Wavelet-domain frequency-mixing transformer unfolding network for low-dose computed tomography image denoising

Introduction

Computed tomography (CT) is widely employed in medical diagnosis, image-guided surgery, and routine physical examinations due to its ability to noninvasively and rapidly capture structural features and pathological conditions (1,2). However, during multiple CT scans, patients may experience cumulative X-ray radiation exposure, which increases the risk of developing cancer (3). To mitigate this risk, clinical radiologists have minimized X-ray doses to the lowest feasible levels in clinical CT scans, adhering to the principle of as low as reasonably achievable (ALARA) (4). Unfortunately, reducing X-ray doses often leads to the significant contamination of CT images by electronic noise and streak artifacts (5-7), which can severely impact subsequent diagnosis and treatment. In clinical practice, particularly in the management of orthopedic conditions such as osteonecrosis of the femoral head, precise visualization of trabecular bone integrity and early necrotic lesions are essential. Low-dose CT (LDCT)-induced noise can obscure subtle structural changes at the bone-soft tissue interface, resulting in delayed diagnosis or inappropriate surgical planning. The available denoising methods frequently oversmooth critical anatomical details or fail to effectively suppress mixed noise distributions (e.g., Poisson-Gaussian noise in clinical LDCT), further complicating the interpretation of high-contrast regions such as cortical-cancellous bone junctions, thereby increasing the risk of misdiagnosis.

There are three primary types of methods for enhancing the quality of LDCT images: projection domain filtering algorithms (8,9), iterative reconstruction algorithms (10,11), and postprocessing methods (12-18). The former two strategies rely on the specific projection and scanning geometries provided by equipment manufacturers. However, projection domain filtering algorithms may lead to blurred details, while iterative reconstruction algorithms incur high computational costs. In contrast, postprocessing methods do not depend on projection data and can directly process reconstructed CT images, offering greater adaptability and thus garnering increased attention. Traditional postprocessing techniques include nonlocal means (NLM) filtering, dictionary learning-based methods, block-matching methods, and diffusion filters. Although these methods are more computationally efficient than iterative reconstruction algorithms, they face challenges in addressing complex noise.

In recent years, the rapid advancement of deep learning has given rise to various postprocessing techniques for CT images, particularly convolutional neural networks (CNNs) (19-23) and transformer-based networks (24-27). These techniques present new opportunities for LDCT image denoising. Although CNNs exhibit impressive feature extraction capabilities, their performance is inherently constrained by limited receptive fields, resulting in an excessive emphasis on local patterns at the expense of global contextual relationships. Additionally, the parameter-dependent and content-independent nature of convolutional filters often leads to undesirable artifacts, such as loss of detail and global blurring in reconstructed images (28). In contrast, transformer-based approaches primarily use standard self-attention mechanisms, which effectively model long-range spatial dependencies but frequently overlook essential local contextual features. Furthermore, traditional U-shaped architectures depend on upsampling and downsampling operations that inevitably result in information loss, particularly in high-frequency regions that are critical for early disease detection.

To address the aforementioned challenges, we propose wavelet-domain frequency-mixing transformer unfolding network (WFTUNet), a novel WFTUNet inspired by the traditional inertial proximal algorithm for nonconvex optimization (iPiano) iterative algorithm, the code for which is available online (https://github.com/xiaoludiver/WFTUNet). Unlike existing transformer-based methods [e.g., TransCT and Uformer (29)], which either overlook local details or are marred by high computational overhead, WFTUNet introduces three key innovations to achieve superior performance:

• Wavelet-domain frequency-mixing transformer proximal mapping module (WFTPMM): this module replaces traditional upsampling and downsampling techniques with the discrete wavelet transform (DWT) and inverse discrete wavelet transform (IDWT), which significantly reduces information loss compared to as Uformer’s pooling operations.
• Frequency-mixing transformer (FMT): this model integrates channel-wise self-attention (CWSA) for global modeling with simple receptive field attention (SRFA) for local refinement. Unlike TransCT, which processes high and low frequencies separately, FMT dynamically fuses multiscale features.
• Multi-input architecture and pixel-by-pixel operation units (POUs): this architecture facilitates multiframe denoising in the feature space, effectively reducing noise and enhancing feature representation. Furthermore, the low-frequency gated fusion module (LGFM) improves the flow of information between the encoder and decoder stages, thereby further enhancing feature representation.

The primary contributions of this work are listed below.

• We developed WFTUNet for LDCT image denoising, effectively leveraging the strengths of iPiano and deep neural networks (DNNs). This approach provides outstanding performance and enhanced interpretability for this task.
• We added a WFTPMM that fully leverages the wavelet transform to significantly minimize the information loss associated with traditional upsampling and downsampling methods. The FMT serves as a crucial component that not only establishes long-range dependencies in images but also effectively captures local information.
• We customized the multi-input architecture and the POU for WFTUNet, implementing a multiframe denoising technique in the image feature space to reduce noise while enhancing feature representation.
• We conducted extensive experiments on LDCT images, in which the proposed WFTUNet achieved remarkable results.

