Diffusion X-ray image denoising

Authors: daniel  sanderson
pablo m olmos
carlos  fernández del cerro
manuel  desco
mónica  abella
gregorio marañón madrid
Venue: MIDL_io_2024_Conference
Published: 2024-01-22 09:57:30

Abstract: X-ray imaging is a cornerstone in medical diagnosis, constituting a significant portion of the radiation dose encountered by patients. Excessive radiation poses health risks, particularly for pediatric patients, but despite the imperative to reduce radiation doses, conventional image processing methods for X-ray denoising often struggle with heuristic parameter calibration and prolonged execution times. Deep Learning solutions have emerged as promising alternatives, but their effectiveness varies, and challenges persist in preserving image quality.
This paper presents an exploration of diffusion models for planar X-ray image denoising, a novel approach that to our knowledge has not been yet investigated in this domain. We perform real time denoising of Poisson noise while preserving image resolution and structural similarity. The results indicate that diffusion models show promise for planar X-ray image denoising, offering a potential improvement in the optimization of diagnostic utility amid dose reduction efforts.

Main paper: 
 introduction and related work: 
 X-ray imaging accounts for 93.7% of the mean radiation dose applied to patients in medical diagnosis (Nuclear, 2010) . A single X-ray chest image implies an approximate effective radiation dose of 0.1 mSv, which is equivalent to 10 days of exposure to ambient radiation. For fluoroscopic interventions or clinical studies requiring several planar X-ray images, doses can significantly build up, posing a risk for the patient, especially for paediatric patients (Luo et al., 2020) . Additionally, large radiation doses can lead to premature hardware failure of the X-ray equipment, due to vaporization of the tube´s anode and breakdown of the tube´s filament. Therefore, it is important to reduce the dose of X-ray acquisitions. However, a reduction in dose implies an increase in image noise, which arises due to the quantic nature of X-rays and the presence of thermal fluctuations in the detector (Ding et al., 2018; Yi and Babyn, 2018) . This hampers the contrast resolution of the image, limiting the diagnostic utility of the radiography and degrading the performance of downstream processing or feature extraction algorithms (Juneja et al., 2023) .Several conventional image processing methods have been proposed to perform denoising of X-ray images. Some of the methods that have shown good results are bilateral filters (Juneja et al., 2023) , total variation (TV) methods (Sagheer and George, 2020) , or 3D Block Matching (Dabov et al., 2007) . However, these methods require heuristic calibration of parameters hindering their generalization and can have long execution times limiting their incorporation into clinical practice (Lin et al., 2023) . To solve these limitations, several Deep Learning (DL) solutions have been proposed for planar X-ray and fluoroscopy image denoising. To date most DL works simulate Gaussian noise, despite being Poisson noise the most relevant type of noise in X-ray images (Ding et al., 2018; Yi and Babyn, 2018) .Depending on the type of data used during training, most DL solutions applied to Xray denoising can be classified into two main categories. The first aim to predict the clean image from the noisy image using simple networks such as DnCNN or Denoising Autoencoders (Juneja et al., 2023; Gondara, 2016) or heuristically designed architectures composed of feature extracting and refinement blocks (Nayak et al., 2023) or of dual denoising networks (Sahu et al., 2023) . The second category of methods either train the network with pairs of noisy images of different noise content (Noise2Noise methods) or aim to predict specific pixels selected either randomly (Krull et al., 2019) or by intensity thresholding (Batson and Royer, 2019 ) (Noise2Self methods). The majority of these solutions minimize MSE estimates of the target, such as the Charbonnier or Frobenius norm, while only a few explore alternative loss functions (Matviychuk et al., 2016) . As MSE estimates only compare pixel wise differences, it is common to obtain results of reduced perceptual quality, generally leading to a loss of spatial resolution (Chung et al., 2022a) or to incomplete noise removal.To better preserve image resolution and texture, generative models