Moner: Motion Correction in Undersampled Radial MRI with Unsupervised Neural Representation
Authors: Qing Wu
Chenhe Du
Xuanyu Tian
Jingyi Yu
Yuyao Zhang
Hongjiang Wei
Venue: ICLR 2025 Spotlight (2025)
Keywords: MRI Reconstruction
Motion Correction
Neural Representation
NeRF
Unsupervised Learning
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: We propose Moner, an unsupervised method that can jointly recover high-quality MR images and estimates  accurate motion from undersampled and rigid motion-corrupted radial MRI measurement data without the need for any extra data.

Abstract: Motion correction (MoCo) in radial MRI is a particularly challenging problem due to the unpredictability of subject movement. Current state-of-the-art (SOTA) MoCo algorithms often rely on extensive high-quality MR images to pre-train neural networks, which constrains the solution space and leads to outstanding image reconstruction results. However, the need for large-scale datasets significantly increases costs and limits model generalization. In this work, we propose Moner, an unsupervised MoCo method that jointly reconstructs artifact-free MR images and estimates accurate motion from undersampled, rigid motion-corrupted k-space data, without requiring any training data. Our core idea is to leverage the continuous prior of implicit neural representation (INR) to constrain this ill-posed inverse problem, facilitating optimal solutions. Specifically, we integrate a quasi-static motion model into the INR, granting its ability to correct subject's motion. To stabilize model optimization, we reformulate radial MRI reconstruction as a back-projection problem using the Fourier-slice theorem. Additionally, we propose a novel coarse-to-fine hash encoding strategy, significantly enhancing MoCo accuracy. Experiments on multiple MRI datasets show our Moner achieves performance comparable to SOTA MoCo techniques on in-domain data, while demonstrating significant improvements on out-of-domain data. The code is available at: https://github.com/iwuqing/Moner

Published as a conference paper at ICLR 2025
MONER: MOTION CORRECTION IN UNDERSAMPLED RADIAL MRI WITH UNSUPERVISED NEURAL REPRE- SENTATION
Qing Wu ShanghaiTech University wuggangeshanghaaitthh..h
Chenhe Du
ShanghaiTech University
duchenhef stneeee@@@haaeee shwngaaiinngggaaeaaaaaaaaaaaaa tech edu, cn
Xuanyu Tian
Jingyi Yu ShanghaïTech University
ShanghaiTech University
clanxylshanghattechheii
edu.
Yuyao Zhang
Hongjiang Wei
ShanghaiTech University
Shanghai Jiao Tong University hongjlang, wei@s jtu. enu.en
thangyy@shaanghiitccc.....
ABSTRACT
Motion correction (MoCo) in radial MRI is a particularly challenging problem due to the unpredictability of subject movement. Current state-of-the-art (SOTA) MoCo algorithms often rely on extensive high-quality MR images to pre-train neural networks, which constrains the solution space and leads to outstanding image reconstruction results. However, the need for large-scale datasets signifi- cantly increases costs and limits model generalization. In this work, propose Moner, an unsupervised MoCo method that jointly reconstructs artifact-free MR images and estimates accurate motion from undersampled, rigid motion-corrupted k-space data, without requiring any training data. Our core idea is to leverage the continuous prior of implicit neural representation (INR) constrain this ill ill-posed inverse problem, facilitating optimal solutions. Specifically, we integrate quasi- static motion model into the INR, granting its ability correct subject's motion. To stabilize model optimization, we reformulate radial MRI reconstruction as a back- pook-projection problem using the Fourier-slice theorem. Additionally, we pro- pose novel coarse-to-fin fine hash encoding strategy, significantly enhancing MoCo accuracy. Experiments on multiple MRI datasets show our Moner achieves perfor- mance comparable SOTA MoCo techniques on in-domain data, while demon- strating significant improvements on out-of-domain data. The code available at: https: / wtw.:/..c.wwwwwa//eeeeeeeeeeeooe wwt.///wwggg///////e/r/
INTRODUCTION
Radial magnetic resonance imaging (MRT) an important technique in medical diagnostics and re- search (Feng, 2022), where measurements, i.e. k-space data, are acquired by lines passing through the center of Fourier space from different views. However, its long acquisition time increases costs and makes it susceptible to motion artifacts (Spieker et al., 2023a). While undersampling can effectively accelerate MRI acquisition, reconstructing artifact-free MR images from under- sampled, motion-corrupted k-space data challenging ill-posed inverse problem due viola- tion of Nyquist's sampling theorem and motion effects. Conventional analytical methods, such as NulFFT (Fessler, 2007), lack effective prior constraints and often fail recover satisfactory MR images, emphasizing the need for advanced motion correction (MoCo) algorithms.
Equal contribution. Corresponding author.

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Published as a conference paper at ICLR 2025
As the emergence of deep learning (DL), supervised methods have significantly improved the qual- ity of MR images (Han et al., 2018; Liu et al., 2020; Sommer et al., 2020). They typically train deep neural networks on large-scale paired MRI datasets to learn inverse mappings from artifact-corrupted images to artifact-free ones. However, such end-to-end learning paradigm neglects motion model- ing, causing severe hallucinations (Singh al., 2024), where visually plausible MRI reconstructions are inconsistent with the acquired k-space data.
Recent works (Singh al., 2024; Levac al., 2023; 2024) propose integrating pre-trained neu- ral networks with model-based optimization. Data priors from the pre-trained networks effectively constrain the solution space, while model-based optimization ensures reliable data consistency us- ing MRI physical models, enabling the recovery of high-quality MR images with high data fidelity. However, pre-training the networks necessitates numerous diagnosis-quality MR images, signifi- cantly increasing reconstruction costs. Although these models demonstrate greater robustness com- pared to supervised methods, they remain vulnerable out-of-domain (OOD) issues. These limita- tions undermine their practicality and reliability in real-world applications.
As an unsupervised DL framework, implicit neural representation (INR) has shown great promise in MRI reconstruction (Shen et al., 2022; Xu et al., 2023; Spieker et al., 2023b). The INR-based methods represent the unknown MR image as a continuous function parameterized by a multilayer perceptron (MLP). By incorporating the differential Fourier transform, the MLP network can be optimized by minimizing prediction errors on the k-space data. The learning bias of the MLP to low-frequency signals (Rahaman al., 2019; Xu al., 2019) facilitates recovery of high-quality MR images under non-ideal conditions. However, the potential of the unsupervised INR framework for the challenging MRI MoCo problem remains unexplored.
In this work, we propose Moner, a novel INR-based approach for addressing rigid motion artifacts in undersampled radial MRI. By leveraging the continuous prior inherent the INR, our Moner can effectively tackle the ill- ill-posed nature of the inverse problem, while presenting several innova- tive designs to enhance reconstructions. First, we introduce quasi-static motion model into the INR, granting its ability to accurately estimate subject's ss rigid motion during MRI acquisition. Then, we reformulate the radial MRI recovery as back-projee pack-projection problem using the Fourier-slice the- orem. This new formulation allows to optimize the INR model on projection data (i.e., Radon transform), mitigating the high dynamic range issues caused by the MRI k-space data and thus sta- bilizing model optimization. Moreover, based on hash encoding technique (Müller et al., 2022), introduce a novel coarse-to- to-sse-t--fine learning strategy, significantly improving MoCo accuracy.
The proposed Moner is an unsupervised DL model, making it adaptable to various MRI scenarios. We evaluate its performance on two public MRI datasets, including fastMRI (Knoll et al., 2020) and MoDL (Aggarwal et al., 2018). The results show our method performs comparably state-of-the- art MoCo techniques on in-domain data, while significantly surpassing them on out-of-domall data. Extensive ablation studies validate the effectiveness the several key components of our Moner.
PRELIMINARIES AND RELATED WORKS
In this section, we first revisit the Fourier-slice theorem discuss the relationship between radial k-space data and projection data (Sec. 12. Then, we briefly review advanced approaches for MRI motion correction (Sec. $2.2) and implicit neural representation (Sec. $2.3).
2.1 RADIAL MRI S-SCC DATA VERSUS PROJECTION DATA
Let f(x,y) v) E Clow represent a complex-valued MR image of w size, then the radial MRI k-space data k(0,w) w) E Caxm can be written as:
k(0,w) ()) = /ʃ f f(x, v) e j2mmm (n con 0+g drdy,
(1)
where A E [0. 2T) represents the acquisition angle. n and m respectively denote the number and size of lines (i.e., k(0,-) :) C", also known as spokes) in the k-space data.
According to the Fourier-slice theorem (Gonzalez, 2009), the 1D inverse Fourier transform (IFT) of the k-space data k00 w) over the variable w is equal to integral projection g00, p) E Ca of the
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MR image along the view 0, defined as:
g(0,p) _ /f f(x,y)
(ang)))((an)) 8(2 sin p)drdy
/ || f(x. v) 32 cos oty sin drdy dww = To- (K
(2)
k0,w)
where (.) denotes the ID IFT operator over the variable w and ö(.) is the Dirac delta function. Eq. 2 implies that the radial MRI solving (i.e., - f) can be theoretically reformulated as a back-projection problem (i.e., g f) using the ID IFT operator. A recent study cardiac MRI acceleration (Catalán et al., 2023) introduces the Fourier-slice theorem to optimize a neural field directly from raw k-space data, showing promising potential. Similarly, but with distinct focus, our work uses the Fourier-slice theorem to reformulate radial MRI as a back-projection problem, fundamentally addressing the high-dynamic range issue and stabilizing optimization.