Related work

Deep learning LDCT image denoising methods

Due to their powerful generalization capabilities, data-driven, deep learning-based methods have gradually supplanted traditional techniques as the predominant approach for LDCT image denoising in recent years. For instance, residual encoder-decoder CNN (RED-CNN) employs residual learning and an encoder-decoder architecture to enhance the quality of reconstructed CT images. The quadratic autoencoder (Q-AE) network is designed to provide improved denoising effectiveness for LDCT images, while denoising CNN (DnCNN) (30) reconstructs noisy images through residual learning to produce denoised outputs. Gradient extraction-based multiscale dense cross network (GE-MDCNet) effectively preserves the detailed features of reconstructed CT images by integrating high-frequency information into the primary network. Additionally, transformers have gained popularity in image restoration due to their content-dependent global receptive fields. For example, TransCT separates high and low-frequency information and uses transformer blocks for feature extraction, whereas CTformer incorporates extended and cyclic displacement modules into the feature-labeling process to refine contextual information. However, it is important to note that the aforementioned CNN-based and transformer-based networks either lack effective local information modeling or are limited in their capacity to model long-range dependencies, both of which are essential for the recovery of LDCT images. Furthermore, these networks do not provide the theoretical guarantees that traditional methods offer, which restricts their potential applications.

Deep unfolding networks

The core concept of deep unfolding networks involves equivalently unfolding traditional iterative optimization algorithms through use of deep learning modules. This approach has been widely applied to various image inverse tasks, including image superresolution (31), image denoising (32,33), and compressive sensing (34,35). In the domain of LDCT image denoising, relaxed projected gradient descent (RPGD) (36) employs a CNN to project the objective function onto the data manifold, thereby enhancing the reconstruction efficiency of the optimization process. The LEARN framework (37) employs a three-layer CNN as a generalized regularization term, where the trainable parameters remain independent at each iteration. The 3p alternating direction method of multipliers (ADMM) method (38) optimizes the ADMM algorithm by employing CNNs to represent the gradient, thereby improving the iteration process and facilitating a parameterized plug-and-play ADMM optimization scheme. The novel block coordinate descent network (BCD-Net) (39) integrates the original BCD-Net (40) into the accelerated proximal gradient method, serving as a fast numerical solver. The proximal forward-backward splitting (PFBS)-analytical and iterative reconstruction (AIR) approach (41) combines deep learning with the PFBS optimization algorithm to provide a deep learning-regularized image reconstruction method for LDCT. Although these methods address the interpretability challenges associated with purely deep learning-based models, most operate in the projection domain and are heavily reliant on projection and scan geometries, which limits their broader applicability. Furthermore, a number of these methods focus solely on optimizing the regularization term within the iterative algorithm, and their network structures tend to be relatively simple, resulting in limited effectiveness in practical clinical LDCT image denoising tasks.

Distinctions between WFTUNet and competing approaches

By unfolding the iPiano optimization algorithm into a deep learning-based module, we developed WFTUNet, which demonstrated exceptional image denoising capabilities and enhanced interpretability as compared to other deep learning-based methods. Table 1 highlights the distinctions between our approach and other deep learning-based techniques in detail. From Table 1 and the discussions in Deep learning LDCT image denoising methods section and Deep unfolding networks section, the following can be concluded.

• CNN-based methods (such as RED-CNN and DnCNN) are constrained by the local receptive field of the convolution kernel, which hampers their ability to model long-range dependencies. Additionally, content-independent parameters can lead to blurred soft tissue contrast.
• Although transformers, such as Uformer and CTformer, can capture global context, they tend to overlook local high-frequency details, and traditional downsampling operations result in information loss.
• Unfolding networks, such as Learned Experts’ Assessment-based Reconstruction Network (LEARN) and PFBS-AIR, primarily rely on projection domain design, depend on specific scanning geometric parameters, and possess relatively simple structures that limit their model representation capability.

To address these challenges, WFTUNet involves three key innovations: (I) the design of a wavelet domain mixing transformer (FMT) to model global and local relationships in the channel and spatial dimensions, respectively, with the traditional sampling being replace with DWT or IDWT to minimize information loss; (II) the introduction of POUs and LGFMs to enhance multiscale feature representation and facilitate cross-level information flow; and (III) the integration of the iPiano optimization framework with deep learning modules for end-to-end interpretable modeling in the image domain, resulting in improved robustness.

From a clinical perspective, the existing methods have limited efficacy in diagnosing orthopedic diseases, such as osteonecrosis of the femoral head. For instance, excessive smoothing of trabecular bone microstructures in CNN-based methods may obscure early necrotic lesions, while artifacts at the bone-soft tissue interface in transformer-based methods can result in surgical planning errors. Recent studies have demonstrated that the accuracy of bone microstructure imaging is highly dependent on the ability of denoising algorithms to preserve high-frequency details (42). WFTUNet significantly enhances the reconstruction accuracy of bone trabecular texture and the delineation of necrotic regions through wavelet-domain multiscale analysis combined with a frequency-hybrid transformer (refer to the experimental results on the clinical dataset in The hip joint CT image dataset section), thereby providing a reliable tool for early diagnosis.

We present this article in accordance with the TRIPOD+AI reporting checklist (available at https://qims.amegroups.com/article/view/10.21037/qims-2024-2368/rc).

