We describe an approach for incorporating prior knowledge into machine learning algorithms. We aim at applications in physics and signal processing in which we know that certain operations must be embedded into the algorithm. Any operation that allows computation of a gradient or sub-gradient towards its inputs is suited for our framework. We derive a maximal error bound for deep nets that demonstrates that inclusion of prior knowledge results in its reduction. Furthermore, we also show experimentally that known operators reduce the number of free parameters. We apply this approach to various tasks ranging from CT image reconstruction over vessel segmentation to the derivation of previously unknown imaging algorithms. As such the concept is widely applicable for many researchers in physics, imaging, and signal processing. We assume that our analysis will support further investigation of known operators in other fields of physics, imaging, and signal processing.

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In this paper, we consider the use of prior knowledge within neural networks. In particular, we investigate the effect of a known transform within the mapping from input data space to the output domain. We demonstrate that use of known transforms is able to change maximal error bounds.In order to explore the effect further, we consider the problem of X-ray material decomposition as an example to incorporate additional prior knowledge. We demonstrate that inclusion of a non-linear function known from the physical properties of the system is able to reduce prediction errors therewith improving prediction quality from SSIM values of 0.54 to 0.88. This approach is applicable to a wide set of applications in physics and signal processing that provide prior knowledge on such transforms. Also maximal error estimation and network understanding could be facilitated within the context of precision learning.

Purpose: In C-arm computed tomography (CT), the field of view (FOV) is often not sufficient to acquire certain anatomical structures, e.g., a full hip or thorax. Proposed methods to extend the FOV use a fixed detector displacement and a 360• scan range to double the radius of the FOV. These trajectories are designed for circular FOVs. However, there are cases in which the required FOV is not circular but rather an ellipsoid. Methods: In this work, the authors show that in fan-beam CT, the use of a dynamically adjusting detector offset can reduce the required scan range when using a noncircular FOV. Furthermore, the authors present an analytic solution to determine the minimal required scan ranges for elliptic FOVs given a certain detector size and an algorithmic approach for arbitrary FOVs. Results: The authors show that the proposed method can result in a substantial reduction of the required scan range. Initial reconstructions of data sets acquired with our new minimal trajectory yielded image quality comparable to reconstructions of data acquired using a fixed detector offset and a full 360• rotation. Conclusions: Our results show a promising reduction of the necessary scan range especially for ellipsoidal objects that extend the FOV. In noncircular FOVs, there exists a set of solutions that allow a trade-off between detector size and scan range. C 2015 American Association of Physicists in Medicine. [http://dx

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