2.2 ADVANCED APPROACHES FOR MRI MOTION CORRECTION
Classical optimization methods use explicit priors, such as TV for local smoothness (Rudin et al., 1992), to address the MRI MoCo problem. However, these handcrafted regularizers fail to fully characterize distribution of images, leading limited performance. With the advancement of high- performance computing and neural networks, DL-baaed methods (Han et al., 2018; Küsstnr et al., 2019; Liu et al., 2020; Sommer et al., 2020; Duffy et al., 2021; Lyu al., 2021: Bao et al., 2022: Lee ct 2021; Oksuz, 2021; Haskell et al., 2019; Cui et al., 2023; Chen et al., 2023a; Singh et al., 2024; Levac et al., 2023; 2024; Klug al., 2024) have significantly outperformed traditional model-based optimization algorithms. Currently, the Score-MoCo proposed by Levac et al. (2023; 2024) is the state-of-the-art (SOTA) MRI MoCo method, benefiting from generative diffusion priors and well- designed optimization strategies. However, these DL-based methods often require extensive high- quality MR images for pre-training neural networks, which significantly increases reconstruction costs and often suffers from the
(OOD) problem (Spieker et al., 2023a). In contrast, our Moner follows an unsupervised paradigm and does not require additional MRI data, significantly enhancing its applicability and generalization across various clinical scenarios.
2.3 IMPLICIT NEURAL REPRESENTATION FOR MRI RECONSTRUCTION
Implicit neural representation (INR) has emerged as a universal unsupervised DL framework for visual inverse problems, such novel view synthesis (Mildenhall et al., 2021) and surface recon- struction (Wang et al., 2021). Its core concept is to leverage the learning bias inherent in neural networks (Rahaman et al., 2019; Xu al., 2019) regularize the underdetermined nature of inverse problems, while incorporating differentiable forward models to simulate physical processes for high data fidelity. Recently, many INRedased MRI reconstruction methods (Feng al., 2023; Shen al., 2022; Xu et al., 2023; Kunz et al., 2023; Feng et al., 2022; Spieker et al., 2023b; Wu et al., 2021; Huang et al., 2023: Chen et al., 2023c; Catalán al., 2023) have been proposed. By using Fourier transforms to simulate MRI acquisition, they can reconstruct high-quality MR images with high data fidelity. However, these -bRebased methods fail to address motion-corrupted radial MRI, since they lack motion modeling. Moreover, these methods either entirely overlook the high dynamic range problem in k-space data or specifically design a relative l2 loss to alleviate it (Huang et al., 2023; Spicker et al., 2023b; Feng et al., 2022). Instead, our Moner incorporates motion model into the INR framework and introduces a new formulation for the radial MRI, fundamentally addressing the MRI MoCo problem.
PROPOSED METHOD
This section introduces Moner model. First, we define new formulation for the rigid motion- corrupted radial MRI (Sec. $3.1). Then, we present a quasi-static motion model within the INR framework and our model optimization pipeline (Sec. $3.2). Finally, propose a novel coarse-to- fine hash encoding that can significantly improve MoCo accuracy (Sec. $3.3). An overview of the proposed Moner model is shown in Fig.
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Σ
A(AA)
6 p) + gio.
Eq. (5)
Loss Function Eq. (7)
Ma
2 IR
IR
(a(*), b(a)) E R
Figure 1: Overview of proposed Moner model. Given any ray r(0;: p) at the 2D canonical space, we uniformly sample multiple coordinates a E rr(:.. p) and generate their version ė at the physical- world space via spatial transform (Eq. 5). Then, the INR network Mo predicts the real a(i) and imaginary b()) parts of the MR images. The projection data g(0.,)) can be obtained using integral projection (Eq. 6). Finally, we jointly optimize the INR Ms and motion parameters (d., Ti) mini- mizing the loss L (Eq. 7) between the estimated g(t: p) and measured g p) projections.
3.1 PROBLEM FORMULATION
Our goal is reconstruct artifact-free MR images f y) from undersampled (i.e., nm < hw in Eq. 1). rigid motion-corrupter radial MRI k-space data k(0,p) p) in an unsupervised manner. There are two key challenges solving the ill-posed inverse problem: How to inject effective priors for narrowing the solution space? 2) How to ensure stable optimization solving?
The SOTA Score-MoCo method (Levac et al., 2023; 2024) represents the MR image discrete matrix and uses dif- fusion priors to solve it, requiring numerous MRI images for pre-training and suffer from the OOD problem. Our Moner instead adopts the inherent prior of the unsupervised INR learn the continuous representation of the MR im- age, in which an implicit function of spatial coordinates for the complex-valued MR image is defined below:
1D IFT Eq. (2)
k(0,w)
g p)
f: (I, 3) R² (a(x), b(x)) R2,
(3)
Figure 2: Illustration of transforming
where denotes any spatial coordinate in 2D canonical space [-1, 1] x [] The variables a(x) and b(x) rep- resent the real and imaginary parts of the complex-valuee MRI image f(x) at position z. We then use an MLP network Ma. which takes any coordinate æ as input and outputs 2D vector corresponding function f. Due to the neural networks' inherent learning bias towards low-frequency signal pat- terns (Rahaman al., 2019; Xu al., 2019), this continuous function f can be well-approximated, enabling the recovery of high-quality MR images.
radial MRI k-space data & into pro- jection data g via the ID IFT T
the real and imaginary parts (a(x), b(x)), to fit the
To achieve stable inverse learning, we introduce the Fourier-slice theorem (Eq. 2) to reformulate radial MRI reconstruction (i.e., & + f) as back-projectto problem (i.e., - f). As illustrated in Fig. we first transform the k-space data ls into projection data g using the ID IFT operator T By employing differentiable projection model (Eq. 6). we then optimize the MI.P network Ms to reconstruct MR images by minimizing prediction errors on the projection data g. Since the value range of the projection data is much narrower than that of the k-space data k. this optimization solving effectively avoids the high dynamic range problem (Feng et al., 2022; Spicker et al., 2023b) and thus stabilizes model optimization.
3.2 QUASI-STATIC MOTION MODEL AND OPTIMIZATION
We introduce quasi-static motion model (Spieker et al., 2023a), a standard model in the field of the MoCo, into the INR framework, extending for estimating the subject's motion. The motion model has two basic assumptions: 1) The motion is rigid, which is common in many clinical MRI acquisitions, such as brain and leg imaging. 2) The motion occurs between the spokes but remains stationary during the fast acquisition of single spoke. This quasi-static assumption is reasonable since scanning a single spoke typically only takes a few milliseconds using radial MRI sequences, such fast low-angle shot (Zhang et al., 2010).
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For any acquisition views 8,, where i f1, ,n), triplet (1e, T2.s, Ty.s), where it, denotes rotation angle, and Ta, and Ty. are shifts along the X-axis and Y-axis. Then, a rotation matrix (1)) E RR2x2 and shift vector Ti E RR2 are defined as:
define corresponding learnable motion
A().) = cos 0 sin 2 sin d2 cos 0;
T
(4)
where the motion triplets {(0,, Te.i, Ty.+) are estimated from scratch.
Fig. 1 demonstrates the workflow to optimize our Moner model. Given the projection data g(O,, p), we first construct a ray r(0,, p) = (<z,y) 2 cos + ysin = o} in the 2D canonical space and uniformly sample set of coordinates
rr(,, p) by pre-defined interval Az. Then, these coordinates are transformed into a physical-world space (i.e., with the presence of the rigid motion) via a spatial transformation operator, defined as:
(5) where ō represent the coordinates at the whosical-world space. Then, the MLP network MM takes these coordinates x as inputs and predicts the corresponding real and imaginary parts (a()), b(ā)) = Mo (i) of the MR image in the real-world space, i.e., with the subject's motion. Finally, we can generate the projection data using a integral projection model, defined by
T A(0,) + Ts VE E E((..))))
g(0,,))
Σ
a(x)
Aar +J j
Σ (Er(O..))
b(x) Az.
(6)
(O.....))
Because both the spatial transformation (Eq. 5) and integral projection model (Eq. 6) are differen- tiable, we can jointly optimize the motion triplets {(1), Ta,is T= and the MLP network Ma using back proo-propagtttonn techniques, such as Adam (Kingma & Ba, 2014), to minimize the loss func- tion measuring errors between the estimated and real projections, defined as:
Σ (IR(G - g(...p)) + (((((,,)) - g(0..)))).
(7)
r(0.,))R
where R. is set of the random sampling ray r(0,,)) p) at each optimization step. R(- and 3{-} respectively are the real and imaginary parts of a complex value. After the model optimization, the high-quality MR image canonical space into the well-trained network Ma.
a + jb can be solved by feeding all voxel coordinates æ at the 2D
3.3 COARSE-TO-FING HASI ENCODING
We use cutting-edge hash encoding (Müller et al., 2022) with two fully-connected layers to imple- ment the MLP network MM. The hash encoding module transforms low-dimensional coordinates This mapping
x into high-dimensional features (v;(x) E R | at multiscale resolutions significantly enhances the MLP network's 's ability to fit high-frequency signals, thereby accelerating model optimization. However, the powerful hash encoding always impedes motion estimation, ulti- mately degrading model performance. This is because solving motion estimation primarily relies on low-frequency structural patterns (e.g., skull and leg bones) rather than high-frequency details (e.g., cerebellum). While using coarse-resolution hash encoding with small L can improve motion estimation by focusing on these low-frequency signals, it also limits the model's capacity to capture high-frequency components, leading loss of image details.
