MRI denoising with a non‐blind deep complex‐valued convolutional neural network

Abstract

MR images with high signal‐to‐noise ratio (SNR) provide more diagnostic information. Various methods for MRI denoising have been developed, but the majority of them operate on the magnitude image and neglect the phase information. Therefore, the goal of this work is to design and implement a complex‐valued convolutional neural network (CNN) for MRI denoising. A complex‐valued CNN incorporating the noise level map (non‐blind $ℂ$DnCNN) was trained with ground truth and simulated noise‐corrupted image pairs. The proposed method was validated using both simulated and in vivo data collected from low‐field scanners. Its denoising performance was quantitively and qualitatively evaluated, and it was compared with the real‐valued CNN and several other algorithms. For the simulated noise‐corrupted testing dataset, the complex‐valued models had superior normalized root‐mean‐square error, peak SNR, structural similarity index, and phase ABSD. By incorporating the noise level map, the non‐blind $ℂ$DnCNN showed better performance in dealing with spatially varying parallel imaging noise. For in vivo low‐field data, the non‐blind $ℂ$DnCNN significantly improved the SNR and visual quality of the image. The proposed non‐blind $ℂ$DnCNN provides an efficient and effective approach for MRI denoising. This is the first application of non‐blind $ℂ$DnCNN to medical imaging. The method holds the potential to enable improved low‐field MRI, facilitating enhanced diagnostic imaging in under‐resourced areas.

A complex‐valued convolutional neural network for MRI denoising (non‐blind $ℂ$DnCNN) was implemented. It outperformed other algorithms in denoising simulated and in vivo data. The method preserves image phase and effectively handles spatially varying parallel imaging noise. It offers an efficient approach for MRI denoising, which is useful for low‐field MRI.

Abbreviations

ABSD

absolute difference

AWGN

additive white Gaussian noise

BM3D

block‐matching and 3D filtering

BN

batch normalization

CNN

convolutional neural network

GAN

generative adversarial network

GRAPPA

generalized autocalibrating partially parallel acquisitions

NA

number of averages

NLM

non‐local means

NRMSE

normalized root‐mean‐square error

PSNR

peak signal‐to‐noise ratio

SENSE

sensitivity encoding

SNR

signal‐to‐noise ratio

SSIM

structural similarity index

$ℂ$Conv

complex‐valued convolution

$ℂ$ReLU

complex‐valued rectified linear unit

1. INTRODUCTION

Signal‐to‐noise ratio (SNR) is crucial for MR image analysis. High‐SNR images enable better visualization of small structures, which not only facilitates the human interpretation but also benefits subsequent processing techniques such as registration and segmentation. ¹ , ² , ³ , ⁴ The SNR is intricately linked to various imaging conditions, including field strength, image resolution, and number of averages (NA). The application of parallel imaging and innovative reconstruction techniques can also impact the spatial distribution of noise. Increasing NA is a common method to boost the SNR, but it results in a longer scan time. On the other hand, low‐field MRI has grown in popularity in recent years ⁵ , ⁶ , ⁷ , ⁸ because of advancements in hardware techniques and acquisition strategies. Because low‐field MRI scanners are substantially less expensive to purchase and install, they enable MRI to reach underserved populations worldwide. However, images acquired on low‐field scanners inherently have low SNR because of the low Boltzmann polarization, which hinders their clinical application. Therefore, efficient and effective denoising is important for increasing MRI availability in the clinical setting.

The noise in a complex MR image is typically modeled as a complex additive white Gaussian noise (AWGN), with zero mean and equal variance for real and imaginary parts. Thus, the pixel intensity in the noise‐corrupted magnitude image follows the Rician distribution. ⁹ , ¹⁰ Numerous MRI denoising methods have been proposed based on decades of research. The non‐local means (NLM) algorithm relies on the non‐local similarity to remove Gaussian noise. ¹¹ , ¹² , ¹³ Wiest‐Daesslé et al. ¹⁴ adapted the NLM algorithm for Rician noise and applied it to diffusion tensor MRI. Coupé et al. ¹⁵ reduced the time complexity of NLM through blockwise implementation and parallel computation. The block‐matching and 3D filtering (BM3D) takes advantage of the enhanced sparsity in the transform domain and uses collaborative filtering to remove noise. ¹⁶ , ¹⁷ , ¹⁸ In recent years, deep learning‐based approaches have shown great success in image denoising. A convolutional neural network (CNN) learns to restore the clean image by training it with a large number of noise‐corrupted and ground‐truth image pairs. Zhang et al. ¹⁹ combined residual learning and batch normalization in their DnCNN model for Gaussian denoising with unknown noise level (i.e., blind denoising). Later, Zhang et al. ²⁰ demonstrated that incorporating the noise level information into the network (i.e., non‐blind denoising) increased its generalizability. Quan et al. ²¹ investigated the potentials of complex‐valued CNNs for natural image denoising. Manjón et al. ²² proposed a two‐stage method that combines blind CNN denoising with the NLM algorithm. Tripathi et al. ²³ employed encoder–decoder structure and residual learning scheme for removing Rician noise from magnitude MR images. Li et al. ²⁴ used a progressive learning strategy, cascading two subnetworks for crude and refinement noise estimation, respectively. Tian et al. ²⁵ developed an MRI denoising method based on the conditional generative adversarial network (GAN). Koonjoo et al. ²⁶ presented an end‐to‐end DL‐based noise‐robust reconstruction method for low‐field MRI data.

