Joint Motion Estimation with Geometric Deformation Correction for Fetal Echo Planar Images Via Deep Learning

Authors: jian  wang
razieh  faghihpirayesh
ali  gholpour
Venue: MIDL_io_2024_Conference
Published: 2024-01-25 21:12:25

Abstract: In this paper, we introduce a novel end-to-end predictive model for efficient fetal motion correction using deep neural networks. Diverging from conventional methods that estimate fetal brain motions and geometric distortions separately, our approach introduces a newly developed joint learning framework that not only reliably estimates various degrees of rigid movements, but also effectively corrects local geometric distortions of fetal brain images. Specifically, we first develop a method to learn rigid motion through a closed-form update integrated into network training. Subsequently, we incorporate a diffeomorphic deformation estimation model to guide the motion correction network, particularly in regions where local distortions and deformations occur. To the best of our knowledge, our study is the first to simultaneously track fetal motion and address geometric deformations in fetal echo-planar images. We validated our model using real fetal functional magnetic resonance imaging data with simulated and real motions. Our method demonstrates significant practical value to measure, track, and correct fetal motion in fetal MRI.

Main paper: 
 introduction: 
 Motion estimation is a critical procedure, meticulously designed to correct image artifacts induced by object motion, especially in the realm of medical image analysis for fetal imaging. Its critical role spans across various fields, including image segmentation (Ebner et al., 2020; Faghihpirayesh et al., 2022) , reconstruction (Gholipour et al., 2010; Kuklisova-Murgasova et al., 2012; Xu et al., 2023) , and pose estimation (Salehi et al., 2018; Golland and Adalsteinsson, 2020) , among others. Effective motion compensation and correction techniques significantly contribute to the overall accuracy and efficiency of fetal imaging studies (Malamateniou et al., 2013) . Estimating fetal motion presents inherent challenges, notably the unpredictable nature of fetal head movements and local geometric distortions due to accumulated imaging errors. To overcome these challenges, existing motion correction methods mostly fall into two distinct categories: i) iterative optimization methods based on mathematical models, and ii) predictive models learned by deep neural networks.One of the initial paradigms of 3D fetal brain reconstruction using motion correction was a three-step model (Rousseau et al., 2006) involving multi-resolution slice alignment for motion correction, intensity non-uniformity correction, and super-resolution reconstruction through scattered data interpolation. To address motion errors from 2D slice misalignments, a slice-to-volume registration (SVR) model was introduced (Jiang et al., 2007) , incorporating a scattered data interpolation method with a novel multi-level B-spline kernel. A motion correction technique was presented in (Kim et al., 2009) to align image stacks based on 2D slice intersections to facilitate the reconstruction of a high isotropic resolution 3D volume. A forward model of slice acquisition and its inverse problem solution were proposed in (Gholipour et al., 2010) . This approach featured a robust M-estimation solution to reduce the effects of corrupted (outlier) slices in a super-resolution reconstruction framework. Building upon these developments, an SVR approach was presented to encompass complete outlier removal through robust statistics based on the expectation-maximization algorithm (Kuklisova-Murgasova et al., 2012) . Tourbier et al. (Tourbier et al., 2015) used total variation regularized SVR, solved using the primal-dual hybrid gradient method. While these advancements have enhanced the accuracy and efficiency of motion correction in image reconstruction, all gradient-based optimization methods for slice realignments have a very limited capture range, often failing in cases of large and rapid fetal movements.Motivated by the progress in deep learning, a spectrum of models has emerged, with a particular focus on advancing predictive motion estimation (Mahendran et al., 2017) . Hou et al. developed a method for predicting 3D rigid transformations of arbitrarily oriented 2D slices (Hou et al., 2018) . This pursuit expanded with the introduction of a real-time fetal motion tracking system, leveraging the strengths of diverse neural networks, including Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs), to predict motion directly from input images (Salehi et al., 2018; Singh et al., 2020; Evan et