To this end, we propose novel coarse-to-fine learning strategy to achieve the balance between mo- tion estimation and fine-detailed image reconstruction. Technically, we first assign mask vectors {a for the features {<) at the multiscale resolutions. Inspired by the studies of camera registrations (Lin al., 2021; Chen et al., 2023b), we piecewise update the mask vectors during the model optimization as follow:
1 E F
i else
i
(
0
E
where À E [0, L] is value proportional to the optimization process. Leveraging the mask vectors 1L we can control periodically the spatial resolution of the hash encoding. Finally, the output feature FLF, is generated by
{ar)
vxx) = [vi on
(V2(x) @ az) (vz( 2 oar)
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where @ denotes the Hadamard product and denotes the concatenation operator.
Through the coarse-to-iie learning strategy, in early optimization our model can only capture the low-frequency global structures due to the coarse resolution hash encoding, which benefits accurate motion estimation. As the iteration continues, its learning ability gradually improves, enabling it to recover high-frequency local image details.
4 EXPERIMENTS
This section explores two key questions: techniques for the rigid motion-corrupted MRI reconstruction? 2) How do the key components of our Moner affect its performance? We conduct extensive experiments to address these questions.
Can our unsupervised Moner outperform the SOTA
4.1 EXPERIMENTAL SETUPS
Datasets The fastMRI (Knoll et al., 2020) and MoDL (Aggarwal al., 2018) datasets used in our experiments. For the fastMRI dataset, we first extract 1,925 brain MR slices with image sizes of 320x320 and image spacing of 1x1 mm² from different subjects along the axial direction. These slices include three contrasts of TIw, T2w, and FLAIR. We then split them into three parts, including 1,800 slices for training set, 100 for validation set, and 25 for test set. The training and validation sets are used solely for training supervised baselines, while our Moner does not access them. For the MoDL dataset, we use 20 T2w brain MR slices with image sizes of 256 x26 and an image spacing of 1 Ix1 mm² from 20 subjects along the sagittal direction for additional test.
Pre-processing We use 2D radial sampling pattern with the golden-angle acquisition scheme. Each spoke has length of 511, corresponding to an imaging FOV of 511x511 mm? The fully sampled radial s-space data thus consists of approximately 720 views. For undersampled MRI, we set acceleration factors (AFs) to 2 and 4x, corresponding to 360 and 180 views of undersampled radial k-space data, respectively. Following the motion simulation pipeline used in previous stud- ies (Levac et al., 2023; 2024; Spicker et al., 2023a), we first divide all spokes into 18 motion stages, where spokes within the same motion stage share the same motion trajectory. Then, we simulate four levels of rigid motion 3 {2, 5. 10, 15) with random rotations of [- 3] and shifts of of mm along the X-axis and Y-axis. The MRI reconstructions under different settings (i.e., different AFs and motion ranges B) are considered as different tasks. Thus, all training and test processes are conducted independently.
Baselines and Metrics Four representative MRI MoCo approaches are compared, including: 1) one analytical algorithm (NuIFFT (Fessler, 2007)), 2) one itcrative optimization algorithm (TV (Rudin al., 1992)), 3) image-based: supervised DI. models (DRN-DCMB (Liu et al., 2020)), and 4) one supervised diffusion-based DL method (Score-MoCo (Levac et al., 2023; 2024))- The Score-MoCo is currently the SOTA model for the MRI MoCo. For the reconstructed MR im- ages, we use peak signal-to-noie ratio (PSNR) and structural similarity index (SSIM) as quantitative evaluation metrics. For the estimated motion parameters, we compute the standard deviation of ab- solute eтrors between estimations and trues, denoted by dg and T More details of the baselines and metrics can be found in Appendix A.2 and A.3.
Implementation Details For our Moner, we employ the hash encoding (Müller et al., 2022) fol- lowed by two fully-connected (FC) layers with a width of 128 to implement the MLP network Mg. The first FC layer is followed by a ReLU activation, while the second one (i.e., the output layer) has no activation. For the hash encoding (Müller et al., 2022) used in our model, we set hyper- parameters as follows: base resolution Nmi = 2, maximal hash table size T = 218 and resolution growth rate = 2. Our coarse-to-fine strategy sets its resolution L from to 16 as optimization pro- gresses. At each iteration, we randomly sample 80 rays (i.e., | 80 in Eq. 7). We use the Adam algorithm with default papper-prramterss (Kingma & Ba, 2014) to optimize the model. The learning rate is initialized to 0.001 and decays by half every 1.000 epochs. The total number of epochs is 4,000. Note that here the of the fostMRI dataset (Knoll et al., 2020) and are kept consistent across all other cases.
are determined based on 5 samples from the training set

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Table 1: Quantitative results (Mean in 00/a+) of motion parameters by compared methods on the fastMRI and MoDL datasets. Results of t-test statistical tests comparing our Moner to baselines are denoted by" "" (p-value 0.01), (p-value 0.05), and (not significant, p-value 2 0.05). Here the "AF" and "MR" represent acceleration rate and motion range, respectively, The best performances are highlighted in bold.
Opt
Sel Self-sup.
Unsup
Dataset
MR
TV
Score-MoCo
Moner (Ours)
+22
0.038
/0.290
0.038 /0.293**
0.009/0.00 00.55
15
0.038 /0.309* 0.038"*0..399
1.038"/00.133"
1.110/0.131
2x
+10
0.0377 /0,4077
0.040**/0.44 /0.470"
0.009/0.305 0.305
+15
0.036" /0.621
0.044 /0.648"
0.010/0.602
fastMRI
1.514** /0.676
0.052 /0.312
0.034/0.060
4:5
1.369 /0.670**
0.062 /0.296
0.047 0.119
4x
+:10
1.699 /0,922
0.047 /0.411
0.042/0.279
115
2.513* /1.520
0.050 /0.532*
0.043/0.812
+:2
0.037"
/0.275 0.037
0.389 /0.409 0.389/0..49
0.009/0.058
+5
"0400*/0.276
0.838* /0.7200
0.009/0.144 14.14
2x
+10
0.042* /0.371
0.748* 11.263**
0.008/0,286
+1
0.040 /0.553
0.096 /1.488*
0.009/0.570
MoDL
+2
0.167 /
1,966 /0.824**
0.019 0.019/0.065 0.065
15
0.276'
/
1.125 -125"~/0.78""
/0.782**
0.021 0.021/0.163 /0.163
4x
+00
0,257 /0.494"
1.642' /1.242
0.022/0.324 00.24
115
0.506 /0.824
1.721" '1.79'"
0.019/0.578 0..78
4.2
MAIN RESULTS
Comparison with SOTAs for MoCo Accuracy Table compares the performance of our Moner with two baselines. The other two baselines excluded as they do explicitly model motion. The TV algorithm, relying on local image smoothness, performs well at a high sampling rate of AF 2x. However, when the AF is increased 4x, further worsening the ill-posed nature of the inverse problem, its performance significantly declines. The Score-MoCo model, pre-trained on the fastMRI dataset, delivers good MoCo results on the same datasets, but struggles with the OOD prob- lem on the unseen MoDL dataset, performing even worse than the TV algorithm. In contrast, our Moner achieves the highest MoCo accuracy on both datasets, benefiting from the robust continuous prior of the INR and well-designed optimization strategy. Moreover, performance across different MRs. For instance, the shift error o, remains below for MRs ranging from +2 to 15 the fastMRI dataset. In summary, the results above confirm the superiority of Moner over SOTA techniques in our MoCo accuracy.
method maintains stable
Comparison with SOTAs for MR Image Table presents the quantitative comparisons (PSNR) of MR images reconstructed by our Moner and the baselines. The SSIM results are provided in Table 7. On the fastMRI dataset, both Score-MoCo and our Moner produce com- parable and satisfactory results, significantly outperforming the other baselines. However, when applied to the unseen MoDL dataset. the Score-MoCo model suffers from severe perfor- mance degradation due to the OOD problem, even falling behind the TV algorithm at AF 4x. This performance trend in MR image re- construction mirrors the results in MoCo ac- curacy. In contrast, our Moner consistently
AF
Time (min.)
AF = 2
6x
#15
-10
5x
5
TV
Moner (Ours)
Score-MoCo
Figure 3: Speed comparison of TV, our Moner, and SOTA diffusion-hased Score-MoCo the fastMRI dataset.
achieves robust and high-quality reconstructions across both datasets. Fig. 4 shows the qualita- tive results. The analytical NulFFT produces severe artifacts. The image- supervised DRN- DCMB yields smooth but fuzzy MRT results. The TVV algorithm, relying local image smoothness,
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Table 2: Quantitative results (Mean-STD in PSNR) of MR images by compared methods on the fastMRI and MoDL datasets. Results of t-test statistical tests comparing our Moner to baselines are denoted by ** (p-value 0.01), (p-value 0.05), and (not significant, p-value 2 0.05). Here the "AF" and "MR" represent acceleration rate and motion range, respectively, The best and second performances are highlighted in bold and underline, respectively.
Analy.
Opt im.
sup.
