Jiayang Wang^{1} and Justin P. Haldar^{1}

^{1}Signal and Image Processing Institute, Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, United States

This work investigates the potential value of combining non-uniform k-space averaging with advanced nonlinear image denoising-reconstruction methods in the context of low-SNR MRI. A new data-driven strategy for optimizing the k-space averaging pattern is proposed, and is then applied to total variation and U-net reconstruction methods. It is observed that non-uniform k-space averaging (with substantially more averaging at the center of k-space) is preferred for both reconstruction approaches, although the distribution of averages varies substantially depending on the noise level and the reconstruction method. We expect that these results will be informative for a wide range of low-SNR MRI applications.

In high-SNR scenarios with k-space undersampling, there has been substantial interest in the data-driven design of optimal k-space undersampling patterns.

It has been known for a long time that non-uniform averaging (with a larger number of averages near the center of k-space and fewer averages at high-frequencies) can outperform uniform averaging for simple methods like apodized Fourier reconstruction.

We consider a scenario in which there are $$$L$$$ possible k-space sampling locations $$$\mathbf{k}_{\ell}$$$, $$$\ell=1,\ldots, L$$$, and we wish to determine the optimal number $$$m_\ell$$$ of repeated measurements (averages) to be obtained at each location, subject to a constraint on the total number of k-space measurements.

For optimization, we assume that we have access to a database of high-SNR datasets from an application of interest, and that we can add simulated noise to mimic the effects of a low-SNR acquisition. We also assume that we are given a denoising-reconstruction procedure (which may depend on the averaging pattern). We then define an optimality criterion as the average error between each high-SNR image and the denoised image obtained by applying a given denoising-reconstruction method to simulated noisy k-space data obtained with a given averaging pattern.

In general, finding an averaging pattern that minimizes this optimality criterion is a difficult integer programming problem with combinatorial complexity. As a result, we consider a continuous relaxation of the optimization problem in which the values $$$m_\ell$$$ now represent the fractional averaging effort applied to the $$$\ell$$$th position in k-space. This is a standard relaxation approach.

We optimize this relaxed optimality criterion using backpropagation.

We used 2400 T2-weighted brain slices from the fastMRI dataset

Optimized averaging was performed for three different noise levels (SNR=0.8, 0.5, and 0.4) as illustrated in Fig. 1, and reconstruction results obtained with optimized averaging patterns were compared against images obtained with conventional uniform averaging. Performance was then assessed using datasets that were not used for training.

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Fig. 1. The top row shows representative ground truth and noisy Fourier reconstructions with uniform averaging for different SNR values. The second row shows zoom-ins to the images from the first row to better depict image details. The bottom row shows U-net reconstructions of this data.

Fig. 2. (left)
A conventional uniform
averaging pattern with
8-averages at every k-space position, and (right) three averaging patterns that
were optimized for TV reconstruction with different SNRs.

Fig. 3. (left) A conventional uniform averaging pattern with
8-averages at every k-space position, and (right) three averaging patterns that
were optimized for U-net reconstruction with different SNRs.

Table 1. Quantitative evaluation
of different averaging approaches. In
each case, the best error metrics are highlighted with bold text.