WaveletDFDS-Net: A Dual Forward Denoising Stream Network for Low-Dose CT Noise Reduction

Abstract

The challenge of denoising low-dose computed tomography (CT) has garnered significant research interest due to the detrimental impact of noise on CT image quality, impeding diagnostic accuracy and image-guided therapies. This paper introduces an innovative approach termed the Wavelet Domain Dual Forward Denoising Stream Network (WaveletDFDS-Net) to address this challenge. This method ingeniously combines convolutional neural networks and Transformers to leverage their complementary capabilities in feature extraction. Additionally, it employs a wavelet transform for efficient image downsampling, thereby preserving critical information while reducing computational requirements. Moreover, we have formulated a distinctive dual-domain compound loss function that significantly enhances the restoration of intricate details. The performance of WaveletDFDS-Net is assessed by comparative experiments conducted on public CT datasets, and results demonstrate its enhanced denoising effect with an SSIM of 0.9269, PSNR of 38.1343 and RMSE of 0.0130, superior to existing methods.

1. Introduction

Computed tomography (CT), which uses an X-ray beam to scan a certain region of the human body, is a widely utilized medical imaging technique, due to its high-resolution output and rapid scanning capability. Unfortunately, the widespread use of CT has sparked concerns about the potential carcinogenic and genetic risks associated with X-ray exposure [1,2]. In response, the last decade has shifted towards minimizing radiation doses in CT scans, adhering to the As Low As Reasonably Achievable (ALARA) principle to mitigate safety hazards [3]. However, this reduction in radiation dose inherently increases noise in the resulting CT images [3]. This noise amplification negatively affects image quality, which poses a challenge to the diagnostic reliability of using the images. In essence, the lower the radiation dose, the greater the noise, and consequently, the less the clinical diagnostic value of CT images [4].

Several denoising algorithms have been developed to address challenges in improving image quality in low-dose CT (LDCT). These can be broadly classified into three types: sinogram filtration, iterative reconstruction, and image post-processing. Sinogram filtration methods [5,6,7] operate on the raw data before image reconstruction, benefiting from the well-known noise properties in this domain. However, they often lead to edge blurring or a loss of resolution, and access to sinogram data may not be available to all researchers. Iterative reconstruction methods [8,9,10] aim to optimize an objective function that combines the statistical characteristics of sinogram data with prior image information. Despite achieving impressive results, these methods are limited by long processing times and the requirement for dedicated hardware, hindering their clinical use. Lastly, image post-processing methods [11,12,13] focus on the suppression of noise in reconstructed images. This approach, unlike the others, faces challenges in accurately determining the noise distribution within the image domain, complicating the achievement of an optimal balance between noise reduction performance and structural detail preservation.

The advent of deep learning, particularly the success of convolutional neural networks (CNNs) in computer vision, has sparked significant advancements in LDCT denoising. Many CNN-based methods, focussing primarily on image post-processing, training end-to-end networks in a supervised manner to learn mappings from LDCT to normal-dose CT (NDCT) images, using a predefined loss function for optimisation. Chen et al. [14] were among the first to demonstrate that a basic CNN could estimate the value of the NDCT Hounsfield Unit (HU) from LDCT patches. Gondara [15] also validated the effectiveness of the CNN-based encoder–decoder structure in medical image denoising. Furthermore, the RED-CNN model proposed by Chen et al. [16], incorporating shortcut connections in a residual encoder–decoder convolutional neural network, surpassed existing traditional image processing methods in Structural Similarity Index Measure (SSIM) and Peak Signal-to-Noise Ratio (PSNR), which were up to 0.0514 and 4.0802 dB, respectively. Additionally, to alleviate the requirement of paired LDCT and NDCT images, the exploration of unsupervised learning in LDCT denoising has been notable. Yang et al. [17] employed a generative adversarial network with the Wasserstein distance (WGAN) and perceptual loss to enhance denoised image quality and reduce over-smoothing. Kang et al. [18] introduced a CycleGAN-based approach using unpaired LDCT and NDCT images for training. In addition, Lee et al. [19] introduced an additional noise extractor network based on CycleGAN [20] to cooperate with its generators and obtained excellent results. However, a limitation of CNN-based models is their reliance on cascaded convolutional layers for high-level feature extraction focusing on local regions, restricting their ability to capture global contextual information due to the limited receptive fields of the convolution operation. This limitation hampers their efficiency in modelling structural similarity across the whole images [21].

In recent years, vision Transformers have gained significant traction in computer vision, demonstrating remarkable achievements [22,23,24]. The core of these Transformers, the self-attention unit, excels at extracting long-range dependencies by computing interactions between any two positions in the input sequence, outperforming CNN models in some extent. Vision Transformers have been increasingly applied to image restoration tasks. For instance, SwinIR [25] successfully adapted the shifted window self-attention mechanism from the Swin Transformer [26] for image restoration with an average PSNR of over 30 dB across several datasets. Uformer [27] utilized non-overlapping window-based self-attention and depth-wise convolution in its feed-forward network, efficiently capturing local context. In the realm of LDCT denoising, several innovative approaches have been developed. Eformer [28] combined a network similar to Uformer with an edge enhancement module, effectively enhancing image quality. CTformer [29] introduced the first dedicated Transformer for LDCT denoising, employing dilation and cyclic shift in Token2Token to broaden the receptive field and gather more extensive contextual information from feature maps. These advancements underscore the Transformers’ superiority in this domain. However, their self-attention mechanism leads to a significant drawback—excessive GPU memory consumption, particularly when processing high-resolution images like CT scans. This results in extended processing times and heightened equipment demands.

