Wavelet-domain sparse Bayesian learning for uncertainty-aware MRI reconstruction

doi:10.11772/j.issn.1001-9081.2025101275

Abstract

Abstract:

Full-sampling Magnetic Resonance Imaging （MRI） requires a long scan time， which not only limits the examination efficiency， but also tends to induce motion artifacts due to subject movement. To address the issues of high parameter sensitivity and inability to quantify result uncertainty in traditional Compressed Sensing MRI （CS-MRI） reconstruction methods， an uncertainty-aware MRI reconstruction method based on wavelet-domain Sparse Bayesian Learning （SBL） was proposed， namely BU-MRI （Bayesian Uncertainty-guided MRI）. Firstly， the advantage of the wavelet transform in multi-resolution image representation was leveraged， and a hierarchical Bayesian probability model was constructed by characterizing the sparsity of MRI images in the wavelet domain as a prior. Secondly， a posterior inference strategy combining Gibbs sampling and marginal likelihood maximization was adopted to achieve effective estimation of high-dimensional sparse coefficients and adaptive hyperparameters updating. Finally， based on the updated model parameters， high-quality images were iteratively reconstructed from undersampled K-space data. Furthermore， pixel-wise posterior confidence intervals could be provided， offering a quantitative assessment of uncertainty in the reconstruction results. Experimental results on both simulated and real MRI data demonstrated that BU-MRI outperformed methods such as Zero-Filled Inverse Discrete Fourier Transform （ZF-IDFT） and k-t Robust Principal Component Analysis （k-t RPCA） in terms of Peak Signal-to-Noise Ratio （PSNR） and Structural Similarity Index Measure （SSIM）. On real cardiac MRI data and brain MRI data with a sampling rate of 0.5， the PSNR of BU-MRI achieved 44.42 dB and 40.37 dB， and SSIM reached 0.976 5 and 0.954 7， respectively. BU-MRI exhibits excellent performance in structural fidelity， error suppression， and frequency-domain consistency. It shows stable convergence and robustness across various sampling rates and noise levels， providing a reliable reconstruction framework with uncertainty quantification capability for clinical MRI.

Key words: Magnetic Resonance Imaging (MRI), Sparse Bayesian Learning (SBL), image reconstruction, wavelet transform, uncertainty quantification, Bayesian inference

摘要：

磁共振成像（MRI）全采样扫描时间长，既制约检测效率，还易因受检者移动产生运动伪影。针对传统压缩感知MRI（CS-MRI）重建方法参数敏感性高、无法量化结果不确定性等问题，提出一种基于小波域稀疏贝叶斯学习（SBL）的非确定性MRI重建方法BU-MRI（Bayesian Uncertainty-guided MRI）。首先利用小波变换对图像进行多分辨率表征的优势，通过刻画MRI图像在小波域中的稀疏性作为先验，构建一个分层贝叶斯概率模型。其次，采用吉布斯（Gibbs）采样与边际似然最大化相结合的后验推断策略，实现对高维稀疏系数的有效估计与超参数的自适应更新。最后，基于更新后的模型参数，从欠采样的K空间数据中迭代恢复出高质量图像。此外，该方法还能够输出像素级的后验置信区间，为重建结果提供定量化的不确定性评估。仿真与真实MRI数据上的实验结果表明，BU-MRI方法的峰值信噪比（PSNR）与结构相似性指数（SSIM）优于零填充逆离散傅里叶变换（ZF-IDFT）和k-t鲁棒主成分分析（k-t RPCA）等方法；而且在真实心脏MRI数据和大脑MRI数据上，当采样率为0.5时，BU-MRI重建数据的PSNR分别达到44.42 dB和40.37 dB，SSIM分别达到0.976 5和0.954 7。BU-MRI方法在结构保真、误差抑制与频域一致性上表现优异，在不同采样率与噪声水平下收敛稳定且鲁棒性良好，能为临床MRI提供可靠且具备不确定性量化能力的重建框架。

关键词: 磁共振成像, 稀疏贝叶斯学习, 图像重建, 小波变换, 不确定性量化, 贝叶斯推断

CLC Number:

TP391.41

Kaiyan CUI, Shuna WEI. Wavelet-domain sparse Bayesian learning for uncertainty-aware MRI reconstruction[J]. Journal of Computer Applications, 2026, 46(5): 1634-1646.

崔凯燕, 魏舒娜. 基于小波域稀疏贝叶斯学习的非确定性MRI重建[J]. 《计算机应用》唯一官方网站, 2026, 46(5): 1634-1646.

