.722*	0.694
SEM	±0.06	±0.11	±0.028	±0.023

SEM: standard error of the mean.

^*Statistically significant improvement over CS as determined by paired $t$-test ($p<0.01$).

^†Volunteers were instructed to cough during the last 30 seconds of acquisition.

Fig. 8.

Comparison of image quality and left-ventricular (LV) functional assessment. (a) EMORe and CS image quality comparison using blood–myocardium edge sharpness and blinded artifact scores averaged across two expert readers, with statistical significance ($p$-value) indicated. (b) LV end-diastolic volume (EDV), (c) LV end-systolic volume (ESV), and (d) LV ejection fraction (EF) derived from EMORe and CS 5D MRI reconstructions compared against reference 2D cine MRI. For LV functional parameters, the top row shows linear regression analysis between 5D and 2D measurements, with the dotted line indicating the line of identity. The bottom row shows the corresponding Bland–Altman plots, where solid lines denote the mean bias and dashed lines indicate the limits of agreement (LOA); reported p-values indicate statistical comparison between 5D and 2D measurements.

Fig. 9 presents sagittal and coronal cardiac frames and the corresponding vertical temporal profiles at end-inspiratory and end-expiratory phases for EMORe and CS reconstructions from three representative volunteer scans. Fig. 10 further shows representative systolic and diastolic cardiac phases at the inspiratory state for the same subjects, facilitating a direct visual comparison between EMORe and CS. Volunteer #11 was instructed to simulate coughing motion during the final 30 seconds of the 5-minute scan, whereas Volunteers #4 and #6 underwent standard free-breathing acquisitions without specific instructions. A video¹ containing all the cine slices presented in Fig. 9 is available under the “Supplementary Files.”

Fig. 9.

Representative sagittal (left) and coronal (right) cine frames at end-inspiratory and end-expiratory phases from CS and EMORe 5D MRI reconstructions of three in vivo volunteers. Corresponding temporal profiles along the $\mathit{y}–\mathit{t}$ dimensions extracted at the dotted yellow lines, with the average blood-myocardium edge sharpness values for each volunteer, are shown for qualitative and quantitative comparison. Arrows highlight comparisons of observed differences between EMORe and CS reconstructions.

†Volunteer instructed to simulate coughing during the final 30 seconds of the scan.

Fig. 10.

Representative cardiac frames at systole (left) and diastole (right) phases in reconstructions shown in Fig 9, comparing compressed sensing (CS) and the proposed EMORe framework at an inspiratory state. For each subject, corresponding cardiac phases are shown for CS (top rows) and EMORe (bottom rows). Arrows highlight regions with residual motion artifacts and blurring in CS reconstructions, which are visibly reduced in EMORe.

†Volunteer instructed to simulate coughing during the final 30 seconds of the scan.

Fig. 11 presents the respiratory (blue) and cardiac (teal) surrogate signals for the scans corresponding to Fig. 9 and Fig. 10; overlaid with the coughing interval (horizontal black bar) and the outlier bin assignment percentage of the corresponding data (vertical red bars). During the forced coughing interval of Volunteer #11, aggressive outlier rejection occurs, consistent with the behavior seen in the phantom study (Fig. 4). In addition, we can observe pronounced outlier rejection during potential bulk motion and irregular breathing episodes in Volunteer #4, and during deep breathing in Volunteer #6. Furthermore, a subtle but consistent outlier rejection coinciding with end-inspiration during otherwise regular free-breathing is observed in all three cases. This behavior can be attributed to physiological variability in free-breathing respiration. In human subjects, the end-expiratory position is typically more stable and reproducible across respiratory cycles, whereas the depth of inspiration exhibits greater cycle-to-cycle variability. This variability is directly observable in the respiratory surrogate signals (blue curves) in Fig. 11, where the inspiratory peaks show larger amplitude fluctuations compared to the relatively consistent end-expiratory plateaus. As a result, a small subset of readouts acquired at extreme end-inspiration corresponds to motion states that are infrequently visited or inconsistent with the dominant end-inspiratory bin. During the E-step, EMORe appropriately assigns these readouts an increased posterior probability of belonging to the outlier bin in order to preserve the fidelity of the primary respiratory motion bins. This behavior is not observed in the simulation study (Fig. 4), as the simulated respiratory motion assumes a fixed inspiratory depth, with variability introduced only in the respiratory period. Additionally, Fig. 5b shows convergence of the reconstructed image and probabilistic bin assignments across iterations.

