FedFDD: Federated Learning with Frequency Domain Decomposition for Low-Dose CT Denoising

Authors: xuhang  chen
zeju  li
zikun  xu
kaijie  xu
chen  qin
Venue: MIDL_io_2024_Conference
Published: 2024-01-31 15:23:38

Abstract: Low-dose computed tomography (LDCT) enables imaging with minimal radiation exposure but typically results in noisy outputs. Deep learning algorithms have been emerging as popular tools for denoising LDCT images, where they typically rely on large data sets requiring data from multiple centers. However, LDCT images collected from different centers (clients) can present significant data heterogeneity, and the sharing of them between clients is also constrained by privacy regulations. In this work, we propose a personalized federated learning (FL) approach for enhancing model generalization across different organ images from multiple local clients while preserving data privacy. Empirically, we find that earlier FL methods tend to underperform single-set models on non-IID LDCT data due to the presence of data heterogeneity characterized by varying frequency patterns. To address this, we introduce a Federated Learning with Frequency Domain Decomposition (FedFDD) approach, which decomposes images into different frequency components and then updates high-frequency signals in an FL setting while preserving local low-frequency characteristics. Specifically, we leverage an adaptive frequency mask with discrete cosine transformation for the frequency domain decomposition. The proposed algorithm is evaluated on LDCT datasets of different organs and our experimental results show that FedFDD can surpass state-of-the-art FL methods as well as both localized and centralized models, especially on challenging LDCT denoising cases. 
Our code is available at https://github.com/xuhang2019/FedFDD.