However, the vast majority of existing MRI denoising methods do not fully exploit the complex‐valued feature of MRI data. The input and output of conventional model‐based approaches, such as NLM and BM3D, are both magnitude images, and phase information is ignored. CNN‐based approaches typically treat the real and imaginary parts as two separate real‐valued channels, similar to the RGB channels of a color image, which may limit the network's ability to extract features from complex‐valued data. Recently, complex‐valued CNNs have received increased attention. Complex‐valued CNNs have several advantages over real‐valued CNNs, including easier optimization, faster learning, and richer representational capacity. ²⁷ , ²⁸ , ²⁹ , ³⁰ Wang et al. ³¹ used the residual learning strategy to accelerate the convergence of their complex‐valued CNN. El‐Rewaidy et al. ³² applied a complex‐valued CNN to reconstruct highly under‐sampled cardiac MRI data. Cole et al. ³³ evaluated the performance of complex‐valued CNNs on phase‐related MRI applications by systematically analyzing the impact of different model design choices. All of these works concentrated on the MRI reconstruction task, and to the best of our knowledge, no previous work attempted to use complex‐valued CNN in the MRI denoising task.

Therefore, in this paper, we designed and implemented non‐blind $ℂ$DnCNN, a complex‐valued CNN for non‐blind MRI denoising. The noise level map was estimated from the noise‐corrupted image and fed into the network. Complex‐valued building blocks were used throughout the network. We trained the network on simulated data and tested it on both simulated and in vivo data. We compared our method to several other denoising algorithms, both quantitatively and qualitatively. In the comparison, both magnitude and phase performance were evaluated. We also investigated the role of the noise level map in dealing with parallel imaging noise that varies spatially.

2. METHODS

2.1. Training datasets

For supervised neural network training, ground truth images are required. Assuming a noise‐free image $m\left(\mathbf{x}\right)$, we generated the noise‐corrupted image $m^{\prime }\left(\mathbf{x}\right)$ by applying random, complex AWGN to $m\left(\mathbf{x}\right)$:

$$m^{\prime }\left(\mathbf{x}\right)\mathbf{=}m\left(\mathbf{x}\right)\mathbf{+}n\left(\mathbf{x};{\sigma }^2\right)$$ (1)

where $n\left(\mathbf{x};{\sigma }^2\right)=n_r\left(\mathbf{x};{\sigma }^2\right)+j·n_i\left(\mathbf{x};{\sigma }^2\right)$ is random, complex AWGN with zero mean and variance ${\sigma }^2$. However, ideal noise‐free images do not exist. The training and validation datasets used in this work were built from the fastMRI brain dataset (https://fastmri.med.nyu.edu/). ³⁴ , ³⁵ The raw fastMRI dataset contains nearly 7000 fully sampled multi‐coil brain MRIs obtained on 1.5 or 3 T scanners, comprising axial T1‐weighted, post‐contrast T1‐weighted, T2‐weighted, and FLAIR images. Detailed descriptions can be found on the project website. A reference study has indicated that the SNR at 0.55 T low‐field scanners is approximately 70% of that at 1.5 T scanners. ³⁶ We randomly selected 2000 T2‐weighted imaging volumes for training the model and selecting model hyperparameters. We strategically reversed the other three modalities for testing the denoising performance across a diverse range of imaging conditions and contrasts. Single‐coil data were reconstructed from multi‐coil data through the use of an adaptive combination method, ³⁷ in which the complex‐valued coil sensitivities were estimated from a locally matched filter. The adaptive combination method achieved near‐optimal SNR while retaining the phase information. The reconstructed images were then center‐cropped to have a matrix size of 320 × 320. During the noise simulation, each image was normalized to have its magnitude between 0 and 1 and its phase unchanged. The noise standard deviation $\sigma$ was sampled from a uniform distribution between 0 and 0.1.