Methods

Overall architecture

Mathematically, the optimization model for the image denoising problem can be formulated as follows:

x=argminx‖y−Ax‖22+λJ(x)[1]

where J(x) represents the regularization term, and A denotes the identity matrix. The iPiano (43) algorithm, which is based on proximal gradient descent (PGD), decomposes the energy function into two sub-problems: gradient descent and proximal mapping. Additionally, an inertial term is introduced in the gradient descent to enhance model convergence and guide the results toward the global optimum. The overall formulation is as follows:

sk=xk−1−ρkΦT(Φxk−1−y)+αk(xk−1−xk−2)[2]

xk=argminx12‖x−sk‖22+λR(x)[3]

where xk is the reconstructed image of the k-th iteration, y is the sampled image, αk is the learnable step size parameters, Φ is the measurement matrix, and ρk is the dynamic adaptive step size. In the proximal mapping of Eq. [3], the L1 norm is traditionally employed for regularization. However, the regulation parameters of the L1 norm are more complex, and its constrained representational capacity limits its applicability in clinical CT images.

To enhance the representation capability of the model and to address this issue, we combine the iPiano algorithm with a deep learning module and, inspired by unfolding networks such as interpretable optimization-inspired deep network (ISTA-Net) (44) and based optimization-inspired Cross-attention unfolding Framework OCTUF (34), propose the WFTUNet. The network consists of K identical stages, each containing a tensor gradient descent module (GDM) and a proximal mapping module. The tensor GDM uses physical degradation information to accelerate the gradient descent process, while the proximal mapping module employs the deep learning module to learn the prior information of the image, which is expressed as follow:

Sk=HGDM(Xk−1,Xk−2)[4]

XK=HWFTPMM(Sk)[5]

where the GDM is trivial and represents the tensor form of Eq. [2]. WFTPMM functions as the proximal mapping module in WFTUNet, corresponding to Eq. [3], and is a wavelet-domain denoiser with learnable parameters. In the initial stage of the model, x0=y. The WFTPMM, in contrast, achieves feature optimization through a wavelet-domain FMT. Unlike traditional unfolding networks, the WFTPMM employs the DWT and IDWT instead of conventional upsampling and downsampling methods, thereby minimizing information loss. Furthermore, the POU, which is designed based on the principles of multiframe denoising, effectively reduces noise interference by using cross-scale feature alignment and averaging.

Wavelet-domain denoiser

The denoiser framework for stage k is illustrated in Figure 1. This framework employs an asymmetric U-shaped structure that consists of residual blocks, DWT, IDWT, FMT, and LGFM. Initially, the method downsamples the noisy input images to scales of one-half and one-fourth using an interpolation operator, resulting in three distinct scales of input images, referred to as s1, s2, and s3. The network processes these multiscale noisy images as inputs, extracting the underlying features at each corresponding scale through the residual blocks, yielding outputs S1, S2, and S3.

Figure 1 Illustration of the proposed WFTUNet. The first row contains the overall architecture, which includes several stages. Each stage corresponds to an iteration in the iPiano algorithm. The second row presents a comprehensive design of the WFTPMM. It includes the RB for extracting multiscale features, the DWT for decomposing low-frequency and high-frequency subbands, and the FMT for effectively modeling both local and global features. Additionally, it incorporates the LGFM to enhance the integration of information across different phases, as well as the IDWT for reconstructing features. BN, batch normalization; C, channels; CWSA, channel-wise self-attention; DWT, discrete wavelet transform; FFN, feed-forward network; FMT, frequency-mixing transformer; GC, group convolution; GDM, gradient descent module; GELU, Gaussian Error Linear Unit; H, height; IDWT, inverse discrete wavelet transform; LH, low-frequency horizontal subband; LN, layer normalization; LGFM, low-frequency gated fusion module; N, H × W; POU, pixel-by-pixel operation unit; RB, residual block; SRSA, simple receptive field attention; W, width; WFTPMM, wavelet-domain frequency-mixing transformer proximal mapping module; WFTUNet, wavelet-domain frequency-mixing transformer unfolding network.

Subsequently, these features (S1, S2, and S3) are input into the encoder module, which consists of the DWT, the FMT module, and the POU. The encoder performs wavelet decomposition, depth feature extraction, and low-frequency pixel-by-pixel operations on the input features. The encoded features are then fed into the decoding module, which includes the IWT and FMT. The decoder module conducts wavelet multisubband integration and depth refinement on the input features. Finally, the depth features output from the decoder module undergo a convolution to obtain the recovered image.

Within each stage, the low-frequency features separated during the encoding process and the corresponding features in the decoder are gated and aggregated via the LGFM module to facilitate the flow of useful information. The POU sums and averages the two parts of the feature inputs to this module on a pixel-by-pixel basis for denoising and feature enhancement.

FMT

Local and global information representation is essential for image-denoising tasks. Therefore, we propose an FMT that fully leverages the complementary strengths of transformer and CNN components to facilitate the deep extraction of both local and global information.

As shown in Figure 1, FMT incorporates multihead CWSA, a feed-forward network (FFN), and SRFA in parallel with CWSA to enhance local representations.

