Back-Projection Filtration Inversion of Discrete Projections

Abstract

AbstractWe present a new, robust discrete back-projection filtration algorithm to reconstruct digital images from close-to-minimal sets of arbitrarily oriented discrete projected views. The discrete projections are in the Mojette format, with either Dirac or Haar pixel sampling. The strong aliasing in the raw image reconstructed by direct back-projection is corrected via a de-convolution using the Fourier transform of the discrete point-spread function (PSF) that was used for the forward projection. The de-convolution is regularised by applying an image-sized digital weighting function to the raw PSF. These weights are obtained from the set of back-projected points that partially tile the image area to be reconstructed. This algorithm produces high quality reconstructions at and even below the Katz sufficiency limit, which defines a minimal criterion for projection sets that permit a unique discrete reconstruction for noise-free data. As the number of input discrete projected views increases, the PSF more fully tiles the discrete region to be reconstructed, the de-convolution and its weighting mask become progressively less important. This algorithm then merges asymptotically with the perfect reconstruction method found by Servières et al in 2004. However the Servières approach, for which the PSF must exactly tile the full area of the reconstructed image, requires O(N 2) uniformly distributed projection angles to reconstruct N ×N data. The independence of each (back-) projected view makes our algorithm robust to random, symmetrically distributed noise. We present, as results, images reconstructed from sets of O(N) projected view angles that are either uniformly distributed, randomly selected, or clustered about orthogonal axes.KeywordsDiscrete tomographyimage reconstruction from discrete projectionsinverse problems