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CLEAN

The original Hogböm CLEAN algorithm can be considered as a pattern recognition procedure whereby the strongest beam pattern was searched for in the residual image. The best match was obtained by cross-correlating the residual image with the PSF, which for uniform weighting can be approximated by looking for the peak in the residual image alone. In the 4-dimensional space used here, it is not clear just what is meant by the peak. One possible definition of the peak is of the length of the vector $\vec{\cal I}_{k}^{}$ : I² + Q² + U² + V². With such a choice, the CLEAN image solver would know nothing about the physical interpretation of the components of $\vec{\cal I}\,$ , in particular that $\vec{\cal I}^{T}_{}$ M $\vec{\cal I}\,$ $\geq$ 0. This is somewhat strange but is analogous to the case of CLEANing separately the dirty images for the different Stokes parameters.

To introduce some physics, one could use the maximum eigenvalue of the coherence matrix: I + $\sqrt{Q^2+U^2+V^2}$ or perhaps the determinant: I² - Q² - U² - V². The latter form is blind to totally polarized emission so the former choice is probably more useful. In either case, one would probably want to add an additional criterion that both eigenvalues be non-negative. Hence a reasonable prescription would be to search for a peak in I + $\sqrt{Q^2+U^2+V^2}$ for those dirty image points for which I² - Q² - U² - V² $\geq$ 0.

For the GI, the PSF can vary considerably with position, and so searching for the peak could be considerably more computationally expensive.

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