al., 2022) .To address non-rigid deformations in fetal MRI, two distinct approaches were presented. These include the patch-to-volume reconstruction technique (Alansary et al., 2017) and the deformable SVR technique (Uus et al., 2020) , both designed to correct non-rigid motion to reconstruct deformable anatomy such as fetal body parts. In the domain of automated fetal MRI reconstruction, Ebner et al. developed a toolkit that featured slice-level outlier rejection, fetal brain localization, and volume reconstruction (Ebner et al., 2020) .Noteworthy progress has been achieved in the domain of sensorless imaging, particularly in 3D volume reconstruction from 2D freehand ultrasound images, leveraging deep implicit representations (Yeung et al., 2021) . Xu et al. incorporated transformers (Vaswani et al., 2017) trained on synthetically transformed data, streamlining automatic relevance detection between slices (Xu et al., 2022) . Advancing this approach, Xu et al. further introduced a reconstruction technique that incorporates implicit neural representations, enhancing image reconstruction performance (Xu et al., 2023) . Evan et al. introduced KeyMorph, an unsupervised deep learning framework designed for robust and interpretable multi-modal medical image registration (Evan et al., 2022) . Utilizing anatomically-consistent keypoints, KeyMorph aims to enhance the registration process and increase alignment accuracy. Moyer et al. introduced a real-time method for tracking rigid motion by employing equivariant neural networks (Moyer et al., 2021) . The effectiveness of this approach in capturing significant rigid motions is derived from the intrinsic rotation-equivariant nature of equivariant filters (Cohen and Welling, 2016) .Despite the notable performance achieved by the aforementioned methods, there is currently a lack of an end-to-end predictive motion correction framework that can adequately address both uncontrollable fetal motion and geometric distortions. This gap in the existing literature motivated our endeavor to develop a comprehensive framework proficient in efficiently managing fetal motions and geometric distortions through deep learning. The core contributions of our proposed motion correction method are summarized in three folds:• Pioneering contribution: Our study introduces the first comprehensive end-to-end predictive motion tracking framework adept at handling both significant rigid motion and geometric distortions in fetal echo planar imaging (EPI).• Significance: Our proposed framework not only achieves motion correction accuracy comparable to current state-of-the-art methods but also ensures faster and more stable convergence. This attribute is of significant value, especially in real-time, automated fetal head steering systems, where frequent and substantial motions occur.• Theoretical advancement for broad applicability: The theoretical tool developed through our joint learning algorithm demonstrates wide applicability, offering advantages in various fetal imaging applications. This includes real-time fetal brain segmentation, facilitating population-based studies of fetal brain development using mean template estimation, and improving EPI image reconstruction with advanced distortion correction techniques. 
methodology: 
 In this section, we first outline the theoretical basis needed to address rigid motions and deformations in fetal imaging. We start by reviewing the principles of rigid motion estimation. Next, we focus on correcting geometric distortions, specifically using Large Deformation Diffeomorphic Metric Mapping (LDDMM), chosen for its effectiveness in creating smooth, invertible mappings that maximally maintain the correctness of topological information (Beg et al., 2005) . Finally, we describe the design of our proposed network architecture and the approach to its training.Rigid motion estimation aims to identify the best translation T and rotation R parameters that define a rigid transformation Q(T , R) between a pairwise images. This process involves minimizing the Euclidean distance d between the source image S and the target image T ,EQUATION)where • is the composition operator that resamples S using the rigid transformation Q(T , R).When this operator is applied to any vector v, it yields a transformed vectorQ(v) of the form Q(v) = Rv + T . Here, R T = R -1 indicating that R is an orthogonal matrix.The transformation function Q can be calculated using low-dimensional representations, such as point clouds or key points of images (Moyer et al., 2021; Evan et al., 2022) . This process is quantified by the energy function E(Q), as shown in the equation,EQUATIONwhere ∥ • ∥ F denotes the Frobenius