2.2. Network architecture

A DnCNN ¹⁹ was used as our backbone network structure. The original DnCNN model was proposed for blind Gaussian denoising of natural images and achieved state‐of‐the‐art performance. The network was designed to use complex‐valued operations rather than splitting the real and imaginary components into two separate channels. Figure 1 shows the proposed non‐blind $ℂ$DnCNN for MRI denoising. The input to the network is a 2D complex‐valued MR image concatenated with a tunable complex‐valued noise level map. The non‐blind $ℂ$DnCNN consists of a series of complex‐valued convolution blocks. Three types of operations were adopted in each block: complex‐valued convolution ($ℂ$Conv), radial batch normalization (BN), and complex‐valued rectified linear unit ($ℂ$ReLU). The $ℂ$Conv operation between the input $\mathbf{d}=\mathbf{a}+j·\mathbf{b}$ and filter $\mathbf{w}\mathbf{=}\mathbf{x}\mathbf{+}j·\mathbf{y}$ can be accomplished by four real‐valued convolutions ³⁰ :

$$\mathbf{w}⊛\mathbf{d}=\left(\mathbf{x}⊛\mathbf{a}-\mathbf{y}⊛\mathbf{b}\right)+j·\left(\mathbf{y}⊛\mathbf{a}+\mathbf{x}⊛\mathbf{b}\right)$$ (2)

where $⊛$ represents the convolution operation. Batch normalization is an important operation to expedite training and stabilize model performance. ³⁸ We adopted radial BN, which maintains the phase information while scaling the magnitude ³² :

$${\mathbf{m}}_{\mathrm{BN}}=\left(\frac{\mathbf{m}-{\mu }_{\mathbf{m}}}{\sqrt{{\sigma }_{\mathbf{m}}^2}}\right)\gamma +\beta +\tau$$ (3)

$${\mathbf{d}}_{\mathrm{BN}}={\mathbf{m}}_{\mathrm{BN}}{e}^{j\mathbf{\theta }}$$ (4)

where $\mathbf{d}\mathbf{=}\mathbf{m}{e}^{j\mathbf{\theta }}$ is the input expressed in its polar form, ${\mu }_{\mathbf{m}}$ and ${\sigma }_{\mathbf{m}}^2$ are the mean and variance of $\mathbf{m}$, $\beta$ and $\gamma$ are trainable parameters, and $\tau$ is a constant to ensure that the normalized ${\mathbf{m}}_{\mathrm{BN}}$ is positive (empirically set to 1). The $ℂ$ReLU function separately activates the real and imaginary components of the input ³⁰ :

$$ℂ\text{ReLU}\left(\mathbf{d}\right)=\text{ReLU}\left(\mathbf{a}\right)+j·\text{ReLU}\left(\mathbf{b}\right)$$ (5)

FIGURE 1.

Architecture of the non‐blind $ℂ$DnCNN for MRI denoising. The input is the complex‐valued noisy image concatenated with the complex‐valued noise level map, and the output is the complex‐valued denoised image. The network consists of twelve 3 × 3 $ℂ$Conv. Each $ℂ$Conv is followed by a radial BN and a $ℂ$ReLU except for the first and last.

The first block of the network is composed of a $ℂ$Conv and a $ℂ$ReLU; the middle blocks are composed of a $ℂ$Conv, a radial BN, and a $ℂ$ReLU; and the last block is composed of a $ℂ$Conv to produce the output image. In order to balance the denoising performance and computational efficiency, all of the $ℂ$Conv kernels have a size of 3 × 3 and a number of channels of 64, and the network has 20 convolution blocks. More information can be found in the Supporting Information (Part A).

The network was implemented in the open‐source machine learning library PyTorch ³⁹ and trained with an L1 loss:

$$\mathcal{L}\left(\mathrm{\Theta }\right)=\frac{1}{N}{\sum }_{i=1}^N\left|ℂ\text{DnCNN}\left(m^{\prime }\left({\mathbf{x}}_i\right);\mathrm{\Theta }\right)-m\left({\mathbf{x}}_i\right)\right|$$ (6)

where $m^{\prime }\left({\mathbf{x}}_i\right)$ is the noise‐corrupted image, $m\left({\mathbf{x}}_i\right)$ is the ground truth image, $N$ is the total number of training pairs, and $\mathrm{\Theta }$ represents trainable parameters in the network. The optimization was carried out by an Adam optimizer ⁴⁰ with an initial learning rate of 0.0001 and momentum parameters ${\beta }_1$ = 0.9 and ${\beta }_2$ = 0.999. The training batch size was fixed to 32. Random flips and random cropping were employed as training augmentation to reduce the possibility of overfitting and improve the model robustness.