Specifically, the input features undergo layer normalization before being fed into the parallel CWSA and SRFA. The CWSA captures global information for both low-frequency and high-frequency subband features along the channel dimension while establishing correspondences between different frequency subbands. In contrast, the SRFA concurrently models local spatial information for both low-frequency and high-frequency subband features through spatial attention with parameter sharing. Since CWSA and SRFA are distinct mechanisms, simple addition is insufficient for effective feature fusion. Therefore, we introduce a learnable parameter α, which adaptively adjusts the output features of SRFA, which are then added to the output features of the CWSA. Finally, the adjusted features undergo layer normalization before being input into the FFN to further capture both local and global contextual information. With the input feature is denoted as x, the entire computation process of FMT can be expressed as follows:

xN=LN(x)[6]

xM=CMSA(xN)+αRFAB(xN)+x[7]

x′=FFN[LN(xM)]+xM[8]

where x′ is the output, and LN is the layer normalization. For CWSA, assuming the number of heads is 1 and the input feature is x, we first use a 1×1 convolution to generate the query (Q), key (K), and value (V), followed by a 3×3 depth-wise convolution to further encode the channel contextual information. Subsequently, the resulting feature matrix undergoes self-attention computation. The calculation process of the C-MSA can be formulated as follows:

x^=Softmax(QKTτ)v[9]

where x^ is the output, and τ is a learnable temperature parameter.

For SRFA, according to the design of previous work (45,46), we let the input be x. First, group convolution (Group Conv) extracts spatial features from the receptive field. Next, average pooling (avg pool) aggregates spatial information from each receptive field, which is followed by a 1×1 convolution that generates the attention map. Finally, the attention map adjusts the importance of spatial features through element-wise multiplication with the preceding features. However, in contrast to the original design of RFAConv, we forego use of the activation function to simplify the module and improve computational efficiency. The computation process of SPFA is as follows:

SRFA=g1×1[AvgPool(x)]×Norm[g3×3(x)][10]

where gi×i is the group convolution of size i, Norm is the batch normalization, and x is the input feature map. For FFN, we employ an innovative design (47) that enhances the local and global contextual interaction capabilities of the FMT by incorporating a gating mechanism and deep convolution, thereby promoting information complementarity.

Pixel-by-pixel operation unit

Multiframe denoising uses the invariance of feature information and the randomness of noise across consecutive frames to align and merge these frames, thereby mitigating noise effects and improving the image signal-to-noise ratio (48,49). Building on this concept, we developed a POU. This unit diminishes noise effects and enhances image quality by aligning, merging, and averaging the input image features on a pixel-by-pixel basis.

Given an input of other scale images after residual block processing, the feature Sn∈ℝC×H×W and the feature LH∈ℝC×H×W obtain low-frequency features after wavelet decomposition. The entire process can be represented as follows:

LHl=(Sn1+LH12,Sn2+LH22,…,Snc+LHi2)[11]

where Sni is the feature map of the ith channel of Sn, LHi is the feature map of the ith channel of LH, and LHl∈ℝC×H×W is the processed output feature.

Since one component of the features input into this unit consists of low-frequency subband features derived from larger-scale features through wavelet transformation while the other component includes image features at the current scale, both parts of the features maintain invariance in feature information and randomness in noise. Consequently, the POU effectively implements a multiframe denoising technique at the image feature level, reducing noise in the features and enhancing feature representation. Furthermore, the dual use of the POU within the deep denoiser network further improves the network’s capability for multiscale feature representation.

LGFM

The inner-stage skip connection effectively enhances information flow; however, the use of DWT and IDWT within the network generates distinct high- and low-frequency subband contextual features between the encoder and decoder (50). Achieving a more efficient skip connection and effectively fusing the intrastage features continues to represent a significant challenge.

As illustrated in the Figure 2, the wavelet transform reveals that the low-frequency subband contains the structural information of the image, while noise and detailed information are primarily located in the high-frequency subband. The differences between the low-frequency subband features in the encoder and decoder are minimal; therefore, it is more efficient to provide the structural information directly to the decoder stage than it is to transmit the noise and detailed information. Based on this observation, we designed the LGFM, the detailed structure of which is depicted in Figure 1. Specifically, the two components of the low-frequency subband features from the corresponding stages in the encoder and decoder are first jointly input into a convolutional layer, which is followed by the implementation of a gating mechanism that fuses both components of the information complementarily to enhance the information flow. By gating the fusion of low-frequency subband features during the encoding and decoding stages, this module not only ensures effective information transfer but also reduces feature discrepancies in the intermediate stages. Given the input features xen and xdn of the encoder and decoder, respectively, the computational process of low frequency gated fusion can be expressed as follows:

x′dn=f1×1c(xen)⊙f1×1c(xdn)[12]

Figure 2 Module architecture of the POU. LH, low-frequency horizontal subband; POU, pixel-by-pixel operation unit; PWA, pixel-by-pixel average; PWM, pixel-by-pixel merge.

where f1×1c denotes the 1×1 convolution, and x′dn denotes the output low-frequency feature after gated fusion.

Loss function

To achieve more robust optimization while maintaining the quality of the reconstructed images, the proposed WFTUNet is trained using Charbonnier (Char) loss (51) and edge loss (52). The total loss function can be defined as follows:

ltotal=lchar+αledge[13]

lchar=‖IDenoised−INDCT‖22+ε2[14]

ledge=‖ΔIDenoised−ΔINDCT‖22+ε2[15]

where IDenoised and INDCT are the reconstructed and NDCT images, respectively; lchar is the Char loss; penalty coefficient k is set to 10−3, where ledge is the edge loss; and Δ represents the Laplacian operator.

Results

We used two datasets for the experiments to compare our method with other competing approaches to assess its performance and computational advantages. Finally, we conducted a comprehensive ablation study to evaluate the effectiveness of various modular design choices. The multidisciplinary team comprised hospital clinicians and university researchers, all qualified with relevant professional certifications and research experience.

