# Cram'er-Rao bound-informed training of neural networks for quantitative MRI

arXiv: https://arxiv.org/abs/2109.10535 · HTML: https://mlsanity.com/paper/2109.10535/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 58 references that are in the map.
- First posted on arXiv 22 Sep 2021.

## Builds on (most cited first, 12 of 12)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Deep Learning for Real-time Gravitational Wave Detection and Parameter Estimation: Results with Advanced LIGO Data](https://mlsanity.com/paper/1711.03121/) (2017)
- [Better than Real: Complex-valued Neural Nets for MRI Fingerprinting](https://mlsanity.com/paper/1707.00070/) (2017)
- [MR fingerprinting Deep RecOnstruction NEtwork (DRONE)](https://mlsanity.com/paper/1710.05267/) (2017)
- [Geometry of Deep Learning for Magnetic Resonance Fingerprinting](https://mlsanity.com/paper/1809.01749/) (2018)
- [Magnetic Resonance Fingerprinting using Recurrent Neural Networks](https://mlsanity.com/paper/1812.08155/) (2018)
- [Rapid three-dimensional multiparametric MRI with quantitative transient-state imaging](https://mlsanity.com/paper/2001.07173/) (2020)
- [Magnetic Resonance Fingerprinting Reconstruction via Spatiotemporal Convolutional Neural Networks](https://mlsanity.com/paper/1807.06356/) (2018)
- [HYDRA: Hybrid Deep Magnetic Resonance Fingerprinting](https://mlsanity.com/paper/1902.02882/) (2019)
- [Dictionary-Free MRI PERK: Parameter Estimation via Regression with Kernels](https://mlsanity.com/paper/1710.02441/) (2017)

## Built on by (most cited first, 3 of 3)

- [PINQI: An End-to-End Physics-Informed Approach to Learned Quantitative MRI Reconstruction](https://mlsanity.com/paper/2306.11023/) (2023)
- [Fisher information flow in artificial neural networks](https://mlsanity.com/paper/2509.02407/) (2025)
- [Bias-Reduced Neural Networks for Parameter Estimation in Quantitative MRI](https://mlsanity.com/paper/2312.11468/) (2023)

## Abstract (arXiv)

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.