The notion of a linogram is introduced. It corresponds to the notion of a sinogram in the conventional representation of projection data in image reconstruction. In the sinogram, points which correspond to rays which go through a fixed point in the cross section to be reconstructed all fall on a sinusoidal curve. In the linogram, however, these points fall on a straight line. Thus, backprojection corresponds to integration along straight lines in the linogram. A theorem is proved expressing the backprojection operator in terms of the Radon transform and simple changes of variables. Consequences of this theorem are discussed. A novel image reconstruction method based on the theorem is presented.

The double Fourier decomposition of the sinogram is obtained by first taking the Fourier transform of each parallel -ray projection and then calculating the coefficients of a Fourier series with respect to angle for each frequency component of the transformed projections.

A method is described for preprocessing projection data prior to image reconstruction in single -photon emission computed tomography. The projection data of the desired spatial distribution of emission activity is blurred by the pointresponse function of the collimator that is used to define the range of directions of gamma -ray photons reaching the detector. The point-response function of the collimator is not stationary, but depends on the distance from the collimator to the point. Conventional methods for deblurring collimator projection data are based on approximating the actual depth -dependent pointresponse function by a spatially-invariant blurring function, so that deconvolution methods can be applied independently to the data at each angle of view. The method described in this paper is based on Fourier analysis of the multi-angular data set as a whole, using special depth-dependent characteristics of the Fourier coefficients to achieve spatially-variant inverse filtering of the data in all views simultaneously. Preliminary results are presented for simulated data with a simple collimator model.

The notion of a linogram corresponds to the notion of a sinogram in the conventional representation of projection data for image reconstruction. In the sinogram, points which correspond to rays through a fixed point in the cross section to be reconstructed all fall on a sinusoidal curve. In the linogram, however, these points fall on a straight line. The implementation of a novel image reconstruction method using this property is discussed. The implementation is of order N (2) log N, where N is proportional to the number of pixels on a side of the reconstruction region. It is demonstrated that the performance of the algorithm is superior to that of the filter backprojection method in computational speed on realistic size problems and is equivalent to filtered backprojection in accuracy of reconstruction.

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