There are many studies on improving the image quality of medical CT slices. In 2023, Li et al. [16] proposed a Progressive Circular Convolutional Neural Network (PCCNN) that introduces a noise transfer model. Transfers noise from low-dose CT to normal-dose CT using denoised and noisy CT images generated from unpaired CT images. The denoising model also contains a progressive module that efficiently removes noise through a multistage wavelet transform without sacrificing high-frequency components such as edges and details. In 2023, Ma et al. [17] proposed a Residual Dense Network with Self-Calibrated Convolution for low-dose CT image denoising. This network fully utilizes the hierarchical features of the original image to obtain a reconstructed image with more details.

Due to the 3D imaging properties of Nano-CT, the reconstruction results are 3D data; the correlations in the 3D still exist between slices apart from the 2D correlations in the slice. For this reason, extracting the feature from the adjacent slices is very effective for capturing and enhancing fine details.

3D network design: In 2017, Liu et al. mentioned the importance of the third dimension and applied a 3D convolution over 2D convolutions for the segmentation of Digital Breast Tomosynthesis and CT [18]. In 2023, Liu et al. [19] proposed an attention mechanism based 3D segmentation network, and refer to this algorithm as the Dilated Hierarchical Decoupled Convolution Network with Attention (ADHDC-Net). In 2023, Cao et al. proposed a 3D lightweight segmentation network for multimodal brain tumor MRI images. It added 3D SA attention to skip connections to address the semantic gap problem with spatial and channel aggregation [20].

3D image denoising: In 2018, You et al. [21] proposed a Structure-Sensitive Multiscale Generative Adversarial Network (SMGAN) that utilizes the information of 3D data to improve image quality, which can effectively preserve the structural and textural information of low-dose CT images and significantly reduce noise and artifacts. In the same year, Shan et al. [22] proposed a Conveying Path-Based Convolutional Encoder-Decoder network used for LDCT noise reduction within the GAN framework. An initial 3D CPCE denoising model is directly obtained by extending a trained 2D CNN. Then, it is fine-tuned to merge 3D spatial information from neighboring slices. The method suppresses image noise and preserves fine image structure. In 2019, Yin et al. [23] proposed a domain-progressive 3D residual convolutional network for LDCT noise reduction, which can reduce CT images from the projection domain to the image domain. In 2020, Li et al. [24] proposed a new 3D self-attention convolutional neural network that utilizes 3D data to capture a wide range of spatial information within and between CT slices. In 2022, Gunduzalp et al. [25] proposed a 3D U-Net based noise reduction method, and the proposed 3D U-NetR architecture was tested on simulated and real chest CT data to achieve better noise reduction. In 2023, Zhou et al. [15] aimed to investigate whether multi-slice input improves the denoising performance compared to single-slice input and whether the 3D network architecture outperforms the 2D version in utilizing multi-slice input.

The existing research on 3D networks provides new ideas and directions to solve the noise reduction of Nano-CT 3D data.

The self-attention mechanism enables the model to automatically learn the relationships between different positions within a sequence. It dynamically adjusts the level of attention each location pays to other locations by calculating the interrelationships between each location in the sequence and all other locations in the sequence [26]. This mechanism allows the model to capture long-distance dependencies, independent of sequence length.

Self-attention is the core idea of Transformer, and the mechanism of self-attention has become a hot research topic and has been explored in various NLP tasks. For example, Wang et al. [27] proposed a convolution-free Token2Token extended visual Transformer for low-dose CT noise removal. The method applied Transformer for noise suppression processing and enhanced the feature optimization process in the model by introducing inflated convolution, which better removes common boundary artifacts. In 2023, Zhu et al. [28] proposed an LDCT noise and artifact suppressing network based on the Swin Transformer. The noise removal sub-network improves the network’s ability to extract relevant image features by using a Swin Transformer with a shift window as an encoder-decoder for global feature fusion. In 2023, Kirillov et al. [29] proposed a robust segment anything model. SAM improves the accuracy and efficiency of image recognition and segmentation by using the Transformer architecture.