To optimize the performance of the Transformer without the constraints of GPU memory, a viable solution is to use low-resolution images, since it reduces the device resource consumption of the models. The wavelet transform, a prevalent technique in image processing, offers a reversible method to halve image size by decomposing signals into different frequency bands, achieving this without any loss of information and thereby reducing computational resource demands. Another significant advantage of the wavelet transform is the ease of handling sub-band signals separately and effectively. Integrating the wavelet transform with deep learning has already shown impressive results in several studies. Bae et al. [30] demonstrated the effectiveness of learning on wavelet sub-bands, introducing the wavelet residual network for image restoration. Guo et al. [31] developed a deep wavelet super-resolution network to retrieve missing details in wavelet sub-bands between low and high-resolution images. Similarly, Liu et al. [32] proposed the multi-level wavelet-CNN (MWCNN) for image restoration, utilizing multi-level wavelet transform for various tasks and obtaining a PSNR of over 32 dB with a run time of less than 0.1 s. In the context of image denoising, the decomposition of noise along with the image allows for tailored noise suppression methods in different sub-bands. Such an approach is anticipated to surpass traditional noise reduction methods that operate directly in the image domain.

In this work, we leverage the wavelet transform as the sampling framework and attenuate noise in the wavelet domain. Drawing inspiration from previous LDCT denoising research, we introduce a novel denoising network that synergizes the strengths of both CNNs and Transformers. Our approach employs the discrete wavelet transform (DWT) for image downsampling and the inverse discrete wavelet transform (IDWT) for upsampling. The network architecture features dual forward denoising streams, which effectively combine the local feature extraction capability of the convolution operation with the fine-grained information connectivity modelling prowess of Transformers. This integration allows for the extraction of CT image features at various levels. Trained under a specifically designed dual-domain loss function, our proposed network, termed WaveletDFDS-Net, demonstrates enhanced performance in exquisite detail restoration, effectively utilizing the complementary advantages of CNNs and Transformers in the context of LDCT denoising. In summary, this paper introduces the following key contributions to LDCT denoising:

• Development of WaveletDFDS-Net: We propose WaveletDFDS-Net, which harnesses the local feature extraction capabilities of the convolution operation and the pixel-level information encoding strength of Transformers and reduces noise in the wavelet domain. This network is designed to efficiently extract LDCT image wavelet features from various levels in parallel, leading to more effective noise suppression in LDCT images and fewer computing resource requirements.
• Dual-domain compound loss function: An efficient dual-domain compound loss function has been formulated to train WaveletDFDS-Net. This function incorporates an additional wavelet domain loss as an auxiliary component to the image domain loss, aiming to achieve high-fidelity detail restoration in the denoising process.
• Superior experimental outcomes: Our experimental evaluations demonstrate that the proposed method outperforms existing well-known denoising techniques. WaveletDF DS-Net not only shows improved performance metrics but also produces images of higher quality, underlining its effectiveness in LDCT denoising applications.

The remainder of this paper is structured as follows: Section 2 provides a comprehensive description of the proposed WaveletDFDS-Net and the dual-domain compound loss mechanism. Section 3 outlines the experimental setup, presents the outcomes of various ablation studies, and compares experimental results. The paper concludes with Section 5, summarizing the key findings and contributions of this research.

2. Methodology

2.1. Problem Formulation

The noise in LDCT is primarily composed of statistical noise, also known as quantum noise, and electronic noise, which arise during the signal acquisition process [33]. To simplify the complex degradation transition from NDCT to LDCT in the image domain, the noisy LDCT image ${\mathbf{I}}_{\mathrm{ld}}\in {\mathbb{R}}^{H\times W}$ can be represented as:

$${\mathbf{I}}_{\mathrm{ld}}=\mathrm{D}\left({\mathbf{I}}_{\mathrm{nd}}\right),$$

(1)

where ${\mathbf{I}}_{\mathrm{nd}}\in {\mathbb{R}}^{H\times W}$ denotes the NDCT image, and $\mathrm{D}:{\mathbb{R}}^{H\times W}\to {\mathbb{R}}^{H\times W}$ symbolizes the degradation process. Consequently, the task is reformulated as finding a denoising function f to minimize the following objective function:

$$\underset{f}{\mathrm{argmin}}\left|\right|f\left({\mathbf{I}}_{\mathrm{ld}}\right)-{\mathbf{I}}_{\mathrm{nd}}\left|\right|.$$

(2)

In our approach, the denoising function is characterized by a neural network, denoted as $f_{\theta }$, where $\theta$ represents the network parameters. This function is obtained through deep learning training techniques.

