Fast Cross Correlation for Limited Angle Tomographic Data

The cross-correlation is a fundamental operation in signal processing, as it is a measure of similarity and a tool to find translations between signals. Its implementation in Fourier space is used for large datasets, as it is faster than the one in real space, however, it does not consider any special properties which signals may have, as is the case of Limited Angle Tomography. The Fourier space of limited angle tomograms, which are reconstructed from a reduced number of projections, has a large number of empty values. As a consequence, most operations needed to calculate the cross-correlation are executed on empty data. To address this issue, we propose the projected Cross Correlation (pCC) method, which calculates the cross-correlation between a reference and a limited angle tomogram more efficiently. To reduce the number of operations, pCC follows a project, cross-correlate, reconstruct process, instead of the typical reconstruct, cross-correlate process. Both methods are equivalent, but the proposed one has lower computational complexity and provides significant speedup for larger tomograms, as we confirm with our experiments. Additionally, we propose the usage of a \(l_1\) penalty on the cross-correlation to improve its sensitivity and its robustness to noise. Our experimental results show that the improvements are achieved with no significant additional computational cost.