SpiNet: A deep neural network for Schatten p-norm regularized medical image reconstruction

Purpose: To propose a generic deep learning based medical image reconstruction model (named as SpiNet) that can enforce any Schatten p-norm regularization with 0 < p <= 2, where the p can be learnt (or fixed) based on the problem at hand. Methods: Model-based deep learning architecture for solving inverse problems consists of two parts, a deep learning based denoiser and an iterative data consistency solver. The former has either L2 norm or L1 norm enforced on it, which are convex and can be easily minimized. This work proposes a method to enforce any p norm on the noise prior where 0 < p <= 2. This is achieved by using Majorization-Minimization algorithm, which upper bounds the cost function with a convex function, thus can be easily minimized. The proposed SpiNet has the capability to work for a fixed p or it can learn p based on the data. The network was tested for solving the inverse problem of reconstructing magnetic resonance (MR) images from undersampled k space data and the results were compared with a popular model-based deep learning architecture MoDL which enforces L2 norm along with other compressive sensing-based algorithms. This comparison between MoDL and proposed SpiNet was performed for undersampling rates (R) of 2x, 4x, 6x, 8x, 12x, 16x, and 20x. Multiple figures of merit such as PSNR, SSIM, and NRMSE were utilized in this comparison. A two-tailed t test was performed for all undersampling rates and for all metrices for proving the superior performance of proposed SpiNet compared to MoDL. For training and testing, the same dataset that was utilized in MoDL implementation was deployed. Results: The results indicate that for all undersampling rates, the proposed SpiNet shows higher PSNR and SSIM and lower NRMSE than MoDL. However, for low undersampling rates of 2x and 4x, there is no significant difference in performance of proposed SpiNet and MoDL in terms of PSNR and NRMSE. This can be expected as the learnt p value is close to 2 (norm enforced by MoDL). For higher undersampling rates >= 6x, SpiNet significantly outperforms MoDL in all metrices with improvement as high as 4 dB in PSNR and 0.5 points in SSIM. Conclusion: As deep learning based medical image reconstruction methods are gaining popularity, the proposed SpiNet provides a generic framework to incorporate Schatten p-norm regularization with 0 <p <= 2 with an added advantage of providing superior performance compared to its counterparts. The proposed SpiNet also has useful addition of Schatten p-norm value in regularization term being automatically chosen based on the available training data.