In the proposed model, we implement the DWT operation as the downsampling layer to process the input image, transforming the input image ${\mathbf{I}}_{\mathrm{ld}}$ into four distinct wavelet sub-images:

$${\mathbf{I}}_{\mathrm{LL}},{\mathbf{I}}_{\mathrm{LH}},{\mathbf{I}}_{\mathrm{HL}},{\mathbf{I}}_{\mathrm{HH}}=\mathrm{DWT}\left({\mathbf{I}}_{\mathrm{ld}}\right),$$

(3)

where ${\mathbf{I}}_{\mathrm{LL}}$, ${\mathbf{I}}_{\mathrm{LH}}$, ${\mathbf{I}}_{\mathrm{HL}}$, ${\mathbf{I}}_{\mathrm{HH}}\in {\mathbb{R}}^{\frac{H}{2}\times \frac{W}{2}}$ represent sub-images capturing various frequency components. This process effectively separates high- and low-frequency information while reducing the resolution by half. An example of DWT decomposition is depicted in Figure 1, illustrating the different sub-images: ${\mathbf{I}}_{\mathrm{LL}}$ represents the low-frequency sub-image, essentially an approximation of the original image; ${\mathbf{I}}_{\mathrm{LH}}$ and ${\mathbf{I}}_{\mathrm{HL}}$ capture horizontal and vertical edge features, respectively; ${\mathbf{I}}_{\mathrm{HH}}$ reflects the diagonal edge feature. The sub-images are then concatenated into a latent feature ${\mathbf{I}}_{\omega }\in {\mathbb{R}}^{\frac{H}{2}\times \frac{W}{2}\times 4}$, upon which the designated network operates to diminish noise in the wavelet domain:

$${\widehat{\mathbf{I}}}_{\omega }=f_{\theta }\left({\mathbf{I}}_{\omega }\right).$$

(4)

In this equation, ${\widehat{\mathbf{I}}}_{\omega }\in {\mathbb{R}}^{\frac{H}{2}\times \frac{W}{2}\times 4}$ signifies the denoised latent feature. The final phase of our model utilizes the IDWT operation as an upsampling layer. This step converts the feature back to its original resolution and reconstructs the denoised image ${\widehat{\mathbf{I}}}_{\mathrm{nd}}\in {\mathbb{R}}^{H\times W}$:

$${\widehat{\mathbf{I}}}_{\mathrm{nd}}=\mathrm{IDWT}\left({\widehat{\mathbf{I}}}_{\omega }\right).$$

(5)

Consequently, the objective function is finally reformulated as:

$$\underset{\theta }{\mathrm{argmin}}\left|\right|{\widehat{\mathbf{I}}}_{\mathrm{nd}}-{\mathbf{I}}_{\mathrm{nd}}\left|\right|.$$

(6)

Given the reversible nature of the DWT, this function can be optimized either in the image domain, the wavelet domain, or a combination of both.

2.2. Network Architecture

As proven by many previous works, the convolution operation has the ability to capture positional information in images, which is lacking in Transformers. Although positional embedding methods have been introduced in Transformers to mitigate this shortfall, they are generally less efficient and more computationally intensive compared to a convolution operation [34,35]. Transformers, on the other hand, excel at encoding pixel-level features from a global sequence, a function that convolution operations struggle with due to their limited receptive fields. Our approach synergises convolution and Transformer operations, harnessing their respective strengths and counterbalancing their weaknesses.

Specifically, the proposed WaveletDFDS-Net, as illustrated in Figure 2, features a distinctive architecture primarily composed of dual forward denoising streams, including a convolution operation branch and a Transformer branch. These streams are intricately designed for efficient feature extraction and aggregation within the wavelet domain. At the network’s inception, a 3 × 3 convolutional layer is utilized to extract the fundamental information as the initial features and elevate the wavelet sub-images to a higher dimensional space. Conversely, towards the end of the network, another 3 × 3 convolutional layer is employed to reduce the dimensionality. This design deviates from conventional methods that predominantly rely on stacking Transformer layers. Instead, the WaveletDFDS-Net innovatively incorporates a CNN branch in parallel to the Transformer layers. Throughout the feature processing phase, information is synergistically integrated between these two branches multiple times. This approach is instrumental in enhancing the learning of the feature representation, ensuring a more robust and efficient noise reduction process.

In our model, the high-dimensional features are processed through dual forward streams. One stream, comprising a ConvBlock with multiple residual blocks, is dedicated to extracting local, coarse features. Concurrently, the other stream, built from several vision Transformer blocks, focuses on modelling pixel relationships and capturing pixel-wise, fine-grained information. The specific structures of the ConvBlock and Transformer are shown in Figure 3 and Figure 4, respectively. This dual-stream structure enables each pathway to specialize in different aspects without mutual interference, mitigating the issue of information dilution often encountered in densely stacked neural networks. The distinct natures of the convolution operation and the Transformer allow for a multi-level information extraction. By merging the outputs from the ConvBlock and the Transformer, a richer feature set is obtained. A subsequent 3 × 3 convolutional layer then adeptly weights and fuses these features, facilitating adaptive redundancy filtering while maintaining dimensional consistency. This enhances the efficiency of subsequent processes. The WaveletDFDS-Net repeats this processing mode thrice, effectively balancing model complexity and performance.