Fig. 11.

Respiratory (blue) and cardiac (teal) surrogate signals from the three volunteers in Fig. 9, overlaid with the assignment percentage to the outlier bin for the corresponding readouts (vertical red bars). The instructed coughing interval of Volunteer #11 is indicated by a horizontal black bar. Magnified panels (bottom row) show 30-second intervals highlighting the relationship between surrogate signal behavior and outlier-bin assignments.

The LV quantification validation results are summarized in Fig. 8(b–d). The measurements from both EMORe and CS 5D MRI demonstrated strong linear agreement with reference short-axis 2D cine MRI, with high correlation coefficients across all parameters. Bland–Altman analysis further showed small mean biases and comparable limits of agreement for both methods, with no statistically significant differences relative to 2D cine ($p>0.05$ for all comparisons). These results indicate that EMORe preserves quantitative functional measurements at a level comparable to CS, while providing improved image quality as shown in Fig. 8. Furthermore, Fig. 12 provides a visual comparison of representative short-axis and 4-chamber views from 2D cine scans and 5D EMORe reconstruction at end-inspiration and end-expiration. EMORe demonstrates comparable image quality. Notably, 5D imaging provides both views from a single acquisition, whereas 2D cine requires separate acquisitions for each view. However, the observed differences in blood-pool contrast between 2D cine and 5D images reflect inhomogeneous blood-pool intensity in 2D scans in presence of contrast enhancement.

Fig. 12.

Visual comparison of representative short-axis (left) and 4-chamber (right) frames at end-inspiratory and end-expiratory from real-time 2D cine and 5D EMORe. 2D cine views are obtained from two separate scans, while 5D EMORe views are interpolated from a single 5D MRI acquisition. The difference in blood-pool contrast between 2D cine and 5D scans (more obvious in 4-chamber view of Volunteer #11) is due to inhomogeneous blood-pool intensity in presence of contrast enhancement.

†Volunteer instructed to simulate coughing during the final 30 seconds of the scan.

Additionally, Fig. 13 presents a representative within-subject comparison for the controlled coughing experiment, showing sagittal cardiac cine frames and corresponding vertical temporal profiles reconstructed with CS and EMORe. For the “no instructed motion” condition, both CS and EMORe yield comparable image quality; however, CS exhibits slightly higher residual motion artifacts than EMORe. In contrast, when instructed coughing is included in the reconstruction, CS exhibits pronounced motion artifacts, highlighted by arrows in both the image frames and temporal profiles. EMORe substantially mitigates these effects, preserving temporal coherence and anatomical details despite the presence of coughing-induced bulk motion. A video¹ containing all the cine slices presented in Fig. 13 is available under the “Supplementary Files.”

Fig. 13.

Controlled within-subject bulk-motion experiment from three scans with instructed coughing in the last 30 seconds. Two 5D reconstructions were generated from the same raw data: a “No Instructed Motion” reconstruction using only data acquired before the coughing period (left), and a “With Instructed Motion” reconstruction using an equal-duration time window that includes instructed coughing (right). Sagittal cardiac cine frames and corresponding vertical temporal profiles are shown for CS and EMORe. Yellow dashed lines indicate the spatial locations used to extract temporal profiles. Arrows highlight motion-induced temporal inconsistencies and motion artifacts, which are more pronounced in CS and substantially reduced by EMORe.