Datasets and implementation details

We used the 2016 NIH-AAPM-Mayo Clinic Low Dose CT Grand Challenge dataset (53) to conduct a comprehensive evaluation of our method, subsequently validating the model on the Hip Joint CT Image Dataset. The 2016 NIH-AAPM-Mayo Clinic Low Dose CT Grand Challenge dataset comprises abdominal CT image data from 10 patients. The data were categorized into full-dose and low-dose (quarter-dose) images, with slice thicknesses of 1 and 3 mm. For our evaluation, we selected data from patient L506, while the data from the remaining nine patients were employed for model training. All 1- and 3-mm slices were used for training, with full-dose slices serving as the target and low-dose slices as the input.

Five patients with osteonecrosis of the femoral head from the Hip Joint CT Image Dataset were recruited for this study. Each patient underwent CT scans that included axial, coronal, and sagittal sequences, with images captured from the first tomographic image of the femoral head to the last. All image data were obtained from the Medical Imaging Center of Luoyang Orthopedic-Traumatological Hospital of Henan Province (Henan Provincial Orthopedic Hospital). This study was conducted in accordance with the Declaration of Helsinki and its subsequent amendments. The study was approved by ethics board of Luoyang Orthopedic-Traumatological Hospital of Henan Province (Henan Provincial Orthopedic Hospital) (No. 2023KYKT0045-01), and individual consent for this retrospective analysis was waived.

Both datasets employed in this study were in Digital Imaging and Communications in Medicine (DICOM) format with a specified pixel size. To augment the data and enhance the model’s generalization, we randomly rotated the training set of images by 90°, 180°, and 270° and further applied horizontal and vertical flips to generate additional training samples.

The proposed WFTUNet is an end-to-end training model that was developed with the PyTorch framework and trained on two RTX 4080 GPUs (Nvidia, Santa Clara, CA, USA). All training was conducted via the Adam optimizer, beginning with an initial learning rate, which was reduced over 200 iterations through a cosine annealing strategy. We preprocessed the training set as described in previous studies (26,27), cropping each CT image into two patches, with a batch size of 6. We established the visualization window for both the Mayo dataset and the Hip Joint CT Image Dataset for visual analysis of the experimental data. We employed three evaluation metrics: peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and root mean squared error (RMSE) to comprehensively assess the denoising performance of the model.

Performance comparisons

To comprehensively evaluate the LDCT denoising performance of our method, we compared it with six classical methods. These included block-matching and 3D filtering (BM3D), a traditional denoising method; DnCNN and iterative residual CNN (IRCNN) (54), which are CNN-based denoising methods; RED-CNN (19), a codec-based denoising method; and Uformer and CTformer, both of which are transformer-based denoising methods. To ensure fairness in our comparisons, we retrained all of the reference methods, maintaining consistency in all parameters with our approach except for the model architecture itself.

Mayo dataset

Table 2 shows the test results of various methods applied to the 3-mm Mayo dataset. The proposed WFTUNet demonstrated optimal performance across all evaluated metrics. In comparison to the classical BM3D denoising algorithm, WFTUNet exhibited significant improvements in all three metrics, with an increase of approximately 2 dB in PSNR and a reduction of about 2.3 in RMSE. The RED-CNN algorithm, which is based on an encoder-decoder structure, performed relatively poorly in terms of the PSNR and SSIM metrics, likely due to its simplistic architecture and the loss of critical image information during the upsampling and downsampling processes. The transformer-based methods, Uformer and CTformer, significantly outperformed other CNN-based approaches across all metrics, while our proposed method achieved optimal performance. This underscores the limitations of traditional upsampling and downsampling techniques, as well as CNN architectures, and highlights the effectiveness of WFTUNet in minimizing information loss and establishing long-range pixel dependencies.

The POU module architecture of WFTUNet, as depicted in Figure 2, provides the foundation for its denoising capabilities. Figures 3,4 illustrate the denoising results for two representative slices from the 3-mm Mayo dataset. Although the other methods achieved varying degrees of noise reduction, their outputs exhibited persistent issues such as blurring and artifacts. In contrast, WFTUNet not only effectively suppressed noise and eliminated artifacts but also preserved structural details and image contrast with remarkable fidelity. The reduced spatial smoothing and mitigated blurring further enhanced visual quality. These improvements, supported by both the architectural design (Figure 2) and the denoised samples (Figure 3), align with the quantitative metrics in Table 2, unequivocally demonstrating the superiority of our approach. Table 3 shows the test results of various models on the 1-mm Mayo dataset. WFTUNet demonstrated optimal performance across all three metrics. The transformer-based Uformer performed slightly worse than did WFTUNet, while the metrics for the other methods were significantly lower. In comparison to the combined experimental results from the 3-mm slices indicate, the 1-mm slices contained more noise, which considerably diminished the denoising capabilities of traditional filtering and CNN-based methods as the noise intensity increased. In contrast, WFTUNet maintained excellent denoising and image reconstruction capabilities even under high noise conditions, owing to its POU and FMT. The denoising results for the 1-mm Mayo slices, illustrated in Figures 5,6, further confirm the superiority of our method.

Figure 3 The denoised results of different models on L506 with 3-mm sample 1. CT, computed tomography; DnCNN, denoising convolutional neural network; IRCNN, iterative residual convolutional neural network; LDCT, low-dose computed tomography; NDCT, normal-dose computed tomography; PSNR, peak signal-to-noise ratio; RED-CNN, residual encoder-decoder convolutional neural network; RMSE, root mean squared error; SSIM, structural similarity index.