Main paper: 
 introduction: 
 Federated learning (FL) is a decentralized approach that can integrate data from multiple local clients for training machine learning models in a privacy-preserving way (Pati et al., 2022) . FL is particularly well-suited for medical imaging tasks that require extensive data, where privacy is a major concern. For example, FL has been discussed for applications including COVID-19 classification (Dayan et al., 2021) , skin lesion classification (Yan et al., 2023) , and human activity recognition (Rieke et al., 2020; Rauniyar et al., 2023) .Machine learning models typically rely on large training data sets for achieving good generalization capability. Low-dose Computed Tomography (LDCT) denoising models have been shown to perform better by expanding the training dataset with samples from different sources and various anatomies (Yang et al., 2023; Immonen et al., 2022) . Nevertheless, a centralized training paradigm may not be always feasible, as local clients may be reluctant to share data due to privacy concerns. Hence, in this study, we propose to investigate a novel LDCT denoising method that can leverage data from diverse clients in a privacy-preserving way, with each personalized local model specializing in different anatomical regions. Related works. Fruitful research has been proposed in the LDCT denoising field. RED CNN (Chen et al., 2017) adopts residual skip connection and small convolution kernels to enable smooth variation on different layers of feature maps for LDCT denoising. Transformerrelated works have also been proposed to utilize the pretrained data (Jing et al., 2022) and enrich the diversity and effectiveness of features (Wang et al., 2023) . Similarly, adversarial learning methods (Wolterink et al., 2017; Han et al., 2022) have also been studied to learn the denoising task. However, few of them addressed the CT denoising issue in the FL settings. Current FL approaches mainly focus on addressing classification or segmentation challenges rather than tasks related to image restoration. FedAvg (McMahan et al., 2017) proposed to iteratively average the model weights from local models following each local update step, assuming that all clients have independent and identically distributed (IID) data, which however is often not the case in real-world scenarios. To mitigate the sub-optimal performance caused by non-IID data, FedProx (Li et al., 2020) integrated a proximal term to the objective function to guide the local updates to align with the global model, thereby improving convergence with heterogeneous data. FedBN (Li et al.) was proposed to freeze the Batch Normalization layer when training the domain-shifted medical data for stabilizing the training and thus improving the averaging model. Besides, MOON (Li et al., 2021) has added a contrastive loss from the latent vector of each client to the global model, which aims to alleviate the global optimal parameter deviation issue by reducing the imbalanced gradient drift.More recently, FL methods in low-level medical imaging tasks have also raised attention, such as in image reconstruction. For instance, FedMRI (Feng et al., 2023b) has proposed to divide the reconstruction model into a global-shared encoder and separate local-preserved decoders with a weighted contrastive regularization, which has been shown to improve the efficiency and accuracy of MRI reconstruction tasks. FedPR (Feng et al., 2023a) further proposed a federated paradigm to only communicate the pre-trained-model-generated prompts and optimize them in an approximate null space of global prompts. Of particular relevance to our work, HyperFed (Yang et al., 2022) has proposed to utilize localized hypernetworks based on simulated geometric parameters and dose levels to guide the CT reconstruction task, i.e., reconstructing the projection data to the imaging data. In contrast, our work focuses on the LDCT denoising with various anatomies in an FL setting, and the design of our proposed model is inherently inspired by the denoising nature. Our contributions. We propose a Federated Learning with Frequency Domain Decomposition (FedFDD) strategy for training the LDCT denoising model in a privacy-preserving way. FedFDD is motivated by the observation that LDCT images share common patterns across different anatomies, particularly in high-frequency components (i.e., noises). Therefore, we aim to leverage that and enhance the learning of the denoising task in the general high-frequency domain using an FL approach while preserving individual low-frequency components (i.e. semantic anatomical structures) (Yang et al., 2022) for each local client.Our contributions are mainly threefold: 1) We introduce a novel dual-path FL strategy with frequency domain decomposition to split the feature space for training and maintain gradient stability during model aggregation under non-IID conditions. 2) We propose to leverage data from different anatomies in LDCT denoising tasks, to mitigate data scarcity in FL settings. 3) We demonstrate that selectively updating high-frequency components in a dual setting significantly enhances the model's performance in LDCT denoising tasks, with competitive performance against state-of-the-art FL methods. Our proposed approach aligns with the nature of noise removal and indicates a promising direction in FL for imaging. 
federated learning problem formulation: 