Unsup
Dataset
MR
NulFFT
TV
DRN-DCMB
Score-MoCo
Moner (Ours)
122
27.16=2.33*
29.53:12.224
29.80:1.744
33.941.2.29
32.64:12.65
+
22.93=2.06"
29.53=22119*
25.63+2.57
33.64+22.22
32.62:+2.61
2x
20.46-2.02*
29.46:t-2.27
22.62422.40*
33.52.12.55*
32.49:+2..78
19.55=2.07* +15
29.44:12.28
21.0742.13*
33.69:1..38*
32.50+2.65
fastMRI
+2
25.99-2.01*
25.12+33.91
29.18+2.44*
32.33++2.39
31,49+2.551
15
22.30=2.15**
25.56.13.72*
25.18:12.53**
32.03-...36'
31.07
4x
+10
19.81=1.96"
25.41=4..05"
22.01+2..36*
32.34++..25*
31.20+
19.20-2.03**
25.1243398
21.22:42.23*
32.22.42.38
31,07:12.74
#2 1..91=
31.33=11111"
29.66
33.344:3.54
34.56:10.92
+5
25.15-
31.32+
25.79
32.45-+4.27'
0..5
2x
1..51=
31.29
23.9811.10"
32.30=3.533
04.57
+15
0.97
31.26+1.005
22.52+1.044"
33.90++.228'
34.3++0911
MoDL
12 27.89=1.08**
30.70
28.78
28.4513.004*
33,50-11033
+5
24.39=0.95"
30.30+2.000
25.67+#0.11*
30.09+
33.24
4x
+10
22.91=1.011*
3..26
23.98
30.33:1..48
33.21
22.40=0.92"
1
22.72:10.96
29.9413.19
33.25
0.2
Eas tMRI
0.381
/ 0.842
/ 0.695
3 0.840
30455 0.900
PSNR/ SSIM
MoDL
23.72/ 0.632
28.43/ 0.854
24.79 / 0.721
29.24 / 0.860
32.33/ 0.945
PSNR / SSIM
NulFFT
TV
DRN-DCMB
Score-MoCo
Moner (Ours)
GT
Figure 4: Qualitative and quantitative results MR images by compared methods on two test samples (#22 and #9) of the fastMRI and MoDL datasets for AF = 2x and MR +5.
reduces motion artifacts but introduces strong cartoon-like features and loses many image details. While Score-MoCo recovers high-quality MR images on the fastMRI dataset, its reconstructions on the unseen MoDL. dataset contain many artifacts due to inaccurate MoCo cuased by the OOD problem. Visually, our Moner consistently achieves high-quality MRI reconstructions across both datasets, further confirming its robustness and superiority over SOTA MoCo methods. Fig. 3 com- pares the reconstruction speeds of our method with two iterative methods (TV and Score-MoCo) using single NVIDIA RTX 4070 Ti GPU the fastMRI dataset. Our Moner achieves the fastest reconstruction speed, being more than 5x and 6x faster than the SOTA Score-MoCi model. Addi- tional visual results provided in Appendix A.4.
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Table 3: Quantitative results of MR images by our Moner ablating the motion model on the fastMRI dataset for AF 2x and MR 15. Results of t-test statistical tests are denoted by (p-value < 0.01), (p-value < 0.05), and (not significant, y-value 2 0.05).
23.86 /0.680
/0:91
PSNE/ SSIM
w/o MoCo
w/ MoCo
GT
Figure 5: Qualitative results of MR images by our Moner ablating the motion model on a sam- ple (#5) the fastMRI dataset for AF 2x and MR +5.
Motion Model
PSNR
SSIM
w/o MoCo w/ MoCo
21.90:12.22 32.37-1.94
0.614.1.
0.935+0.014
Table 4: Quantitative results of MR images and motion parameters by our Moner with different forward models on the fastMRI dataset for AF =2x and MR = #5. Results of t-test statistical tests denoted by (p-value 0.01), (p- value < 0.05), and (p-value 2 0.05).
0.800
9.936
SSID
NuFFT
Projection
GT
MR Image
Mction
Forw. Model
Figure 6: Qualitative results of MR images by our Moner with different forward models sample (#4) the fastRI dataset for AF 2x
PSNR
00/0,
NuFFT Projection
31.38122.1 32.37=1194
0.108 0.108*</0.456 0.009/0.041
/0,456
and MR = #5.
4.3 ABLATION STUDIES
Influence of Quasi-static Motion Model We first explore the effectiveness of the quasi-static motion model in our Moner. Specifically, we remove the motion model while keeping other settings unchanged for fair comparison. Fig. shows the reconstructed MR images. Clearly, without the motion model, our Moner fails to produce satisfactory reconstructions, with many artifacts caused by motion. contrast, the Moner with the motion model reconstructs clean images with fine details. Table 3 presents the quantitative results, showing that the motion model contributes improvement of over 10 dB in PSNR and 0.3 SSIM. This ablation study demonstrates that the motion model plays indispensable role in the motion-corrupted MRI problem.
significant
Influence of Forward Model for Optimization We then explore the influence of the forward model used in our optimization process. Current MRI MoCo methods (Singh et al., 2024; Levac al., 2023; 2024) typically leverage NuFFT (Fessler, 2007) as the forward model for optimization, which leads to high dynamic range problems. Some additional optimization tricks (e.g., normaliz- ing the loss function) are required (Feng et al., 2022: Spieker al., 2023b). contrast, our Moner introduces the Fourier-slice theorem to reformulate MRI reconstruction back back-projection prob- lem, allowing the use of a differential projection model. As shown Table 4, the projection model outperforms NuFFT in both MoCo accuracy and MRI reconstruction quality. The reconstructed MR images are shown in Fig. 6. From the visual comparison, the projection model produces superior image details compared the NuFFT model. This ablation study confirms the effectiveness of the proposed projection- based optimization pipeline in improving model performance
Influence of Coarse2fine Hash Encoding We finally investigate the influence of coarse-to-fine hash encoding on model performance. We compare with the naive hash encoding at coarse (L=6) and fine (L=16) resolutions, keeping other model configurations the same for fair evaluation. Fig. 8 shows the estimated motion parameters. Both the naive coarse (L=6) hash encoding and our coarse-to-fine strategy accurately predict motion parameters. However, the naive fine (L=16) hash encoding fails to correct the motion, confirming our argument that low-free wow-frequency image information is more crucial for solving MoCo. Fig. 7 displays the reconstructed MR images. The naive hash

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Table 5: Quantitative results of MR images by our Moner with different hash encodings on the fastMR dataset for AF 2x and MR =5. Results of t-test statistical tests are denoted by (p-value < 0.01), (p-value < 0.05). and (p-value 2 0.05).
00438
26 0.877
58 0:909
PSNR/ /SSI/SSIM
Fine (L=16)
Coarse (L=6)
Coarse2fine
GT
Hash Encoding
PSNR
Figure 7: Qualitative results of MR images by our Moner with different hash encodings on sample (#2) the fastMRI dataset for AF = 2x and MR = 15.
Fine (L=16)
23.69+33.2"
Coarse (L=16)
30.77::2.21
Coarse2fine
32.37+-1.94
encoding (both fine and coarse) cannot produce satisfactory MR images. The fine encoding degrades reconstruction due to incorrect motion estimation, while the coarse encoding loses image details due to its limited learning capacity. In contrast, our coarse-to-fine strategy achieves excellent visual results. The quantitative results shown Table also demonstrate the superiority of our method over the naive hash encoding.
5 CONCLUSION
This work proposes Moner, novel method address rigid motion-corrupted, undersam- pled radial MRI reconstruction. The proposed Moner an unsupervised DL model, which eliminates the need for external training data and thus can flexibly adapt to different MRI acquisition protocols, such different accel- cration rates. Our Moner makes several key innovations that significantly improve MRI constructions, including integrating a motion model into the INR framework, presenting new formulation for radial MRI reconstruc- tion, and introducing a coarse-to- co-se-to-fine hash cod- ing approach. Empirical evaluations on sev- eral MRI datasets demonstrate that our Moner achieves SOTA performance in terms of both efficiency and image quality.
Rotation Angle (9)
(mm)
5.0
2.5
Index of Stages
Index of Stages
--Fine (L=16)
Coarse (L=6) Coarse2fine GT
Figure 8: Motion trajectory by our Moner with different hash encodings a sample (#5) of the fastMRI dataset for AF = 2x and MR #5
Limitation While the proposed Moner demonstrates promising MoCo performance, it has several limitations. First, our Moner is currently designed for 2D MRI, while 3D MRI MoCo more prac- tical, as subject's movements occur in 3D space. Secondly, the quasi-static motion model assumes rigid motion between acquisition frames, but it falls short in simulating certain complex motion scenarios. These include motion occurring within acquisition frames, non-rigid motion, and intri- cate spin-history effects. However, we emphasize that extending it to 3D is feasible with advanced INR architectures, such as K K-plane (Fridovich-Keil et al., 2023) and Hexplane (Cao & Johnson, 2023). Also, for non-rigid motion, Moner can be extended by incorporating deformation field mod- eling (Reed et al., 2021). Additionally, our Moner primarily focuses on radial MRI by leveraging the naive Fourier-slice theorem. Adapting it to handle more diverse MRI sampling patterns (e.g., Cartesian and spiral) also presents a potential direction for future work.
Acknowlegement This study was supported by the National Natural Science Foundation of China (grant 62471296& & 62071299), 1299), the National Key R&D Program of China (grant 2024YFC242 and MoE Key Lab of Intelligent Perception and Human-Machine Collaboration (ShanghaiTech Uni- versity). Also, the authors thank Bowen Li, Jie Feng, and Guoyan Lao for their help with 3D MRI data acquisition.
