Minimum square-deviation tomographic reconstruction from few projections

Abstract

The problem of tomographic reconstruction from a few discrete projections is addressed. When the projection data are discrete and few in number, the image formed by the convolution back-projection algorithm may not be consistent with the observed projections and is known to exhibit artifacts. Hence, the problem formulated here is one of finding an image that is closest to a nominal and is consistent with the projection data and other convex constraints such as positivity. The measure of closeness used is a Hilbert space norm, typically a weighted sum/integral of squares, with weights used to reflect expected deviation from the nominal in different regions. In the absence of constraints, this approach leads to a direct, noniterative algorithm (based on a simple matrix-vector computation) for construction of the image. When additional convex constraints such as positivity and upper-bounds need to be enforced on the reconstructed image to improve resolution, a quadratically convergent Newton algorithm is suggested.<>