Although the self-attention mechanism dramatically improves network performance optimization, it requires high computational resources due to its higher complexity. Currently, there are not many studies on lightweight 3D networks and self-attention mechanisms; later, we introduce in detail the lightweight 3D network proposed in this paper and design a 3D axial attention mechanism to improve noise reduction performance on Nano-CT images.

Inspired by the successful application of 3D-Unet in medical image segmentation [33], we constructed AAD-ResNet using 3D convolution. Fig. 2 presents the network architecture, containing encoding and decoding paths, each with three resolution levels. The encoder and decoder contain 3D axial attention mechanisms and residual dense blocks. This network design can efficiently capture contextual information and preserve local details to achieve better performance in Nano-CT noise reduction.

Three-dimensional axial attention mechanism: The self-attention mechanism enables associations between a target pixel and any other pixel. In many Natural Language Processing (NLP) applications, Transformer has been demonstrated to encode long-distance dependencies [26]. It is due to the self-attention mechanism that reveals the correlations between the input sequences. This involves considerable computation, especially when the feature map size is large, and can lead to a significant efficiency decrease. To cope with this computational demand while efficiently integrating global information and preserving long-distance dependencies, we propose a three-dimensional axial attention mechanism that improves efficiency by constraining attention on specific axial, as shown in Fig. 3.

For the computation of the axial attention mechanism, we need to compute Query, Key, and Value for all depths of all columns in each row. These can be realized by linear changes: (2)Qrow=X×WQKrow=X×WKVrow=X×WVwhere X is the input block data with dimension n×m×d. WQ, WK, WV are the weight matrices of Query, Key, Value. In the example of a particular axis, these can be realized by linear changes: (3)Attentionrow=softmax(QrowKrowTdk)Vrow

Typically, the self-attention mechanism needs to computes the relationships between all the pixels in the image when dealing with 3D data, which usually involves a large weight matrix, thus increasing the number of parameters. Pixel position encoding is generally obtained by network training, which also increases the number of parameters of the model. While our proposed 3D axial attention mechanism only compute the relationship between elements in one direction. The computational complexity is reduced by handling rows, columns, and depths and combining their attention separately. It effectively reduces the number of parameters and computation of the model while maintaining the performance of the model. This mechanism is crucial when dealing with large-scale 3D data, as it can improve computational efficiency without sacrificing performance.

For example, suppose the dimension of the input image block is n×m×d, to which we apply the attention mechanism. The traditional self-attention mechanism would use all elements of the input for computation and the computational complexity is O(nmd×nmd). In contrast, the axial attention mechanism has a computational complexity of O(nmd+nmd+nmd) by combining the combination of the attention mechanisms on each axis (rows, columns, and depth). After applying the axial attention mechanism, the computational complexity is reduced from O(nmd×nmd) to O(nmd+nmd+nmd), which significantly reduces the computational complexity, it is significant when n, m, and d are all large [34].

Residual dense module: This paper uses residual dense blocks to extract many local properties. Specifically, we use three 3 × 3 × 3 convolutions to construct a third-order dense connection. The residual connection [35] adaptively fuses the previous and current local features to better learn these features. By incorporating residual connection and dense connection [36], the proposed model can learn image features more deeply and improve the noise reduction performance of the network. In addition, since our network applied dense blocks, each layer can directly access the previous layers, reducing the redundancy of parameters and improving the network’s training efficiency [37].

The network structure of each section is shown below. Each part of our model uses 3D convolution to enhance the correlation between slices. The specific parameters of the network are shown in Table 1. At feature extraction, we use two convolution blocks with kernel size of 3 × 3 × 3 and the filter is 64. We use convolution with kernel size of 1 × 2 × 2 and stride of 1 × 2 × 2 for down-sampling. Similarly, the transposed convolution with kernel size 1 × 2 × 2 and stride 1 × 2 × 2 is used for up-sampling. We use a 3D axial attention mechanism and residual dense block in the encoder and decoder.

