Deep supervised dictionary learning by algorithm unrolling—Application to fast 2D dynamic MR image reconstruction
Authors: Kofler Andreas , Pali Marie‐Christine , Schaeffter Tobias , Kolbitsch Christoph 
Published: December 2022

Background

Unrolled neural networks (NNs) have been extensively applied to different image reconstruction problems across all imaging modalities. A key component of the latter is that they allow for physics‐informed learning of the regularization method, which is parametrized by the NN. However, due to the lack of understanding of deep NNs from a theoretical point of view, unrolled NNs are still black‐boxes when the regularizers are given by deep NNs, for example, U‐Nets.

Purpose

Dictionarylearning (DL) is a well‐established regularization method, which is based on learning a transform to sparsely approximate the signals of interest. Typically, DL‐based image reconstruction either employs a dictionary, which was pretrained on a set of patches which were extracted from ground‐truth images or a dictionary which is jointly trained during the reconstruction. However, in both cases, the used DL‐algorithms are not designed to take into account the reconstruction problem or the underlying physical model, which describes the imaging process. In this work, we propose a DL‐algorithm based on unrolled NNs to overcome these limitations.

Methods

We construct an unrolled NN, which corresponds to an unrolled DL‐based reconstruction algorithm and train the unrolled NN to optimize its weights, that is, the atoms of the dictionary, by back‐propagation in a supervised manner. Further, we propose a new way to employ a 2D dictionary in the spatio‐temporal domain. We tested and evaluated the method on an accelerated cardiac cine MR image reconstruction problem using 216/36/36 dynamic images for training, validation, and testing and compared it to two well‐known state‐of‐the‐art approaches for cardiac cine MRI based on deep iterative CNNs. Further, we analyze the obtained dictionaries in terms of dictionary‐coherence and structure of the atoms. Last, we compare the reported methods in terms of stability by applying them to an entirely different dataset consisting of 49 different test images.

Results

The investigated physics‐informed DL‐approach yields significantly more accurate reconstructions compared to the DL‐method, which uses dictionaries obtained by decoupled pretraining, thereby providing an improvement of up to 4.90 dB in terms of PSNR and 5% in terms of SSIM. Further, the proposed spatio‐temporal 2D dictionary outperforms the 1D and 3D dictionaries by preventing smoothing of image details while still accurately removing undersampling artifacts and noise resulting in an increase of up to 1.10 dB in terms of PSNR and 4% in terms of SSIM. Although being surpassed by the CNNs on the first dataset, the proposed NNs‐based DL method is more stable compared to the latter approach and yields comparable results on the second dataset. Last, it has the advantage of being entirely interpretable in each component.

Conclusions

The presented physics‐informed NN can be used as training algorithm for a classical and interpretable data‐driven regularization method based on a learned dictionary, which can then not only be linked to the considered data but also to the reconstruction method that the NN defines.