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Convolution Backprojection
The convolution backprojecton (CBP) algorithm for three-dimensional reconstruction classifies as a transform technique. If has been used for medical imaging of the human brain for the past several decades. Significant advantages of this method include (a) its noniterative character, (b) availability of analytical results on convergence of the solution with respect to the projection data, and (c) established error-estimates. A disadvantage to be noted is the large number of projections normally required for good accuracy. In engineering applications, this translates to costly experimentation, and nonviability of recording data in unsteady experiments the use of CBP continues to be seen in steady flow experiments, particularly when the region to mapped is physically small in size. The statement of the CBP algorithm is presented below.
Let the path integral equation be written as
where  is projection data recorded in the experiments and  is the function to be computed by inverting the above equation In practice, the function is a field variable such as density, void fraction, attenuation coefficient, refractive index, and temperature. The symbols  stand for the ray position, view angle, position within the object to be reconstructed, and the polar angle, respectively (fig. 4.50).The integration is performed with respect to the variable  along the chord  of the ray defined be  Following Herman [28], the projection slice theorem can be employed in the form
Where the overbar indicates the fourier transform and  is the spatial frequency. In words, this theorem states the equivalence of the one-dimensional Fourier transform of  with respect to s and the two-dimensional Fourier transform of  with respect to r and  A two–dimensional Fourier inversion of this theorem leads to the well-known Radon transform
where
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