Basis pursuit is a compressed sensing optimization in which the l1-norm is minimized subject to model error constraints. Here we use a deep neural network prior instead of l1-regularization. Using known noise statistics, we jointly learn the prior and reconstruct images without access to ground-truth data. During training, we use alternating minimization across an unrolled iterative network and jointly solve for the neural network weights and training set image reconstructions. At inference, we fix the weights and pass the measurements through the network. We compare reconstruction performance between unsupervised and supervised (i.e. with ground-truth) methods. We hypothesize this technique could be used to learn reconstruction when ground-truth data are unavailable, such as in high-resolution dynamic MRI. Deep learning in tandem with model-based iterative optimization [2] - [6] , i.e. model-based deep learning, has shown great promise at solving imaging-based inverse problems beyond the capabilities of compressed sensing [7] . These networks typically require hundreds to thousands of examples for training, consisting of pairs of corrupted measurements and the desired ground-truth image. The reconstruction is then trained in an end-to-end fashion, in which data are reconstructed with the network and compared to the ground-truth result. in many cases, collecting a large set of fully sampled data for training is expensive, impractical, or impossible. In this work, we present an approach to model-based deep learning without access to ground-truth data [8] - [10] . We take advantage of (known) noise statistics for each training example and formulate the problem as an extension of basis pursuit denoising [11] with a deep convolutional neural network (CNN) prior in place of image sparsity. During training, we jointly solve for the CNN weights and the reconstructed training set images. At inference time, we fix the weights and pass the measured data through the network. As proof of principle, we apply the technique to undersampled, multi-channel magnetic resonance imaging (MRI). We compare our Deep Basis Pursuit (DBP) formulation with and without supervised learning, as well as to MoDL [6] , a recently proposed unrolled modelbased network that uses ground-truth data for training. We show that in the unsupervised setting, we are able to approach the image reconstruction quality of supervised learning, thus opening the door to applications where collecting fully sampled data is not possible. There are strong connections to iterative optimization and unrolled deep networks [10] , [18] , [19] . Jointly optimizing over the images and weights could be viewed a non-linear extension to dictionary learning. Nonetheless, there is a cost in reconstruction error when moving to unsupervised learning, highlighting the importance of a large training data set to offset the missing ground-truth information. The choice of measurement loss function and data SNR may also greatly impact the quality. Fortunately, in many practical settings there is an abundance of undersampled or corrupted measurement data available for training. In conclusion, the combination of basis pursuit denoising and deep learning can take advantage of undersampled data and provide a means to learn model-based deep learning reconstructions without access to groundtruth images.
We present an unsupervised deep learning reconstruction for imaging inverse problems that combines neural networks with model-based constraints.
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