The network was trained and tested in the PyTorch framework on the AMAX workstation with an Intel Xeon Gold 5118 CPU and 64 GB of available memory. Used three Graphics Processing Units GeForce RTX 3090Ti. An Adam optimizer with a 1 × 10⁻⁴ learning rate was used to minimize the objective function during the network training. In the network parameter training optimization experiments, the batch size was set to 48, the epoch was set to 100, and it took 36 h to complete the training on 14,297 pairs of real datasets.

The proposed method uses deep networks to deal with the problem of Nano-CT image noise, which is independent of the statistical properties of the noise [28]. It attempts to learn from many data pairs to find the mapping relationship between short exposure time CT images and long exposure time. Therefore, Nano-CT denoising can be summarized as the optimal approximation of the objective function, and the denoising process can be expressed as: (4)arg min∑iL[f(xi)−yi] (5)LMSE=1mnd∑a=1m∑b=1n∑c=1d(f(a,b,c)−g(a,b,c))2where f(x) is the output of the model, y is the labelled image, and LMSE is the loss function.

To ensure the accuracy and generalization of the model usually requires a large amount of data to train the model. We have accumulated two types of sample data (fossil and coral) based on Nano-CT. We acquire real data under different exposure time scanning conditions. The reconstruction algorithm obtains the 3D volume data to construct the dataset. The scanning parameters are shown in Table 2, the scanning parameter tube voltage is set to 80 kV, and the power of the ray source is set to 3 W. The long exposure time is set to 3 s, and the short exposure time is set to 0.1 s. Different quality CT images are acquired by this acquisition method. The scans that are taken at long exposure times are of high quality and are used as labels for network training. The scans that are taken at short exposure times are of poor quality, contain a lot of noise and artifacts, and are used as training data.

In addition, due to the size of the Nano-CT reconstructed volume data, which is 512 × 512 × 1024, the memory consumption is high during processing. Due to the limitation of computer hardware, it is inefficient and infeasible to process complete 3D CT images directly. Therefore, we consider overlapping and cropping the full 3D volume data into block data. This strategy is effective and efficient and can significantly increase the training dataset. In network training, models trained on small image blocks converge faster and use less GPU memory, while models trained on larger image blocks have better image quality and take longer to train. Considering the model convergence, memory usage, model performance, and testing time, the method in this paper selects 64 × 64 × 64 block data for network training. The consecutive 128 layers in the volume data are randomly chosen as the test data (the test data size is 512 × 512 × 128). Each dataset contains 14,297 training data and 2,523 test data.

The image recovery task widely uses the evaluation metrics Peak Signal Noise Ratio (PSNR) and Structural Similarity (SSIM) to evaluate the method performance. The formulae for PSNR and SSIM are defined as follows: (6)PSNR=10⋅log10⁡(N⋅(2n−1)2∑i=1N(Xi−Yi)2)

(7)SSIM(x,y)=(2μxμy+C1)(2σxy+C2)(μx2+μy2+C1)(σx2+σy2+C2)where x is the output image of the network and y is the ground truth, μx and σx2 are the mean and variance of the output image, μy and σy2 are the mean and variance of the ground truth, and σxy is the variance between the ground truth and the output image. PSNR are generally used to evaluate the deviation of the output image from the ground truth. The larger PSNR means better image quality. The larger SSIM indicates that the output image is visually closer to the ground truth.

To analyze and judge the denoising effect of the methods proposed in this paper, we choose 3D UnetR, ADHDC-Net, and 3D-Unet as the comparison methods. We use their officially released codes as much as possible to make a fairer comparison with the existing methods. We train and test the network according to the training parameters given in the related literature.