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REFERENCES
Hemant K Aggarwal, Merry P Mani, and Mathews Jacob. Modl: Model-based deep learning archi- tecture for inverse problems. IEEE transactions on medical imaging, 38(2):394-445 -405, 2018.
Qinjia Bao, Yalei Chen, Chongxin Bai, Pingan Li, Kewen Liu, Zhao Li, Zhi Zhang, Jie Wang, and Chaoyang Liu. Retrospective motion correction for preclinical/clincaa magnetic resonance imag- ing based on a conditional generative adversarial network with entropy loss. NMR in Biomedicine, 35 :5(12), 2022.
Ang Cao and Justin Johnson. Hexplane: A fast representation for dynamic scenes. In Proceedings of the IEEE/CVE Conference on Computer Vision and Puttern Recognition, pp. 130-141, 2023.
Tabita Catalán, Matías Courdurier, Axel Osses, René Botnar, Francisco Sahli Costabal, and Claudia Prieto. Unsupervised reconstruction of accelerated cardiac cine mri using neural fields. arXiv preprint arXiv:2307..0. 0tt.:22007..3333, 2023.
Lixuan Chen, Xuanyu Tian, Jiangjie Wu, Ruimin Feng, Guoyan Lao, Yuyao Zhang, and Hongjang Wei. Jsmoco: Joint coil sensitivity and motion correction parallel mri with a self-calibrating score-based diffusion model. arXiv preprint arXiv:2310.0962. 2023a.
Yue Chen, Xingyu Chen, Xuan Wang. Qi Zhang, Yu Guo, Ying Shan, and Fei Wang. Local-to- global registration for bundle-adjusting neural radiance fields. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8264-8273, 2023b.
Zixuan Chen, Lingxiao Yang, Jian-Huang Lai, and Xiaohua Xie. Cunerf: Cube-based neural ra- diance field for zero-shot medical image arbitrary-scale super resolution. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 21185-21195, 2023c
Hyungjin Chung and Jong Chul Ye. Deep diffusion image prior for efficient ood adaptation in 3d inverse problems. arXiv preprint PrXXiv2407.1064// 2024.
Long Cui, Yang Song, Yida Wang, Rui Wang. Dongmei Wu, Haibin Xie, Jianqi Li, and Guang Yang. Motion artifact reduction for magnetic resonance imaging with deep learning and k-space analysis. PloS one, 18()):0078666, 2023.
Ben A Duffy, Lu Zhao, Farshid Sepehrband. Joyce Min, Danny JJ Wang, Yonggang Shi, Arthur W Toga, Hosung Kim, Alzheimer's ' Disease Neuroimaging Initiative, et al. Retrospective motion artifact correction of structural mri images using deep learning improves the quality of cortical surface reconstructions. Neuroimage, 230:117755 2021.
Jie Feng, Ruimin Feng, Qing Wu, Zhiyong Zhang, Yuyao Zhang, and Hongjiang Wei. Spatiotem- poral implicit neural representation for unsupervised dynamic mri reconstruction. arXiv preprint arXi::2301.001277 2022.
Li Feng. Golden Golden-angle radial mri: basics, advances, and applications. Journal of Magnetic Reso- nance Imaging, 36(1):45-66 62, 2022.
Ruimin Feng, Qing Wu, Jie Feng, Huajun She, Chunlei Liu, Yuyao Zhang, and Hongjiang Wei. Imjense: scan-specific implicit representation for joint coil sensitivity and image estimation in parallel mri. IEEE Transactions Medical Imaging, 2023.
Jeffrey A Fessler. On nufft-based gridding for non-cartesian mri. Journal of magnetic resonance, 188(2):191-155, 2007.
Sara Fridovich-Keil, Giacomo Meanti, Frederik Rahbæk Warburg, Benjamin Recht, and Angjoo Kanazawa. K-planes: Explicit radiance fields in space, time, and appearance. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12479-12488 2023.
Rafael C Gonzalez. Digital image processing. Pearson education india, 2009.
Yoseob Han, Jaejun Yoo, Hak Hee Kim, Hee Jung Shin, Kyunghyun Sung, and Jong Chul Ye. Deep learning with domain adaptation for accelerated projection-rrcoonstuuttiio mr. Magnetic resonance in medicine, 1063):1189-1255, 2018

---

Published as a conference paper at ICLR 2025
Melissa W Haskell, Stephen F Cauley, Berkin Bilgic, Julian Hossbach, Daniel N Splitthoff, Josef Pfeuffer, Kawin Setsompop, and Lawrence L Wald. Network accelerated motion estimation and reduction (namer): convolutional neural network guided retrospective motion correction using separable motion model. Magnetic resonance in medicine, 82((): 1454-)4155,,,,,,, 2019.
Wenqi Huang. Hongwei Bran Li, Jiazhen Pan, Gastao Cruz, Daniel Rueckert, and Kerstin Ham- mernik. Neural implicit k-space for binning-free non-cartesian cardiac mr imaging. In Alejandro Frangi, Marleen de Bruijne, Demian Wassermann, and Nassir Navab (eds.). Information Pro- cessing in Medical Imaging, pp. 548-560, Cham, 2023. Springer Nature Switzerland. ISBN 978-3-031-33444--2
Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6990 69800 2014.
Tobit Klug, Kun Wang, Stefan Ruschke, and Reinhard Heckel. Motionttt: 2d test-time-training motion estimation for 3d motion corrected mri. arXiv preprint arXiv:2409.09311 09340090077, 2024.
Florian Knoll, Jure Zbontar, Anuroop Sriram, Matthew J Muckley, Mary Bruno, Aaron Defazio, Mare Parente, Krzysztof J Geras, Joe Katsnelson, Hersh Chandarana, al. fastmri: A publicly available raw k-space and dicom dataset of knee images for accelerated mr image reconstruction using machine learning. Radiology: Artificial Intelligence, 2(1)xe190007, 2020.
Johannes Kunz, Stefan Ruschke, and Reinhard Heckel. Implicit neural networks with fourier- feature inputs for free-breathing cardiac mri reconstruction. arXiv preprint arXiv:2305.06822. 2023.
Thomas Küstner, Karim Armanious, Jiahuan Yang, Bin Yang, Fritz Schick, and Sergios Gatidis.
Retrospective correction of motion-affected mr images using deep learning frameworks. Magnetic resonance in medicine, 82(4):1527-1540, 2019.
Jongyeon Lee, Byungjai Kim, and Hyun Park. Mc2-net: motion correction network for multi- contrast brain mri. Magnetic Resonance Medicine, 86(2):1077-1022, 2021.
Brett Levac, Ajil Jalal, and Jonathan I Tamir. Accelerated motion correction for mri using score- based generative models. In 2023 IEEE 20th International Symposium Biomedical Imaging (ISBI), pp. 1-5. IEEE, 2023.
Brett Levac, Sidharth Kumar, Ajil Jalal, and Jonathan Tamir. Accelerated motion correction with deep generative diffusion models. Magnetic Resonance in Medicine, 92(2):853-868 2024.
Chen-Hsuan Lin, Wei-Chiu Ma, Antonio Torralba, and Simon Lucey. Barf: Bundle-adjustnn neural radiance fields. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 5741-5751, 2021.
Junchi Liu, Mehmet Kocak, Mark Supanich, and Jie Deng. Motion artifacts reduction in brain mri by means of deep residual network with densely connected multi-resolltion blocks (dmn-demb). Magnetic resonance imaging, 71:69-79, 2020.
Qing Lyu, Hongming Shan, Yibin Xie, Alan C Kwan, Yuka Otaki, Keücchir Kuronuma, Debiao Li, and Ge Wang. Cine cardiac mri motion artifact reduction using a recurrent neural network. IEEE Transactions Medical Imaging, (0(8)21770221 2021.
Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes neural radiance fields for view synthesis. Communications of the ACM, 65(1):99-106, 2021.
Thomas Müller, Alex Evans, Christoph Schied, and Alexander Keller. Instant neural graphics prim- itives with a multiresolution hash encoding. ACM transactions on graphics (TOG), (4): 2022.
Ilkay Oksuz. Brain mri artefact detection and correction using convolutional neural networks. Com- puler methods and programs in biomedicine, 199: 105909, 2021.
12

---

Published as a conference paper at ICLR 2025
Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks. In International conference machine learning. pp. 5301-5310. PMLR, 2019.
Albert W Reed, Hyojin Kim, Rushil Anirudh, Aditya Mohan, Kyle Champley, Jingu Kang, and Suren Jayasuriya. Dynamic ct reconstruction from limited views with implicit neural representa- tions and parametric motion fields. Proceedings of the IEEE/CVE International Conference on Computer Vision, pp. 2258-2268. 2021.
Leonid Rudin, Stanley Osher, and Emad Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: nonlinear phenomena, 60(1-4):259-266) 1992.
Liyue Shen, John Pauly, and Lei Xing. Nerp: implicit neural representation learning with prior embedding for sparsely sampled image reconstruction. IEEE Transactions on Neural Networks and Learning Systems, 35(1):770-782, 2022.
Nalini M Singh, Neel Dey, Malte Hoffmann, Bruce Fischl, Elfar Adalsteinsson, Robert Frost, Adrian V Dalca, and Polina Golland. Data consistent deep rigid mri motion correction. In Medical imaging with deep learning, pp. 368-381. PMLR, 2024.
K Sommer, A Saalbach, T Brosch, C Hall, NM Cross, and JB Andre. Correction of motion artifacts using multiscale fully convolutional neural network. American Journal of Neuroradiology, 41 (3):416 423,416-423,, 2020.