Figures/Tables 16

Tab. 1 Description of main symbols

符号	含义
$x$	真实图像数据向量化后的列向量
$y$	原始的观测数据向量
$m a s k$	K空间欠采样掩码矩阵
$A, B$	线性变换矩阵
$n$	噪声矩阵
$F$	傅里叶变换矩阵
$W$	小波变换矩阵
$Y$	完整K空间数据
$m$	掩码向量化后的列向量
$X$	真实图像数据矩阵
$ε$	观测到的噪声向量
$Φ$	传感矩阵
$α$	先验分布的超参数向量
$Δ$	超参数向量构成的对角精度矩阵
$Λ$	分块的先验精度矩阵
$δ$	中间计算变量，正比于传感矩阵的共轭转置与观测向量的乘积
$μ ̂ x$	后验均值向量
$Σ ̂ x$	后验协方差矩阵对角线元素
$I$	单位阵

Tab. 1 Description of main symbols

符号	含义
$x$	真实图像数据向量化后的列向量
$y$	原始的观测数据向量
$m a s k$	K空间欠采样掩码矩阵
$A, B$	线性变换矩阵
$n$	噪声矩阵
$F$	傅里叶变换矩阵
$W$	小波变换矩阵
$Y$	完整K空间数据
$m$	掩码向量化后的列向量
$X$	真实图像数据矩阵
$ε$	观测到的噪声向量
$Φ$	传感矩阵
$α$	先验分布的超参数向量
$Δ$	超参数向量构成的对角精度矩阵
$Λ$	分块的先验精度矩阵
$δ$	中间计算变量，正比于传感矩阵的共轭转置与观测向量的乘积
$μ ̂ x$	后验均值向量
$Σ ̂ x$	后验协方差矩阵对角线元素
$I$	单位阵

Fig. 1 Radial sampling diagram

Tab. 2 Comparison of PSNR and SSIM for simulated image dataset at different noise levels and sampling rates

高斯噪声标准差	采样率p=0.20		采样率p=0.25		采样率p=0.30		采样率p=0.40		采样率p=0.50
高斯噪声标准差	PSNR/dB	SSIM	PSNR/dB	SSIM	PSNR/dB	SSIM	PSNR/dB	SSIM	PSNR/dB	SSIM
$10 - 2$	28.75	0.465 0	29.99	0.495 8	31.01	0.523 1	32.15	0.541 7	33.20	0.558 2
$10 - 3$	28.87	0.475 1	30.23	0.511 3	31.20	0.537 9	32.54	0.574 3	34.02	0.610 1
$10 - 4$	28.58	0.463 9	30.17	0.510 5	31.02	0.526 9	32.57	0.575 5	33.95	0.607 2

Tab. 2 Comparison of PSNR and SSIM for simulated image dataset at different noise levels and sampling rates

高斯噪声标准差	采样率p=0.20		采样率p=0.25		采样率p=0.30		采样率p=0.40		采样率p=0.50
高斯噪声标准差	PSNR/dB	SSIM	PSNR/dB	SSIM	PSNR/dB	SSIM	PSNR/dB	SSIM	PSNR/dB	SSIM
$10 - 2$	28.75	0.465 0	29.99	0.495 8	31.01	0.523 1	32.15	0.541 7	33.20	0.558 2
$10 - 3$	28.87	0.475 1	30.23	0.511 3	31.20	0.537 9	32.54	0.574 3	34.02	0.610 1
$10 - 4$	28.58	0.463 9	30.17	0.510 5	31.02	0.526 9	32.57	0.575 5	33.95	0.607 2

Fig. 2 PSNRs and SSIMs of BU-MRI on simulated image dataset at different noise levels and sampling rates

Fig. 3 Uncertainty quantification of BU-MRI reconstruction using pixel-wise posterior confidence intervals on simulated image datasets

Fig. 4 Iteration convergence curves of PSNR and SSIM for BU-MRI on simulated image dataset at different sampling rates and noise levels

Fig. 5 Reconstruction comparison of simulated images by different methods on simulated image dataset

Fig. 6 Comparison of residual heatmaps obtained by different methods under multiple noise levels on simulated image dataset

Fig. 7 PSNR and SSIM vs. sampling rate for BU-MRI method on real cardiac MRI images