such as Generative Adversarial Networks (GANs) have been recently been applied to medical image denoising. However, GANs suffer from convergence issues, mode collapse and vanishing gradients, greatly hindering their training (Skandarani et al., 2023) . Recently, diffusion models have outperformed GANs (Dhariwal and Nichol, 2021) , and have further improved image quality with notably simple models. These models apply a Noise2Noise training strategy, simulating noise in a self-supervised fashion at different noise levels and predicting the residual. Diffusion models are recently being applied to CT denoising (Xia et al., 2022; Liu et al., 2023) , but to our knowledge they have not yet been applied to denoising of planar X-ray and/or fluoroscopy images.In this work, we propose a denoising method based on diffusion models for planar X-ray imaging. The method is trained with a small database to mimic the conditions of clinical scenarios where images are difficult to obtain and is evaluated on images contaminated with Poisson noise. 
materials and methods: 
 The proposed method is based on the original implementation of Denoising Diffusion Probabilistic Models (DDPMs) (Ho et al., 2020) , which are designed for generative modelling by using a DL network to sequentially remove noise in a residual fashion.Figure 1 shows the workflow of the proposed method, DDPM-X, that consists of two stages: I) a diffusion model is trained with real clinical data for image generation by pro-gressively eliminating Gaussian noise starting from pure Gaussian noise and II), the method identifies the step of the generative pipeline from which to start denoising real images. This is achieved by identifying the specific denoising step within the generative pipeline that corresponds to the equivalent noise level of the noisy image. For this stage we used real images with simulated noise. Evaluation was done on images contaminated with Gaussian noise, for which the DDPM-X was trained, and Poisson noise, which better models the noise found in X-ray images.As shown in Figure 1 , the diffusion model used in DDPM-X consists of three elements: a Forward Diffusion Kernel (FDK), a DL network, and a Reverse Diffusion Kernel (RDK). The FDK iteratively applies a diffusion process to corrupt a clean image x 0 ∈ ℜ H×W with Additive White Gaussian Noise (AWGN) for a set of timesteps t ∈ [0, T ], as shown in Equation 1.q(x t |x t-1 ) = N (x t ; 1 -β t x t-1 , β t I) (1)where x t is the noise corrupted image for timestep t and β t is a hyperparameter. The FDK is applied in a single step for any random t as follows:q(x 0:t |x 0 ) = q(x 1 , x 2 , ..., x t |x 0 ) M arkov = T t=1 q(x t |x t-1 ) = N (x t ; √ α t x 0 , (1 -α t )I) (2)where α t = 1 -β t and α t = t s=0 α s . β t may be updated following any differentiable function that ensures that √ α T ≈ 0. We use the linear function presented in (Ho et al., 2020) : β t = βe-βs T t + β s , with β s = 10 -6 and β e = 0.02. The value of β s was selected to ensure that evaluation noise levels corresponded to realistic X-ray doses.The RDK generates an image by iteratively reversing the forward diffusion. Given that the reversal of a Gaussian diffusion process is also Gaussian, the p.d.f of the data can be recovered by marginalization of the individual Markov steps:p θ (x 0 ) marginal = p θ (x 0:T )∂x 1:T markov = p(x T ) T -1 t=0 p θ (x t-1 |x t )∂x 1:T (3)where θ ∈ ℜ Θ are the parameters of the DL network, p(x T ) ∼ N (0, I), andp θ (x t-1 |x t ) = N (x t-1 ; µ θ (x t , t), Σ θ (x t )) is a step from the RDK, being µ θ ≈ x t -η θ and η θ the Gaussian noise prediced by the network. For simplicity, we used a fixed small variance Σ θ = β t . At each step of the RDK, we clipped µ 0 to the [-1,1] intensity range (Saharia et al., 2022) .To perform denoising we follow a similar approach to the Come Closer Diffuse Faster algorithm (CCDF) (Chung et al., 2022b) . The RDK is applied from t = t ′ to t = 0 (Fig. 1 ), where t ′ is the denoising timestep obtained from an estimate of the noise level of the image. The approach followed to compute the denoising timestep t ′ varies depending on the probabilistic model used to simulate noise. In this work we consider two noise models: Gaussian noise, for which the DL network has been specifically trained, and Poisson noise, which is the type of noise inherent to X-rays due to their quantic nature. Gaussian noise is simulated with equation 2 for a specific timestep t, and therefore t ′ = t. Poisson