norm, and S, T represent the low-dimensional representations of the source and target respectively. We derive the closed-form solution for both translation and rotation parameters,EQUATIONwhere U ΣV * = S • T T , U and V * are real orthogonal matrices, Σ is a diagonal matrix with non-negative real numbers on the diagonal. Setting the determinant of R to 1 guarantees that it accurately reflects a rigid transformation.In this section, we provide an overview of the LDDMM algorithm for image registration (Beg et al., 2005) . This algorithm is employed to address deformable geometric distortions between the rigid motion-corrected image S • Q(T , R) and the target image T . For simplicity, we denote Ŝ as S • Q(T , R).Given both Ŝ and T , defined on a d-dimensional torus domain Γ = R d /Z d ( Ŝ(x), T (x) : Γ → R), the objective of diffeomorphic image registration is to find the shortest path to generate time-varying diffeomorphisms {ψ t (x)} : t ∈ [0, 1] such that Ŝ • ψ 1 is similar to T . This is typically solved by minimizing an explicit energy function over the initial transformation fields v 0 (Vialard et al., 2012) , which can be expressed as follows,EQUATIONwhere σ 2 is noise variance in images and L : V → V * is a symmetric, positive-definite differential operator that maps a tangent vector v t ∈ V into its dual space as a momentum vector m t ∈ V * . This is typically denoted as m t = Lv t or v t = Km t , with K being an inverse operator of L. The notation (•, •) denotes the pairing of a momentum vector with a tangent vector, which is similar to an inner product.The geodesic shooting process states that the geodesic path {ψ t } can be uniquely determined by integrating a given initial velocity v 0 forward in time by using the Euler-Poincaré differential (EPDiff) equation (Arnol'd, 1966; Miller et al., 2006) as:EQUATIONwhere the operator D denotes a Jacobian matrix and • represents element-wise matrix multiplication. Here, div is the divergence.We develop a deep learning model that explicitly estimates both rigid motion and geometric distortions by integrating a combination of Eq. ( 2) and Eq. ( 4) into our objective function.We show that such joint estimation results in improved accuracy and robustness of the Eq. ( 6)Generated Θ Φ v o (Φ) Q(Θ) S T S ∘ Q(Θ) ∘ ψ(Φ) S ∘ Q(Θ)Eq. ( 5) model. Our framework consists of two modules: an enhanced rigid motion correction neural network that corrects rigid motions with closed-form update, and an unsupervised learning of geometric distortion correction through an image registration network. An illustration of our joint learning framework is shown in Fig. 1 . Below, we provide a detailed description of our network architecture and objective function.Enhanced motion correction network. Let Θ = (T , R) represent the encoder parameters that learn rigid parameters and key features from image spaces, with Q(Θ) denoting the transformation function yielded by the learned rigid parameters from the key points of images. The rigid correction loss is computed between the aligned and target images. Θ is characterized using one of two backbone structures, (i) a 7-layer equivariant neural network to effectively capture rotation-equivaraint representations using 3D steerable CNNs (Weiler et al., 2018; Moyer et al., 2021) , and (ii) a conventional 9-layer CNN as employed in Key-Morph (Evan et al., 2022) .Geometric distortion correction based on image registration network. Let Φ denote the parameters of an encoder-decoder in our geometric learning network, where ψ(Φ) represents the deformation fields and v 0 (Φ) represents the velocity fields learned by the network. Except for estimating deformations by LDDMM (Hinkle, 2018), we also provide another deep learning model that learns stationary velocity fields (Balakrishnan et al., 2019) meanwhile maintains a comparable model accuracy. Advanced predictive image registration models, including TransMorph (Chen et al., 2022) and DiffuseMorph (Kim et al., 2022) , can easily be integrated into our framework.Joint correction network and objective function. Our joint correction network (JCN) comes in two versions: JCN: EF, which employs equivariant filters, and JCN: KM, based on the CNN architecture that is similar to KeyMorph. Our total objective function, which serves as the network's training loss, combines both the motion correction component (Eq. ( 2)) and the geometric deformation estimation (Eq. ( 4)), formulated as follows, 3), & Eq. ( 5). ( 6)l(Θ, Φ) = λ∥S • Q(Θ) -T ∥ F l rigid + 1 σ 2 ∥S • Q(Θ) • ψ(Φ) -T ∥ 2 2 + (Lv 0 (Φ), v 0 (Φ)) lgeo + reg(Θ, Φ), s.t. Eq. (Here, reg(•) is a regularization term constrained on network parameters, and λ is a weighting factor balancing the effects of both networks. Other image dissimilarity terms, such as normalized cross correlation (Avants et al., 2008) and mutual information (Wells III et al., 1996; Wang et al., 2023) can be adopted in our framework. 
