Standard Image Recovery Methods In The Iterative Data Refinement Framework

Abstract

Iterative Data Refinement (IDR) is a general procedure for estimating data that would have been collected by an ideal measuring device from data that were collected by an actual measuring device. An example is in Computerized Tomography (CT), where we have a mathematical procedure to reconstruct the x-ray attenuation coefficient at individual points inside the human body from (the ideal) data obtained by passing monoenergetic x-rays through the body and measuring the percentage of energy that gets through. Unfortunately, x-ray tubes deliver polyenergetic x-rays and the actual measurements only approximate what is assumed by the mathematics of CT, resulting in images whose quality is noticeably worse that those reconstructed from ideal data. This is one of the applications where the efficacy of IDR has been demonstrated: it can be used to estimate data that would be obtained from the ideal monoenergetic x-ray tube from the data that are obtained from the actual polyenergetic x-ray tube. In fact, IDR is general enough to encompass such well-accepted image recovery methods as the Gerchberg-Saxton algorithm and the error reduction and hybrid input-output methods of Fienup. The generalizations provided by IDR give new insights into the nature of such algorithms and, in particular, allow us to introduce the notion of relaxation into them, resulting in many cases in a much improved computational behavior.