2.3. Loss Function

In this work, two loss functions were employed for denoising LDCT images: the L1 loss and the Structural Similarity Index Measure (SSIM) loss [36]. The SSIM loss, formulated as follows, plays a crucial role in evaluating image quality:

$$L_{\mathrm{SSIM}}(\mathbf{X},\mathbf{Y})=\frac{1}{N}\sum _{i=1}^N\parallel 1-\mathrm{SSIM}({\mathbf{X}}_{\mathrm{i}},{\mathbf{Y}}_{\mathrm{i}})\parallel ,$$

(7)

$$\mathrm{SSIM}(\mathbf{X},\mathbf{Y})=\frac{(2{\mu }_{\mathbf{X}}{\mu }_{\mathbf{Y}}+C_1)(2{\sigma }_{\mathbf{XY}}+C_2)}{({\mu }_{\mathbf{X}}^2+{\mu }_{\mathbf{Y}}^2+C_1)({\sigma }_{\mathbf{X}}^2+{\sigma }_{\mathbf{Y}}^2+C_2)},$$

(8)

where $\mu$ and $\sigma$ represent the mean and standard deviation of the images, respectively, and ${\sigma }_{\mathbf{XY}}^2$ denotes the covariance between $\mathbf{X}$ and $\mathbf{Y}$. $C_1$ and $C_2$ are constants that stabilize the division with weak denominators. This SSIM loss is particularly effective in maintaining the structural integrity and similarity between the denoised and original images, which is vital in medical imaging applications like LDCT. The integration of L1 loss complements the SSIM by focusing on pixel-wise differences, thus ensuring both global structural fidelity and local accuracy in the denoised images.

To train the proposed model, we initially utilized the L1 loss as the primary loss function, aiming to minimize the L1 distance between the predicted output ${\widehat{\mathbf{I}}}_{\mathrm{nd}}$ and the ground truth ${\mathbf{I}}_{\mathrm{nd}}$. However, relying solely on the image domain loss did not fully harness the model’s capabilities. The wavelet transform offered the unique ability to reduce noise in various sub-bands, applying tailored methods based on the specific characteristics of each sub-image. To enhance denoising performance, we formulated a dual-domain compound loss function that integrated the image domain L1 loss with a wavelet-domain detail restoration loss. Given that the low-frequency sub-image ${\mathbf{I}}_{\mathrm{LL}}$ retained most of the original image’s features and details, we incorporated an SSIM loss specifically in this sub-band to bolster structure information learning, as SSIM effectively quantifies the structural similarity between two images. The wavelet domain loss thus served as an auxiliary regulatory component relative to the image domain loss.

In summary, the compound loss function was defined as:

$$\begin{array}{c}\hfill L_{\mathrm{compound}}=L_1({\widehat{\mathbf{I}}}_{\mathrm{nd}},{\mathbf{I}}_{\mathrm{nd}})+\lambda \times L_{\mathrm{SSIM}}({\widehat{\mathbf{I}}}_{\mathrm{LL}},{\mathbf{I}}_{\mathrm{LL}}).\end{array}$$

(9)

In this equation, ${\widehat{\mathbf{I}}}_{\mathrm{LL}}$ denotes the low-frequency sub-image of predicted output ${\widehat{\mathbf{I}}}_{\mathrm{nd}}$, and ${\mathbf{I}}_{\mathrm{LL}}$ corresponds to the low-frequency sub-image of NDCT image ${\mathbf{I}}_{\mathrm{nd}}$. The parameter $\lambda$ represents the weight of the total wavelet domain loss. In our experiments, we set $\lambda$ = 0.05.

3. Experiments

3.1. Implementation Details

In this study, we standardized the feature channels across all layers at 64. Each ConvBlock comprised two residual blocks, while the Transformer segment included two Swin Transformer blocks adopted from the Swin Transformer model [26]. Various combinations were explored, with findings detailed in Section 3.4. Both the window size and the number of attention heads within the Transformer were set to eight for convenient operation, and the MLP dimension count quadrupled the number of feature channels used as default in the paper [26].

All our experiments were conducted on a workstation equipped with an Intel(R) i9-9900K CPU, 64 GB RAM and dual NVIDIA GeForce RTX 2080Ti graphics cards, utilizing Pytorch 1.8.1. We trained our model for 100,000 iterations using the Adam optimizer with ${\beta }_1=0.9$, ${\beta }_2=0.99$, with a batch size of four. The initial learning rate was set to 0.001.

3.2. Dataset Description

In this study, we used the publicly accessible dataset, Low-Dose CT Image and Projection Data [37], to evaluate our proposed method. This dataset includes a variety of exam types: 99 head scans, 100 chest scans, and 100 abdomen scans. All these scans were acquired at standard dose levels, and each case was processed to include a second simulated lower-dose projection dataset—head and abdomen scans were provided at 25% of the normal dose, and chest scans at 10%. In our experiments, we randomly selected 2000 slices from 20 patients for each exam type. These images were then normalized to a range of [0, 1], using exam type-specific window settings: [0, 80] HU for head scans, [−1000, 1000] HU for chest scans, and [−300, 300] HU for abdomen scans. Moreover, the large background of all of the head CT images was removed and only the centre $320\times 320$ was preserved. Then, all datasets were divided into training, validation, and test sets, where the training sets were used to train the proposed network, the validation sets were used to monitor the networks’ performance during training, and the test sets were used to evaluate the networks after training.