experimental setup: 
 Data. Our study includes 1, 881 pairs of 3D EPIs from fMRI time series of 15 subjects who underwent fetal MRI scans using a Siemens 3T scanner (Skyra or Prisma) between August 2015 and September 2021. Approved by the institutional review board, the study obtained written informed consent from all participants. The dataset covers a gestational age range from 22.57 to 38.14 weeks (mean 32.39 weeks). Imaging parameters included a slice thickness of 2 to 3mm, TR of 2 to 5.6 seconds (mean 3.1 seconds), TE of 0.03 to 0.08 seconds (mean 0.04 seconds), and a FA of 90 degrees. Fetal brains were extracted using a real-time deep learning segmentation method (Faghihpirayesh et al., 2023) , resampled to 96 3 with a voxel resolution of 3mm 3 , and underwent intensity normalization.Baselines & Evaluation Metrics. We compared various motion correction methods, including conventional iterative rigid registration in ITK to estimate rigid transformation as 3D versors (Ibanez et al., 2003) , and rigid motion estimation by deep learning methods such as DeepPose (Salehi et al., 2018) and KeyMorph (Evan et al., 2022) . We compared their performance with our joint models JCN: KM and JCN. To demonstrate the benefit of our joint learning, we report the best Dice score of disjoint approaches, where rigid motion estimation is treated as a preprocessing step before geometric distortion estimation.Performance was evaluated through visual comparison and by computing translational and angular errors on simulated motions. We also manually added translations (in mm) and rotations (in degrees) to real fMRI scans with natural fetal motions, altering their maximum motion values in three directions. To better demonstrate our model's stability, we compared the Dice coefficient by computing the overlapping regions between the aligned image and the target across three levels of motion1 . Hyperparameters. We set the dimension for low-dimensional key points to 128 when computing the close-form update of rigid transformation in Eq. ( 3) and set α = 3 for the operator L with using 5 as time steps of Euler integration for geodesic shooting (Eq. ( 5)).The noise variance was fixed at σ = 0.02. For network training, we set batch size as 4, weight decay as 0.0001 for a L 2 regularization, and a learning rate of η = 1e -5 for 300 epochs with Adam optimizer. We divided the data into training (1283 pairs from 11 subjects), validation (299 pairs from 2 subjects), and testing (299 pairs from 2 subjects). The best-performing networks were saved based on validation performance across all models. All experiments were conducted with an NVIDIA RTX A6000 GPU. 
results: 
 Fig. 2 presents a motion correction study for pairwise fMRI scans characterized by significant motions. It shows that the conventional and DeepPose methods fail to correctly align images, while our approach excels, closely matching the target image. Our method effectively handles both rigid motion and local geometric deformations, leveraging estimated velocity fields for more accurate correction. Fig. 3 illustrates the comparative analysis of motion correction errors in translation and rotation across different methods. It highlights the enhanced performance of our proposed technique in correcting fetal motions, surpassing the state-of-the-art methods.Fig. 4 shows the dice coefficient comparisons for fMRI time sequences under various motion conditions. It demonstrates that our method consistently achieves higher dice scores, regardless of the motion degree. This highlights the robustness and stability of our model, particularly in challenging scenarios with significant motion occurrences. 
conclusion: 
 This paper presents a pioneering predictive model for fetal motion correction using deep neural networks. In contrast to conventional approaches that independently estimate fetal brain motions and geometric distortions, our proposed method adopts an efficient and robust joint learning framework. This framework excels in achieving optimal and consistent performance across various degrees of fetal motions. The model demonstrates notable effectiveness on fMRI data with both simulated and real motions, showcasing significant potential in real-time tracking and steering systems for fetal head motion. An intriguing avenue for future exploration involves applying our developed model to image reconstruction for EPI data, which poses more challenging fetal motion artifacts. 
 