noise is simulated using equation 4, which includes a small Gaussian noise η ∼ N (0, I) scaled by σ 2 = 10 to emulate electronic noise, as in (Gao et al., 2023) .EQUATIONwhere Y = Ie -Y log , Y log is the noiseless image, and I 0 (ϵ)∂ϵ = I represents the flood image.Due to the signal dependency of Poisson noise, the denoising timestep t ′ is estimated from the maximum noise variance found in the image, as follows:Ŷ log = I * 1 n N i=1 P 99 (max(Y i log )), I ∈ R H×W (5a)where N correspond to the size of the training dataset. The percentile is applied to avoid the contribution of high intensity artificial details present in the images such as medical annotations. Then, Ẑlog is computed from Ŷ = Ie -Ŷlog by using Equation 4, and the denoising timestep t ′ is estimated as follows:EQUATIONCalculating the timestep t ′ using Equation 6 ensures that the model removes the noise of maximum variance. It must be noticed that obtaining Z log requires knowing the dose I. For real noisy images, I can be estimated from the X-ray acquisition parameters, or by using noise estimation methods (Turajlić and Karahodzic, 2017) . To ensure an equivalent noise level between Gaussian and Poisson noise, Gaussian noise was simulated by taking the timestep t from Equation 2 as the denoising timestep t ′ estimated for Poisson noise.We used a U-Net composed of five pairs of downsampling and upsampling blocks with SiLU activation functions, each built of 2 Resnet layers, and an attention block of 8 heads. The number of output channels per downsampling and attention block were duplicated from 128 to 512 every two blocks. The network was conditioned on the timestep which was given to each block as a sinusoidal embedding preprocessed by an MLP block of two layers.The model was trained for 100 epochs with mixed precision, using a learning rate of 10 -4 , an AdamW optimizer, and a cosine schedule to achieve super convergence (Smith and Topin, 2019). The MSE loss function L(θ) = E[∥η -η θ (x t , t)∥ 2 ] was used to predict the Gaussian noise η of the image at a timestep t randomly drawn from a uniform distribution. A random horizontal flip was applied to the images to perform data augmentation. Training was performed on a RTX 3090 GPU of 24 GB, and took 500 s per epoch, while inference took 0.25s per timestep. All code was implemented on Pytorch based on Fastai (Howard and Gugger, 2020) and Diffusers from Hugging Face (von Platen et al., 2022) .The proposed method was evaluated for Poisson noise and Gaussian equivalent noise. Noisy images were obtained for high dose with I = 5 × 10 4 and √ 1 -α 3 = 9.6 × 10 -3 for Poisson and Gaussian noise, respectively, and for low dose with I = 9×10 3 and √ 1 -α 9 = 9.6×10 -2 , for Poisson and Gaussian noise, respectively. The high dose corresponded to an estimated denoising step of t ′ = 3 and the low dose to t ′ = 9. We randomly selected 1225 images from the NIH Chest X ray database (Wang et al., 2017) , splitted into a training set of 1125 images and a validation set of 100 images. Images were resized from 1024x1024 to 512x512 and normalized to the [-1,1] intensity range, as in (Matviychuk et al., 2016) . These images were taken as the noiseless image Y log from Equation 4.The evaluation of Poisson contaminated images was compared with four well-known algorithms: Block Matching and 3D filtering (BM3D) (Dabov et al., 2007) ; Neighbor2Neighbor (Nei2Nei) (Huang et al., 2021a) ; Dual GAN (DU-GAN) (Huang et al., 2021b) , and the same UNet architecture used by DDPM-X trained in a supervised fashion with the MSE loss. As BM3D is designed for Gaussian noise, to be fair we preprocessed the images with the Anscombe transform to convert Poisson noise into Gaussian of variance 1 and normalized them to the [0,1] intensity range as in (Bodduna and Weickert, 2019) .To evaluate the performance of the models, we applied three metrics commonly used in denoising: Peak Signal to Noise Ratio (PSNR), to evaluate pixelwise differences, Learned Perceptual Image Patch Similarity (LPIPS), to evaluate visual quality, and Structural Similarity Index Measure (SSIM), to evaluate both distortion and visual quality (Blau and Michaeli, 2018) . The absolute difference (AD) was obtained as the difference between the metrics computed for the target and the denoised image, and the relative difference (RD) as the ratio of AD and the metric computed for the target and noisy image. Visual evaluation was done after a simple post-processing pipeline, consisting of a Contrast Limited Adaptative Histogram Equalization (CLAHE) with size tile of 70 píxels and clip limit of 0.0001, and a Laplacian Pyramid of 3 levels. We additionally performed a preliminary evaluation of our method with real noisy data acquired at a low dose. This evaluation is found in Appendix A. 
