Joint Learning of Measurement Matrix and Signal Reconstruction via Deep Learning

In this work, we propose an automatic sensing and reconstruction scheme based on deep learning within the compressive sensing (CS) framework. Classical CS utilizes pre-determined linear projections in the form of random measurements and convex optimization with a known sparsity basis to reconstruct signals. Here, we develop a data-driven approach to learn both the measurement matrix and the inverse reconstruction scheme for a given class of signals, such as images. The developed deep learning approach paves the way for end-to-end learning and reconstruction of signals with the aid of cascaded fully connected and multistage convolutional layers with a weighted loss function in an adversarial learning framework. Results obtained over the CIFAR-10 image database show that the proposed deep learning architectures provide higher peak signal-to-noise ratio (PSNR) levels, and, hence, learn better measurement matrices than that of randomly selected, specifically designed to reduce average coherence with a given basis, or state-of-the-art data driven approaches. The learned measurement matrices achieve higher PSNR compared to random or designed matrices not only when they are utilized in the proposed data-driven approach but also when used in
$\ell _1$
based recovery. The reconstruction performance on the test dataset improves as more training samples are utilized. Quantitative results for sparsity level analysis, incremental measurement design, and various training scenarios are provided.