Kanghyun Ryu^{1}, Cagan Alkan^{2}, and Shreyas S. Vasanawala^{1}

^{1}Radiology, Stanford University, Stanford, CA, United States, ^{2}Electrical Engineering, Stanford, Stanford, CA, United States

In this study, we propose a novel refinement method using auto-calibrated k-space null-space kernel to reduce the k-space errors and enable reconstruction of improved high-frequency image details and textures. The refinement algorithm can be easily plugged in after DL-based reconstructions. We show that our method enables the reconstruction of sharper images with significantly improved high-frequency components measured by HFEN and GMSD while maintaining overall error in the image measured by PSNR and SSIM.

In this work, we propose a novel refinement method using auto-calibrated k-space null-space kernel [ref] to reduce the k-space errors and enable reconstruction of improved high-frequency image details and textures. The proposed scheme constrains the DL output to satisfy the neighborhood relationship in the frequency space (k-space) which can be easily calibrated in the auto-calibration (ACS) lines, and corrects the under-estimation in the peripheral region of the k-space as well as reduce structured k-space errors. Note that auto-calibrated k-space null space kernel contains additional information such as coil sensitivities, image features, limited spatial support, slowly varying phase, and etc [5-8].

We show that our method enables the reconstruction of sharper images with significantly improved high-frequency components measured by HFEN [9] and GMSD [10] while maintaining overall error in the image measured by PSNR and SSIM.

For this study, we used the “Brain dataset” (T1 post contrast enhanced), and the “Knee dataset” (with proton density weighted + proton density weighted with fat suppression). For the Brain dataset, we used the dataset marked as “validation” which we split the data randomly into (Train=212, Val=32, Test=42 subjects). For the Knee dataset, we split the data into (Train=108, Val=30, Test=62 subjects). For both dataset, equi-spaced undersampling (R=4,6) were used for the work.

Two state-of the art unrolled neural networks were used – Variational Network [ref], MoDL [ref]. For Variational Network (VarNet) we used 12 unrolled blocks, and MoDL with 10 unrolled block.

For the examples in the study, we used SPIRiT [5] with virtual coil conjugate augmentation [11] as the method for calibrating the null-space kernel. We also tested SURE-based ESPIRiT method for the null-space kernel [7], which offers faster convergence but lacks information of the slowly varying phase.

The proposed refinement requires three inputs: acquired under-sampled k-space, null-space kernel that is calibrated from the ACS-line of the undersampled k-space, and the DL reconstructed k-space. The overall scheme of the proposed refinement method and inputs required for the refinement are demonstrated in Fig.1.

With these inputs, we solve the following optimization problem:

$$\operatorname{argmin}_{\boldsymbol{k}}\|\mathcal{D} k-y\|^{2}+\lambda_1\|(\mathcal{G}-I) k\|^{2} + \lambda_2\|\mathcal{D}_c (k - \tilde{k})\|^{2}$$

Where $$$\mathcal{D}_c$$$ is an operator that selects un-acquired k-space locations, $$$\mathcal{G}-I$$$ auto-calibrated null-space kernel, $$$\tilde{k}$$$ the DL-estimated k-space, assuming that the DL estimation has estimated k-space with some degree of precision in mean squared error. For solving the optimization problem, we used the conjugate gradient (CG) algorithm with 300 iterations to ensure convergence.

Figure 3 shows the result of refinement on a PDw knee dataset. First column is a result reconstructed from MoDl with R=6. Second column is the corresponding refinement result, and third column is the reference image from fully sampled k-space. Likewise, sharper details in the ligament and improved texture can be observed.

Figure 4 shows the results of quantitative evaluation for acceleration factor of 6. The quantitative metrics are improved with the refinement for both DL methods (VarNet, MoDL).

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Figure 1. The proposed refinement method. The refinement is performed on the output k-space from the DL reconstruction. The resulting k-space is refined with the auto calibrated kernel from ACS line, by performing the optimization problem below.

Figure 2. Result of refinement on MoDL reconstruction on R=6 Brain T1 Post dataset. Improvement of texture and details can be seen. Also quantitative metrics (PSNR, SSIM, HFEN, GMSD) show improvements

Figure 3. Result of refinement on MoDL reconstruction on R=6 Knee PDw dataset. Improvement of texture and details can be seen. Also quantitative metrics (PSNR, SSIM, HFEN, GMSD) show improvements

Figure 4. Evaluation of quantitative metrics on two datasets with two different DL networks. The refined results show significant improvements on the two high-frequency error metrics (HFEN, GMSD) shown inside the dashed line.

DOI: https://doi.org/10.58530/2022/3478