Fully three-dimensional reconstruction from data collected on concentric cubes in Fourier space: implementation and a sample application to MRI

Abstract

An algorithm is proposed for rapid and accurate reconstruction from data collected in Fourier space at points arranged on a grid of concentric cubes. The Fourier transform of the object to be reconstructed is decomposed into the sum of three functions by subdividing its domain into three non-overlapping mutually orthogonal double pyramids. Each of the three functions is zero-valued outside one of the double pyramids and has values inside that double pyramid which are the same as those of Fourier transform of the object to be reconstructed at the same points. Inverse Fourier transforms of these individual functions can be calculated using the chirp z-transform. The outputs of these inverse transforms for the three functions are estimates of their values at points of the same rectangular grid. The function to be reconstructed is estimated for this grid by adding together the three inverse transforms. The whole process has computational complexity of the same order as required for the 3D fast Fourier transform and so (for medically relevant sizes of the data set) it is faster than backprojection into the same size rectangular grid. The design of the algorithm ensures that no interpolations are needed, in contrast to methods involving backprojection with their unavoidable interpolations. As an application, a 3D data collection method for MRI has been designed which directly samples the Fourier transform of the object to be reconstructed on concentric cubes as needed for the algorithm.