First, we show the test results for the fossil data, where we randomly show the xOy and xOz planes of the test data. As shown in Fig. 4, from left to right, long exposure time image, short exposure time image, UnetR, ADHDCNet, 3D-Unet, and the proposed method. From Fig. 4, it can be seen that all the methods can reduce the noise in the Nano-CT images with different degrees. Fig. 3 demonstrates the deviation between the noise reduction results of each method and the label image. All processing methods can suppress the noise in the Nano-CT image. However, each method also loses part of the image information while removing noise, especially the loss of image contour information is more serious. In contrast, the method proposed in this paper avoids this phenomenon as much as possible.

To better identify the low-contrast regions and further compare the results of each method in terms of detailed structural processing effects, we mark the region of interest (ROI) with a red box in Fig. 4 to observe better and compare the correction effects of each method. More obvious splicing gaps exist in the comparison methods UnetR and ADHDCNet processing results. Our method can effectively improve the image quality of a 0.1-s exposure scan to a level close to a 3-s exposure, preserving the image structure and detail information.

In addition, evaluation metrics are used to evaluate the results of each method quantitatively. As shown in Table 3, the optimal and suboptimal solutions of the quantization results are highlighted in the table, where the optimal solutions are shown in bold and the suboptimal solutions are underlined. The quantitative results are consistent with the visual perception, and the proposed method has the highest PSNR and SSIM among all the test results. Based on the average of the test results, we calculate the percentage change in metric values of the proposed method relative to the other methods. Specifically, the proposed method improves the PSNR and SSIM of the noise reduction results by 1.62% and 0.14% compared to UnetR, 22.96% PSNR and 1.32% SSIM compared to ADHDCNet, and 7.22% PSNR and 0.1% SSIM compared to Unet3D.

Next, we show the denoising results of different methods on the coral data. As above, we show the xOy plane and xOz plane of the coral volume data. Similarly, our method denoises while minimizing the loss of image information. Moreover, we can see from Fig. 5 that the denoising results of UnetR have serious blurring of image information in the ROIs and more obvious splicing gaps in the xOz plane. ADHDCNet can suppress noise but still produces new artifacts in the low-contrast region. The 3D-Unet method can achieve noise suppression, but the processing results also have blurring, resulting in some image information loss. The proposed method can achieve better noise suppression and retain image details.

The quantitative results of each method for coral data denoising are presented in Table 4, using the same approach as in Table 3 to highlight the optimal and suboptimal solutions. The quantitative results are consistent with the visual perception, and the AAD-ResNet method proposed in this chapter achieves the highest PSNR and SSIM for both denoising results for the coral test set among all the test results, outperforming the other noise reduction methods in terms of numerical results.

Fig. 6 gives the gray scale distribution of the labeled positions in Figs. 4 and 5. The denoising results of the proposed method are closer to the fluctuation of the label image at the edge position, which indicates that the proposed method has good edge preservation ability. The gray level distribution of the rest of the compared methods still has a part of the deviation from the label.

In addition, we compare the number of network parameters and test times of these four methods. As shown in Table 5, the number of parameters and test time of the proposed method are closer to that of the UnetR model, and the ADHDCNet model has the least number of parameters, but the test time is slightly longer. The 3D-Unet model has the most significant parameters and the longest test time. Combining the above objective and quantitative analyses, the following conclusion can be drawn: the 3D-Unet model has similar noise reduction performance as our method, but our method has the advantages of small number of parameters and fast test time.

In this section, we design ablation experiments to demonstrate that each module in the proposed method is helpful for the denoising task by analyzing the noise reduction effect of the 3D axial attention mechanism and residual dense blocks. As shown in Table 6, we validate each module on the coral dataset, while considering the values of the evaluation metrics with the number of network parameters, and verify that each module is helpful for the denoising task through ablation experiments. The network removes the dense residual block, PSNR decreases by 12.94%, and SSIM decreases by 0.7%. Removal of the axial attention block decreases PSNR by 2.61% and SSIM by 0.1%.

The above results show that the residual dense block can fully utilize the global features, which can effectively improve the ability of the network to remove noise. More features are essential in image recovery to maintain detailed information in the original image.