 Appendix: Our code is released online, https://github.com/bchimagine/JointMotionTracking 


Decision: Since the reviewers had mixed feelings about this paper, I carefully read it as well as the rebuttal.  

First, a number of issues have been answered by the authors, but only on OpenReview not in the paper, which is not ideal for readers.  

Then, I agree with reviewer TFsQ (and to a lesser extent WKwa) that the evaluation of this method lacks clear ablations. Comparing against different rigid methods is interesting but not central, since the authors did not propose a new rigid framework, but rather took existing ones and incorporated them into their joint optimisation. Here, since the main contribution is the joint optimisation of rigid/elastic registration, the main experiment should compare each step against models separately trained for rigid/elastic registration. The authors have done a small experiment, but the results are only included in the caption of a visual result figure, which is not enough in my opinion. I also think the figure suggested by reviewer WKwa should have been included, at least in the Appendix.

Also, there are some important missing/unclear information: How do the authors derive equation (3) ? What is the architecture of the non-linear deformation network ? What are the rationales behind the 2nd term in (4) ? 

Finally, I also spotted some notation problems: $\mathcal{R}$ and $\mathcal{T}$ are directly computed from $\bar{S}, \bar{T}$ so I don’t think $Q$ should depend on $\bar{S}, \bar{T}$ anymore. $V$ is used for different variables in sections 2.1/2.2. Then, in section 2.3 $\Theta$ both represents the encoder parameters and its outputs $\Theta=(\bar{S}, \bar{T}, R, T)$. $\alpha$ has not been properly introduced.

Overall, I recommend to accept this paper, because it tackles an important problem with wide applications (not restricted to fatal imaging) and is likely to foster interesting discussions at MIDL. However, I think the evaluation flaws are too fatal to recommend an oral presentation. The authors are encouraged to include as many changes as possible in the final version.

Reviews: review: paper_summary: paper_summary: This paper proposes a deep learning framework for jointly tracking rigid motion corruption and geometric distortion between slices in fetal EPI. The framework is comprised of two modules: one that estimates rigid motion parameters from key points detected from source and target images and another that estimates deformation and velocity fields characterizing the geometric distortion from the rigid motion-adjusted source and target images. The networks are trained jointly via a loss that combines (1) a loss encouraging the rigidly transformed source to match the target and (2) an energy function over the distortion transformation fields. The proposed method is tested on fetal EPI time series data with real and simulated fetal movements and is compared to rigid image registration as well as several deep learning baselines.
main_review: main_review: I was originally at strong accept and keep my rating, as the authors implemented all my questions/changes with the exception of visualizing the intermediate output of the rigid-correction network that was trained jointly with the deformation network.

This is related but not identical to reviewer TFsQ’s suggestion to visualize a rigid correction-only network. As mentioned in the authors’ rebuttal, the KeyMorph and DeepPose models are already presented as baselines, which I believe is sufficient for showing the contribution relative to rigid-only models. My suggestion to show the intermediate output of the jointly trained model would provide an additional layer of understanding when compared to the already present rigid-only baselines in the paper by showing how the joint training affects the rigid components. I agree with the authors that this is fine to leave for a journal extension.