Tab. 3 Performance comparison of different methods on real cardiac MRI images

方法	p=0.20		p=0.25		p=0.30		p=0.40		p=0.50
方法	PSNR/dB	SSIM	PSNR/dB	SSIM	PSNR/dB	SSIM	PSNR/dB	SSIM	PSNR/dB	SSIM
ZF-IDFT	30.30	0.767 4	32.16	0.821 5	34.43	0.871 9	37.74	0.913 3	39.92	0.951 8
k-t RPCA	33.93	0.916 5	35.32	0.938 6	37.67	0.947 8	38.47	0.952 5	38.72	0.953 2
TNN	34.42	0.934 3	36.68	0.950 2	37.16	0.954 9	40.18	0.967 2	41.64	0.972 7
MNN	27.60	0.846 5	30.40	0.900 0	34.64	0.953 8	35.92	0.962 5	35.85	0.957 0
TMNN	35.46	0.953 6	37.02	0.961 1	36.46	0.961 1	39.09	0.970 5	40.47	0.975 2
BU-MRI	33.31	0.770 6	35.79	0.858 1	37.93	0.908 6	41.41	0.956 3	44.42	0.976 5

Fig. 8 Reconstruction comparison of real cardiac MRI images by different methods

Fig. 9 Comparison of residual heatmaps by different methods on real cardiac MRI images with p=0.50

Tab. 4 Comparison of brain MRI reconstruction performance between DIFF-MRI and BU-MRI at different undersampling rates

p	DIFF-MRI		BU-MRI
p	PSNR/dB	SSIM	PSNR/dB	SSIM
0.20	32.18	0.904 3	31.16	0.744 6
0.25	33.14	0.916 8	33.03	0.819 5
0.30	34.06	0.926 5	34.47	0.861 9
0.40	35.67	0.940 2	37.30	0.919 3
0.50	36.73	0.948 2	40.37	0.954 7

Fig. 10 Comparison of brain MRI reconstruction between DIFF-MRI and BU-MRI at different undersampling rates

Fig. 11 Comparison of residual maps for brain MRI reconstruction between DIFF-MRI and BU-MRI at different undersampling rates

Tab. 5 Performance comparison of different parameters

$a$	$b$	$c$	$d$	PSNR/dB	SSIM
0	0	$m n / 2 - K$	0	48.26	0.970 4
0	0	$m n / 2 - K$	$10 - 3$	48.14	0.970 1
0	0	$m n - K$	0	45.17	0.940 6
0	0	$m n - K$	$10 - 3$	45.85	0.948 2
0	$10 - 3$	$m n / 2 - K$	0	28.55	0.329 3
0	$10 - 3$	$m n / 2 - K$	$10 - 3$	29.04	0.343 7
0	$10 - 3$	$m n - K$	0	26.66	0.268 0
0	$10 - 3$	$m n - K$	$10 - 3$	26.36	0.259 9
$10 - 3$	0	$m n / 2 - K$	0	47.99	0.968 1
$10 - 3$	0	$m n / 2 - K$	$10 - 3$	48.21	0.970 2
$10 - 3$	0	$m n - K$	0	44.83	0.937 8
$10 - 3$	0	$m n - K$	$10 - 3$	45.59	0.942 6
$10 - 3$	$10 - 3$	$m n / 2 - K$	0	28.85	0.337 6
$10 - 3$	$10 - 3$	$m n / 2 - K$	$10 - 3$	28.72	0.335 3
$10 - 3$	$10 - 3$	$m n - K$	0	26.59	0.269 7
$10 - 3$	$10 - 3$	$m n - K$	$10 - 3$	26.28	0.261 9

Tab. 5 Performance comparison of different parameters

$a$	$b$	$c$	$d$	PSNR/dB	SSIM
0	0	$m n / 2 - K$	0	48.26	0.970 4
0	0	$m n / 2 - K$	$10 - 3$	48.14	0.970 1
0	0	$m n - K$	0	45.17	0.940 6
0	0	$m n - K$	$10 - 3$	45.85	0.948 2
0	$10 - 3$	$m n / 2 - K$	0	28.55	0.329 3
0	$10 - 3$	$m n / 2 - K$	$10 - 3$	29.04	0.343 7
0	$10 - 3$	$m n - K$	0	26.66	0.268 0
0	$10 - 3$	$m n - K$	$10 - 3$	26.36	0.259 9
$10 - 3$	0	$m n / 2 - K$	0	47.99	0.968 1
$10 - 3$	0	$m n / 2 - K$	$10 - 3$	48.21	0.970 2
$10 - 3$	0	$m n - K$	0	44.83	0.937 8
$10 - 3$	0	$m n - K$	$10 - 3$	45.59	0.942 6
$10 - 3$	$10 - 3$	$m n / 2 - K$	0	28.85	0.337 6
$10 - 3$	$10 - 3$	$m n / 2 - K$	$10 - 3$	28.72	0.335 3
$10 - 3$	$10 - 3$	$m n - K$	0	26.59	0.269 7
$10 - 3$	$10 - 3$	$m n - K$	$10 - 3$	26.28	0.261 9

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