Synopsis

Keywords: Image Reconstruction, Image Reconstruction

Model-based deep learning algorithms offer high quality reconstructions for accelerated acquisitions. Training the regularization parameter λ can lead to instabilities during training. In this work, we evaluated effect of fixing λ parameter while training. We observed no difference in image quality when the network was trained with a fixed λ parameter when the fixed value was equal to the value learned from training. We observed that IQ is dependent on the fixed λ values used during training. Furthermore, we observed that tuning the λ parameter during inference adapts the framework to the SNR of the testing dataset, yielding improved performance.

Introduction

Model-based deep-learning algorithms have shown great promise in reconstructing accelerated MRI acquisitions^1,2. These algorithms alternate between the neural network denoiser and the data consistency blocks. The regularization parameter λ balances the relative contribution of these two blocks to the reconstructed image. When this parameter is trained, we have observed some instabilities (Fig.1), similar to a previous report³. One way to mitigate this problem is to use a fixed λ value during training. However, fixing this parameter can lead to a decrease in performance as it may not be the optimal value. In this work, we evaluate the effect of fixing the λ value during training on the reconstructed image quality (IQ). We also evaluate the effect of varying this parameter during inference as a way to compensate for fixing it during training.

Methods

MoDL Framework: For this study, we used the MoDL algorithm². The under-sampled MRI acquisition can be modelled as $$$b = A(x) + n$$$, where $$$b$$$ is the undersampled data, $$$A$$$ is the forward operator which incorporates coil sensitivities, sampling mask and Fourier transform and $$$n$$$ is white Gaussian noise. MoDL framework is formulated as follows²:
$$arg⁡min_x \frac{λ}{2} ‖A(x)-b‖_2^2 + ‖N_w (x)‖_2^2$$
where the first term is the data-consistency term and the second term is the deep-learning-based denoiser. Here $$$N$$$ is the learned CNN that depends on learned parameters $$$w$$$.

Data Acquisition: T1, T2, and FLAIR data were acquired on human subjects on Vantage Orian 1.5T and Vantage Galan 3T systems (Canon Medical Systems Corporation, Tochigi, Japan) using a 16-channel head/neck and 32-channel head coil. All the datasets were acquired with informed consent and IRB approval.

Experiments: A total of 3000 2D datasets were used for training, and 700 datasets were used for testing. The training data was generated by undersampling the fully-sampled dataset with a variable-density sampling pattern with an acceleration factor of 4. A single complex CNN network was trained for both 1.5T and 3T data with a complex-MSE loss function and Adam optimizer. At first, MoDL was trained with a learnable l value. Then, MoDL was trained with a fixed l value. We evaluated the performance of MoDL for different fixed λ values. The network was trained with a fixed l value equal to the trained lambda (TL) value, for TL/2, and TL*2. In total, four different networks were trained: 1) λ is trainable, 2) λ is fixed and equal to trained λ, 3) λ is fixed and equal to TL/2, and 4) λ is fixed and equal to TL*2. Additionally, the λ value was also varied at inference for each of the four trained networks. The different inference λ values were TL, 2*TL, and TL/2. PSNR and SSIM metrics were used to analyze IQ.

Results

We observed that the network with λ as a trainable parameter and the network with fixed λ equal to trained λ have similar performance both quantitatively (PSNR: 39.27 and 39.20 respectively, SSIM: 0.95 for both) and qualitatively as seen in Fig2a-b (first two rows) and Fig.3 respectively. Fig.4 shows the IQ from networks trained with different fixed λ values for 1.5T(top row) and 3T(bottom row) datasets. We observed that the network yielded similar performance for different λ values for low SNR(1.5T). However, for high SNR datasets, artifacts were observed in the network with training lambda = TL/2. IQ improved with the increase in λ value. Within the range of lambda values in this study, inference λ value higher than that used in training yielded sharper images (Fig.5a) for 3T datasets. This is also reflected in higher SSIM and PSNR values (Fig.2c-d). In contrast, for 1.5T datasets, the inference λ equal to λ during training gave the best performance. Higher inference λ resulted in noise amplification (Fig.5b) leading to a decrease in PSNR (Fig.2c).

Discussion

The network performance is not affected when the λ parameter is fixed when the fixed value was equal to the value learned from training, thus mitigating the concerns from the instabilities when the parameter is trained. The networks with different fixed λ parameters yield different image quality. To address this issue, a few networks with different fixed λ parameters can be trained to find the optimal value. Some of the IQ degradation due to fixing the λ during training can be compensated by tuning the λ during inference. When the SNR of the testing data is low, higher weighting needs to be given to the ML denoiser, whereas when the SNR of the testing data is high, higher weighting given to the data-consistency term gives optimal results. Thus, tuning the λ parameter at inference can improve performance for a specific acquisition thereby offering flexibility to train a single network for different acquisition settings.

Conclusion

In this work, we evaluated the effect of using a fixed λ parameter during training. We also demonstrated that tuning this parameter during inference to adapt to the SNR of the testing dataset can further improve performance.

Acknowledgements

No acknowledgement found.