The Compton camera has been proposed as an alternative to the Anger camera in SPECT. The advantage of the Compton camera is its high geometric efficiency due to electronic collimation. The Compton camera collects projections that are integrals over cone surfaces. Although some progress has been made toward image reconstruction from cone projections, at present no filtered backprojection algorithm exists. This paper investigates a simple 2D version of the imaging problem. An analytical formula is developed for 2D reconstruction from data acquired by a 1D Compton camera that consists of two linear detectors, one behind the other. Coincidence photon detection allows the localization of the 2D source distribution to two lines in the shape of a "V" with the vertex on the front detector. A set of "V' projection data can be divided into subsets whose elements can be viewed as line-integrals of the original image added with its mirrored shear transformation. If the detector has infinite extent, reconstruction of the original image is possible using data from only one such subset. Computer simulations were performed to verify the newly developed algorithm.

Background Magnetic resonance imaging (MRI) is an effective auxiliary diagnostic method in clinical medicine, but it has always suffered from the problem of long acquisition time. Compressed sensing and parallel imaging are two common techniques to accelerate MRI reconstruction. Recently, deep learning provides a new direction for MRI, while most of them require a large number of data pairs for training. However, there are many scenarios where fully sampled k-space data cannot be obtained, which will seriously hinder the application of supervised learning. Therefore, deep learning without fully sampled data is indispensable. Main text In this review, we first introduce the forward model of MRI as a classic inverse problem, and briefly discuss the connection of traditional iterative methods to deep learning. Next, we will explain how to train reconstruction network without fully sampled data from the perspective of obtaining prior information. Conclusion Although the reviewed methods are used for MRI reconstruction, they can also be extended to other areas where ground-truth is not available. Furthermore, we may anticipate that the combination of traditional methods and deep learning will produce better reconstruction results.

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