2.3. Noise level map

To cope with images at different noise levels, specifying the noise standard deviation $\sigma$ is required for most conventional model‐based denoising techniques, such as NLM and BM3D. In practice, the noise standard deviation $\widehat{\sigma }$ can be estimated from the k‐space or image data. We adopted a commonly used wavelet‐based approach ⁴¹ in this work. Specifically, we used the estimate_sigma function from the scikit‐image Python package. ⁴² In order to incorporate this information into our $ℂ$DnCNN for MRI denoising, we built a complex‐valued noise level map with its size matching the input MR image and concatenated it with the input image. All pixels in the noise level map were set to ${\widehat{\sigma }}_{\mathrm{avg}}+j·{\widehat{\sigma }}_{\mathrm{avg}}$, where ${\widehat{\sigma }}_{\mathrm{avg}}$ was the average of $\widehat{\sigma }$’s estimated from the real and imaginary parts of the input image:

$${\widehat{\sigma }}_{\mathrm{avg}}=\frac{\text{estimate}\_\text{sigma}\left(\text{Real}\left\{m^{\prime }\left(\mathbf{x}\right)\right\}\right)+\text{estimate}\_\text{sigma}\left(\text{Imag}\left\{m^{\prime }\left(\mathbf{x}\right)\right\}\right)}{2}$$ (7)

Because $ℂ$DnCNN is fully convolutional, it inherently provides the flexibility to deal with spatially non‐uniform noise. For parallel MRI, such as sensitivity encoding (SENSE) ⁴³ and generalized autocalibrating partially parallel acquisitions (GRAPPA), ⁴⁴ the noise is amplified by the geometry factor (g‐factor). The g‐factor is determined by the coil geometry and changes across the image. By weighting the uniform noise level map with a g‐factor map, the network is capable of handling the spatially varying noise.

2.4. Evaluation

We first created a simulated testing dataset from the fastMRI brain dataset to evaluate the performance of the proposed denoising method. To avoid overlap between the training and testing subsets, another 200 T2‐weighted imaging volumes were chosen at random. Simulated complex AWGN with zero mean and standard deviation between 0 and 0.1 was added to the testing data. We compared our method with other denoising algorithms including NLM, BM3D (https://webpages.tuni.fi/foi/GCF-BM3D/), real‐valued DnCNN, $ℂ$DnCNN without noise level map (blind). The NLM and BM3D algorithms operated on the magnitude images, whereas the $ℂ$DnCNN and non‐blind $ℂ$DnCNN operated on the complex‐valued images, and the real‐valued DnCNN treated the real and imaginary components as two separate channels. For quantitative assessment, the normalized root‐mean‐square error (NRMSE), peak signal‐to‐noise ratio (PSNR), and structural similarity index (SSIM) ⁴⁵ of the magnitude images were calculated. In the following definitions, $x$ denotes the output image with size $m\times n$, $y$ denotes the reference image with the same size, and $\left|·\right|$ means taking the magnitude.

$$\text{NRMSE}\left(\left|x\right|,\left|y\right|\right)=\sqrt{\frac{\mathrm{MSE}\left(\left|x\right|,\left|y\right|\right)}{\mathrm{MSE}\left(\left|y\right|,0\right)}}$$ (8)

$$\text{PSNR}\left(\left|x\right|,\left|y\right|\right)=20{\log }_{10}\left(\frac{\max \left(\left|x\right|\right)}{\sqrt{\mathrm{MSE}\left(\left|x\right|,\left|y\right|\right)}}\right)$$ (9)

$$\text{SSIM}\left(\left|x\right|,\left|y\right|\right)=\frac{\left(2{\mu }_{\left|x\right|}{\mu }_{\left|y\right|}+c_1\right)\left(2{\sigma }_{\left|x\right|\left|y\right|}+c_2\right)}{\left({\mu }_{\left|x\right|}^2+{\mu }_{\left|y\right|}^2+c_1\right)\left({\sigma }_{\left|x\right|}^2+{\sigma }_{\left|y\right|}^2+c_2\right)}$$ (10)