results: 
 Table 1 shows that our method is powerful enough to achieve significant denoising for a wide range of noise levels. Table 2 shows that DDPM-X achieved the best quantitative performance of all methods for most. We can see that for most metrics, and specially for low doses, the best results were achived by DDPM-X for Gaussian noise, for which the network was trained. However, the performance for Poisson noise was similar, with percentually small differences in RD (below 3% in the worst of the cases) and from a qualitative point of view, these differences do not significantly hinder the visualization of the denoised image (Figure 2 ). The denosing timesteps for which the best quantitative results were obtained shows an error in the estimations of t ′ = ±1 for our method (t ′ in Table 2 ).Visual results in Figure 2 show that DDPM-X can effectively handle the different Poisson noise levels found in the images due to the signal dependency of Poisson noise, while preserving spatial resolution. However, for low doses it was unable to restore details of low contrast resolution that had been masked by noise, and in some cases it introduced a slight spatial distortion and/or small details as shown by the red arrows. BM3D introduces smoothing and artificial textures, being more noticeable for low doses, while Nei2Nei loses low contrast details for high doses and fails to achieve complete denoising for low doses. The vanilla UNet blurs the images for both high and low doses, while the DU-GAN preserves spatial resolution for high doses but introduces smoothing for low doses.Figure 3 shows that for denoising timesteps above the optimum, the value of the metrics for DDPM-X on Poisson noise is not significantly affected. However, visual evaluation shows smoothing and hallucinations (Figure 4 ). For smaller timesteps, images preserve spatial content despite suffering from incomplete noise removal.  
discussion: 
 In this work we have proposed DDPM-X, a method for planar X-ray image denoising based on a diffusion model. Although the network was trained on Gaussian noise, results suggest that the diffusion model can be also applied to Poisson denoising without any modification or fine-tuning of the network. Given the noise level conditioning of the network, the user can regulate the amount of denoising ad hoc. An initial estimate of the noise level, which is used to define the denoising timestep, may be obtained from the SVD decomposition of the image (Turajlić and Karahodzic, 2017) , or from the X-ray dose associated to the acquisition parameters. However, the selection of the appropriate denoising timestep can be critical as overly large values can introduce hallucinations. Given that the preservation of spatial content is of uttermost importance in the medical field, it is therefore preferable to cautiously use smaller values. In the future, we will explore the inclusion of data consistency models to constrain the generative power of these models and reduce the risk of content distortion.Evaluation showed that the metrics failed to detect the appearance of hallucinations. In the future we will explore alternative metrics such as the Edge Preservation Index (EPI) (Sagheer and George, 2020), and we will evaluate them patch-wise to account for small local spatial distortions. On the other hand, the low differences in metric values between our method and the baselines did not seemingly correlate with the noticeable visual differences observed. This discrepancy is likely caused by the difference in contrast between the denoised image used to evaluate the metrics and the contrast-enhanced images used for visualization.The proposed method can be efficiently trained with a small database of down to approximately 1100 images, enabling its application to real clinical scenarios which often lack large databases. Inference can be done in less than 3 seconds allowing its real-time application. The method could be further sped up by estimating the variance Σ θ of the reverse diffusion path rather than taking a fixed value (Song et al., 2020) . © 2024 CC-BY 4.0, D. Sanderson, P. M.Olmos, C.F.d. Cerro, M. Desco & M. Abella. 
 