IV. Discussion

In the phantom experiments, EMORe image quality was indistinguishable from CS in the absence of motion outliers, as measured by PSNR, SSIM, and edge sharpness. Yet even without bulk motion outliers, EMORe achieved statistically significant improvement in the Brier score by adaptively refining a soft binning; this highlights that some inaccuracies in SG-based binning may occur without bulk motion, due to irregular breathing and beat-to-beat variations. With increasing simulated bulk motion outliers, CS reconstructions exhibited significant degradation, whereas EMORe maintained higher fidelity, reflecting its robustness against sporadic motion artifacts. In extreme cases with more than 40% motion outliers, although EMORe maintained an advantage over CS, the overall image quality of EMORe also degrades substantially, highlighting the limitations of the proposed method under widespread data corruption. Importantly, EMORe’s effective performance does not depend on the temporal location of the bulk motion in the scan. Rather, it depends on the total fraction of motion outliers, the type of bulk motion, and also on the quality of the remaining data. EMORe works best for sporadic bulk motion, where the participant returns to the original state after the movement, e.g., coughing, sneezing, or twitching. In contrast, if a subject undergoes a persistent position change and remains in a shifted motion state for the remainder of the acquisition, the impact on the performance depends on the relative amount of valid data acquired before and after the shift. In the phantom study, it should be noted that bulk motion was confined to a finite number of motion episodes, while the remaining readouts remained motion-consistent. In the in vivo study, three volunteers were instructed to mimic coughing in the last 30 seconds of a 5-minute scan, while the rest of the scan remained unguided.

Visual comparisons in Fig. 3 further support these quantitative findings in the phantom study, showing that EMORe reconstructions exhibit clearer anatomical boundaries and reduced motion artifacts relative to CS, particularly at higher outlier levels. Moreover, the surrogate signals overlaid with outlier bin assignments in Fig. 4 demonstrate EMORe’s capability to aggressively reject outlier data during simulated bulk motion instances, resulting in improved Brier scores and preserving the fidelity of data in the valid motion state bins throughout the scan. However, it can be observed that not all readouts acquired during bulk motion episodes are rejected, as indicated by the white gaps between the red bars in Fig. 4. As highlighted in Fig. 6, EMORe’s outlier rejection is inherently more effective for central k-space (low-spatial-frequency) readouts, which are most sensitive to motion [4]. In contrast, outer k-space (high-spatial-frequency) readouts from different motion states tend to be more similar, making them harder to distinguish from valid data. Although corruption in outer k-space contributes less to gross motion artifacts, it can still lead to residual blurring of fine anatomical details. In addition, it can be observed that blurring in EMORe reconstructed images increases with increasing simulated outlier fraction. This phenomenon occurs due to two factors acting in parallel. First, a higher outlier fraction leads to an increase in residual outer k-space outliers. Second, increased outlier fraction results in greater data rejection, effectively increasing the acceleration rate, which further contributes to image blurring.

Moreover, the diverse bulk motion simulations produce varying effects on the respiratory and cardiac surrogate signals, as highlighted in the magnified 30-second intervals in Fig. 4. In the magnified interval of the 10% simulated outlier case, bulk motion predominantly perturbs the respiratory signal, while the cardiac signal remains relatively consistent. In contrast, in the magnified interval of the 20% case, both respiratory and cardiac surrogate signals exhibit noticeable perturbations. Interestingly, in the magnified interval of the 40% case, both surrogate signals appear largely unperturbed despite the presence of an 11-second simulated bulk motion episode. These observations highlight a key limitation of surrogate signal–based motion characterization, i.e., it may fail to capture diverse types of bulk motion. In contrast, EMORe leverages k-space data consistency within motion bins rather than relying solely on surrogate signals.

From the in vivo study, EMORe reconstructions showed statistically significant improvement in blood–myocardium edge sharpness, compared to CS recovery without adaptive refinement of bin assignments. Likewise, the expert reviewers’ 1−5 Likert scores, averaged across cine image pairs ($N=26$), improved by 0.45 via EMORe processing, showing enhanced robustness to motion artifacts. Together, these improvements suggest practical value in clinical application.

