PUERT: Probabilistic Under-Sampling and Explicable Reconstruction Network for CS-MRI

Abstract

Compressed Sensing MRI (CS-MRI) aims at reconstructing de-aliased images from sub-Nyquist sampling k-space data to accelerate MR Imaging, thus presenting two basic issues, i.e., where to sample and how to reconstruct. To deal with both problems simultaneously, we propose a novel end-to-end Probabilistic Under-sampling and Explicable Reconstruction neTwork, dubbed PUERT, to jointly optimize the sampling pattern and the reconstruction network. Instead of learning a deterministic mask, the proposed sampling subnet explores an optimal probabilistic sub-sampling pattern, which describes independent Bernoulli random variables at each possible sampling point, thus retaining robustness and stochastics for a more reliable CS reconstruction. A dynamic gradient estimation strategy is further introduced to gradually approximate the binarization function in backward propagation, which efficiently preserves the gradient information and further improves the reconstruction quality. Moreover, in our reconstruction subnet, we adopt a model-based network design scheme with high efficiency and interpretability, which is shown to assist in further exploitation for the sampling subnet. Extensive experiments on two widely used MRI datasets demonstrate that our proposed PUERT not only achieves state-of-the-art results in terms of both quantitative metrics and visual quality but also yields a sub-sampling pattern and a reconstruction model that are both customized to training data. 1

Learing Sampling and Reconstruction Using Bregman Iteration for CS-MRI

“…LOUPE [34] and PUERT [22] both assume that each binary sampling element is an independent Bernoulli random variable and learn a probabilistic sampling pattern rather than a deterministic mask. PUERT introduces an effective gradient estimation strategy when using the binarization function,and uses DUN to make full use of the intrinsic structural features of mask knowledge at each stage to realize the combination of deep learning and traditional models.…”

“…Finding an effective way to solve the PM step is crucial for CS problems. Since the Prox R operator is similar to denoising, a lot of work [22,29] has been done to learn this operator using a denoising network to obtain adaptive regularized R. Similarly, in this paper, the traditional Equation (16b) formula is converted to the form of a deep network to build a deep unfolding network (Equation ( 17)). In this way, the lower-level optimization problem is transformed into a phase reconstruction problem, and each stage adds the residual of the previous stage data items to the next stage of training.…”

“…In order to ensure that the learned sampling mask reaches the desired sampling ratio, we encourage the sampling mask to achieve the desired sampling rate by imposing a penalty term on the gradient. Since the true weight of the sampling mask is not required, only the relative weight size needs to be obtained, and a penalty factor is directly applied to the gradient of the back-propagation, such as shown in Equation (22), where α is the penalty factor.…”

“…It is worth noting that our mask also has a better structure than LOUPE in the zero-filled state, which can speed up the inference of the model. We compare the proposed reconstruction model with five representative state-ofthe-art methods, namely U-Net [17], ADMM-Net [29], ISTA-Net [15], PUERT [22], and LOUPE [34]. The first three methods are reconstructed networks trained under a certain fixed sampling mask, while the latter two methods jointly optimize the learnable sampling mask and reconstruction network parameters.…”

“…Specifically, based on the strategy of the accumulation gradient [20,21], we develop a sampling sub-network, learn the weight of each point, and decide whether to sample according to the size of the weight. Like PUERT [22], we adopt a dynamic gradient estimation strategy that progressively approximates the binarization function, which effectively preserves the gradient information and further improves the recovery quality. In order to be consistent with the conclusions of CS theory, our model is an optimization model with data term constraints.…”