 The main goal of our FL method is to utilize datasets from different clients in a privacypreserving way. Specifically, we aim to construct a model that outperforms localized models, i.e., models trained on local datasets. In order to prevent privacy leakage, we follow the setting that data from different clients cannot be communicated (i.e., different hospitals do not allow patient data transmission). Given that there are N clients with their own datasets D i , i = 1, 2, ..., N and the optimization loss function as L, we want to achieve:EQUATIONwhere ω global denotes parameters of a global model, and p k is the weight of each local dataset, defined as (Xu et al., 2023) mentioned that the conventional mini-batch gradient descent method aimed to update the model parameters at j + 1 time step on the client k byp k = ||D k ||/ N i=1 ||D k || where ||D k || is the size of the k-th dataset. Previous researchω j+1 k = ω j k -η∇L(D k ; ω j k ), where η is the learning rate and ∇ calculates the gradient w.r.t ω j k . In this way, the global model parameters can be aggregated by ω j+1 global = N i=1 p i ω j+1 k . However, the non-IID property of D k would generate varying directions of ∇L(D k ; ω j k ) and the weighted operation would therefore drift the global parameter to a sub-optimal solution (McMahan et al., 2017) .In our LDCT denoising setting with images of various anatomies from different clients, we observed that the mainstream FL scheme suffered from the drifted direction caused by data of different anatomical regions. Therefore, we aim to alleviate the misleading impact of the varying anatomical structures on the denoising task and guide the model to learn the denoising essence. To achieve this, we propose to decompose the denoising task into two paths: one path consists of part of the model that updates the local parameters ω anatomy corresponding to refining each anatomical structure, and the other specifically deals with the noise reduction part with model weights of ω denoise . In this way, the global model will only be updated with the gradient direction of the denoising part ∇L(D k ; ω j k,denoise ), without the negative influence from the aggregation of the drifted item ∇L(D k ; ω j k,anatomy ). The model parameter at the client k can then be viewed as ω k = ω k,anatomy ∪ ω k,denoise . Therefore, the optimization problem can be formulated as:EQUATION)During the aggregation at time step j + 1, ω j+1 denoise and ω j+1 k are updated as ω j+1 denoise = N i=1 p i ω j+1 k,denoise and ω j+1 k = ω j+1 denoise ∪ ω j+1 k,anatomy .Based on the above formulation, we propose a dual-path FL strategy for LDCT denoising, as shown in Fig. 1 . Noise in LDCT can be represented using various noise models including quantum noise from X-ray (Yang et al., 2023) , normally distributed stochastic process noise (Li et al., 2023) , and speckle and streak (Yang et al., 2020) noise, which are commonly corresponding to the high-frequency component in the images. On the other hand, the anatomical structures can be represented by the low-frequency component of the images. Motivated by this, we propose to decompose the effects of anatomical structures and noises through frequency domain decomposition, thereby updating a global denoising model component with high-frequency data across different anatomies while preserving personalized local low-frequency model components for each client, as seen in Fig. 1 . In detail, for the frequency domain decomposition, we propose to design a mask to decompose the image and utilize an FL strategy to learn the model parameters for the denoising part, i.e., ω denoise . Specifically, the Discrete Cosine Transformation (DCT) is adopted for the frequency decomposition due to its real-value property (Fig. 2 ). The majority of the informative part of the image can then be efficiently condensed into a small number of coefficients with the DCT, especially in the low-frequency coefficients, distributed in the upper-left corner of the DCT frequency domain. The low-frequency part mainly includes semantic-related components of the images (i.e., the structure of anatomies), and the noise is mainly distributed into the remainder area (Fig. 2 ). Therefore, we use a Bernoulli binary mask to separate the semantic component (the low-frequency component) from the noisy area (the high-frequency component) inspired by (Yue et al., 2021) . This is achieved by setting a low-frequency threshold r l , within which coefficients are preserved (mask value 1), while those beyond it are determined by a Bernoulli distribution based on their normalized Euclidean distance r u,v to retain potentially useful high-frequency information. The mask can be formulated as:EQUATIONWith the defined binary adaptive frequency mask (Appendix A) M and an input X, the low frequency image part X L and high frequency part X H can be derived as:X L = F -1 (F(X)⊙ M ), X H = F -1 (F(X) ⊙ (1 -M ).As both the DCT and the inverse DCT process are linear, we can have X = X L + X H , which allows for the linear fusion strategy.Fig. 2 demonstrates that the low-frequency components pixel intensity distribution (Lowf) of LDCT correlates well with the distribution of the NDCT image (i.e., clear, fulldose imaging), while high-frequency components (High-f) predominantly exhibit noise with an IID distribution. Motivated by this, we propose to tackle the non-IID distribution characteristics of the problem by decomposing the model into dual paths, where one network branch (parameterized by ω k,anatomy ) is designed to preserve the anatomy locally and the other branch aims to tackle the noise part using FL, leveraging its effectiveness in dealing with IID data (i.e., high-frequency components in this case).An illustration of the FedFDD model is shown in Fig. 1 . Both the low f and high f backbone models are inherited from the RED-CNN model (Chen et al., 2017) . The first convolutional layer outputs a single-channel feature map taking both the frequency component and the original LDCT image as input. This layer can be viewed as a feature fusion layer. The frequency component, exclusively treated as a residual item added to the output of the branch, is expected to drive the branch to learn the intrinsic features within each branch (i.e., anatomies for low frequency and denoising for high frequency). During the aggregation, the low-frequency path (the blue branch) is reserved for each local client, and the high-frequency path (the orange branch) is aggregated. The objective is defined by the Mean Squared Error (MSE) between the reconstructed image and the NDCT standard. The model is trained in an end-to-end way. 
datasets and implementation details: 