where $\mathrm{MSE}\left(\left|x\right|,\left|y\right|\right)=\frac{{\sum }_{i=0}^{m-1}{\sum }_{j=0}^{n-1}{\left(\left|x_{\mathit{ij}}\right|-\left|y_{\mathit{ij}}\right|\right)}^2}{\mathit{mn}}$ is the mean‐square error between $\left|x\right|$ and $\left|y\right|$, ${\mu }_{\left|x\right|}$ and ${\mu }_{\left|y\right|}$ are the means of $\left|x\right|$ and $\left|y\right|$, ${\sigma }_{\left|x\right|}^2$ and ${\sigma }_{\left|y\right|}^2$ are the variances of $\left|x\right|$ and $\left|y\right|$, ${\sigma }_{\left|x\right|\left|y\right|}^2$ is the cross‐correlation of $\left|x\right|$ and $\left|y\right|$, and $c_1=0.01$ and $c_2=0.03$ are regularization constants. Both $\left|x\right|$ and $\left|y\right|$ were normalized with respect to $\left|y\right|$ before metric calculation. For three CNN‐based methods generating complex‐valued output images, the absolute difference (ABSD) of the phase images was also calculated to test whether the phase information was altered:

$$\text{ABSD}\left(\angle x,\angle y\right)=\frac{{\sum }_{i=0}^{m-1}{\sum }_{j=0}^{n-1}\left|\angle x_{\mathit{ij}}-\angle y_{\mathit{ij}}\right|}{\mathit{mn}}$$ (11)

where $\angle ·$ means taking the phase. For a fair comparison between different CNN‐based methods, all CNNs had the same number of trainable parameters. Manual tuning was performed to determine the optimal combination of hyperparameters for each model.

The role of the noise level map in dealing with spatially non‐uniform parallel imaging noise was also explored. We first utilized the fully sampled multi‐coil k‐space data from the raw fastMRI dataset to generate the coil sensitivity map for g‐factor calculation. Complex AWGN was then added to each coil. We assume, for simplicity, that the standard deviation of Gaussian noise at each coil has the same value and there is no correlation across coils. ⁴⁶ The noise‐corrupted k‐space data were subsampled and reconstructed using SENSE. To address the spatially varying noise, the uniform noise level map was weighted by the g‐factor map. In practice, the coil sensitivity map and g‐factor map can be acquired during the preparation phase with no additional scan time cost.

To test the generalizability of non‐blind $ℂ$DnCNN, its performance on different contrasts and anatomies was also assessed. T1‐weighted and FLAIR brain images and proton density (PD) weighted knee images from the fastMRI dataset were randomly selected to form a testing dataset out of the training distribution. Additionally, the network was applied to the low‐field M4Raw dataset ⁴⁷ and local in vivo data collected on a prototype 0.55 T MR scanner with high‐performance gradients (ramped‐down MAGNETOM Aera, Siemens Healthcare, Erlangen, Germany). The M4Raw dataset contains multi‐contrast and multi‐coil MRI data collected using a 0.3 T MR system. A detailed description can be found on the project website (https://github.com/mylyu/M4Raw). We randomly selected 5 T2‐weighted volumes from the dataset for testing the network performance. We also applied retrospective undersampling and SENSE reconstruction on the multi‐coil data to test the network performance on spatially varying noise. The local in vivo data were acquired with a SPRING‐RIO TSE sequence ⁴⁸ with TR/TE_eff, 3000/105 ms; echo train length, 9; FOV, 230 × 230 mm²; voxel size, 0.70 × 0.70 × 4 mm³. Written informed consent from all subjects was given. To evaluate the image quality without reference, the SNR for white matter (WM) and gray matter (GM) were computed based on manually defined regions of interest (ROIs):

$$\mathrm{SNR}=0.66\times \frac{{\mu }_{\mathrm{ROI}}}{{\sigma }_{\mathrm{air}}}$$ (12)

where ${\mu }_{\mathrm{ROI}}$ is the mean intensity of the ROI, ${\sigma }_{\mathrm{air}}$ is the standard deviation of the air region, and 0.66 is the correction factor for the Rayleigh distribution of the noise in the magnitude image. ⁴⁹ More information can be found in the Supporting Information (Part D).

3. RESULTS

Figure 2A shows the performance of different denoising algorithms on the simulated noise‐corrupted dataset at different noise levels. When the noise standard deviation $\sigma$ was larger than 0.04, three CNN‐based algorithms outperformed NLM and BM3D. Compared with real‐valued DnCNN with two‐channel input, the output of $ℂ$DnCNN and non‐blind $ℂ$DnCNN showed superior NRMSE, PSNR, and SSIM over the entire range of 0 to 0.1. Representative images are displayed in Figure 2B. Compared with other methods, the output of $ℂ$DnCNN and non‐blind $ℂ$DnCNN showed reduced noise and less visual blurring. Figure 3A shows the phase difference for CNN‐based methods. To eliminate the impact of random phase in the background, a mask covering the brain region was generated from the reference magnitude image using Otsu's thresholding ⁵⁰ and convex hull operation, as shown in Figure 3B. Representative phase images are displayed in Figure 3C. The phase of the output images did not deviate significantly from the reference phase. $ℂ$DnCNN and non‐blind $ℂ$DnCNN showed comparable performance and outperformed the real‐valued DnCNN in terms of preserving the phase information.