Reviewer TFsQ’s suggestion to show the Dice losses of the separately and jointly trained networks is a good one, and I believe is now incorporated in Figure 2 caption, though not in the Dice plots summarized across the dataset. I think this is fine as an ablation study for now but would encourage the authors to make this limitation clearer and mention the ablation results in the main body, not just the figure captions.
strength_weakness: strength_weakness: - There is a small missing ablation study and metrics that are not computed in the current version of the draft that would further improve the paper
- Paper is missing commentary on the computational cost of the geodesic shooting
- Overall, these are small weaknesses and I do not believe these are critical for acceptance.
questions: questions: - What is the regularizer used in Equation (6)?
- Does Figure 3 combine the translation errors in all axes and the angular errors in all axes?
paper_summary: paper_summary: The manuscript presents a method for fetal pose estimation. The method combines rigid and deformable registration in a jointly optimizing end-to-end architecture. It is compared to a number of existing methods, using a dataset of 15 subjects for training and evaluation.
main_review: main_review: The authors have unfortunately not added the separate rigid correction network as a baseline in their revision. I would argue this is the most important baseline of all, as the joint optimization of the two networks is the main novelty of the work. Hence, it is not possible to judge the benefit of their contribution from the current version of the manuscript. Additionally, the authors have not updated the visualization in figure 4 to do an apples-to-apples comparison, still comparing Dice scores of their deformable method with Dice scores of rigid transformations. This is not an appropriate representation of the state of the art and misrepresents the benefit of the proposed method.

The work clearly has potential, but the current version of the manuscript is not in a publishable state. A number of problems with the paper have been addressed in the OpenReview comments to both my own and the other reviews, but readers should not be required to search for the OpenReview page in order to get the full picture of the work. Hence, I unfortunately have to downgrade my initial assessment.
strength_weakness: strength_weakness: The authors state "We show that such joint estimation results in improved accuracy and robustness of the model", but I do not think they adequately show this in the results. The rigid model-only (i.e. without the subsequent deformable model and loss) is not added as a baseline, so it is hard to tell whether the joint loss caused the strong performance, or whether this is simply the result of a well-tuned network for rigid motion estimation.

The method is trained and evaluated on a large number of image pairs from 15 subjects. However, it is not clear how the train/test split was done. The authors state they used 15% of the data for evaluation, but 15% of 15 subjects is 2.25, which is impossible. The authors should clarify whether they split on a subject-level, or split on an image pair-level.

The comparison of dice overlap seems inappropriate to me, as the authors compare their deformable registration model with rigid and affine ones. It would make more sense to compare to a baseline pipeline where the rigid model is trained first, and the deformable network is trained after (i.e. not end-to-end and without a joint loss), and both are applied subsequently. It is extremely common to perform affine or rigid pre-alignment before applying a deformable registration method, so this would match the typical application of such methods.
questions: questions: See: the points in weaknesses
paper_summary: paper_summary: This paper presents an algorithm that mixes rigid and non-rigid transformations to deform fetal MRI images for motion correction. The use of rigid transformation and LDDMM based deformations together in a deep learning setup is interesting, and seems to yield impressing results, as compared to previous methods.
main_review: main_review: Very good, I do not have any further comments.                                                                                                                                                            .
strength_weakness: strength_weakness: I did not see any major weaknesses in this work. What I could suggest as improvements would be a more extensive survey of recent papers on LDDMM and ML. Only a few are cited on this topic, and i think there are more relevant works out there that could be interesting to relate to this one.
questions: questions: I don't have any pressing questions to be answered.

Comments: title: Clarifications.
Wrong pdf uploaded?
Please implement the promised changes
comment: **Conventional method.**
The description of the iterative conventional model was included in the Baselines of Experiment section. 

