We propose a novel framework for uncertainty quantification using compressed sensing magnetic resonance image reconstruction. The problem is formulated within a Bayesian framework as a linear inverse problem, with prior distributions assigned to the unknown model parameters. Specifically, the image to be reconstructed is assumed to be sparse in a given basis. We develop a general framework applicable to any basis and as examples, we test the sparsity of the image in its (1) spatial gradients using a total variation prior model, and in its (2) wavelet transform. A Markov chain Monte Carlo (MCMC) method, based on a split-and-augmented Gibbs sampler, is then employed to sample from the posterior distribution of the unknown parameters. The non-differentiable conditional distributions are efficiently sampled using a proximal MCMC method. The proposed algorithms are validated on both single-coil and multi-coil datasets using various k-space sub-sampling patterns and ratios. The results demonstrate the superior performance of each proposed approach in reconstructing images compared to its counterpart optimisation-based method. Moreover, our framework effectively quantifies uncertainty, showing a notable correlation between estimated uncertainty maps and error maps computed using ground truth and reconstructed images, compared with existing deep learning-based methods.

翻译：我们提出了一种新颖的框架，用于在压缩感知磁共振图像重建中实现不确定性量化。该问题在贝叶斯框架内被表述为线性逆问题，并为未知模型参数分配了先验分布。具体而言，假设待重建图像在给定基下具有稀疏性。我们开发了一个适用于任意基的通用框架，并以（1）利用总变分先验模型的空间梯度稀疏性以及（2）小波变换稀疏性为例进行测试。随后，采用基于分裂-增广吉布斯采样器的马尔可夫链蒙特卡洛（MCMC）方法，从未知参数的后验分布中进行采样。利用近端MCMC方法高效地采样不可微的条件分布。所提出的算法在单线圈和多线圈数据集上，通过不同的k空间子采样模式和采样率进行了验证。结果表明，与基于优化的对应方法相比，每种提出的方法在图像重建方面均展现出优越性能。此外，我们的框架能够有效量化不确定性，与基于深度学习的方法相比，估计的不确定性图与利用真实图像和重建图像计算的误差图之间表现出显著的相关性。