13.7.1 The CLEAN method

The standard deconvolution technique, CLEAN relies on such a qualitative constraint: it assumes that the sky brightness is essentially a ensemble of point sources (the sky is dark, but full of stars). The algorithm which derives from such an assumption is straightforward. It is a simple ``matching pursuit''

1.

Initialize a Residual map to the Dirty map

2.

Initialize a Clean component list to zero.

3.

Assume strongest feature in Residual map originates from a point source

4.

Add a fraction $\gamma$ (the Loop Gain) of this point source to the Clean component list, remove the same fraction, convolved with the dirty beam, from the Residual map.

5.

If the strongest feature in the Residual map is larger than some threshold, go back to point 3 (each such step is called an iteration).

6.

If the strongest feature is below threshold, of if the number of iterations N_iter is too large, go to point 7.

7.

Convolve the Clean component list by a properly chosen Clean Beam (this is called the restoration step).

8.

add to the result the Residual map to obtain the Clean Map.

The CLEAN algorithm as a number of free parameters. The loop gain controls the convergence of the method. In theory, $0 < \gamma < 2$, but in practice one should use $\gamma \simeq 0.1 - 0.2$, depending on sidelobe levels, source structure and dynamic range. While high values of $\gamma$ would in principle give faster convergence, since the remaining flux is $\propto (1-\gamma)^{N_{\mathrm{iter}}}$ if the object is made of a single point source, deviations from an ideal convolution equation force to use significantly lower values in order to avoid non linear amplifications of errors. Such deviations from the ideal convolution equation are unavoidable because of thermal noise, and also of phase and amplitude errors which distort the dirty beam.

The threshold for convergence and number of iterations define to which accuracy the deconvolution proceeds. It is common practice to CLEAN down to about the noise level or slightly below. However, in case of strong sources, the residuals may be dominated by dynamic range limitations.

The clean beam used in the restoration step plays an important role. It is usually selected as a 2-D Gaussian, which allows the convolution to be computed by a simple Fourier transform, although other choices could be possible. The size of the clean beam is a key parameter. It should be selected to match the (inner part of) the dirty beam, otherwise the flux density estimates may be incorrect. To understand this problem, let us note first that the units of the dirty image are undefined. Simply, a 1 Jy isolated point source appears with a peak value of 1 in the dirty map. This is no longer true (because of sidelobes) if there is more than one point source, or a fortiori, an extended source. The unit of the clean image is well defined: it is Jy per beam, which can easily be converted to brightness temperature from the effective clean beam solid angle and the observing wavelength. Now, assume the source being observed is just composed of 2 separate point sources of equal flux, and that the dirty beam is essentially a Gaussian. Let us clean the dirty image in such a way that only 1 of the 2 point sources is actually included in the clean component list. If we restore the clean image with a clean beam which is, e.g. twice smaller than the original dirty beam, the final result will undoubtedly be odd. The second source would appear extended and have a larger flux than the first one. No such problem appears if the clean beam matches the dirty beam. Admittedly, the above example shows a problem which results from a combination of two effects: an inappropriate choice for the clean beam, and an insufficient deconvolution. However, the second problem always exists to some extent, because of noise in the original data set. Hence, to minimize errors, it is important to match the clean and dirty beams.

Note that in some circumstances, there may be no proper choice. An example is a dirty beam with narrow central peak on top of a broad ``shoulder''. Small scale structures will be properly reconstructed, but larger ones not.

The last step in the CLEAN method plays a double role. On one hand, it protects against insufficient deconvolution. Furthermore, since the residual image should be essentially noise if the deconvolution has converged, it allows noise estimate on the cleaned image.