3.3. Evaluation Metrics

To facilitate a thorough comparison, we employed three quantitative metrics, including the SSIM, Peak Signal-to-Noise Ratio (PSNR), and Root-Mean-Square Error (RMSE), for evaluating the image quality of various compared denoising methods. Among the three metrics, SSIM and RMSE primarily gauge pixel-wise similarity, whereas PSNR quantifies the ratio between the maximal potential signal value and the interfering noise, thereby assessing signal representation accuracy. Specifically, the SSIM, as represented by Equation (8), is meant to compare the brightness, contrast, and structure between two images, while the PSNR, usually measured in decibels (dB), is defined as the following function:

$$\mathrm{PSNR}(\mathbf{X},\mathbf{Y})=10lg\frac{MaxValu{e}^2}{\mathrm{MSE}(\mathbf{X},\mathbf{Y})},$$

(10)

$$\mathrm{MSE}(\mathbf{X},\mathbf{Y})=\frac{1}{n}\sum _{i=1}^n{[{\mathbf{X}}_i-{\mathbf{Y}}_i]}^2,$$

(11)

where $MaxValue$ is the largest possible pixel value, n is the total number of pixels, and MSE calculates the mean squared error of two images. Furthermore, the RMSE is the square root of the MSE, which mainly reflects the average deviation between images:

$$\mathrm{RMSE}(\mathbf{X},\mathbf{Y})=\sqrt{\mathrm{MSE}(\mathbf{X},\mathbf{Y})}.$$

(12)

Optimal noise reduction performance is indicated by higher values of SSIM and PSNR, coupled with a lower RMSE.

3.4. Model and Performance Trade-Offs

We firstly conducted several experiments on the chest dataset to evaluate the efficiency of WaveletDFDS-Net with various configurations.

Table 1. Comparison of model size and performance among different combinations.

${\mathit{N}}_{\mathit{c}}$	${\mathit{N}}_{\mathit{t}}$	Params	Time	GPU	FLOPs	SSIM	PSNR	RMSE
1	1	0.60 M	0.08 s	930 MB	29.40 G	0.9215	37.9093	0.0133
1	2	0.75 M	0.15 s	1555 MB	29.45 G	0.9213	37.9363	0.0133
1	4	1.05 M	0.31 s	2805 MB	29.55 G	0.9164	37.6980	0.0137
2	2	0.97 M	0.16 s	1699 MB	43.98 G	0.9224	38.0251	0.0132
2	4	1.27 M	0.31 s	2949 MB	44.08 G	0.9196	37.8288	0.0135

$N_c$ and $N_t$ refer to the number of residual blocks per ConvBlock and the number of Transformer blocks per Transformer, respectively. Model size is evaluated by the number of network parameters (Params), GPU memory usage (GPU), and floating point operations (FLOPs), while performance is reflected by the model inference time per image (Time) and three types of evaluation metrics.

illustrates that as the number of residual blocks $N_c$ per ConvBlock increased, there was a significant rise in the number of network parameters and floating-point operations (FLOPs). Concurrently, the number of Transformer blocks $N_t$ per Transformer markedly impacted inference time per image and GPU memory usage. Although an expanded model demonstrated enhanced information learning capability, this did not necessarily translate into a higher PSNR. In certain configurations, the PSNR improvement was minimal and did not justify the increased computational resource consumption. To balance model size and denoising performance, we settled on two residual blocks $N_c$ and two Transformer blocks $N_t$ for each unit, which remains the standard unless noted otherwise in the following sections.

3.5. Ablation Study

3.5.1. Ablation of Denoising Stream

To verify the effectiveness of the proposed convolution operation and Transformer parallel denoising stream strategy, we performed the ablation experiments on the chest dataset to compare the performance of the proposed network with the settings of only the Transformer stream ($N_c=0$) and only the convolution stream ($N_t=0$). Experimental results are shown in

Table 2. Performance comparison between different denoising strategies.

Denoising Stream	Params	Time	GPU	FLOPs	SSIM	PSNR	RMSE
Transformer-only	0.42 M	0.15 s	1411 MB	7.67 G	0.9200	37.6779	0.0137
Conv-only	0.56 M	0.20 s	353 MB	36.63 G	0.91980	37.6308	0.0138
Both	0.97 M	0.16 s	1699 MB	43.98 G	0.9224	38.0251	0.0132

. We observe that denoising only using convolution operation was more time-consuming, while the addition of transformer stream could accelerate the inference to some extent. Moreover, the performance of the network with only a single denoising branch was similar, which was significantly lower than that of the network with two branches, although this combination increased requirements in terms of the number of network parameters, GPU memory, and FLOPs. Apparently, these results fully validated the necessity and effectiveness of the combination of convolution operation and Transformer, since this combination allowed the network to extract features from different levels and remove noise to a greater extent.