 The LDCT and NDCT datasets (Moen et al., 2021) are licensed by The Cancer Image Archive (TCIA) team. Original data is collected by the Mayo Clinic, containing the abdomen, chest, and head regions. We simulated 36 patients with different anatomies as three clients (Client 1: 12 Abdomen, Client 2: 12 Chest and Client 3: 12 head ). We provide detailed data settings in Appendix B. We divide those datasets into 60% (training), 10% (validation), and 30% (testing). The modified dual-path network is trained with an MSE loss with Adam optimizer for a total of 200 epochs. The learning rate is initialized as 10 -4 and decayed per 3000 iterations. We stop the training process early if the validation loss fails to decrease for 10 consecutive epochs (Qian et al., 2024) . The r l of the mask is 0.45. The Hounsfield Units (HU) window of the CT images is [-160, 240] and the images are normalized to [0, 1] by the minimum of -1024 and a maximum of 3072 (Yang et al., 2022; Bera and Biswas, 2023) . Patch training is adopted with patch size 64 × 64 and a total of 16 per image following (Bera and Biswas, 2021) .Limitation of current FL approaches. Contrary to expectations, state-of-the-art FL methods (e.g., FedAvg, FedProx, MOON, and FedBN) do not surpass localized or centralized training in LDCT denoising tasks, as evidenced by Table 1 . Typically, FL is expected to excel by leveraging diverse client data, but the non-IID nature of the LDCT dataset and notable domain shift (Fig. 2 ) impede this advantage. Current FL approaches mostly do not address these specific challenges and thereby exhibit constrained performance in such scenarios. In contrast, centralized training benefits from comprehensive data exposure, facilitating superior generalization, whereas the effectiveness of localized training is curtailed by its limited dataset scope. This discrepancy is particularly marked in chest dataset comparisons, underscoring the limitations of existing FL approaches in handling non-IID distributions and domain variability. Comparison results. We compare our proposed FedFDD with state-of-the-art FL algorithms in Table 1 . Our method outperforms other methods in terms of PSNR and SSIM in B ) where all FL approaches do not outperform the localized training. Despite that, our approach can achieve a higher SSIM compared to baseline methods. In contrast, other advanced FL algorithms underperform the localized training on all three datasets. Furthermore, we visualize the denoised images and their corresponding error maps in Fig. 3 . We find that the error produced by FedFDD is significantly less than that of others and the denoised image has well maintained the structural details and textures. Effects of FL strategy in different frequencies. Fig. 4 presents the experiments ascertaining the benefits of exclusively updating the high-frequency components. Here, FedFDD all represents a learning strategy that updates both the parameters of the dualpath model during aggregation and FedFDD lf means that we only update the low-frequency path with the high-frequency path frozen during aggregation. Our proposed approach, i.e., FedFDD hf , which only updates the high-frequency components, achieves the best performance across all metrics. This indicates that our strategy of focusing on high-frequency updates and freezing the low-frequency components is effective. In contrast, FedFDD lf approach shows a slight deterioration in performance compared with FedFDD all . This is likely due to the reason that the merge of low-frequency components from different clients could introduce inconsistencies, leading to a potential loss of specific structural details and thereby affecting the overall image quality. The better performance of FedFDD hf compared with FedFDD all also indicates that the proposed targeted update strategy, focusing on high-frequency components, can be more effective than a holistic update.Effects of varying frequency split thresholds. We show the model performance with varied thresholds r l 's in Fig. 4 . Recall that higher r l refers to less information in the high frequency images. When the threshold is low, the low-frequency component (intrinsic anatomical area) leaks to high frequency updating procedure, which results in a deterioration of the model performance. Besides, there is a slight peak around 0.4 to 0.5, suggesting the optimal value within the range. When r l > 0.6, the performance slightly decreases because the model would benefit less from FL. 
conclusions: 
 In this study, we proposed the FedFDD model and demonstrated its effectiveness on the LDCT denoising task across different anatomical images. We proposed an FL strategy with frequency domain decomposition, where only the high-frequency components of the network are updated during the aggregation process. This ensures that the intrinsic characteristics of the low-frequency components are preserved locally. Our method achieves up to 1.2 dB improvement compared with the state-of-the-art FL algorithms. Notably, it also outperforms both single-set and centralized training, particularly in more noisy scenarios (e.g., on Chest data). In the future, we plan to validate FedFDD in an out-of-federation setting for data from unseen clients. FL training, we froze each client's decoder and enabled the communication of the encoder. Ultimately, each client has its own decoder and a universal encoder. PromptFL: Hyperparameter Prompts (Yang et al., 2022; Feng et al., 2023a) :As discussed in Related Works, HyperFed (Yang et al., 2022) proposed a personalized FL approach using localized hypernetwork of the CT scanning physical properties for the reconstruction of CT projection imaging. However, our task mainly focuses on LDCT denoising, where we cannot access the physical parameters of the imaging, which therefore limits us from using that for the hypernetwork training. On the other hand, FedPR (Feng et al., 2023a) used visual prompts in the null space of global prompt for the FL paradigm, which is not directly feasible in our case.However, inspired by both the HyperFed and FedPR, we propose to adapt them to our task by introducing the client-specific information as hyperparameters/prompts in the task. Specifically, we proposed to introduce the CLIP representation vector of our client information as prompts to inform the denoising process. The CLIP vector of "This is an image of {anatomy} low dose CT" (anatomy can be chest, abdomen or head) is projected into a 96-dim vector, in line with the number of channels of the network bottleneck. Then the bottleneck feature maps are weighed with the softmaxed prompt vector in the channel dimension. The whole operation can be regarded as an attention mechanism prioritizing the specific channels based on the prompts. The denoising network was then trained with the guided information from the prompts. 
 
