Source-linked AI summary
Better than Real: Complex-valued Neural Nets for MRI Fingerprinting
Patrick Virtue, Stella X. Yu, Michael Lustig
TL;DR
MRI fingerprinting must recover tissue parameters from complex-valued signals, while dictionary matching becomes slow and poorly scalable as the parameter space grows. The paper trains neural networks on simulator-generated signal-parameter pairs, introduces complex-valued networks with cardioid activations, and reports greater effectiveness than real-valued networks. The approach is evaluated for pixel-wise fingerprinting, with full-image predictions for under-sampled MRI left for future work.
Problem
Dictionary-based MRI fingerprinting is slow and does not scale with larger tissue-parameter spaces or more parameters.
Method
The paper trains neural networks on synthetic MRI simulator data and develops complex-valued networks with a phase-sensitive cardioid activation.
Results
Complex-valued neural networks outperform 2-channel real-valued networks in almost all experiments, while neural inverse mapping outperforms nearest-neighbor matching for T1 and T2.
Takeaways & Limitations
Complex-valued networks can use information in complex MRI data more effectively than treating signals as arbitrary two-channel real data.
Takeaways & Limitations
The study focuses on pixel-wise fingerprinting reconstruction and leaves full-image predictions for under-sampled MRI fingerprinting to future work.
Abstract
from arXiv · showhide
The task of MRI fingerprinting is to identify tissue parameters from complex-valued MRI signals. The prevalent approach is dictionary based, where a test MRI signal is compared to stored MRI signals with known tissue parameters and the most similar signals and tissue parameters retrieved. Such an approach does not scale with the number of parameters and is rather slow when the tissue parameter space is large. Our first novel contribution is to use deep learning as an efficient nonlinear inverse mapping approach. We generate synthetic (tissue, MRI) data from an MRI simulator, and use them to train a deep net to map the MRI signal to the tissue parameters directly. Our second novel contribution is to develop a complex-valued neural network with new cardioid activation functions. Our results demonstrate that complex-valued neural nets could be much more accurate than real-valued neural nets at complex-valued MRI fingerprinting.
1. INTRODUCTION
MRI fingerprinting infers quantitative tissue and magnetic-field parameters from complex-valued signals, but dictionary matching becomes inefficient as the parameter space grows. The paper instead learns the inverse mapping with neural networks and investigates complex-valued models that preserve phase information.
- Motivation: MRI fingerprinting simultaneously estimates quantitative tissue parameters from one scan, unlike traditional MRI’s multiple qualitative-contrast scans.It can also provide information about B0 and B1 imperfections in the applied magnetic field.
- Inverse mapping: A predetermined sequence of flip angles and repetition times produces tissue-dependent signal signatures that can be decoded into parameters at each pixel.The simulator converts parameter tuples into MRI temporal signals for training, while scanners provide signals at test time.
- Computational challenge: Nearest-neighbor dictionary matching scales linearly with dictionary size, becoming infeasible at finer parameter resolution or with additional tissue parameters.Prior work improves matching efficiency with SVD and group matching, but remains dictionary-based.
- Neural inverse mapping: Feedforward neural networks learn the nonlinear inverse mapping with computational efficiency that does not scale with the number of training examples.The approach replaces non-parametric nearest-neighbor matching with a parameterized model trained on simulator-generated data.
- Complex-valued modeling: Complex-valued neural networks are investigated because two-channel real representations do not preserve the phase information captured by complex algebra.The paper reports that complex-valued networks are more effective than real-valued networks for MRI fingerprinting.
2. COMPLEX-VALUED NEURAL NETWORKS
Complex-valued neural networks address the mismatch between complex MRI signals and standard real-valued nonlinearities. The section introduces a phase-sensitive cardioid activation and Wirtinger-calculus optimization for complex networks.
- Complex network research must address the poor transfer of standard real-valued nonlinear layers to complex-valued networks.
- The complex cardioid activation attenuates output magnitude according to input phase while preserving output phase.
- On real-valued inputs, the complex cardioid reduces to the ReLU activation function.
- Wirtinger, or CR, calculus enables gradient descent on functions that are not complex differentiable but are differentiable in their real and imaginary components.
- Complex-network weights are optimized against a real-valued loss using complex gradient descent and the complex cogradient as the steepest-descent direction.
3. MRI FINGERPRINTING EXPERIMENTS
Experiments trained neural networks on simulated complex MRI signals and evaluated pixel-wise parameter reconstruction on clean and noisy numerical-phantom data. They compared computational scaling, nearest-neighbor and neural inverse mapping, and real- versus complex-valued architectures.
- Training and testing: 100,000 simulated MRI points were used for training, sampled with the baseline dictionary’s T1, T2, B0 density.Signals were simulated with the Bloch equations and had length 500.
- Training and testing: Testing used a numerical MRI phantom with specified T1, T2, and proton-density values, a B0 ramp from -60 Hz to 60 Hz, and clean or noisy signals.The experiments omitted B1 inhomogeneity, although an additional neural network could incorporate it.
- Model comparisons: The fully connected architecture was repeated for each desired output label, including T1, T2, and B0.The compared implementations included two-channel real-valued ReLU networks, wider real-valued networks, and complex-valued networks with cardioid, separable sigmoid, or siglog activations.
- Computational efficiency: A dictionary approach became infeasible as outputs increased: relative to T1,T2, FLOPs rose 171× for T1,T2,B0 and 3,585× for T1,T2,B0,B1.Figure 4 compares floating-point operations for computing parameters for one pixel on a logarithmic scale.
- Prediction accuracy: Neural inverse mapping outperformed nearest-neighbor matching for T1 and T2, while nearest-neighbor matching predicted B0 more accurately.The results compare prediction accuracy under clean and pSNR=40 noise conditions.
- Complex-valued models: Complex-valued networks outperformed two-channel real-valued networks in almost all experiments, and cardioid significantly outperformed separable sigmoid and siglog activations.The reported advantage was not explained by the doubled capacity of the wider real-valued model.