3.5.2. Ablation of DWT

To test the denoising performance of WaveletDFDS-Net in the image domain and wavelet domain, we conducted comparative experiments on the chest dataset. As shown in

Table 3. Performance comparison of WaveletDFDS-Net with or without the DWT.

DWT	Params	Time	GPU	FLOPs	SSIM	PSNR	RMSE
✕	0.97 M	1.88 s	6787 MB	175.02 G	0.9212	37.7222	0.0136
✓	0.97 M	0.16 s	1699 MB	43.98 G	0.9224	38.0251	0.0132

, it is obvious that the proposed network trained in the image domain took up more computing resources. Except the number of network parameters, the use of the wavelet transform dramatically reduced the execution time, GPU memory consumption, and FLOPs of the network while improving the SSIM, PSNR, and RMSE. Therefore, the power of the wavelet transform was that it not only significantly reduced the consumption of computing resources by efficiently halving the image resolution but also helped to detect noise in different frequency bands and obtain a better denoising effect.

3.5.3. Ablation of Loss Function

To prove the effect of the proposed dual-domain compound loss function, we compared the performance of the proposed network trained under three types of loss functions, including the L1 loss, image-domain compound loss, and dual-domain compound loss. As shown in

Table 4. Performance comparison of WaveletDFDS-Net trained with different loss functions.

Loss Type	SSIM	PSNR	RMSE
L1 (image)	0.9224	38.0251	0.0132
L1 (image) + ${\mathrm{L}}_{\mathrm{SIMM}}$ (image)	0.9255	38.0833	0.0131
L1 (image) + ${\mathrm{L}}_{\mathrm{SIMM}}$ (wavelet)	0.9269	38.1343	0.0130

, the L1 loss was the main component of the three loss functions, all of which were calculated in the image domain. Incorporating the SSIM loss either in the image or wavelet domain enhanced all metrics. However, compared with the image domain SSIM loss that evenly learned the information of different frequency bands, the wavelet-domain SSIM loss, which guided the network to heavily focus on the low-frequency band with more noise components, improved metrics more. Consequently, for optimal denoising performance, the dual-domain compound loss function emerged as the preferred optimization objective for subsequent experiments.

3.6. Experimental Results

Comparison methods: We compared our method with several well-known low-dose CT denoising methods, including BM3D [38] and K-SVD [39], which are the most popular image-based denoising methods, RED-CNN [16], MAP-NN [40], QAE [41], and EDCNN [42], which are CNN-based methods, and TransCT [21] and CTformer [29], which are Transformer-based methods. The training parameters of these competing methods were set according to the recommendations of the original papers.

Here, we firstly tested the execution speed of these comparison methods other than BM3D and K-SVD (non-deep-learning-based methods executing on CPU).

Table 5. Comparison of inference time between different methods.

Method	Time
BM3D	-
K-SVD	-
RED-CNN	0.2412 s
MAP-NN	0.0615 s
QAE	0.0210 s
EDCNN	0.0161 s
TransCT	0.0094 s
CTformer	0.3624 s
WaveletDFDS-Net (Ours)	0.1564 s

shows the average inference time of 200 images with a resolution of 512 × 512. We observe that TransCT and CTformer were the fastest and slowest models, with average inference times of 0.0094 s and 0.3624 s per image, respectively. In addition, the proposed network was in the middle among these methods, with lower inference times than those of RED-CNN and CTformer but higher than those of MAP-NN, QAE, EDCNN, and TransCT, with approximately 0.1564 s per image processed. To sum up, even though WaveletDFDS-Net applied the DWT, its model efficiency still lagged behind that of some approaches.

Noise reduction performance: The left section of

Table 6. Quantitative comparison with well-known low-dose CT denoising methods on the head, chest, and abdomen datasets.

Methods	Head (25% Dose)			Chest (10% Dose)			Abdomen (25% Dose)
	SSIM	PSNR	RMSE	SSIM	PSNR	RMSE	SSIM	PSNR	RMSE
Before Denoising	0.8605	23.5636	0.0682	0.6244	29.4161	0.0382	0.7489	24.9297	0.0649
BM3D [38]	0.8706	24.6219	0.0606	0.8315	34.4676	0.0228	0.7967	27.6156	0.0504
K-SVD [39]	0.8589	24.1619	0.0631	0.7065	31.6084	0.0295	0.7811	26.8750	0.0518
RED-CNN [16]	0.8845	28.0380	0.0406	0.9126	37.1635	0.0146	0.8228	30.2898	0.0337
MAP-NN [40]	0.8877	28.0046	0.0407	0.9191	37.6004	0.0138	0.8265	30.2510	0.0338
QAE [41]	0.8874	28.0660	0.0404	0.9186	37.5780	0.0139	0.8256	30.3197	0.0336
EDCNN [42]	0.8870	27.9533	0.0409	0.9148	37.4787	0.0140	0.8261	30.1952	0.0340
TransCT [21]	0.8453	26.3517	0.0493	0.9112	36.9757	0.0149	0.7995	29.4216	0.0375
CTformer [29]	0.8758	27.6840	0.0422	0.8641	35.4648	0.0179	0.8005	29.4902	0.0368
WaveletDFDS-Net (Ours)	0.8890	28.0718	0.0403	0.9269	38.1343	0.0130	0.8297	30.3680	0.0334