Veronika Spieker, Hannah Eichhorn, Kerstin Hammernik, Daniel Rueckert, Christine Preibisch, Dimitrios C Karampinos, and Julia A Schnabel. Deep learning for retrospective motion correction in mri: a comprehensive review. JEEE Transactions on Medical Imaging, 2023a.
Veronika Spieker, Wenqi Huang, Hannah Eichhorn, Jonathan Stelter, Kilian Weiss, Veronika A Zimmer, Rickmer F Braren, Dimitrios Karampinos, Kerstin Hammernik, and Julia A Schn- abel. Iconik: Generating respiratory-resolved abdominal reconstructions using neural im- plicit representations in k-space. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 183-192. Springer, 2023b.
Peng Wang, Lingjie Liu, Yuan Liu, Christian Theobalt, Taku Komura, and Wenping Wang. Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction. Advances in Neural Information Processing Systems, 01-27171--21888 2021.
Qing Wu, Yuwei Li, Lan Xu, Ruiming Feng, Hongjiang Wei, Qing Yang, Boliang Yu, Xiaozhao Liu, Jingyi Yu, and Yuyao Zhang. Irem: High-resolution magnetic resonance image reconstruction via implicit neural representation. In Medical Image Computing and Computer Assisted Intervention- MICCAI 2021: 24th International Conference, Strasbourg, France, September 27 77-October 1, 2021, Proceedings, Part VI 24, pp. 65-74. Springer, 2021.
Junshen Xu, Daniel Moyer, Borjan Gagoski, Juan Eugenio Iglesias, Ellen Grant, Polina Golland, and Elfar Adalsteinsson. Nesvor: implicit neural representation for slice-to-volume reconstruction in mri. IEEE transactions on medical imaging, 1206)11777 2023.
Zhi-Qin John Xu, Yaoyu Zhang, Tao Luo, Yanyang Xiao, and Zheng Ma. Frequency principle: Fourier analysis sheds light on deep neural networks. arXiv preprint arXiv: 1901. (1900,0652,, 2019.
Shuo Zhang, Kai Tobias Block, and Jens Frahm. Magnetic resonance imaging in real time: advances using radial flash. Journal of Magnetic Resonance Imaging, 31(1):001-19, 2010.
A APPENDIX
A.1 AN EXTENSION OF OUR MONER FOR 3D RADIAL MRI
Method Our Moner is currently designed for 2D radial MRL Here, demonstrate its extension to
3D radial MRI. Using a 3D stack-of-radit sampling (Feng, 2022), the 2D inverse Fourier transform of spoke array from a specific view corresponds to the respective 2D projections, as illustrated in
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2D IFT
Stack-of-Rase -f--adii k-space data
2D Projections in spatial domain
Figure 9: Illustration of the relationship between stack-of-radial k-space data and 2D projections in the spatial domain. The 2D inverse Fourier transform (IFT) of a spoke array (orange) from a specific view corresponds to the respective 2D projections (purple).
Fig. 9. This allows Moner be extended to 3D radial MRI by solving parallel back-projectie problem in 3D space. Moreover, we learn motion parameter (1), yp, th Te Ty, Ty) for each motion state to account for subject's movements occurring in 3D space. Specifically, rotation matrix A(0, 4, 1)) E I333 and a shift vector T E R are defined as below:
coss ub
sin 8b
(
COS 4p
0
sin
A(8, 1)
sin
cos 0
0 sin y
1
cos 0
sin 1 cos
0
cos yp
0
sin 1
(10)
Tz
T
y
By integrating the rotation matrix A(O, ip, 1)) and shift vector into the spatial transformation (Eq. 5), our Moner can effectively model and correct the rigid motion in 3D space.
Results on Simulated Data To test the effectiveness of Moner in 3D radial MRI, conduct a simulation study on 3D brain MR image with dimensions of x40 240 x 240 acquired by a 3T Siemens MRI scanner. We use 3D stack-of-radial sampling pattern with the golden-angle acquisi- tion scheme. Detailed parameters are as follows: FOV = 283 283 x833 image spacing 1x1x1 mm², and total spoke views 720. The AFs are set and 4x, corresponding
360 and 180 views, respectively. We also simulate two level of rigid motion B = (5, 10} with random rotations of [- 3 and shifts [-3, (]] mm along the X-axis, Y-axis and 7-axis. Note that here we consider 120 motion states, i... our model totally estimates 120x6 = 720 motion parameters.
Fig. 10 presents the qualitative comparisons. NulFFT struggles to achieve satisfactory MRI recon- structions, producing noticeable artifacts. In contrast, method produces results that are visually close to the GTs, preserving both global structures and local details. Quantitative results Table further validate the significant improvements of our method over traditional NulFFT. Additionally, Fig. 10 shows that our Moner accurately estimates 3D motion parameters.
Table 6: Quantitative results of 3D MR image by NulFFT and our Moner the 3D brain data. Results of t-test statistical tests comparing our Moner to NulFFT are denoted by (p-value < <0.1)), (p- value 0.05), and (not significant, p-value 0.05).
NuIFFT
Moner (Ours)
MR
PSNR
SSIM
PSNR
SSIM
+5
23.64**
0.499
31.58
0.844
2x
+10
21.77**
0.435*
31.64
0.843
+5
23.29**
0.439**
30.68
0.810
4x
+10
2
0.391
30.70
0.813
Results on Real-Worid Data we collect two Tlw brain 3D seans from a single subject using a 3.0 T United Imaging Health- care (UTH) uMR 790 scanner. To obtain motion-free data and motion-cormupted data, the subject
To evaluate the effectiveness of our Moner real-world MRI data,
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AF 2x, 110
AF 1R=
Axial
Sagi ttall
Coronal
NulFFT
Moner (Ours)
GT
NuiFFT
Moner (Ours)
GT
Figure 10: Qualitative results of 3D MR images (Axial, Sagittal, and Coronal views) by NulFFT and our Moner on the 3D brain data for AF = 2x, MR = = 10 and AF 3x, MR= 10.
X-axis Rotation Angle (*)
Y-axis Rotation Angle (*)
Z-axis Rotation Angle (*)
10.
10.
10.
MM MM m/
MMMMMM MM
M
MM
-10
Index of Stages
Index of Stages
Index of Stages
X-axis Shift (mm)
Y-axis Shift (mm)
Z-axis Shift (mm)
101
WAM MMMM MAl
MM M MM
M
6-
10
-30
30
60
50
120
60
120
80
120
Index Stages
Index Stages
Index of Stages
Our estimation
GT
Figure 11: Qualitative results of motion parameter estimation by our Moner on the 3D brain data for AF = 4x and MR = + 10.
instructed remain still or make abrupt movements (head shaking motion) three times during acquisition. The used radial Spoiled gradient echo sequence parameters we used are as follows: TR 7.1 ms; TE = ms; flip angle 17%; matrix size 120x 120x 120; FOV 240x240X mm²; number of spokes = 65,000. Note that the data acquisition is approved by the institutional review board. Our Moner model does not access any prior motion knowledge. instead assumes 650 motion stages and thus estimates unique i.e., 650x = 3,900 motion parameters.
Fig. 12 shows the visualizations. Due to the 3D reconstruction task, only NulFFT included as an compared method. The result of NulFFT suffers severely from motion artifacts, losing nearly a lots of image details. In contrast, our Moner's reconstruction closely resembles the motion- free reference in global structures and successfully recovers tissue details, such as the lateral ventricles, as highlighted in the zoomed-in view. Moreover, we shows the visualizations of motion trajectory estimated by Moner in Fig. 13. The estimated trajectory is generally consistent with the motion pattern. summary, this study preliminarily validates the effectiveness of our method on rorl-world MRI data.
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Axial
Sagittal
Coronal
NulFFT
Moner (Ours)
Motion-free Reference
Figure 12: On real-world Tlw human brain 3D MRI volume with dimensions of 120x 120x 120 and voxel size of 2x2x2 mm3, acquired using UIH uMR 790 scanner, our Moner effectively reduces motion artifacts and recovers both global structures and detailed image features.
X-axis Rotation Angle (*)
Y-axis Rotation Angle (*)
Z-axis Rotation Angle (9)
-5
-5
-5
-30
-10
-10
100
200
300
400
100
200
300
400
100
200
300
400
Time (s)
Time (s)
Time (s)
X-axis Shift (mm)
Y-axis Shift (mm)
Z-axis Shift (mm)
20
20
10
10-
10-
-20
400 -20
100
300
400
300
300
400
Time (s)
Time (s)
Time (s)
Figure 13: Visualizations of motion trajectory estimated by our Moner real-world TIw human bran 3D MRI volume with dimensions of 120x 120x 120 and voxel size of 2x2x2 mm3 acquired using UIH uMR 790 scanner. Scanning single spoke requires 7.1 ms. The whole acquisition (6,5000 spokes in total) thus takes about 461 s. The subject is instructed to make abrupt movements (head shaking motion) three times during the acquisition. The estimated trajectory is generally consistent with the motion pattern.
A.2 ADDITIONAL DETAILS OF METRICS
MR Image Quality In our evaluation, two commonly used visual Pestrics-PSRR and SSIM are employed to assess the quality of reconstructed MR images. These metrics are implemented using How-
the Python library skimage (https: //github. com/ ever, since the GT and reconstructed images may not at the same space and PSNR is pixel-wise metric, perform rigid registration align the reconstructed MR images with the GT images before calculating these metrics. This registration is conducted using the Python library ants hattps://github.com wttpp///...oo/aa///////....
