Fast Backprojection Operator for Synchrotron Tomographic Data

Reduction of computational time in high resolution image reconstruction is essential in basic research and applications as well. This reduction is important for different types of traditional non diffractive tomography in medical diagnosis as well as for applications in nanomaterials research, related to modern technologies. Alternatives to alleviate the computationally intense part of each iteration of iterative methods in tomographic reconstruction have all been based on interpolation over a regular grid in the Fourier domain or in fast nonuniform Fourier transforms. Both approaches speed up substantially the computation of each iteration of classical algorithms, but are not suitable for being used in a large class of more advanced faster algorithms: incremental methods such as OS-EM, BRAMLA or BSREM, among others, cannot benefit from these techniques. The backprojection is a stacking operator, known to be the adjoint of the Radon transform. As a mapping \(\mathcal{B}\), the backprojection can be recast as a convolution operator, in a different coordinate system, which is an improvement in accelerating the computation of \(\mathcal{B}\). In this work, we propose several analytical representations for the operator \(\mathcal{B}\), in order to find a fast algorithm.