Figure 4 The denoised results of different models on L506 with 3-mm sample 2. CT, computed tomography; DnCNN, denoising convolutional neural network; IRCNN, iterative residual convolutional neural network; LDCT, low-dose computed tomography; NDCT, normal-dose computed tomography; PSNR, peak signal-to-noise ratio; RED-CNN, residual encoder-decoder convolutional neural network; RMSE, root mean squared error; SSIM, structural similarity index.

Figure 5 The denoised results of different models on L506 with 1-mm sample 1. CT, computed tomography; DnCNN, denoising convolutional neural network; IRCNN, iterative residual convolutional neural network; LDCT, low-dose computed tomography; NDCT, normal-dose computed tomography; PSNR, peak signal-to-noise ratio; RED-CNN, residual encoder-decoder convolutional neural network; RMSE, root mean squared error; SSIM, structural similarity index.

Figure 6 The denoised results of different models on L506 with 1-mm sample 2. CT, computed tomography; DnCNN, denoising convolutional neural network; IRCNN, iterative residual convolutional neural network; LDCT, low-dose computed tomography; NDCT, normal-dose computed tomography; PSNR, peak signal-to-noise ratio; RED-CNN, residual encoder-decoder convolutional neural network; RMSE, root mean squared error; SSIM, structural similarity index.

The regions of interest (ROIs) in an image contain critical details essential for accurate diagnosis, making it imperative to evaluate these regions. Figures 7,8 illustrate the comparison of ROI methods in the 1- and 3-mm slices, respectively. All the algorithms compared achieved varying degrees of denoising; however, at the junction of bone and surrounding soft tissue, WFTUNet excelled in preserving detail and enhancing tissue boundaries. In the soft tissue region, CTformer and Uformer demonstrated improved detail retention but do not match WFTUNet in terms of tissue contrast or edge sharpness. These results not only confirm WFTUNet’s superiority in noise reduction but also highlight its exceptional ability to preserve image edges and structural details.

Figure 7 The denoised results of different models on L506 with 1-mm sample 3. CT, computed tomography; DnCNN, denoising convolutional neural network; IRCNN, iterative residual convolutional neural network; LDCT, low-dose computed tomography; NDCT, normal-dose computed tomography; PSNR, peak signal-to-noise ratio; RED-CNN, residual encoder-decoder convolutional neural network; RMSE, root mean squared error; ROI, region of interest; SSIM, structural similarity index.

Figure 8 The denoised results of different models on L506 with 3-mm sample 3. CT, computed tomography; DnCNN, denoising convolutional neural network; IRCNN, iterative residual convolutional neural network; LDCT, low-dose computed tomography; NDCT, normal-dose computed tomography; PSNR, peak signal-to-noise ratio; RED-CNN, residual encoder-decoder convolutional neural network; RMSE, root mean squared error; ROI, region of interest; SSIM, structural similarity index.

In the 3-mm Mayo abdominal CT dataset, the PSNR of WFTUNet reached 33.87 dB (Table 2), representing an increase of 0.33 dB over Uformer. This improvement has significant implications for the diagnosis of abdominal diseases, such as liver lesions and kidney stones. As illustrated in Figures 3,4, WFTUNet outperformed other methods in preserving organ boundaries (e.g., the liver-spleen interface) and microcalcified structures (indicated by arrows), thereby reducing the risk of missed lesion detection that can result from the excessive smoothing characteristic of traditional methods. In the Hip Joint CT Image Dataset (Table 4), WFTUNet demonstrated advantages in detail recovery (4.27 compared to Uformer’s 4.13) and diagnostic acceptability (4.33 compared to Uformer’s 4.05), suggesting that it may directly address the clinical need for visualization of trabecular structures and the early detection of necrosis.

The Hip Joint CT Image Dataset

To verify the denoising effect and generalization capabilities of the proposed algorithm, we conducted evaluations with the Hip Joint CT Image Dataset. Due to the absence of a corresponding noise-free reference image for real clinical CT images, we employed a blind reader study (BLIS) and subjective visual evaluation to determine the algorithm’s effectiveness.

In the BLIS study, we randomly selected 20 LDCT images from different scan sites of the hip joint CT image from a selection patients, along with the corresponding images processed by the different denoising models. Five experienced radiologists evaluated the denoising results of each model based on the following criteria: noise reduction, detail reproduction, diagnostic acceptability, and overall visual quality. The scores ranged from 0 (worst) to 5 (best). We calculated the average scores from the five radiologists for each criterion, and the results are presented in Table 4. Our proposed method significantly outperformed other denoising algorithms across all scoring criteria.

The five radiologists who participated in the BLIS all had more than 10 years of experience in orthopedic imaging diagnosis, and three of them focused on the osteonecrosis of the femoral head. The Cohen kappa coefficient was used to quantify interobserver consistency, and the results for each assessment index were as follows: noise suppression, κ =0.82; detail recovery, κ =0.79; and diagnostic acceptability, κ =0.81. According to the criteria of Landis and Koch, these results were almost identical (κ >0.8), indicating that the subjective score is highly reliable.