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Rotstion 15
Y-axis shift
GT Image at Canonical Space
Reconstruction at Non-canonical Space
Figure 14: Illustration of the linear rigid transformation between GT canonical space and recon- structed image non-canonical space. Using the conventional & metric (Eq. 11), the rotation and shift errors are 15 and 12.5, respectively. However, when applying our MoCo metries (Eq. 12 and Eq. 13), both rotation and shift errors are measured to O.
MoCo Accuracy To measure the MoCo accuracy, many works (Levac et al., 2023; 2024; Chen al., 2023a) typically calculate the L distance, defined by
(1)
10 0.l. (TT) 11 Σ T -
(11)
where 0, and T; denote the GTs, while Ú, and f; represent the predictions. However, the & metric cannot accurately assess MoCo performance. In particular, a linear rigid transformation may exist between the GT image and the reconstructed image, meaning they are not aligned in the same space. While this linear transformation does not degrade image quality, it can lead large l errors. As shown in Fig. 14, the GT and reconstructed images are visually very similar, yet the & MoCo errors are significantly high (rotation = 15", shift = 12.5). To more accurately evaluate MoCo performance, propose calculating the standard deviation of the absolute errors between GTs and predictions, which is defined as below:
1 Ad,
0g
[Ad m0)3, with pe
(12)
n i=1
D ATi -17), with Pr
Ar: n
(13)
n
A lower value of the proposed metric denotes greater MoCo accuracy. The intuition behind this metric that if the transformation between the GT and reconstructed images is linear, the errors in the predicted MoCo parameters across different spokes are identical, iee, A Ad and ATI = AT = = AT,, resulting in a standard deviation of O. Compared to the conventional & metric, our new metric provides more accurate assessment of MoCo accuracy.
A0,
A.3 ADDITIONAL. DETAILS OF BASELINES
We compare our Moner model with four representative MRI MoCo algorithms. Here, provide the implementation details of these baselines to improve the reproducibility of this work.
NuIFFT Non-uniform inverse fast Fourier transform (NuIFFT) (Fessler, 2007) is an analytical reconstruction algorithm designed for MRI with non-uniform sampling patterns, such as radial and Poisson sampling. It first uses an interpolation algorithm (e.g.. linear) generate uniform k-space data, followed by applying the IFFT operator to reconstruct the final MR images. We implement using the function KbNu ftAdjoint from the Python library forch (https: github. wwt.://muukkkee/oooooootttr..e.
TV
Total variation (TV) (Rudin et al., 1992) is a widely used explicit regularizer for various ill- posed inverse reconstruction problems. For the MRI MoCo problem, solve the following opti-
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mization problem:
f ".' = arg min - f. 2 T (R (fim)) +
(14)
where T() is the Fourier transform, and R(.)) denotes the rigid transformation (rotation and shift) of the image f according to the motion parameter m, with weight À set to 1e-3. We solve the optimization problem using the Py Torch automatic differentiation framework, with a learning rate of le-3 and optimization epoch of 200. The Adam algorithm (Kingma & Ba, 2014), with default settings, is employed. Here the hyper-parameten were tuned using 5 samples from the training set of the fastMRI dataset (Knoll al., 2020) and were kept consistent across all other cases.
DRN-DCMB Deep residual network with densely connected multi-resolution blocks (DRN- DCMB) (Liu et al., 2020) is supervised end-to-end DL model for the MRI MoCo reconstruction. trains deep neural networks on paired MRI datasets to learn the inverse mapping from low-quality MR images to high-quality ones. Following the original paper, implement and train DRN-DCMB using the training and validation sets from the fastMRI dataset (Knoll et al., 2020). It is important note that an independent model is trained for each MRI acquisition setting, i.e. independently train& 8 models for 8 cases (2 AFs x 4 MRs) in our experiments.
Score-MoCo Levac et al. (2023; 2024) proposed Score-MoCo, SOTA approach for the rigid motion-corrupted MRI. This approach pre-trains a score-based generative model to provide high- quality prior images. During the inference phase, the model optimizes both the motion param- eters and the underlying image, ultimately searching for reconstruction results that satisfy both data consistency and the distribution prior. We reproduce the results based their official code (https: //github match our experimental settings. As the official pre-trained model is trained on only T2 brain MRI images, we take the diffusion model trained on the fastMRI (Knoll et al., 2020) BRAIN data from Chung Ye (2024). The saper-prrameters were tuned using 5 samples from the training set of the fastMRI dataset (Knoll et al., 2020) and were kept consistent across all other cases.
com/utcsilabe wwwcssiaaamooooooooooooooo score mri) with appropriate modifications
Table 7: Quantitative results (Men-+ in SSIM) of MR images by compared methods on the fastMRI and MoDL datasets. Results of t-test statistical tests comparing our Moner to baselines are denoted by ** (p-value < 0.01), (p-value < 0.05), and (not significant, p-value 2 0.05). Here the "AF" and "MR" represent acceleration rate and motion range, respectively. The best and second performances are highlighted in bold and underline, respectively.
Analy
Optim im.
Unsup.
Dataset
AF
MR
NuIFFT
TV
DRN-DCMB
Score-MoCo
Moner (Ours)
+2
0.537+0.057
0.860=0.090"
0.836=0. 102
0.866=+0.045**
0.915
15
0.36410.065* 0.85910.090
0716=0.104"
0.849-10.038**
0.914110067
2x
100
0.2744..0074*
0.85910.091
0.624=0.118
0.851:10.042
0.915=10.067
+1
0.007
0.859-10.011
0.532=0.102
0.857=00.011*
0.913=10.071
fastMRI
+2
0.460+00.042*
0.613+0.202*
0.812-0.096
0.813+00.52"*
0,888+0.077
+5
0.308+0.0055**
0.645+0.203*
0.715-0.116
0.800+000000*
0.879-0.078
4x
10
0.221 +0.067"
0.639+:.2100
0.570=0.124**
0.809+0.055
0.881
115
0.206:10.066"
0.610100200 2000
0.535=0.112
0.804:10.05
0.881:10.079
12
0.513-10.048*
0.88810.0155*
0.723
0.766:10.I 156"
0.952.10.006
0.336+0.0041
0.887+0.016
0.613=0.019
0.716:10.18
0.952
2x
100
0.256+:0.041
0.887+0.016
0.549=0.040
0.718410. 1800"
0.952-10.007
15
0.246+00.041
0.886=0.015
0.356=0.029
0.795=+0.043*
0.949=1.0100
MoDL
+2
0.420+0.040**
0.861
+0.063
0.696-
0.552+0.1155
0.938+00.00 0..11
+5
0.257+0.076
0.830+0 106
0.628
14"
0.933 + 0.011
4x
+10
0.197+
0.835+0. 100**
0.032-0.035*
0.643
0.932
115
0.81110.115" 0.
0.371
0.643-10.168
0.933-10.007
A.4 ADDITIONAL VISUAL RESULTS
Fig. 15, Fig. 16, Fig. 17, and Fig. 18 show additional reconstructed MR images. The proposed Moner method obtains the SOTA reconstructions.
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NuiFFT
DRN-DCMB
Score-MoCo
(Ours)
AF-2x, MR=12
0.2
AF-2x MR=
AF=2x MR-x10
AF=2x, MR-
0.2
Figure 15: Qualitative results of MR images by compared methods on four test samples of the fastMRI for AF ===2 2x and various MR (MR +2, +5, 110, +15).
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Published as a conference paper at ICLR 2025
NuIFFT
DRN-DCMB
Score-MoCo
(Ours)
AF=4x, MR=12
0.2
AF-4x, MR=
AF=4x, MR-x10
AF=4x, 1R-
0.2
Figure 16: Qualitative results of MR images by compared methods on four test samples of the
fastMRI for AF 4x and various MR (MR +2, +5, 110, 15)).
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Published as a conference paper at ICLR 2025
NuiFFT
DRN-DCMB
(Ours)
AF-2x, MR=12
0.2
AF-2x, MR=
AF=2x MR-x10
AF=2x, MR-
0.2
Figure 17: Qualitative results of MR images by compared methods on four test samples of the MoDL for AF and various MR (MR +22, +5, 100 15)..
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Published as a conference paper at ICLR 2025
NuiFFT
DRN-DCMB
(Ours)
AF-4x, MR=12
0.2
AF-4x, MR=
AF=4x MR-x10
AF=4x, 1R-
0.2
Figure 18: Qualitative results of MR images by compared methods on four test samples of the MoDL
for AF 4x and various MR (MR +2, +5, 100 15).
