Synopsis

Keywords: Image Reconstruction, Diffusion/other diffusion imaging techniques

The diagnostic value of diffusion-weighted MR images is often degraded by their inherently low signal-to-noise ratio (SNR), especially for high b-values. In this context, the application of learning-based denoising methods is difficult since most methods require noise-free target images for training. We show how to denoise and evaluate diffusion-weighted MR images in a self-supervised manner by exploiting an adapted version of Stein’s unbiased risk estimator and specific properties of the data. Both quantitative and qualitative evaluations indicate increased performance over state-of-the-art unsupervised denoising methods.

Introduction

Diffusion-weighted imaging (DWI) is an important tool for detecting and characterizing lesions. However, one major drawback of DWI is that the images, especially those acquired with higher b-values, suffer from inherently low signal-to-noise ratio (SNR) due to the diffusion weighting and long echo times necessary to accommodate the additional gradient pulses¹. As a result, in some body regions such as the prostate, individual repetitions are hardly usable for diagnosis. The state-of-the-art practice is to acquire repeated acquisitions and compute an average over these repetitions to improve SNR. However, acquiring many repetitions is time-consuming and the averaged result is prone to motion artifacts. Therefore, the number of repetitions and thus the improvement of the SNR is limited. In this work, we show how to leverage the redundancy in repeated acquisitions for self-supervised denoising using deep learning and an extended version of Stein's unbiased risk estimator (SURE)².

Methods

SURE was first proposed by Charles Stein in 1981 and provides a statistical method to estimate the mean squared error (MSE) between the unknown mean $$$\boldsymbol{x}$$$ of a Gaussian distributed signal $$$\boldsymbol{y}$$$ and its estimate $$$\hat{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{y})$$$. This can be adapted to an image denoising problem as shown by Metzler et al.³. Here, the goal is to reconstruct an unknown noise-free image $$$\boldsymbol{x}$$$ corrupted by Gaussian noise $$$\boldsymbol{\eta}$$$ from a noisy image $$$\boldsymbol{y}=\boldsymbol{x}+\boldsymbol{\eta}$$$. Since the noise is additive and has zero mean, the unknown noise-free image $$$\boldsymbol{x}$$$ can be considered as the mean vector of the noisy image $$$\boldsymbol{y}$$$.
The original SURE expression assumed the presence of spatially invariant Gaussian noise. To properly address the spatially variant noise enhancement in reconstructed MR images, we use an adapted SURE approach that incorporates a noise map $$$\boldsymbol{\sigma}$$$ indicating the standard deviation of the noise for every pixel as proposed in⁴.
Consequently, SURE can be used as a loss function to train a neural network $$$\boldsymbol{f}(\boldsymbol{y})$$$ that receives noisy measurements $$$\boldsymbol{y}$$$ as input and predicts an estimate of $$$\boldsymbol{x}$$$ as output by minimizing SURE, i.e., the estimated MSE as follows:
$$\frac{1}{D}\parallel\boldsymbol{f}(\boldsymbol{y})-\boldsymbol{x}\parallel^{2}=\frac{1}{D}\parallel \boldsymbol{f}(\boldsymbol{y})-\boldsymbol{y}\parallel^{2}-\frac{1}{D}\sum\nolimits_{d=1}^D\sigma_d^2+\frac{2}{D}\sum\nolimits_{d=1}^D\sigma_d^2\frac{\partial f_d (\boldsymbol{y})}{\partial y_d} ,$$
where D is the number of pixels.

Considering that SURE cannot be applied to magnitude averages due to their non-Gaussian noise distribution, we performed a phase correction as described on the individual repetitions⁵. This allows us to compute averages over the complex-valued images without signal loss while also preserving the zero-centered Gaussian noise distribution. Representative results are illustrated in Figure 1.
Since multiple repetitions are acquired for each slice and diffusion direction, the required spatially resolved noise map $$$\boldsymbol{\sigma}$$$ incorporated in the SURE loss can simply be generated by calculating the standard deviation between the repetitions for each pixel, as shown in Figure 2.

A total of 319 prostate image volumes with a b-value of 800 and four diffusion directions with ten repetitions each were acquired from volunteers using various 1.5 and 3 T scanners (MAGNETOM, Siemens Healthcare, Erlangen, Germany) and standard acquisition protocols.
The DnCNN-based architecture⁶ in Figure 3 was implemented in PyTorch and trained with images from 4,073 slices using SURE as an unsupervised loss function. Images of the four different diffusion directions were denoised separately before calculating the trace image via geometric mean. We compare our method with two state-of-the-art unsupervised denoising methods: the deep-learning-based approach Noise2Noise⁷, as well as the non-learning-based approach BM3D⁸.
To be able to quantitatively compare the results of the different methods, we calculated estimates of the MSE and peak signal-to-noise-ratio (PSNR) between the denoised images and the original images for 1,349 test slices without access to noise-free ground-truth images using SURE as a substitute for the MSE.

Results and Discussion

The SURE-based quantitative evaluation results are presented in Table 1. Representative result images of the different methods are shown in Figure 4. Based on the quantitative results, our method performs considerably better than the state-of-the-art unsupervised denoising methods Noise2Noise and BM3D. It is important to note that due to the lack of ground truth, the presented results are SURE-based estimates of the actual metrics. However, in cases where the noise model can be determined precisely this estimate is very accurate. The quantitative findings are further supported by the representative result images. While Noise2Noise and BM3D tend to produce over-smoothed images, the results obtained with SURE appear less blurred and retain structures that may be anticipated from the noisy input.

Conclusion

We demonstrated how to perform learning-based fully self-supervised denoising in prostate DWI, outperforming established denoising methods like BM3D. However, the proposed approach is not limited to prostate DWI but can also be applied to other body regions. Our method can be used to either improve the image quality by increasing the SNR with the same number of repetitions or to reach equivalent image quality with fewer repetitions, resulting in reduced scan time.

Acknowledgements

No acknowledgement found.