Gel'fand-Graev's Reconstruction Formula in the 3D Real Space - a Framework towards a General Interior Tomography Theory

In [1-4], I. M. Gel'fand and M. I. Graev proposed inversion formulas for x-ray transforms in different spaces. In particular, Gel’fand-Graev’s inversion formula [1] is a fundamental relationship linking projection data to the Hilbert transform of an image to be reconstructed. This finding was re-discovered in the CT field; see [5-9]. It has wide applications, including local reconstruction [10-11], backprojection filtration (BPF) [12], interior tomography [13-17], and limited-angle tomography [18]. For a survey, see [19, 20]. Despite its high information density, Gel’fand-Graev’s inversion formula [1] was cast in high dimensions and specialized terms, and difficult to follow for a well-trained engineer. In this poster, we represent this formula and its proof for the 1D x-ray transform in a 3D real space for easy access and further extension.