Fourier-based forward and back projectors for iterative tomographic image reconstruction

Forward projectors are needed for iterative image reconstruction algorithms for CT or PET image reconstruction. The major cost of iterative reconstruction algorithms is computing the forward and backward projections. Conventionally, this is done in the image domain as computing the lengths of intersections between each X-ray ray and each image voxel. These operations are the major computational bottleneck in acquire forward and backward projections (since we have to loop all the pixels for each projection ray). We implemented a Fourier based forward projector for 2D (aiming at 3D) transmission tomographic imaging (such as CT). This projector speeds up the iterative reconstruction algorithms, without much lost of image quality. By using Fourier-based forward and backprojection, we can reduce the computational cost from O(N) of the line-intersection method to O(1/2N log N) [Fes02][KS88], where N is the total number of pixels to be evaluated. This is related to non uniform FFT [Fes02]. The key step influencing the image quality and computational time of the Fourier based projectors is the interpolation between polar and Cartesian frequency samples. In the project, we compared several different interpolation methods. The ultimate goal is to use this fast Fourier forward projection in the transmission tomography reconstruction (like CT). In these iterative algorithms, we have many such forward and backward loops. The projectors have to be accurate enough so that the errors do not accumulate as iteration goes. We also test the performance of the forward back projectors in such one loop.