 Appendix:  


Decision: After revision, there is reviewer consensus that the work can be accepted. Despite this positive outcome, overall enthusiasm was moderate largely due to small dataset sizes and strong focus on synthetic experiments. The real data experiments on an anthropomorphic phantom are considerably less convincing and small scale, making this paper more borderline that it seems to be from just considering the review scores.

Reviews: review: paper_summary: paper_summary: In this work, the authors propose a diffusion probabilistic generative model for Chest X-Ray image denoising. The authors show that their approach surpasses the performance of baselines.
main_review: main_review: Adding the variance matching strategy to the DDPM-based planar x-ray image denoising model indeed enhances the value of this work. Furthermore, the authors have notably improved the writing and incorporated baseline comparisons and supplementary experiments to illustrate the strengths and weaknesses of their approach. While the experiments could potentially include a downstream task and discuss the runtime of the methodologies, in its current form, I still believe that this paper will be an important contribution to MIDL 2024. Therefore, I am raising my rating to 4 and recommend accepting this paper.
strength_weakness: strength_weakness: 1. This paper can be benefited from better writing, especially in the Stage II of their approach.
2. This paper may need some elaboration on the more mathematical explanations.
3. Baselines can be expanded.
questions: questions: Mentioned in the detailed comments section.
paper_summary: paper_summary: In the broadly practical usage of X-ray, and the sufferings that it faces in maintaining the high quality of image nowadays, the authors aspired to use a diffusion model to denoise the X-ray images. The results show real time denoising of Poisson noise while preserving image resolution and structure similarity
main_review: main_review: Thanks the authors for the classification! The authors have addressed all my concerns regarding its novelty and clarity. In addition, the authors seem to have missed some relevant literature. Specifically, they don't discuss learning-based methods for denosing tasks, missing out on several relevant citations, e.g. “Structurally-Sensitive Multi-Scale Deep Neural Network for Low-Dose CT Denoising”, and “CT super-resolution GAN constrained by the identical, residual, and cycle learning ensemble (GAN-CIRCLE)”. These methods are relevant to the method proposed in this paper. These relevant papers should be discussed in the papers.
strength_weakness: strength_weakness: - The introduction part of the article is a combination of Introduction and Related works. The authors should really split it into two parts for better clarification.
- The authors imitate two different sorts of noises: Poisson and Gaussian. While the previous works are mostly focused on Gaussian Noises, the authors didn’t provide the reasons for choosing Poisson Noise to denoise.
- The proposed method was only tested on one dataset: NIH Chest X ray database, with a random selection of 1225 images in total. There are two concerns regarding this: 1. The authors should try on different datasets. 2. Why not use the full dataset instead of sampling?
- Are the training timestep and the testing timestep identical? e.g. 3 and 4 in low dose, and 8 in high dose?
Is there any possibility to combine the two models regarding Gaussian and Poisson noises together to perform a better result? So that in real use, we don’t have to choose from these two.
questions: questions: - The introduction part of the article is a combination of Introduction and Related works. The authors should really split it into two parts for better clarification.
- The authors imitate two different sorts of noises: Poisson and Gaussian. While the previous works are mostly focused on Gaussian Noises, the authors didn’t provide the reasons for choosing Poisson Noise to denoise.
- The proposed method was only tested on one dataset: NIH Chest X ray database, with a random selection of 1225 images in total. There are two concerns regarding this: 1. The authors should try on different datasets. 2. Why not use the full dataset instead of sampling?
- Are the training timestep and the testing timestep identical? e.g. 3 and 4 in low dose, and 8 in high dose?
Is there any possibility to combine the two models regarding Gaussian and Poisson noises together to perform a better result? So that in real use, we don’t have to choose from these two.
paper_summary: paper_summary: In this paper, a diffusion-model-based method is proposed to denoise X-ray. The author stated that this is the first study that applies the denoise diffusion model to X-ray denoise. The validation experiment on a simulated dataset demonstrates the proposed method outperforms the two existing methods.
main_review: main_review: The authors clarified the novelty, used an anthropomorphic phantom as a surrogate, and added the experiments to compare with the state-of-the-art GAN methods, which makes me increase the rating. Thanks the authors for their efforts.
strength_weakness: strength_weakness: 1. It seems there already is some work that applies the diffusion model on X-ray denoise. such as 
"Cascaded Latent Diffusion Models for High-Resolution Chest X-ray Synthesis" https://link.springer.com/chapter/10.1007/978-3-031-33380-4_14

2. The validation was carried out on simulated data. Although the simulation process has strong theoretical support, the simulated data cannot represent the distribution of real X-ray data.
questions: questions: The author may need to soften the statement that this is the first work that applies the diffusion model to X-ray denoise. 

The author also needs to discuss the contribution of this paper compared with existing methods, including X-ray denoise and other similar topics.

Comments: title: Update
Update
Thank you for your feedback
Official comment by Authors part 2
comment: **DU-GAN and vanilla UNet have been included in the manuscript**. The UNet selected was the same as the one used by the proposed method, trained in a supervised fashion with the MSE loss. We have included the quantitative results in Table 2 and the qualitative results in Figure 2.
**DU-GAN and vanilla UNet have been included in the manuscript**. The UNet selected was the same as the one used by the proposed method, trained in a supervised fashion with the MSE loss. We have included the quantitative results in Table 2 and the qualitative results in Figure 2.
Greetings 

We have still not received an answer to our rebuttal. We believe we have addressed all points, and we would like to know whether if we will have enough time to address further comments.

Thank you
Regards
Dear reviewers,

We believe we have adressed all the comments of the reviewers. We have rewritten the Materials and Methods section to clarify Stage I and Stage II of our method, and have included an introductory paragraph to facilitate the comprehension of that section. We have also expanded the Discussion section to include the issues raised by the reviewers. Aditionally, in our rebuttal we have presented results regarding extra baselines and evaluation databases which will be included in the manuscript as soon as the experiments are completed.

In light of these changes, we await for your feedback.
> Q: Decrease of performance in the last denoising timesteps.

A: The reason is that the model introduces over-smoothing and removes textures. Furthermore, the greater the number of timesteps the model is allowed to act, the more hallucinations it introduces. This was explained in the Results section as follows: *Figure 3 shows that for denoising timesteps above the optimum, the value of the metrics for DDPM-X on Poisson noise is not significantly affected. However, visual evaluation shows smoothing and hallucinations (Figure 4).*
Further suggestions on how to improve this discussion are welcomed.

> Q: The author also needs to discuss the contribution of this paper compared with existing methods, including X-ray denoise and other similar topics.