Figures 9,10 illustrate the overall visual effects of the hip joint CT image and the results from its zoomed ROI. Although all methods exhibited varying degrees of denoising effectiveness, the overall results indicated that RED-CNN, IRCNN, and DnCNN produced noticeable blurring. Notably, DnCNN produced slight artifacts in certain areas, and both CTformer and Uformer involved some loss of soft tissue detail. In contrast, WFTUNet effectively preserved tissue contours and performed capably in regions with similar background structures. Within the ROI, WFTUNet not only eliminated noise but also maintained the boundaries of bone and soft tissue more effectively than did the other methods. It distinctly delineated cancellous and cortical bone and visualized the intricate trabecular bone structures. By preserving the natural morphology of soft tissues, WFTUNet enhanced the differentiation between various tissue types and significantly improved the overall image clarity and contrast. It can be concluded that WFTUNet offers the best visual effect, robustness, and generalization, making it suitable for clinical LDCT image denoising.

Figure 9 The denoised results of different models on the Hip Joint CT Image Dataset with sample 1. CT, computed tomography; DnCNN, denoising convolutional neural network; IRCNN, iterative residual convolutional neural network; LDCT, low-dose computed tomography; RED-CNN, residual encoder-decoder convolutional neural network; ROI, region of interest.

Figure 10 The denoised results of different models on the Hip Joint CT Image Dataset with sample 2. CT, computed tomography; DnCNN, denoising convolutional neural network; IRCNN, iterative residual convolutional neural network; LDCT, low-dose computed tomography; RED-CNN, residual encoder-decoder convolutional neural network; ROI, region of interest.

Ablation studies

We further validated the effectiveness of each module component through ablation experiments. First, we investigated the impact of the number of reconstruction stages on model performance and then analyzed of each component’s contribution to confirm its significance. Additionally, we evaluated the influence of the loss function on model performance and the computational complexity of the model. To ensure fairness in the experiments, the training parameters for all ablation experiments remained consistent with those of the previously mentioned settings, and all experiments were conducted on the 3-mm Mayo dataset.

Performance effect of phase number

To investigate the effect of the number of reconstruction stages on model performance, we analyzed the denoising performance across 2, 4, 6, and 8 stages, as presented in Table 5. As the number of stages increased, the model’s denoising performance gradually improved, confirming the efficacy of our iterative unfolding network. However, this enhancement in performance was accompanied by an increase in computational complexity. To strike a balance between performance and complexity, we ultimately selected 6 stages as the default configuration.

Performance effect of the WFTUNet components

To analyze the effectiveness of GDM in our WFTUNet, we removed the GDM component from each stage, resulting in a network similar to a vanilla neural network. The removal of GDM led to a decrease in performance across three metrics, as shown in Table 6. This suggests that our interpretable design, based on the optimization algorithm, contributes to performance improvement.

We conducted experiments to investigate the impact of each modular component in WFTUNet. Specifically, we first used a single input-output U-shaped network as a baseline. In this configuration, the wavelet transform and inverse transform (①) are replaced by traditional upsampling and downsampling methods, the FMT (②) is substituted with a standard residual block, the multi-input architecture and POU (③) are omitted, and the LGFM (④) is replaced by a simple skip connection. Each modular component is then sequentially added or substituted in the baseline to create the remaining variants, as shown in Table 7. After the application of DWT and IDWT, all metrics showed significant improvement, particularly in the reduction of mean squared error, indicating that this component minimizes information loss. The multi-input architecture and POU enhanced overall image quality, as measured by the PSNR, SSIM, and RMSE metrics. The introduction of the FMT significantly enhanced these metrics and further improved image quality. Additionally, the inclusion of the LGFM effectively boosted PSNR and RMSE. Ultimately, WFTUNet, with all components integrated, achieves an optimal balance of PSNR, SSIM, and RMSE, confirming the effectiveness of the proposed architecture.

Performance effect of loss function

To analyze the impact of the loss function on model performance, we trained and tested the proposed method using three different loss functions: standard loss, Car loss, and Char + edge loss. The results are shown in Table 8. The model demonstrated optimal performance with the Char + edge loss. This finding indicates that combining Char loss with edge loss weighting is more effective in enhancing the denoising quality of CT images during model training optimization.

Complexity analysis

To evaluate the computational complexity of the model, we compared the proposed method with several competing approaches based on the number of parameters and runtime. The runtime was measured with original CT images with a resolution of 512×512, and the average of 500 repetitions was considered to be the final result. As shown in Table 9, although the number of parameters in our WFTUNet increased, its running time remained slightly lower than that of Uformer. Overall, WFTUNet is capable of reconstructing high-quality images quickly and efficiently on GPUs.

The single inference time of WFTUNet on 512×512 resolution CT images was 0.47 seconds (using an RTX 4080, Nvidia), which satisfies the real-time diagnostic requirements for orthopedic emergency images (typically <1 second per slice). Although the parameter count (28.69 m) was higher than that of Uformer (20.63 m), WFTUNet’s PSNR was significantly superior in both the abdominal and orthopedic datasets (see Tables 2,4). This indicates that the model does not compromise its ability to retain critical diagnostic information while achieving high processing speeds.

