Learning from existing datasets has the potential to improve reconstruction quality. However, deep learning based methods typically require many clean fully-sampled datasets as ground truths. Such datasets can be hard to come by, especially in applications where rapid scans are desired. Here, we propose a method based on convolutional sparse coding that can learn a convolutional dictionary from under-sampled datasets for sparse reconstruction. Recent works have shown close connections between deep learning and convolutional sparse coding. The benefit of convolutional sparse coding is that it has a well-defined forward model, and can be easily extended to incorporate physical models during training. We extend convolutional sparse coding to incorporate the under-sampling forward model. We show that the dictionary learned from under-sampled datasets is similar to the dictionary learned from fully-sampled datasets, and improves upon wavelet transform for l1 regularized reconstruction in terms of mean-squared error.
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