Among the 13 volunteers, 3 were instructed to perform intermittent coughing to provide a controlled evaluation of EMORe’s ability to identify and discard data readouts corrupted by sporadic bulk motion. Results in Fig. 11 showed aggressive rejection of corrupted readouts during the known coughing episode. For Volunteer #4, an irregular breathing pattern, and for Volunteer #6, interspersed deep breaths, led to more frequent assignment of readouts to the outlier bin. Consistent with the trends observed in the simulation studies (Fig. 4), the magnified panels of Volunteer #4 and Volunteer #6 show that perturbations in the cardiac signal coincide with the irregularities in the respiratory signal. In contrast, in the initial portion of the magnified interval for Volunteer #11, pronounced outlier rejection occurs despite no visible abnormalities in either surrogate signal, mirroring simulation scenarios where bulk motion is not reflected in the surrogate signals. As this study was conducted in healthy volunteers, no arrhythmic events were expected or observed. Nevertheless, EMORe is designed to identify and reject inconsistent readouts irrespective of their origin, and therefore has the potential to mitigate artifacts arising from irregular cardiac rhythms. A dedicated study incorporating arrhythmic conditions is warranted to systematically evaluate this capability.

The adaptive binning in the EMORe framework is achieved at a modest computational overhead, as reflected in the reconstruction time increases of approximately 24% and 16% for phantom and in vivo datasets, respectively. Moreover, EMORe required the selection of additional hyperparameters, including the prior weights ${\alpha }_g$ and ${\alpha }_o$, as well as the outlier rejection threshold $\tau$, to balance the trade-offs among artifact suppression, outlier rejection, and algorithm stability. While this initial evaluation study is limited to 3D Cartesian sampling, the mathematical formulation is agnostic of the sampling trajectory. Future work will explore the integration of EMORe with non-Cartesian trajectories, such as golden-angle radial or spiral sampling. These trajectories more densely sample the center of the k-space and may yield more reliable binning correction and outlier rejection of the readouts. Adapting the framework would involve replacing the standard Fourier transform in the forward operator ${\mathit{A}}_{(n,k)}$ with a Non-Uniform Fast Fourier Transform (NUFFT). While the use of NUFFT increases the computational cost per iteration, the underlying EMORe remains identical and stands to benefit from the sampling properties of non-Cartesian data.

Future directions include replacing the M-step in EMORe with a deep learning-based reconstruction method to potentially further improve the image quality. Additionally, extending EMORe to in vivo 5D flow studies could enhance quantitative flow analysis and broaden its clinical applicability. Furthermore, evaluating EMORe in patient populations with irregular breathing or arrhythmias would provide further validation of its robustness in clinical settings.

V. Conclusion

EMORe enables motion-robust imaging in free-running, free-breathing self-gated 5D cardiac MRI. The robustness derives from adaptive, probabilistic, retrospective data binning and outlier rejection implemented in an expectation-maximization framework. The EMORe approach adaptively mitigates binning inaccuracies caused by either motion estimation techniques or sporadic bulk motion. For 3D imaging resolved in both cardiac and respiratory phases, the technique offers improved image sharpness and motion artifact suppression, compared to conventional methods.

Biography

Syed M. Arshad was born in Lahore, Pakistan. He received the B.S. degree with honors in electrical engineering from the University of Engineering and Technology (UET), Lahore, Pakistan, in 2019, and the M.S. degree in electrical and computer engineering from The Ohio State University, Columbus, OH, USA, in 2024.

He is currently a Ph.D. Candidate with The Ohio State University, Columbus, OH, USA. His research focuses on motion-robust whole-heart magnetic resonance imaging and is supported by National Institutes of Health (NIH)-funded projects and the OSU Presidential Fellowship (2025–2026). He was an MR Feature Development Research Scientist Intern with Canon Medical Research USA, Inc. He has contributed to advanced reconstruction techniques for free-running volumetric cardiovascular MRI, including work published in Magnetic Resonance in Medicine. His broader research interests include advanced optimization methods and machine learning for medical image reconstruction and processing.

Mr. Arshad is a Trainee Member of the Society for Cardiovascular Magnetic Resonance (SCMR) and the International Society for Magnetic Resonance in Medicine (ISMRM). He was a recipient of the 2024 Graduate Associate Leadership Award (GALA) from The Ohio State University and the 2019 IEEE IAS Humanitarian Project Award.