presents the experimental results on the head CT dataset. We observe that the proposed method was superior to all compared methods, with the SSIM and PSNR achieving the highest value and the RMSE the lowest. Figure 5 visualizes the noise reduction effects across different models on this dataset. Notably, BM3D and K-SVD, unlike other compared methods which successfully reduced noise, were less effective in noise reduction, resulting in images still marred by significant noise. In contrast, images denoised by TransCT and CTformer exhibited only slight noise. RED-CNN and EDCNN, while effective in noise reduction, tended to over-smooth the boundaries of distinct soft tissues, thus diminishing clinical diagnostic value. Other comparative methods like MAP-NN and QAE showed similar denoising performance. However, WaveletDFDS-Net stood out for its superior restoration of details, offering a higher quality and fidelity. As shown in Figure 6, the zoomed images over a region of interest have clearer contrasts in the details. The evaluation metrics for the images presented in Figure 5 are detailed in the left section of Table 7.

The middle section of Table 6 displays the experimental results on the chest CT dataset. Analysis of the metric values revealed that WaveletDFDS-Net achieved superior evaluation metrics, surpassing all other models. Figure 7 depicts the image restoration results of the various methods on that dataset. BM3D primarily smoothed the noise, leaving noticeable traces in the denoised images. K-SVD and CTformer exhibited limited noise reduction capabilities, with their outputs retaining significant noise residues in the whole tissue. Some other methods like RED-CNN, EDCNN, and TransCT showed similar denoising effects with higher SSIM and PSNR and lower RMSE than BM3D, K-SVD, and CTformer but tended to produce blurry images. MAP-NN and QAE demonstrated improved denoising performance and image quality. Nevertheless, WaveletDFDS-Net further enhanced the denoising efficiency and restored images with quality closest to NDCT images. Figure 8 enlarges the partial details of the region marked by the red box in Figure 7, which provides a better observation. The quantitative results corresponding to these observations are presented in the middle section of Table 7, related to Figure 7.

The right section of Table 6 presents the experimental results conducted on the abdomen CT dataset, where our proposed model demonstrated superior performance. Figure 9 offers a qualitative comparison of noise reduction in abdomen CT images. K-SVD showed the least effective noise reduction, with its outputs retaining significant noise distributed throughout the organs. Similar to findings in the head and chest datasets, images denoised by BM3D, TransCT, and CTformer exhibited minor noise spots. In addition, other compared methods, while displaying lower noise levels and better evaluation metrics, still faced the issue of blurriness. In contrast, the proposed model trained with the compound loss not only attained higher metric values but also minimized image error, thereby restoring clear feature details. All these fine details could be more obviously observed in the zoomed images of Figure 10. The evaluation metrics for the images in Figure 9 are detailed in the right section of Table 7.

Table 7. Quantitative results of different denoising methods for Figure 5, Figure 7, and Figure 9.

Methods	Figure 5			Figure 7			Figure 9
	SSIM	PSNR	RMSE	SSIM	PSNR	RMSE	SSIM	PSNR	RMSE
Before Denoising	0.8747	23.9964	0.0631	0.7496	32.8246	0.0228	0.8719	30.1738	0.0310
BM3D [38]	0.8853	25.1456	0.0553	0.9260	38.4559	0.0119	0.9153	33.6569	0.0208
K-SVD [39]	0.8751	24.7358	0.0580	0.8135	34.8253	0.0181	0.8906	31.6105	0.0263
RED-CNN [16]	0.8973	28.3396	0.0383	0.9431	39.3966	0.0107	0.9217	34.6028	0.0186
MAP-NN [40]	0.9002	28.3003	0.0385	0.9477	39.6670	0.0104	0.9233	34.5802	0.0187
QAE [41]	0.8998	28.3502	0.0382	0.9457	39.5115	0.0106	0.9222	34.5999	0.0186
EDCNN [42]	0.8994	28.2622	0.0386	0.9448	39.4922	0.0106	0.9227	34.4535	0.0189
TransCT [21]	0.8617	26.6672	0.0464	0.9428	39.1229	0.0111	0.9110	33.5411	0.0210
CTformer [29]	0.8918	28.0242	0.0397	0.9190	38.0347	0.0125	0.9043	33.8039	0.0204
WaveletDFDS-Net (Ours)	0.9018	28.3540	0.0382	0.9543	40.1704	0.0098	0.9246	34.6519	0.0185

Table 7. Quantitative results of different denoising methods for Figure 5, Figure 7, and Figure 9.