A: The current study does not focus on X-ray image generation but denoising. Presently, we're unaware of any research that employs diffusion models exclusively for denoising planar X-ray images. Other works focus solely on generating high-resolution X-ray images, lacking actual image denoising capabilities, particularly against Poisson noise. Therefore, we believe that this work may be of great interest for the medical imaging community.
> Q: Already existing realted work, such as "Cascaded Latent Diffusion Models for High-Resolution Chest X-ray Synthesis"

A: We apologize for the confusion. We have rewriten Section 3 to clarify the novelty of our method. The work that the reviewer points to only focuses on high-resolution X-ray image generation and does not perform denoising of real images. Denoising is the working mechanism of diffusion models but should not be confused with actual image denoising, especially when dealing with Poisson noise. In the future, we may consider introducing into Stage 1 of our work some of the strategies presented in that paper. However, to our knowledge, there are still no publications regarding the pure denoising of planar X-ray images utilizing diffusion models.

> Q: The validation was carried out on simulated data, which cannot represent the distribution of real X-ray data.

A: We would like to note that all denoising experiments in Stage II and hence the results in Tables 1 and 2 have been carried out using real X-ray data with simulated noise. Validation of Stage 1 was only carried out to demonstrate that the generative model had been correctly trained, but all data used in Stage 2 was obtained from the NIH Chest Xray database. We have removed Figure 1 to avoid confusion. Nevertheless, we agree with the reviewer that the algorithm should be tested on real noise. However, it is not straightforward to gain access to human X-rays given the ethical concerns, and to our knowledge, there is no public database hosting non-simulated noisy images, neither for CT nor planar X-rays. Therefore, we have used an anthropomorphic phantom as a surrogate. Target images were acquired at 100kV and 4 mAs, while noisy images were acquired at the same voltage and 0.8 mAs. [As can be seen](https://imgbox.com/LcJkaHZq) , results are slightly blurry, especially in the lung region. This may be because the phantom image has perfect borders of high resolution and deviates from the data distribution on which the model was trained. Quantitative results are almost identical than for simulated data excepting PSNR, which is surprisingly low likely due to a non perfect alignment of the phantom for the low and high dose acquisitions. Despite this, we believe that the results are highly positive.

| Denoise timestep | LPIPS    | MSSIM     | SSIM      | PSNR      |
|------------------|----------|-----------|-----------|-----------|
| 4.00             | 0.02     | 99.77     | **98.15** | 27.87     |
| 3.00             | **0.02** | **99.78** | 98.15     | **27.88** |
| 2.00             | 0.02     | 99.77     | 98.00     | 27.87     |

**Results on real data have been added to the Appendix.**

> Q: GAN baseline.

A: We have included DU-GAN [1], which was originally developed for CT Denoising. We have kept all training hyperparameters unchanged. Given the limited time of the rebuttal period, we only show the validation for low dose, but we will include the high dose case by the end of the discussion period.

From a [qualitative point of view](https://imgbox.com/CGjvOYj8), DU-GAN achieves better denoising than Nei2Nei, and although it does not smooth the image as much as BM3D, it still introduces blurring, specially in low contrast borders. This is likely the cause for the high value of LPIPS shown in the table below. Importantly, DU-GAN introduces halluciantions, as indicated by the red arrows.

| Model   | t' | LPIPS    | SSIM      | PSNR      |
|---------|----|----------|-----------|-----------|
| BM3D    | -  | 0.05     | 97.81     | 34.58     |
| Nei2Nei | -  | 0.03     | 96.85     | 36.45     |
| DU-GAN  | -  | 0.04     | 97.61     | 37.04     |
| DDPM-Gaussian | 8  | **0.02** | 97.33     | **37.27** |
| DDPM-Poisson | 8  | 0.02     | **98.05** | 36.89     |

**DU-GAN baseline will be included once the validation for the high dose case is completed**

> Q: High performance of baselines.

A: Metric values are high because they are being evaluated on the whole image. Given that images are big and have a non-negligible amount of scatter and, therefore a low contrast resolution, it is not surprising that the metrics fail to reflect how truly inferior the performance of the baselines is. As shown in Figure 2 of the paper, differences in performance become especially noticeable from a visual point of view when zooming into the contrast-enhanced image, as any radiologist is expected to do. Errors in low contrast resolution details may go unnoticed by the metrics, but are of great importance for the radiologists. Differences in metric values would be larger if evaluated on the enhanced images, what we decided not to do to avoid any interference of the post-processing algorithm. Nevertheless, we could change this if the reviewer deems it appropriate.