**Disjoint approaches.**
Sorry for the confusion. As addressed in the previous response, we intended to report the quantitative results here. Now, we have added the brief comparison in the new manuscript as well. A more comprehensive analysis and visualization between the two types of models will be conducted in an extended paper of this work.
Looking at the revised manuscript, I don't see any mention or description of the disjoint baseline, nor any description of the original "conventional" baseline. Could it be that you have uploaded the wrong pdf file?
Thanks for your kind reminder! We will upload the revised manuscript ASAP.
I would like to remind the authors that, this year, the revised manuscript should be submitted along with the responses to the reviewers. It is better to highlight these changes in red for improved readability. Unless I missed something, it seems like this is not the case for this submission. The camera-ready manuscript will be submitted later, upon acceptance.
We greatly thank the positive feedback from the reviewer! We will conduct a thorough review of existing deformable image registration models based on deep neural networks, such as TransMorph [1] and Diffusemorph [2]. We will also enrich the literature review of  motion tracking in both the medical image analysis and the computer vision domain in the extended journal version of our paper.

**References**
[1]. Chen, Junyu, et al. "Transmorph: Transformer for unsupervised medical image registration." Medical image analysis 82 (2022): 102615.
[2]. Kim, Boah, Inhwa Han, and Jong Chul Ye. "Diffusemorph: Unsupervised deformable image registration using diffusion model." European conference on computer vision. Cham: Springer Nature Switzerland, 2022.
We greatly thank the reviewer for the comments and feedback on our manuscript! Below, we present detailed responses to each point raised.

**1. Dataset and Baselines**  We appreciate the reviewer's attention to detail regarding the dataset and baselines. In response, we will enhance the clarity of our manuscript by explicitly describing the dataset split—comprising training (1283 pairs from 11 subjects), validation (299 pairs from 2 subjects), and testing (299 pairs from 2 subjects). We used an iterative rigid image registration model, as conventional one, for estimating 3D versors, encompassing both translation and rotation. This model, implemented in C++ using ITK, will be clarified in the revised manuscript.

**2. Joint Learning** To address the reviewer's concern, we would like to clarify that we show the results of models that focus solely on rigid motion estimation. For instance, we have KeyMorph, DeepPose for rigid motion only, and JCN-based models that include joint geometric distortion correction (Fig.2 ~ 4). Both models are interrelated—the geometrically corrected data is fed back as augmented data to enhance the accuracy of rigid motion correction. This, in turn, aligns better with the geometric distortion network, facilitating the correction of local distortions in corresponding positions of the fetal brain. We will clearly depict this mutually beneficial relationship in our main figure (Fig. 1). We report the accuracy of both joint and disjoint approaches here. The dice evaluation results for disjoint approaches (best performance on rigid and geometric distortion) are 0.843/0.905, while for our joint approaches (best performance on rigid and geometric distortion), the results are 0.912/0.971.

We are willing to incorporate  clarifications and results above  into the revised manuscript.
We thank the valuable feedback from the reviewer and carefully address each point as follows,

**1. Regularization.**
We incorporate weight decay as L-2 regularization during the training of our network.

**2. Computational Efficiency.**
Thanks for highlighting the importance of computational efficiency. The computational time for motion estimation on average of testing data is 0.491 seconds per 3D pair (96^3). We agree that larger integration steps can impact computational time. In our joint learning framework, we've optimized our approach by using a fast version of LDDMM (specifically, lagoMorph[1] on GPU) with the time steps parameter set to 5. Additionally, we will also provide another fast shooting method with stationary velocity fields that maintains a same-level of accuracy in our code. 

**3. Result Visualization.**
It is a great suggestion to display intermediate results between two joint models. We will carefully address this and add clear visualization of geometric distortion correction process by presenting velocity fields in the experiment section of an extended journal version of this work. The title of Fig. 3 will be revised to ensure clarity regarding the inclusion of average translation and angular errors for all directions.

**4. Improvements of Evaluation.**
We agree with the reviewer's suggestion regarding structural-wise or voxel-wise evaluation metrics. We plan to develop such metrics and conduct a thorough evaluation in an extended version of the journal paper dedicated to this work.

**References**
[1]. Hinkle, Jacob D. Lagomorph. No. Lagomorph. Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States), 2018.