Methods	Figure 5			Figure 7			Figure 9
	SSIM	PSNR	RMSE	SSIM	PSNR	RMSE	SSIM	PSNR	RMSE
Before Denoising	0.8747	23.9964	0.0631	0.7496	32.8246	0.0228	0.8719	30.1738	0.0310
BM3D [38]	0.8853	25.1456	0.0553	0.9260	38.4559	0.0119	0.9153	33.6569	0.0208
K-SVD [39]	0.8751	24.7358	0.0580	0.8135	34.8253	0.0181	0.8906	31.6105	0.0263
RED-CNN [16]	0.8973	28.3396	0.0383	0.9431	39.3966	0.0107	0.9217	34.6028	0.0186
MAP-NN [40]	0.9002	28.3003	0.0385	0.9477	39.6670	0.0104	0.9233	34.5802	0.0187
QAE [41]	0.8998	28.3502	0.0382	0.9457	39.5115	0.0106	0.9222	34.5999	0.0186
EDCNN [42]	0.8994	28.2622	0.0386	0.9448	39.4922	0.0106	0.9227	34.4535	0.0189
TransCT [21]	0.8617	26.6672	0.0464	0.9428	39.1229	0.0111	0.9110	33.5411	0.0210
CTformer [29]	0.8918	28.0242	0.0397	0.9190	38.0347	0.0125	0.9043	33.8039	0.0204
WaveletDFDS-Net (Ours)	0.9018	28.3540	0.0382	0.9543	40.1704	0.0098	0.9246	34.6519	0.0185

4. Discussion

We fully compared the efficiency and performance of WaveletDFDS-Net with several well-known LDCT denoising methods and quantitatively and visually presented the experimental results in the above sections. As one of the classical image-denoising algorithms, BM3D groups patches by searching for similar regions in the image and performs collaborative filtering by the group to reduce noise. Although excellent achievements on natural image tasks have been obtained, the drawbacks are also obvious. For images with uneven noise distribution or pixels entangled with noise, BM3D has a limited denoising effect, whose results always suffer from noise residue or blurring. Similarly, classically, K-SVD is a dictionary learning method that applies an SVD decomposition of images and selects the term with the minimum error as the updated dictionary parameter to iteratively optimize until the noise level converges. However, in our case, since the noise of LDCT was introduced in the projection domain and converted into a more complex distribution in the image domain, the capacity of the SVD decomposition was insufficient, resulting in a minimum noise reduction effect of K-SVD in our experiments.

Different from the above two methods, the deep learning strategy had a more powerful denoising ability. RED-CNN, as an early model, constructs an encoder–decoder structure with residual connections that demonstrated far better performance than K-SVD and BM3D through end-to-end training, as shown in the experimental results in Table 6. MAP-NN adaptively performs progressive noise reduction by looping through a convolutional network and takes a discriminator as one of the supervisors to train its network. These improvements contribute to increased evaluation metrics. Based on Red-CNN, QAE employs quadratic neurons (executing a quadratic operation on the input data) instead of the original inner product for high-order nonlinear sparse representation with a reasonable model complexity. As the efficiency shown in Table 5 and performance shown in Table 6, QAE comprehensively outperformed RED-CNN. EDCNN focuses on the extraction of image edge information and introduces an edge enhancement module in the first layer of the dense connection network to improve the effect of image edge restoration. Our experiments confirmed that EDCNN restored clearer boundaries of distinct organs than other models. However, all of these methods were inferior to WaveletDFDS-Net in denoising performance due to the inherent receptive field limitations of convolution.

TransCT and CTformer are two kinds of Transformer-based LDCT noise reduction models; the former separates the high-frequency content and low-frequency content of LDCT images and then mainly denoises the high-frequency part, while the latter designs a token-to-token learning strategy that encompasses local contextual information via token rearrangement rather than convolution operation. Although neither requires much computing resources, the performance is not satisfactory. Obviously, even though Transformer has the long-range feature extraction capabilities that CNN lacks, it does not reach its potential if it is used incorrectly. In contrast to these comparison methods, WaveletDFDS-Net builds two parallel branches based on Transformer and convolution operations, respectively, to learn the noise distribution at the fine-gained level and local coarse level on multiple wavelet bands to improve noise reduction performance in a cooperative manner. WaveletDFDS-Net shows promising potential to avoid the misrepresentation of anatomical structures in images and ultimately lead to better patient outcomes through more accurate diagnoses and treatments.

While the proposed methods demonstrate superior denoising capabilities, some aspects could be improved. Firstly, the concurrent architecture of CNNs and Transformers does not fully exploit the synergistic potential between these technologies, resulting in modest improvements in efficiency and experimental metrics. Secondly, the training of WaveletDFDS-Net relies on a supervised learning strategy requiring a substantial dataset of paired LDCT and NDCT images, which are challenging to acquire in real-world clinical settings, thus restricting practical applicability.

5. Conclusions

In this paper, we introduced WaveletDFDS-Net, a dual forward denoising stream network for low-dose CT noise reduction. This network synergized the local feature extraction prowess of the convolution operation and the exquisite information connectivity capabilities of Transformers to extract multi-level image features, enhancing the learning of feature representation, resulting in more robust and efficient denoising performance. Moreover, WaveletDFDS-Net employed the wavelet transform as the sampling unit to reduce the image size without any information loss, and then processed image features in the wavelet domain. Furthermore, we also devised a unique dual-domain loss function to enhance detail restoration. Experimental results across three different types of CT datasets demonstrated that our method outperformed the compared baseline models in both evaluation metrics and visual quality. Future work will focus on developing an unsupervised version of WaveletDFDS-Net to mitigate the dependency on paired training datasets and broaden its applicability in clinical environments.