A convolution backprojection formula for three-dimensional vector tomography

Three-dimensional (3-D) tomographic measurements of vector flow fields are possible using acoustical, optical, and magnetic resonance imaging. A general mathematical framework for this type of data acquisition called the probe transform has been studied by the author in previous work. It has been shown that the irrotational field component can be reconstructed using only one probe direction, and that the solenoidal field component can be reconstructed using only two probe directions. Previously, the reconstruction formulas were of the backprojection-convolution type; that is, first backproject then convolve in 3-D. In this paper a convolution-backprojection formula, which has significant computational savings, is developed. Numerical simulations are also given to demonstrate the performance of this new approach