Abstract

Different filters may be applied for accurate reconstruction of a three-dimensional (3-D) image from two-dimensional (2-D) parallel projections with the use of filtered back projection. The set of such valid filters is characterized in Fourier space by a general filter equation. We demonstrate that all the solutions to this filter equation may be found by the application of additive or multiplicative corrections to arbitrary functions. One may find the object-space convolution kernel by taking the inverse 2-D Fourier transform of a valid filter. This kernel contains singularities, and the problem of deriving explicit expressions by means of limits of regular functions is discussed in detail. Another approach to 3-D reconstruction is to form planar integrals from the projection data and to use 3-D Radon inversion procedures. Such an approach is shown here to be equivalent to filtered backprojection, where the corresponding filter is closely related to the weighting scheme used in the formation of the planar integrals.

© 1993 Optical Society of America

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