Discussion

Key findings and technical implications

We developed WFTUNet, a WFTUNet that addresses the critical challenge of balancing noise suppression and structural fidelity in LDCT denoising. By integrating wavelet decomposition (DWT and IDWT) to minimize high-frequency information loss, a FMT for joint global-local feature modeling, and an iPiano-inspired unfolding architecture for interpretability, WFTUNet achieved state-of-the-art performance on both abdominal and orthopedic datasets. The quantitative results (Tables 2,3) demonstrate its superiority, with 33.87 dB PSNR on 3-mm Mayo data—a 0.33 dB improvement over Uformer—while clinical evaluations (Table 4) highlight its diagnostic value in preserving trabecular bone patterns (detail score: 4.27). The ablation studies (Table 7) further validated the necessity of each component, showing that wavelet processing reduced RMSE by 0.42 as compared to traditional pooling and that the hybrid loss (Char + edge) enhanced edge preservation. These innovations collectively resolve the limitations of existing methods, including CNNs’ restricted receptive fields and transformers’ pooling-induced detail loss.

Clinical relevance in orthopedic imaging

WFTUNet’s clinical significance is underscored by its ability to retain subtle bone–soft tissue interfaces and trabecular microstructures in the hip joint CT image (Figures 9,10), which are essential for early diagnosis of conditions such as osteonecrosis of the femoral head. Radiologists rated WFTUNet significantly higher in diagnostic acceptability (4.33 vs. Uformer’s 4.05), which can be attributed to its edge-aware loss function and multiscale feature alignment via POUs. The model’s real-time inference speed (0.47 s per 512×512 slice) meets clinical demands, while its robustness against mixed Poisson-Gaussian noise ensures reliability in low-dose settings. This aligns with recent studies emphasizing the importance of high-frequency detail preservation in bone imaging, positioning WFTUNet as a practical tool for dose reduction without compromising diagnostic accuracy.

Comparative advantages over existing methods

Compared to CNN-based methods (i.e., RED-CNN and DnCNN), WFTUNet overcomes local receptive field constraints through FMT’s CWSA, achieving a 3.7-dB PSNR gain over RED-CNN (Table 2). Unlike transformer-based approaches (i.e., Uformer and CTformer), which suffer from information loss during pooling, WFTUNet’s wavelet decomposition preserves multiscale features, yielding a 0.33-dB PSNR advantage over Uformer. Additionally, the physics-guided unfolding framework bridges iterative optimization and deep learning, enhancing interpretability—a limitation of purely data-driven models such as IRCNN. The modular design (e.g., LGFM for cross-stage feature fusion) further ensures adaptability to diverse noise distributions, addressing the oversmoothing artifacts prevalent in traditional methods (Figures 3-6).

Limitations and future directions

Despite its strengths, WFTUNet’s current scope is limited to abdominal and orthopedic CT data, and its generalization to other modalities (e.g., magnetic resonance imaging) requires further validation. Meanwhile, real-time processing on high-end GPUs (RTX 4080, Nvidia) and deployment on resource-constrained devices may demand lightweight adaptations, such as pruning or quantization. The clinical dataset’s small sample size (5 cases of the osteonecrosis of the femoral head) also suggest that larger multicenter trials are needed. To guide clinical adoption, future work will explore 3D extensions for volumetric denoising and integration with generative models for handling ultra-low-dose regimes while maintaining the interpretable unfolding framework.

Conclusions

In this paper, we propose WFTUNet, which effectively extends the iPiano optimization algorithm into a DNN framework, achieving high performance while maintaining model interpretability. Specifically, to enhance the quality of denoised images, WFTUNet employs the innovative WFTPMM. This module fully leverages the characteristics of the wavelet transform to effectively mitigate the information loss associated with traditional upsampling and downsampling methods. The FMT, a key component of the module, improves the extraction of both local and global features. Moreover, a customized multi-input architecture and POU for WFTUNet provide multiframe denoising in the image feature space, thereby enhancing feature representation while reducing noise. Extensive experiments demonstrated that WFTUNet is capable of outstanding performance in LDCT image denoising, with the results indicating significant advantages in artifact suppression and structural information preservation.

In the future, we will investigate the application of WFTUNet in 3D LDCT imaging for simultaneous in-plane denoising and through-plane deblurring.

Acknowledgments

None.

Footnote

Reporting Checklist: The authors have completed the TRIPOD+AI reporting checklist. Available at https://qims.amegroups.com/article/view/10.21037/qims-2024-2368/rc

Data Sharing Statement: Available at https://qims.amegroups.com/article/view/10.21037/qims-2024-2368/dss

Funding: This study was supported by the Science and Technology Major Project of Henan Province (No. 221100310200), the Henan Province Science and Technology Research Project (No. 252102310486), the National Natural Science Foundation of China (Nos. 12071345 and 11701418), and the Programs for Science and Technology Development of Henan Province (No. 212102210511).

Conflicts of Interest: All authors have completed the ICMJE uniform disclosure form (available at https://qims.amegroups.com/article/view/10.21037/qims-2024-2368/coif). The authors have no conflicts of interest to declare.

Ethical Statement: The authors are accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. This study was conducted in accordance with the Declaration of Helsinki and its subsequent amendments. The study was approved by ethics board of Luoyang Orthopedic-Traumatological Hospital of Henan Province (Henan Provincial Orthopedic Hospital) (No. 2023KYKT0045-01), and individual consent for this retrospective analysis was waived.

Open Access Statement: This is an Open Access article distributed in accordance with the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International License (CC BY-NC-ND 4.0), which permits the non-commercial replication and distribution of the article with the strict proviso that no changes or edits are made and the original work is properly cited (including links to both the formal publication through the relevant DOI and the license). See: https://creativecommons.org/licenses/by-nc-nd/4.0/.

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