22


Reviews:
[{"review_id": "8NXDvsuh0R", "reviewer": "Reviewer_rXFf", "summary": "This paper proposes a novel INR that includes motion modeling, allowing for rigid motion-corrected reconstructions. The motion is modeled using translation and rotation parameters to deform a canonical image space, and then transform the the image using the Fourier slice theorem to compute the data consistency loss with the acquired k-space data.\nEvaluation is conducted on simulated motion-corrupted brain data (2 public datasets) and the individual proposed components (motion model, Fourier slice theorem, coarse-to-fine has encding) ablated.", "questions": "- For the motion simulation, are all spokes corrupted by motion? Or how many spokes are chosen to be affected? Realistic motion patterns in brain would be individual lines, but this is not specified. Can you generally provide more details on the motion simulation, e.g. was only the physical movement considered / what motion patterns were the inspiration for this simulation / were any secondary motion effects considered? \n- In line 151 it is mentioned, that the methods overlook the high dynamic range problem, yet it is considered in some of the mentioned works (e.g. Huang 2023)?\n- The low-frequency bias mentioned in the introduction (l.182) is highly dependent on the network settings (activation function, etc.), yet this is not mentioned in the implementation details? \n- I would suggest to add significance testing to support the result claims within the text.\n- In general, while the simulation evaluation is nice for a proof-of-concept, the models performance on real-world data would be very useful to evaluate its actual capabilities and highly recommended for any future work.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The following strengths can be observed within the paper:\n- The inclusion of a motion model for MR reconstruction is sound and by leveraging the Fourier slice theorem, the inclusion directly in image-space allows simple motion modeling.\n- The work includes extensive evaluation, including several comparison methods as well as reasonable ablation studies.\n- The paper is nicely structured.", "weaknesses": "The present work poses some major and minor weaknesses:\nMajor:\n- The evaluation is exclusively conducted on simulated motion-corrupted data, which raises the question of its actual applciability in real settings. Particularly considering that motion artefacts not only arise from the physical movement, but also from resulting field inhomogenieties, etc., the simulation procedure might be too simplified and the model is likely to simply adapt to these parameters. If the work is aimed at MR motion-correction methods within a clinical workflow, testing the method on in-vivo data is mandatory.\n- In this work, it is claimed multiple times, that the Fourier slice theorem is a crucial part of their work and this works novelty. Yet, the Fourier slice theorem was already introduced a while ago (Catalan 2023 et al,  \t\nhttps://doi.org/10.48550/arXiv.2307.14363) and used in several works after. While it is definitely aiding the method, it should be cited and not included as novelty. \n- Within the results, \"significant improvement\" (l., 365/l.431) is often claimed, the standard deviation (e.g. in table 2) and statistical tests should be reported to support this statement. Similarly for the improvement claim regarding Table 4: Statistical testing would be helpful, since the standard deviation reported opens up questions the actual improvement\n\nMinor:\n- For full reproducibility, it would be nice to mention all post-processing steps conducted before quantitative evaluation (nect to registration mentioned in A1, e.g. if normalization was conducted, etc.?)\n- There are several typos which can be corrected (line326, 504,...)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HYijTtYFHr", "reviewer": "Reviewer_63Er", "summary": "The paper proposes Moner, an unsupervised neural representation framework for motion correction (MoCo) in undersampled radial MRI, utilizing implicit neural representations (INR) to model motion without pre-training on high-quality MR datasets. Key innovations include a quasi-static motion model integrated into INR, reformulating MRI recovery as a back-projection problem via the Fourier-slice theorem, and a coarse-to-fine hash encoding strategy. Extensive experiments on fastMRI and MoDL datasets show that Moner not only achieves high-quality reconstructions but also demonstrates robustness across diverse acquisition conditions.", "questions": "1. Could the authors elaborate on potential strategies to adapt Moner for 3D MRI data or non-radial sampling patterns?\n1. How might Moner’s performance change if the motion were non-rigid or occurred within acquisition frames?\n1. What specific benefits does the coarse-to-fine hash encoding offer over traditional encoding techniques for motion correction?\n1. What specific attributes of Moner contribute to its improved robustness on OOD data compared to Score-MoCo? Does the implicit neural representation (INR) framework offer an inherent advantage over diffusion-based priors in terms of adaptability to different MRI datasets?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* Unsupervised Approach: Unlike many MoCo methods that require large pre-trained datasets, Moner’s unsupervised framework enhances generalizability and applicability across different MRI modalities.\n* Motion Estimation and Stability: The quasi-static motion model and back-projection formulation stabilize model optimization, effectively reducing artifacts even in challenging motion-corrupted scenarios.\n* Efficiency and Robustness: Compared to existing methods like Score-MoCo, Moner achieves superior out-of-domain (OOD) performance and faster reconstruction speeds, making it highly efficient and scalable.", "weaknesses": "* Extension to 3D and Non-Radial Sampling Patterns: The current model primarily addresses 2D radial MRI, and while extensibility to 3D MRI is suggested, it remains untested. The same applies to Cartesian or spiral sampling patterns, which are common in clinical MRI.\n* Reliance on Motion Assumptions: The quasi-static motion model assumes rigid motion between acquisition frames. Further clarification on its limitations in more complex motion settings could strengthen the study.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mznsugt9Tz", "reviewer": "Reviewer_mfmU", "summary": "This paper proposed an unsupervised motion correction method for undersampled radial MRI reconstruction, which fits implicit neural representation (INR) of the image to the observation (undersampled k-space with motion) at test time. MRI reconstruction with motion correction is a challenging task. Though applying INR to medical image reconstruction (e.g. MRI and CT) is not new, its application to motion correction is underexplored. This paper presents its great potential in solving the problem.", "questions": "- How were data from different coils handled in this paper? Were data from all coils fed into the model simultaneously during the optimization?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper is overall well-written, presenting an interesting method valuable to the field.\n- A good background introduction that let readers know what radial MRI reconstruction is.\n- Methods were well represented, with motivation explained for each design choice.\n- Extensive experiments and ablation studies prove the efficacy of the proposed method and individual design choices.", "weaknesses": "- Experiments were only conducted on brain MRI datasets with similar contrast/MR sequence. Readers would be interested in how good the proposed method is on MRI images of different contrasts or taken at different body parts. Specifically, will the optimization still converge stably with the same optimization strategy (e.g. hash encoding) and hyperparameter choice? In addition, how does noise level affect the optimization process?\n- How will trans-plane motion affect the stability of optimization?\n- In Fig. 8, I notice that the reconstructed image looks blurry and lacks some thin structures like vessels. In other figures, images are too small to observe such details. I suggest authors show zoomed-in images from both baselines and proposed method for better comparison.\n- Authors may want to report FOV and pixel spacing to let readers get a sense of the magnitude of motion ranges used in this paper.\n- It is surprising to see that a supervised method (DRN-DCMB) is worse than an unsupervised method. In Fig. 4, DRN-DCMB results look very blurry, making me doubt if it is the SOTA supervised method. Authors may consider alternative baselines such as [1].\n\n[1] Singh NM, Iglesias JE, Adalsteinsson E, Dalca AV, Golland P. Joint frequency and image space learning for MRI reconstruction and analysis. The journal of machine learning for biomedical imaging. 2022 Jun;2022.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "I2qGaj4BhA", "reviewer": "Reviewer_kTjY", "summary": "The paper proposes to use implicit neural representations (INRs) to perform unsupervised joint reconstruction and motion estimation from undersampled and motion corrupted 2D radial MRI measurements.\n\nThe method achieves the best motion estimation results across datasets compared to classical and supervised deep learning-based methods for motion estimation. \nIn terms of reconstructed image quality, the method falls behind supervised deep learning for in-distribution performance but exhibits much more robustness against out-of-distribution examples. \nIn terms of computation time the method is 5-6 times faster than the diffusion model baseline requiring 3-5 minutes per 2D slice.", "questions": "1.\tIs formulating the loss in the projection space specific to the radial sampling trajectory or can it also be used for e.g. Cartesian sampling as well?\n2. It would be helpful to add scores to the qualitative comparisons in e.g. Figure 4 to provide a better feeling how differences in the scores translate to perceivable differences.\n3. Why is the diffusion model baseline categorized as self-supervised e.g. in line 309? As the model requires fully sampled data for training I would consider it as a supervised baseline. \n4. How many spokes are acquired for the acceleration factors of 2 and 4 respectively?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\t(Originality) It is the first paper that explores the use of untrained neural networks (here in the form of INRs) for joint reconstruction and motion estimation. The proposed extensions in terms of an adapted hash encoding optimization schedule and a loss function in the projection space are interesting. \n2.\t(Significance) The investigated problem of motion correction in (radial) MRI is important. The lack of training data from all possible domains required for the supervised baselines and the resulting lack of out-of-distribution robustness is a real problem and methods that are more robust are important. \n3.\t(Quality/clarity) Overall the paper is well written, and the experiments seem to be of good quality. However, some additional information and explanations are required in my opinion (see weaknesses). If those can be addressed adequately, I'm willing to raise the score.", "weaknesses": "1.\tIt is not clear how the motion trajectories, e.g., in Figure 7 are simulated – more information should be provided. Are there fixed intervals after which motion changes direction? In between those events does every spoke has its own motion state meaning that 300 spokes result intro estimating 300*3=900 unknowns? How does this model change when the number of spokes changes due to a different acceleration factor?\n2.\tSection 4.1 does not contain enough information about the dataset used in the experiments. More information should be provided either in the main body or the appendix. 25 test slices is not much depending on how the data is selected. Are the 25 slices the entire slices from e.g. two subjects or are they 25 mid-slices from 25 subjects? The fastMRI dataset is very diverse. Are the subjects for training and testing taken from the same contrasts, e.g., T2 weighted or FLAIR? What is the main difference of the MoDL dataset to the data used from the fastMRI dataset? This information is important to assess the severity of the distribution shift. From Figure 4 it seems that the main difference is the view being axial and coronal respectively. Are there any other differences?\n3. The comparison of the computational speed would benefit from a more detailed discussion. Where do the differences in terms of speed come from? How can the TV method with 200 epochs be slower than the INR with 4000 epochs? What is responsible for the speed difference compared to the diffusion model – the size of the networks or the number of steps? (What is the size of the MLP that is used?)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)