 Appendix:  


Decision: Two of the three reviewers support accepting this work as a poster and believe it will generate meaningful discussions. I believe the rebuttal has addressed most of the important comments. The authors are requested to revise the camera-ready version in accordance with the feedback provided in the rebuttal.

Reviews: review: paper_summary: paper_summary: The paper studies the problem of denoising LDCT images in a privacy preserving federated learning framework. The paper specifically explores the problem of non-IID nature and heterogeneity of data in different participating sites in the medical setting. The key idea of the paper is that the image can be decomposed in low frequency and high frequency parts, and through analysis the authors show that the low frequency components capture the semantics of the image and the high frequency components capture the noise in the LDCT that needs to be removed. Hence, the proposed methods factorized the model in 2 components: local and global, followed by a merge layer. The local component is not shared in the federated framework and only the global component is shared and updated with the gradients corresponding to the high freq component of the images. The experiments show the benefits of the proposed solution over previous state of the art methods and lays the foundation to utilize larger datasets for training the denoising models leading to better results.
main_review: main_review: The authors addresses some concerns regarding the paper, including different data splits and more baselines and hence I upgrade my rating to reflect the improvements in the paper with the reviewers comments.
strength_weakness: strength_weakness: * In the experiment section, a very relevant previous work which also proposes a similar approach of having local and global model parameters has not been discussed or used as a baseline even though it is mentioned in the related works. 
* While the promise of increasing the dataset available by combining data from different anatomical structures is interesting, the experiment design is not robust enough to back the conclusion. The data is split in different simulated sites by anatomical structure and does not explain if this is a common setting for the problem. I would assume that each site would have data from different anatomical structures and hence it would have been great if the key idea was validated for different simulated splits than just 1.
* The description of the dataset is incomplete. Two papers with different datasets have been referenced without any statistics of the final dataset used in the study.
questions: questions: * Results for more splits such that each site has data from multiple anatomies will be helpful to validate the generalizability of the described approach.
* A discussion on how the work is different from the HyperFed paper and how the proposed method compares with that.
paper_summary: paper_summary: This paper proposes Federated Learning with Frequency Domain Decomposition (FedFDD), a novel personalized FL approach for denoising Low-Dose Computed Tomography (LDCT) images, which tackles the challenges of data heterogeneity and privacy concerns. By decomposing images into frequency components and selectively updating high-frequency signals, FedFDD effectively enhances model generalization across various organs. Utilizing an adaptive frequency mask for decomposition, the method outperforms existing FL in LDCT denoising tasks, particularly in complex cases.
main_review: main_review: This paper introduces a novel approach to isolating task-specific image components backed by solid analysis and demonstrates its ability to outperform centralized models. The authors have addressed my concerns satisfactorily, and I maintain my initial acceptance score.
strength_weakness: strength_weakness: - The overall evaluation is thorough; however, I am curious as to why it lacks comparison with other reconstruction based denoising baseline, such as HyperFed [1]. Is there a specific rationale for excluding this type of method from the analysis?

