A Direct convolution algorithm for image reconstruction

In image reconstruction algorithms, the choices of filter functions and interpolating functions are very important for the computational speed and the quality of the image reconstructed, especially, for fan-beam geometry, the occurrence of the singular integral operator may lead to some great oscillations compared to the original image. In this paper we will give a direct convolution algorithm which needs not the complex computations occuring in the Fourier transform, then using a circle integral we obtain a stable computational program. Different from all other previous window functions used by many pioneer researchers, in our algorithm* we choose a window function similar to Gabor’s window function 1 $$e^{ - \frac{{x^2 }}{2}} $$ , which can be regarded as the approximation to the inverse Fourier transform of a locally integrable frequency function. Also we point out that such reconstruction algorithm procedures can be used to deal with the SPECT projection data with constant attenuation.