**We have expanded the paper with this discussion.**

---
[1] Zhizhong Huang et al. Du-gan: Generative adversarial networks with dual-domain u-net-based discriminators for low-dose ct denoising. 2022
> Q: Split Introduction into Introduction and Related works.

A: We thank the reviewer for the suggestion. Nevertheless, given that the Introduction section would become too short, we have kept it as a single section, but we have rewriten it for better clarification.

> Q: The authors imitate two different sorts of noises: Poisson and Gaussian. While the previous works are mostly focused on Gaussian Noises, the authors didn’t provide the reasons for choosing Poisson Noise to denoise.

A: We thank the reviewer for helping us to clarify this point. In this work, we simulate noise by a combination of Poisson and Gaussian probabilistic models to account for the quantic nature of X-rays and the presence of thermal fluctuations in the detector, respectively [1,2]. We have added the following sentence to the Introduction: *However, a reduction in dose implies an increase in image noise, which arises due to the quantic nature of X-rays and the presence of thermal fluctuations in the detector*. [...] *Poisson noise, which is the type of noise inherent to X-rays due to their quantic nature [...] is simulated using equation 4, which includes a small Gaussian noise $\eta ∼ N (0, I)$ scaled by $\sigma_2 = 10$ to emulate electronic noise.*

> Q: The proposed method was only tested on one dataset: the NIH Chest X-ray database, with a random selection of 1225 images in total. There are two concerns regarding this: 1. The authors should try on different datasets. 2. Why not use the full dataset instead of sampling?

A: 
1. We agree with the reviewer on the importance of validating this method on a different dataset. Therefore, we have selected 22 noise-free pneumonia images from the [COVID19, Pneumonia and Normal Chest X-ray PA Dataset](https://data.mendeley.com/datasets/jctsfj2sfn/1), to show the robustenss of our method against pathologies not included in the training set. From the [qualitative results](https://imgbox.com/DCO8tDke) we can reach the same conclusions as for the NIH Chest X-ray dataset, i.e., that BM3D oversmooths the results, Nei2Nei does not achive complete denoising, and that DDPM-X preserves spatial resolution.
We are working on the metrics values and eventually include them in future work.

2. One of the main goals of this work is to achieve denoising by training on a small database. Deep Learning models require large databases which are generally difficult to obtain in clinical scenarios. Additionally, pre-trained models available in the medical imaging field usually have a poor transferability and the efficacy of their reuse is limited [3]. To facilitate the implementation of these models in the clinics, demonstrating their utility when trained with small databases is of utmost importance.

> Q: Are the training timestep and the testing timestep identical? e.g. 3 and 4 in low dose, and 8 in high dose? Is there any possibility to combine the two models regarding Gaussian and Poisson noises to perform a better result? So that in real use, we don’t have to choose from these two.

A: 
1. In the Evaluation section's second paragraph, we indicate the parameters used to simulate Poisson and Gaussian noise for the high and low doses in Table 2. We use the flood value *I* for Poisson noise, while for Gaussian noise, we use the noising timestep *t* from the Diffusion FDK. For the low dose case, Gaussian noise was simulated for t=9, while for the high dose it was simulated for t=3. Therefore, as discussed in the text, the denoising timestep t’ did not always coincide with the noising timestep t. We hope we make this clearer with the following modification of paragraph 2 of the Evaluation section: 
*Noisy images were obtained for a high dose ($I = 5 × 10^4$ and $\sqrt{(1 − α_3)} = 9.6 × 10^{-3}$ for Poisson and Gaussian noise respectively) which corresponded to an estimated denoising step of t′ = 3, and for a low dose ($I = 9 × 10^3$ and $\sqrt{(1 − α_9)}= 9.6 × 10^{-2}$), which corresponded to an estimated denoising step of $t^{'} = 9$.*

Any comments are welcome to clarify this point further. If the reviewer considers it helpful for the reader, we may include this information in Table 2.

2. We apologize for the misunderstanding. A single model is used to denoise Gaussian and Poisson noise; therefore, no a priori knowledge is required to apply it, except the noise level. We have added and introductory paragraph in the Materials and Methods section to better explain our method and to hopefully clarify this point.

---
[1] Qiaoqiao Ding et al. Statistical image reconstruction using mixed poisson-gaussian noise model for x-ray ct. 2018.

[2] Xin Yi et al. Sharpness-aware low-dose ct denoising using conditional generative adversarial network. 2018

[3] Yuncheng Yang et al. Pick the best pre-trained model: Towards transferability estimation for medical image segmentation. 2023
We thank the reviewer for their constructive feedback. We have rewritten the Materials and Methods section to better explain all stages of our method.

