Interpolation-Free Scanning And Sampling Scheme For Tomographic Reconstructions

Interpolation -free scanning and sampling scheme for tomographic reconstructionsK.D. Donohue and J. SaniieDepartment of Electrical and Computer Engineering, Illinois Institute of TechnologyChicago, Illinois 60616AbstractIn this paper a sampling scheme is developed for computer tomography (CT) systems thateliminates the need for interpolation. A set of projection angles along with their corre-ponding sampling rates are derived from the geometry of the Cartesian grid such that no in-terpolation is required to calculate the final image points for the display grid. A dis-cussion is presented on the choice of an optimal set of projection angles that will main-tain a resolution comparable to a sampling scheme of regular measurement geometry, whileminimizing the computational load. The interpolation -free scanning and sampling (IFSS)scheme developed here is compared to a typical sampling scheme of regular measurementgeometry through a computer simulation.IntroductionThe field of computerized tomography (CT) has generated much interest in reconstructingtwo -dimensional images from their sampled projections. This type of imaging process can bethought of in three stages. The first is the data acquisition stage. This determines howthe object will be scanned and exactly what samples will be taken. The second stage, whichis the reconstruction algorithm, will process these samples to obtain image points. Final-ly, the reconstructed image must be displayed. This is usually done on a Cartesian grid.For data acquisition, the two basic projections used are the fan -beam and parallel -bear.projections.' For either type of projection, the actual position of the detectors will de-termine the arrangement of samples in the Fourier domain. The reconstruction algorithmmust be adapted to accept the particular set of samples determined by the scanning andsampling geometry. In the last stage, the algorithm must take the processed samples andperform some kind of interpolation to estimate the value of the discrete points on theCartesian grid.Given the general configuration for parallel or fan -beam projections, different samplingschemes can be created by variations of the projection angles and sampling rates on eachprojection. This paper presents a derivation of a set of special projection angles with acorresponding sampling rate for parallel -beam projections that will cause the reconstruc-tion algorithm to calculate points that directly contribute to the image. The samplingscheme is derived from the geometry of the Cartesian grid so that the samples fed into thealgorithm contain more information about the points of interest than the arbitrary samplesthat must be interpolated for the display. The error and the computational load associatedwith interpolation are eliminated with this technique.In a typical parallel -beam scanner, an array of evenly spaced detectors positioned alonga straight line (projection axis) measures the attenuation of a beam, which has traversedthrough the object (Figure 1). This is done for every evenly spaced position along theprojection axis. Then the whole axis is rotated about the center of the object by a fixedangular increment and the measurements are repeated. This type of geometry is referred toas a polar raster because the Fourier space in this case is sampled in an arrangement