FIGURE 2.

(A) Performance of different denoising methods on the simulated testing dataset at different noise levels. The average NRMSE, PSNR, and SSIM were evaluated for each method. (B) Representative magnitude image results of different denoising methods. The top left is the ground truth image, and the bottom left is the simulated noise‐corrupted image with a noise standard deviation of 0.06. The output of $ℂ$DnCNN and non‐blind $ℂ$DnCNN showed reduced noise and less visual blurring compared to other methods.

FIGURE 3.

(A) Phase difference of DnCNN, $ℂ$DnCNN, and non‐blind $ℂ$DnCNN on the simulated testing dataset at different noise levels. The average phase ABSD was evaluated for each network. (B) Procedures for generating the brain mask to eliminate the impact of background random phase in metric calculation. (C) Representative phase image results of different methods. The top left is the ground truth image, and the top center is the simulated noise‐corrupted image with a noise standard deviation of 0.06. The bottom row is the output of DnCNN, $ℂ$DnCNN, and non‐blind $ℂ$DnCNN, from left to right. The complex‐valued models preserved the phase information better than the real‐valued model.

Figure 4 gives an example showing the network performance on spatially varying parallel imaging artifacts. The raw k‐space data from 16 coils was retrospectively undersampled by a factor of 4. The noise in the SENSE reconstructed image was amplified by the g‐factor due to the coil geometry. With scaled g‐factor map as the noise level map, non‐blind $ℂ$DnCNN successfully reduced the noise and showed the greatest image quality, whereas other methods failed at the center regions with large g‐factor.

FIGURE 4.

Example of non‐blind $ℂ$DnCNN on spatially varying parallel imaging artifacts. The top left is the ground truth image. The top center is the simulated noise‐corrupted image reconstructed by SENSE with a subsampling ratio of 4. The top right is the g‐factor map from the SENSE reconstruction. The bottom left is the output of $ℂ$DnCNN without noise level map. The bottom center is the output of non‐blind $ℂ$DnCNN with a uniform noise level map. The bottom right is the output of non‐blind $ℂ$DnCNN with the scaled g‐factor map as the noise level map. The non‐blind $ℂ$DnCNN with the scaled g‐factor map successfully reduced the noise at the center regions with large g‐factor.

Figure 5 shows the generalizability of non‐blind $ℂ$DnCNN on data out of the training distribution. The testing images had contrasts and anatomies that were different from the training dataset. The output images were less noisy and showed superior metrics, demonstrating that the model was able to generalize under these circumstances and did not overfit the training data. The denoising performance for images with pathology can be found in the Supporting Information (Part C). Figure 6 shows the performance of different denoising methods on the low‐field dataset M4Raw. The blind $ℂ$DnCNN and non‐blind $ℂ$DnCNN showed better SNRs for WM and GM compared with other methods. Figure 7 gives an example from the M4Raw dataset showing the network performance on spatially varying parallel imaging artifacts. The 4‐coil data were undersampled by a factor of 2 and reconstructed by SENSE. The noise in the SENSE reconstructed image was amplified compared to the fully sampled reconstruction. The non‐blind $ℂ$DnCNN with the scaled g‐factor map showed the best denoising performance at regions with large g‐factor, demonstrating its generalizability in different coil geometry. Figure 8A shows the performance of different denoising methods on local in vivo data collected from a 0.55 T low‐field scanner. It can be observed that all methods reduced the noise and increased the SNRs for WM and GM. The blind $ℂ$DnCNN and non‐blind $ℂ$DnCNN showed superior performance compared with other methods. Representative images are displayed in Figure 8B. The output of $ℂ$DnCNN and non‐blind $ℂ$DnCNN showed sharper structures and less noise. Figure 9 shows the output of non‐blind $ℂ$DnCNN with different NAs. The network showed its generalizability as it was able to enhance the overall image quality at different noise levels. When NA increased, small brain structures became more observable in the output image.

FIGURE 5.

Performance of non‐blind $ℂ$DnCNN on modalities or anatomies out of the training distribution. The top row is a T1‐weighted brain image, the middle row is a FLAIR brain image, and the bottom row is a PD‐weighted knee image. From left to right, each column is the ground truth image, the simulated noise‐corrupted image, and the output of non‐blind $ℂ$DnCNN, respectively.