[1] Ziyuan Yang, Wenjun Xia, Zexin Lu, Yingyu Chen, Xiaoxiao Li, and Yi Zhang. Hypernetwork-based Personalized Federated Learning for Multi-Institutional CT Imaging, June 2022.
questions: questions: Could the authors offer a concise explanation of how the proposed method manages to achieve superior results, even when compared to a centralized setting?
paper_summary: paper_summary: The author introduce a Federated Learning with Frequency Domain Decomposition (FedFDD) approach, which decomposes images into different frequency components and then updates high-frequency signals in an FL setting while preserving local low-frequency characteristics.
main_review: main_review: I would like to see the comparison with the related works that I have mentioned, not limited to the differences in method design but also the experimental results, because the current version lacks a review and comparison of related works.
strength_weakness: strength_weakness: I noticed that the author has reviewed many related FL works on natural data, e.g., FedBN, MOON, etc.
However, there are many medical FL works that have been published, especially on the low-level task, reconstruction. 
Please see the incomplete list here:
Learning Federated Visual Prompt in Null Space for MRI Reconstruction, CVPR 2023.
Specificity-Preserving Federated Learning for MR Image Reconstruction TMI 2022.
I strongly recommend the author to carefully comparsion those works in your paper.
questions: questions: see the Weaknesses

Comments: title: Response to Reviewer oGSF
paper is open for discussions
Response to Reviewer u6nd
Response to Reviewer EtbZ
Response to Reviewer oGSF
Comment to all reviewers summarising the revision
comment: Thank you for your suggestion. We have updated Appendix C Table 2 (also below) with the comparison with representative baseline methods in the new data setting. It is observed that in the new data setting where client 1 contains data with multiple anatomies, our proposed approach FedFDD can still achieve an overall better performance compared to other baseline approaches. This indicates the good generalizability of our method on alternative data splits.
***
| Method | PSNR↑ (Chest) | SSIM↑ (Chest) | PSNR↑ (Abdomen) | SSIM↑ (Abdomen) | PSNR↑ (Head) | SSIM↑ (Head) |
| --- | --- | --- | --- | --- | --- | --- |
| LDCT | 15.3388 | 0.6857 | 28.3165 | 0.8213 | 43.8707 | 0.9652 |
| Localized | 22.0033 | 0.7445 | 32.1983 | 0.8776 | **44.3414** | 0.9756 |
| Centralized | 22.4649 | 0.7534 | 32.4423 | 0.8809 | 43.6559 |**0.9816** |
| FedAvg | 21.5235 | 0.7492 | 31.6691 | 0.8733 | 43.1827 | 0.9767 |
| FedBN | 21.8890 | 0.7495 | 31.6205 | 0.8701 | 42.2720 | 0.9786 |
| FedMRI | 22.2771 | 0.7499 | 32.2095 | 0.8780 | 42.5368 | 0.9788 |
| **FedFDD** | **22.6056** | **0.7581** | **32.4497**| **0.8821** | 43.0423 | 0.9791 |
Thanks for addressing the review comments.

For the new data setting in Appendix, can you provide the comparison with baseline values similar to table 1 on page 7 in main article.
Dear Reviewers The authors have submitted their rebuttal addressing the raised questions. The paper remains open for further discussion and engagement.
Q1: **Comparison with other medical FL works:** *“I would like to see the comparison with the related works that I have mentioned, not limited to the differences in method design but also the experimental results…”*

Response:

Thank you for the reviewer’s suggestion. We have now included the comparisons of our FedFDD method with current state-of-the-art medical FL studies, e.g., the FedMRI and FedPR, in the Introduction of the revised manuscript. Despite these well-designed models with advanced optimizing methods (e.g., null space) to control the bias within different clients during communication, our work focuses on the LDCT denoising with various anatomies in an FL setting, and the design of our proposed model is inherently inspired by the denoising nature. This can be exemplified throughout the extensive comparison and ablation study.

