HOGBOM

The simplest strategy of the iterative search was introduced by Högbom (1974). It works as follows

1. Localization of the strongest intensity pixel in the current residual map: $\ensuremath{\mathrm{max}}(\ensuremath{\left\vert I_\ensuremath{\mathrm{res}} \right\vert})$.
2. Add $\gamma.\ensuremath{\mathrm{max}}(\ensuremath{\left\vert I_\ensuremath{\mathrm{res}} \right\vert})$ and its spatial position to the clean component list.
3. Convolution of $\gamma.\ensuremath{\mathrm{max}}(\ensuremath{\left\vert I_\ensuremath{\mathrm{res}} \right\vert})$ by the dirty beam.
4. Subtract the resulting convolution from the residual map in order to clean out the side lobes associated to the localized clean component.

$\gamma$ is the loop gain. It controls the convergence of the method. In theory, $0 < \gamma < 2$. $\gamma =1$ would in principle give faster convergence, since the remaining flux at one position is $\propto (1-\gamma)^{n_{{\mathrm{comp}}}}$, where $n_{\mathrm{comp}}$ is the number of clean components found at this position. But, in practice, one should use $\gamma \simeq 0.1 - 0.2$, depending on sidelobe levels, source structure and dynamic range. Indeed, deviations (such as thermal noise, phase noise or calibration errors) from an ideal convolution equation force to use low gain values in order to avoid non linear amplifications of errors.

An important property of HOGBOM algorithm is that only the inner quarter of the dirty image can be properly cleaned when dirty beam and images are computed on the same spatial grid. Indeed, the subtraction of the dirty sidelobes associated to any clean component is possible only in the spatial extent of the dirty beam image. When the user defines a support (a priori knowledge), the cleaned region becomes even smaller than the inner quarter of the dirty map.