FIGURE 6.

(A) Performance of different denoising methods on the low‐field M4Raw dataset. The average WM and GM SNRs of each testing volume were evaluated for each method. (B) Representative image results of different denoising methods. The output of $ℂ$DnCNN and non‐blind $ℂ$DnCNN showed clear structures and better noise reduction compared to other methods. The asterisks indicate statistically significant differences between the methods (p < 0.05).

FIGURE 7.

Example from the M4Raw dataset of non‐blind $ℂ$DnCNN on spatially varying parallel imaging artifacts. The top left is the fully‐sampled image. The top center is the undersampled image reconstructed by SENSE with a subsampling ratio of 2. The top right is the g‐factor map from the SENSE reconstruction. The bottom left is the output of $ℂ$DnCNN without noise level map. The bottom center is the output of non‐blind $ℂ$DnCNN with a uniform noise level map. The bottom right is the output of non‐blind $ℂ$DnCNN with the scaled g‐factor map as the noise level map. The non‐blind $ℂ$DnCNN with scaled g‐factor map showed the best denoising performance at regions with large g‐factor.

FIGURE 8.

(A) Performance of different denoising methods on in vivo images collected from a low‐field scanner with different NAs. The average WM and GM SNRs were evaluated for each method. (B) Representative image results of different denoising methods. The top left is the original noisy image acquired with NA of 3. The output of $ℂ$DnCNN and non‐blind $ℂ$DnCNN showed sharper structures and less noise compared to other methods.

FIGURE 9.

Performance of non‐blind $ℂ$DnCNN at different noise levels. The top row is the original noisy image, and each column was acquired with NA of 1, 2, 3, 4, and 5, from left to right. The bottom row is the output of $ℂ$DnCNN. The small structures (yellow and cyan arrows) became sharp and visible as NA increased.

The total training time for non‐blind $ℂ$DnCNN on a system with an NVIDIA Titan Xp GPU, an Intel Xeon 3.3 GHz CPU, and 128 GB RAM was roughly 42 h. We measured the computational cost of different algorithms on the same system. Table 1 summarizes the inference time and memory required for denoising one slice with a size of 320 × 320. Note that the time for estimating the noise standard deviation was also counted. The non‐blind $ℂ$DnCNN was able to process one slice in less than 1.5 s on the CPU. The inference time could be significantly shortened with GPU acceleration.

TABLE 1.

Computational cost of different algorithms for denoising one slice with size 320 × 320.

	Inference time/slice (GPU)	Inference time/slice (CPU)	Memory usage
NLM	‐	456 ms	85 MB
BM3D	‐	2562 ms	188 MB
DnCNN	27 ms	303 ms	288 MB
$ℂ$DnCNN	79 ms	1297 ms	302 MB
non‐blind $ℂ$DnCNN	154 ms	1472 ms	305 MB

Note: NLM and BM3D were tested only on CPU.

4. DISCUSSION

In this study, we presented non‐blind $ℂ$DnCNN, a network for MRI denoising that leverages complex‐valued building blocks and noise level information to improve denoising performance in various settings. The proposed method achieved superior performance on both simulated and in vivo testing data compared with other algorithms.

The utilization of complex‐valued operations allows the network to better exploit the complex‐valued MRI data and preserve the phase information. For NLM and BM3D, the denoising is directly performed on the magnitude image. The phase information is lost and cannot be recovered after denoising. Thus, the phase performance for these approaches was not examined. For real‐valued CNNs, the input image is split into real and imaginary channels, and real‐valued operations are then applied on these channels. The output image is obtained by combining the two separate output channels, and the reconstructed phase may be changed. For complex‐valued CNNs, the input/output, learned convolutional kernels, and latent features are all in complex‐valued representations, enabling the network to make use of the valuable information contained in the phase map. The superior metrics achieved by complex‐valued CNNs demonstrate that integrating the phase information is beneficial for the denoising process.

The DnCNN network architecture is used in this work, because the original DnCNN model attained remarkable performance on Gaussian denoising. Recently, the U‐Net architecture ⁵¹ is of growing interest in solving problems like medical image segmentation and reconstruction. ³² , ³³ , ⁵² , ⁵³ , ⁵⁴ The downscaling/upscaling blocks in the U‐Net‐based model effectively increase the network receptive field and allow the network to utilize both global and regional features. We also applied our strategies on U‐Net‐based networks and evaluated their denoising performance. More information can be found in the Supporting Information (Part B).