In addition,  we have also conducted **two additional experiments** suggested by the core ideas from the papers "Specificity-Preserving Federated Learning for MR Image Reconstruction"  and "Learning Federated Visual Prompt in Null Space for MRI Reconstruction”.

1. FedMRI. We adapted the idea of freezing the decoder (client-specific) and globally sharing the encoder as in the FedMRI approach to our task. Specifically, we built this on the base network architecture, i.e., RED-CNN, which comprises an encoder and decoder. During the FL training, we froze each client's decoder and enabled the communication of the encoder. Ultimately, each client has its own decoder and a universal encoder.
2. PromptFL: As the physical parameters in HyperFed are not accessible in our LDCT denoising task, and the visual prompt in the null space of the global prompt in FedPR is not directly feasible in our case either, we propose to adapt them to our task by introducing the client-specific information as hyperparameters/prompts in the task. Specifically, we proposed to introduce the CLIP representation vector of our client information as prompts to inform the denoising process. The CLIP vector of *"This is an image of {anatomy} low dose CT"* ({anatomy} can be *chest*, *abdomen* or *head*) is projected into a 96-dim vector, in line with the number of channels of the network bottleneck. Then the bottleneck feature maps are weighed with the softmaxed prompt vector in the channel dimension. The whole operation can be regarded as an attention mechanism prioritizing the specific channels based on the prompts. The denoising network was then trained with the guided information from the prompts.

The comparison results are shown in Table 1 in the revised manuscript. We find that the two methods show better performance than the baseline FedAvg on the Chest and Abdomen. This demonstrates the efficacy of these federated learning schemes  (Feng et al.,2023a,b). However, our proposed FedFDD still shows better overall performance than the compared methods as it is specifically designed for the denoising task. A more detailed description of the baseline methods is provided in Appendix D in the revised manuscript.
Q1: ***Difference with HyperFed:*** *“...I am curious as to why it lacks comparison with other reconstruction based denoising baseline, such as HyperFed. Is there a specific rationale for excluding this type of method from the analysis?”**

Response:

HyperFed focused on CT reconstruction (projection data to imaging data) tasks guided via a hypernetwork with geometry parameters and dose levels. Our approach leans on low dose CT denoising tasks (noisy imaging data to clean imaging data). The core assumption of HyperFed is the access to the scanning and geometry parameters, which are not necessary for image-level denoising tasks. Therefore, the two methods are not directly comparable.

However, to address the concern of the method comparison against hypernetwork, we add an experiment to incorporate the unique client information, here, the anatomy as attention to guide our denoising task (can refer to Appendix D). Specifically, we incorporate the anatomy information as text embedding to guide the denoising task. We find the proposed FedFDD fits the denoising task better.
***
Q2: **Explanation of the superior performance compared to a centralized setting:***“…how the proposed method manages to achieve superior results, even when compared to a centralized setting?”*

Response:

The superior performance of FedFDD, even when compared to centralized settings, can be attributed to its novel approach to mitigating domain or feature shift issues (caused by different anatomies) inherent in LDCT images, which is however not specifically considered in the centralized setting and other baselines. FedFDD leverages the inherent characteristics of denoising tasks, particularly through the adaptive management of frequency components. Therefore it achieves overall better performance compared to the baselines, particularly in the more challenging case of Chest and Abdomen CT denoising.
Q1: **Comparison with similar related work (HyperFed):***“... a very relevant previous work … has not been discussed or used as a baseline even though it is mentioned in the related works” “A discussion on how the work is different from the HyperFed paper and how the proposed method compares with that.”**

Response:

