Neural networks are increasingly used to estimate parameters in quantitative MRI, in particular in magnetic resonance fingerprinting. Their advantages over the gold standard non-linear least square fitting are their superior speed and their immunity to the non-convexity of many fitting problems. We find, however, that in heterogeneous parameter spaces, i.e. in spaces in which the variance of the estimated parameters varies considerably, good performance is hard to achieve and requires arduous tweaking of the loss function, hyper parameters, and the distribution of the training data in parameter space. Here, we address these issues with a theoretically well-founded loss function: the Cram\'er-Rao bound (CRB) provides a theoretical lower bound for the variance of an unbiased estimator and we propose to normalize the squared error with respective CRB. With this normalization, we balance the contributions of hard-to-estimate and not-so-hard-to-estimate parameters and areas in parameter space, and avoid a dominance of the former in the overall training loss. Further, the CRB-based loss function equals one for a maximally-efficient unbiased estimator, which we consider the ideal estimator. Hence, the proposed CRB-based loss function provides an absolute evaluation metric. We compare a network trained with the CRB-based loss with a network trained with the commonly used means squared error loss and demonstrate the advantages of the former in numerical, phantom, and in vivo experiments.
翻译:神经网络越来越多地被用来估计定量磁共振成像仪的参数,特别是磁共振指纹。它们对金标准非线性最差的适配的优势在于其超速和豁免于许多不相容问题的不相容性。然而,我们发现,在不同的参数空间,即估计参数差异大不相同的空间,良好的性能很难实现,并且需要在参数空间内对损失功能、超强参数和培训数据的分布进行艰苦的调整。在这里,我们用理论上有充分依据的损失函数来处理这些问题:Cram\'er-Rao绑定(CRB)为公正的天平选者的差异提供了一个较低的理论约束,我们提议与各自的CRB平方错误正常化。随着这种正常化,我们平衡了难以估计和不难估计参数和参数空间内区域的贡献,避免了前者在总体培训损失中的主导地位。此外,基于Cram\'er-Rao绑定(Cram_er-Rao)的亏损功能在理论上比低偏差的理论约束,我们建议将先前的网络与经过培训的平局的平局的平局的平局性计算模型的计算结果,我们认为,我们认为,我们所使用的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局性平局的平局的平局的平局的平局的平局的平局的平局的计算功能提供了一种理想的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局性功能,我们认为,我们的理想的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局