One major concern about CNN‐based denoising techniques is the blurring effect introduced by the network. To mitigate this drawback, we chose L1 loss instead of L2 loss, because previous studies have shown that using L2 loss is prone to produce over‐smoothed restored images. ⁵⁵ , ⁵⁶ , ⁵⁷ Incorporating the noise level into the network provides the possibility to control the balance between noise suppression and detail retention. However, in our experience, it is still challenging to balance the trade‐off by simply adjusting the estimated noise standard deviation. In the future work, more advanced methods, such as using attention mechanisms ⁵⁸ to better integrate the noise level map, could be explored to achieve this goal. The spatially non‐uniform noise level map allows the network to remove the spatially dependent parallel imaging noise. This advantage improves its clinical feasibility because parallel imaging is frequently used to accelerate scans. In our experiment, we used a wavelet‐based method to estimate the noise level for real and imaginary parts separately and adopted naïve averaging to get the final estimation of $\widehat{\sigma }$. To address parallel imaging noise, we weighted the uniform noise level map with the g‐factor map. There exist numerous MRI noise estimation schemes relying on wavelet domain analysis, local mutual information, or median absolute deviation estimator. ⁵⁹ , ⁶⁰ , ⁶¹ We expect that applying these techniques will give a more accurate noise level map and further improve the network performance. Training the non‐blind $ℂ$DnCNN with GAN ⁶² , ⁶³ is an alternative way to reduce blurring in the network output. However, a GAN scheme for complex‐valued networks remains to be investigated.

Overfitting is another issue for supervised learning models. Because of the limited size of the training set, the model might fit too closely to the training data. In such situation, the network begins to memorize irrelevant information, for example, the brain anatomy or image contrast, instead of finding a general strategy for denoising. This is a critical problem for medical image processing because the spurious structures or subtle artifacts created by the network can severely affect the diagnosis. To alleviate overfitting, image augmentations were employed to increase the diversity of the training set. The validation loss was also monitored after each epoch during the training stage. Additional tests on data out of the training distribution showed that the method is generalizable. The network also showed promising results on in vivo data acquired on a low‐field scanner, demonstrating its ability to boost the SNR of low‐field MR images and its potential to reduce the acquisition time of low‐field MRI. However, more in vivo experiments on the low‐field scanner are needed to validate the network robustness on images of different body regions with different scan parameters.

One potential limitation in our approach is that the current methodology focuses on employing deep learning solely for denoising after reconstructing the MR images. Several studies have tried to mitigate noise during the reconstruction to improve the reconstructed image quality. ⁶⁴ , ⁶⁵ These methods naturally embed the phase modulation and may better exploit the raw complex‐valued imaging data. The integration of denoising within the reconstruction process could be investigated to further enhance the overall imaging pipeline for future work.

4.1. Conclusion

We have shown that the proposed $ℂ$DnCNN has superior denoising performance compared with the real‐valued DnCNN and several other algorithms. The addition of noise level map provides it the ability to remove spatially varying parallel imaging noise. It offers rapid and significant SNR improvements, which is useful for low‐field MRI.

DISCLOSURE

Dr. Campbell‐Washburn is an investigator on a cooperative research and development agreement (CRADA) with Siemens Healthcare that includes the development of 0.55 T MRI.

Supporting information

Data S1 Supporting Information.

Figure S1 Performance of DnCNN, $ℂ$DnCNN, and non‐blind $ℂ$DnCNN as a function of network depth on the simulated testing dataset at different noise levels. The average NRMSE, PSNR, and SSIM were evaluated for each method. The performance of all networks improved with an increase in network depth.

Figure S2 Performance of U‐Net, $ℂ$U‐Net, and non‐blind $ℂ$U‐Net on the simulated testing dataset at different noise levels. The average NRMSE, PSNR, and SSIM were evaluated for each method.

Figure S3 Performance of non‐blind $ℂ$DnCNN on images with pathology. The top row is a FLAIR brain image with a white matter lesion (yellow arrow), and the bottom row is a knee image with bone lesion (yellow arrow). From left to right, each column is the ground truth image, the simulated noise‐corrupted image, and the output of non‐blind $ℂ$DnCNN, respectively.

Figure S4 Example of the manually defined ROIs for SNR calculation (white rectangles for air regions, cyan rectangles for GM regions, and yellow rectangles for WM regions) and the mean intensity and standard deviation of each ROI.

Dou Q, Wang Z, Feng X, Campbell‐Washburn AE, Mugler JP III, Meyer CH. MRI denoising with a non‐blind deep complex‐valued convolutional neural network. NMR in Biomedicine. 2025;38(1):e5291. doi: 10.1002/nbm.5291

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.