We assume that the reviewer is referring to HyperFed (Yang et al. 2022) as the relevant previous work. HyperFed focused on CT reconstruction (projection data to imaging data) tasks guided via a hypernetwork with geometry parameters and dose levels. Our approach leans on low dose CT denoising tasks (noisy imaging data to clean imaging data). To address the concern of our method’s comparison against hypernetwork, we added an experiment to incorporate the unique client-specific client information to adapt the method to our task (can refer to Appendix D in the revised manuscript). Specifically, we incorporate the anatomy information as text embedding to guide the denoising task. The quantitative comparison results are shown in Table 1 in the revised manuscript. We find the proposed FedFDD fits the denoising task better.
***
Q2: **Clarification on the simulated split and generalization experiments for more splits:***“It would be good to describe the typical real-world setting and how it motivated the simulated split used in the study.” “...The data is split in different simulated sites by anatomical structure…and hence it would have been great if the key idea was validated for different simulated splits than just 1” “Results for more splits such that each client has data from multiple anatomies will be helpful to validate the generalizability of the described approach.”*

Response:

A common assumption in low dose CT is that each client only accesses specific anatomical information, as described in the work of HyperFed (Yang et al. 2022). We argue that in real practice if an institute has LDCT images of more than one anatomical part, it can still be divided into different specialized clients and utilize our federated learning scheme to foster better performance.

Moreover, to further address the reviewer’s concerns and validate the generalizability of our approach, we conducted an experiment where client 1 includes two anatomies (chest, abdomen 1), client 2 includes abdomen (mutually exclusive with client 1) and client 3 includes head. The results demonstrate a comparable conclusion with the initial experiments (PSNR: 22.6056 vs 22.6209; SSIM: 0.7581 vs 0.7586). We think that the experiment demonstrates the robustness and generalization ability of our method. The added results are included in Appendix C in the revised manuscript.
***
Q3: ***More details of the dataset:*** “*Please provide more details about the dataset in terms of size and construction.*” “*The description of the dataset is incomplete. Two papers with different datasets have been referenced without any statistics of the final dataset used in the study.*”

Response:

We have modified the corresponding part and further enriched the dataset-related content in Appendix B of the revised manuscript. The 2016 dataset (McCollough, 2016) is a subset of the LDCT and NDCT datasets (Moen et al. 2021). Usage of the latter needs an additional license and approval. To clarify the manuscript, we removed the misleading reference.
We thank all the reviewers for their insightful review and constructive feedback on our manuscript. Reviewers have highlighted the importance of the addressed problem (Reviewer EtbZ), commended the novelty and motivation behind our approaches (Reviewer oGSF, EtbZ), and acknowledged the significant improvements over state-of-the-art methods (Reviewer u6nd). As emphasized by Reviewer EtbZ, our work makes a substantial contribution to the problem of LDCT and offers insights for various applications.

In addition to the point-to-point responses to reviewers, we would like to emphasize the clarifications of two main concerns to resolve misunderstandings.

1. Comparison with HyperFed. The presented FedFDD and HyperFed have different application scenarios. FedFDD focuses on denoising low-dose CT images, while HyperFed aims to reconstruct CT images from raw data like Sinograms. The core assumption of HyperFed is access to the scanning and geometry parameters, which are not necessary for image-level denoising tasks. Therefore, the two methods are not directly comparable. In the revised manuscript, we incorporate the concept of employing a client-specific feature vector from HyperFed into our task and include the findings. We observe that FedFDD outperforms this approach.
2. Comparison with other federated learning strategies for low-level tasks. In the revised manuscript, we extend our literature review of current federated learning algorithms. We add the comparison with a state-of-the-art federated learning method proposed for MRI reconstruction, e.g. FedMRI from TMI 2022. We extend our discussion with related works and find that FedFDD shows overall the best results in our task.

We appreciate the insightful feedback from the reviewers, which we've integrated into the updated manuscript. Notable modifications, marked in color, have been made during this rebuttal period:

1. We add a comparison method inspired by HyperFed.
2. We add a comparison method based on specificity-preserving federated learning.
3. We add an experimental setting where a client contains more than one anatomical structure.
4. We extend our related works with more state-of-the-art federated learning approaches in medical imaging.