> Q: I wasn't able to grasp the procedure for finding the Gaussian noise equivalent in Stage 2. Do you initiate the diffusion procedure not from $x_T$ but from some $x_{t-1}$? For the sake of completeness, unless there is a misunderstanding, could you formally describe how to find the equivalence between $ Z_{log}$ and some $x_{t-1}$?

A: We agree with the reviewer that the description of Stage II may be convoluted, and we welcome the suggestions to make our explanations clearer. In this work, Gaussian noise is being simulated with the Forward Diffusion Kernel (FDK) for a specific timestep t<T. Therefore, we take that same timestep as the starting point for the denoising pipeline. In this way, we reduce the number of timesteps from T to t and highly speed up inference. We have rewritten the text as follows: *we follow a similar aproach to the Come Closer Diffuse Faster algorithm (CCDF). The RDK is applied from t = t′ to t = 0 (Fig. 1), where t′ is the denoising timestep obtained from an estimate of the noise level of the image. [...] Gaussian noise is simulated with equation 2 for a specific timestep t, and therefore t′ = t.*
Furthermore, we have included a formal description of the pipeline followed to find the denoising timestep for Poisson noise, which is shown hereinafter.

*Due to the signal dependency of Poisson noise, the denoising timestep $t'$ is estimated from the maximum noise variance found in the image, as follows*:
\begin{equation}
    \hat{Y_{log}}=\mathbf{I} * \frac{1}{n} \sum^N_{i=1} P_{99} (\max(Y^i_{log})) , \quad \mathbf{I} \in \mathbb{R}^{h\times w}
\end{equation}
*where $N$ correspond to the size of the training dataset. The percentile is applied to avoid the contribution of high intensity artificial details present in the images such as medical annotations. Then, $\hat{Z_{log}} $ is computed from $\hat{Y}=Ie^{-\hat{Y_{log}}}$ by using Equation 4, and the denoising timestep $t'$ is estimated as follows*:
\begin{equation}
    t' := 1- \overline{\alpha_t'} \thickapprox Var (\hat{Z}_{log})
\end{equation}


> Q: What does the parameter t correspond to in the second paragraph of Section 2.3? Does it describe where exactly you start the RDK?

A: We apologize for the typo. As the reviewer has correctly identified, parameter t should be t’ and it corresponds to the denoising timestep, or equivalently, to the starting point in the RDK.

> Q: Can you also run a U-Net and vanilla Noise-2-Noise baseline? What was your motivation for running the Nei2Nei model as the baseline?

A: The reason for selecting Nei2Nei as one of the baselines was that, out of the supervised DL methods tested on Poisson noise, it is one of the most recent (2021) and a well-established supervised denoising method (more than 200 citations).

We are working on adding a U-Net baseline and will show some preliminary results by the end of the discussion period. We tried to evaluate Noise2Noise but found that the official repository is not reproducible. To compensate for this, we show the results for DU-GAN [1], a more recent baseline requested by another reviewer that likely shows better results than Noise2Noise or a vanilla U-Net. Here we only show the validation for low dose, but we will include the high dose case by the end of the discussion period.

From a [qualitative point of view](https://imgbox.com/CGjvOYj8), DU-GAN achieves better denoising than Nei2Nei, and although it does not smooth the image as much as BM3D, it still introduces blurring, specially in low contrast borders. This is likely the cause for the high value of LPIPS shown in the table below. Importantly, DU-GAN introduces halluciantions, as indicated by the red arrows.

| Model   | t' | LPIPS    | SSIM      | PSNR      |
|---------|----|----------|-----------|-----------|
| BM3D    | -  | 0.05     | 97.81     | 34.58     |
| Nei2Nei | -  | 0.03     | 96.85     | 36.45     |
| DU-GAN  | -  | 0.04     | 97.61     | 37.04     |
| DDPM-Gaussian | 8  | **0.02** | 97.33     | **37.27** |
| DDPM-Poisson | 8  | 0.02     | **98.05** | 36.89     |

**DU-GAN baseline will be included in the manuscript once the validation for the high dose case is completed.**

> Q: Code release.

A: Currently, we are working on a complete version of the code that includes multiple functionalities before releasing the code.

> Q: Display the SSIM metric with 4 decimal places for convenience.

A: We decided to use 2 decimal places to facilitate the tables' readability and avoid clutter. If 4 decimal places are to be used, we believe they should be added for all metrics, not only the SSIM. Nevertheless, if the reviewer deems it appropriate, we will implement this suggestion upon reply.


[1] Zhizhong Huang et al. Du-gan: Generative adversarial networks with dual-domain u-net-based discriminators for low-dose ct denoising. 2022