Two-dimensional reconstructions from one-dimensional data by maximum entropy

Abstract

The maximum entropy method (MEM) is a powerful information–theoretical approach to the inversion of many types of data in science and engineering. It has been used for reconstructing images in such areas as radioastronomy^1,2, medical tomography³ and crystallography⁴. Practical data are always corrupted by noise and are usually incomplete: for example, there may be missing projections or missing Fourier components. Also, the instrumental response function may be incompletely known⁵. These factors can lead to severely ill-posed inverse problems. The method is used here in a novel technique for analysing time-series data representing a set of non-interacting damped oscillators. The time series is simultaneously analysed in both frequency and decay to obtain positive two-dimensional reconstructions on the frequency–decay plane. The method is shown to be superior to conventional techniques in that it allows frequency and decay information to be more reliably extracted from the data. Such data occur in areas such as Fourier transform nuclear magnetic resonance (FTNMR), which is used here as an example. In the FTNMR case the decay is the inverse of the effective spin–spin relaxation time.