Parallel-beam Tomography Research Articles

O(N/sup 2/log/sub 2/N) filtered backprojection reconstruction algorithm for tomography

We present a new fast reconstruction algorithm for parallel beam tomography. The new algorithm is an accelerated version of the standard filtered backprojection (FBP) reconstruction, and uses a hierarchical decomposition of the backprojection operation to reduce the computational cost from O(N(3)) to O(N(2)log(2 )N). We discuss the choice of the various parameters that affect the behavior of the algorithm, and present numerical studies that demonstrate the cost versus distortion tradeoff. Comparisons with Fourier reconstruction algorithms and a multilevel inversion algorithm by Brandt et al., both of which also have O(N(2)log(2)N) cost, suggest that the proposed hierarchical scheme has a superior cost versus distortion performance. It offers RMS reconstruction errors comparable to the FBP with considerable speedup. For an example with a 512 x 512-pixel image and 1024 views, the speedup achieved with a particular implementation is over 40 fold, with reconstructions visually indistinguishable from the FBP.

A fast and accurate Fourier algorithm for iterative parallel-beam tomography

We use a series-expansion approach and an operator framework to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction. This algorithm is applicable for parallel-ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small. The conjugate gradient (CG) algorithm is used to minimize a regularized, spectrally weighted least-squares criterion, and we prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the fast Fourier transform (FFT). The proposed algorithm requires O(N(2)logN) floating-point operations per iteration to reconstruct an NxN image from P view angles, as compared to O(N (2)P) floating-point operations per iteration for iterative convolution-backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse- and limited-angle tomography under realistic sampling scenarios. Although the proposed algorithm cannot explicitly account for noise with nonstationary statistics, additional simulations demonstrate that for low to moderate levels of nonstationary noise, the quality of reconstruction is almost unaffected by assuming that the noise is stationary.

An alternative approach to direct Fourier reconstruction in parallel-beam tomography

In this work, we elaborate on some of the features of the direct Fourier reconstruction method, which has not spurred much interest in the literature. We searched for the possibility of an interpolation in the object space prior to Fourier transformation of the projection. To implement this idea, the central slice theorem based on the finite Fourier transform is reconsidered for a density distribution of point weights placed over a square grid. Some industrial applications carried out on a scanner with a Cs-137 source are given to demonstrate and verify our algorithm. It is observed that the image resolution is improved over that obtainable by the application of the basic direct Fourier transform method.

An interesting feature of the radon transform

It is shown that the reconstructed image in two-dimensional, parallel beam tomography has an imaginary component which, while zero for perfect data, is of necessity nonzero with real world data. The imaginary component can be used in at least two ways. The first is as an indicator function of the efficacy of corrective measures taken to correct flaws in the original data or in the processing technique. The second is as an essential ingredient in the many complex number operations that are used to process noisy data: crosscorrelations, autocorrelations, spectral analyses, and so forth.

Polynomial Based Algorithms for Computed Tomography II

This paper is concerned with two-dimensional parallel beam tomography when the radiographs are given in arbitrary directions. As in the first paper of this series [SIAM J. Appl. Math., 43 (1983), pp. 157–185], we concentrate on reconstruction algorithms whose point response is a radial polynomial.We develop representations of such polynomials in terms of ridge functions and relate them to the expected mean square error for reconstructions from radiographs perturbed by white noise.

Polynomial Based Algorithms for Computed Tomography

This paper is concerned with two-dimensional parallel beam tomography. We investigate the computational effectiveness of the general approach outlined in a previous paper by one of us. That approach was based on reconstructions by means of convolution with a polynomial kernel. Here we consider only the case of uniformly spaced views to take advantage of the symmetry. In this case we give a class of algorithms and calculate several numerical examples. We also obtain estimates on the expected mean square error when the data is perturbed by white noise. These estimates lead to several surprising applications.

Characterization of tomographic reconstructions which commute with rigid motions

This note considers linear reconstruction operators for parallel beam tomography when the number of radiographs is finite. It is shown that if such reconstructions are continuous and commute with rigid motions in an appropriate sense, then they must be representable as convolution operators with a polynomial kernel whose degree depends on the number of radiographs.