Recent deep learning approaches focus on improving quantitative scores of dedicated benchmarks, and therefore only reduce the observation-related (aleatoric) uncertainty. However, the model-immanent (epistemic) uncertainty is less frequently systematically analyzed. In this work, we introduce a Bayesian variational framework to quantify the epistemic uncertainty. To this end, we solve the linear inverse problem of undersampled MRI reconstruction in a variational setting. The associated energy functional is composed of a data fidelity term and the total deep variation (TDV) as a learned parametric regularizer. To estimate the epistemic uncertainty we draw the parameters of the TDV regularizer from a multivariate Gaussian distribution, whose mean and covariance matrix are learned in a stochastic optimal control problem. In several numerical experiments, we demonstrate that our approach yields competitive results for undersampled MRI reconstruction. Moreover, we can accurately quantify the pixelwise epistemic uncertainty, which can serve radiologists as an additional resource to visualize reconstruction reliability.
翻译:最近深层次的学习方法侧重于改进专用基准的量化分数,因此只能减少与观测有关的(代数)不确定性。然而,模型性(流行性)不确定性分析不那么经常地进行系统化分析。在这项工作中,我们引入了巴伊西亚变异框架,以量化认知性不确定性。为此,我们解决了在变异环境中未充分抽样的MRI重建的线性反问题。相关的能源功能由数据忠诚术语和作为学习参数的全深度变异(TDV)组成。我们从多变量高斯分布中提取TDV常规化器的参数,其中和共变异矩阵是在随机最佳控制问题中学习的。在几个数字实验中,我们证明我们的方法为未得到充分抽样的MRI重建产生竞争性结果。此外,我们可以准确地量化微量的缩微误数不确定性,它可以作为可视化重建可靠性的额外资源,供放射学家使用。