Clean algorithm

One way of linearizing the problem is to use the method of least squares in an iterative linear differential correction technique (McCalla, 1967). This approach has been used by Taylor et al. (1980) to solve the problem of modeling two-dimensional electrophoresis gel separations of protein mixtures. One may also treat the components—in the present case spectral lines—one at a time, approximating each by a linear least-squares fit. Once fitted, a component may be subtracted from the data, the next component fitted, and so forth. To refine the overall fit, individual components may be added separately back to the data, refitted, and again removed. This approach is the basis of the CLEAN algorithm that is employed to remove antenna-pattern sidelobes in radio-astronomy imagery (Hogbom, 1974) and is also the basis of a method that may be used to deal with other two-dimensional problems (Lutin et al., 1978 Jansson et al, 1983). 

This general model, usually referred to as dark spectmm, allows a variety of reconstmction methods to be employed (see Sect. 6). The CLEAN algorithm, proposed originally for the reconstmction of two-dimensional maps in radio astronomy [86], utilizes essentially the same signal properties. It is noteworthy that the dark spectmm model is especially well suited to multidimensional NMR spectroscopy. 

Fig. 9 The principle of the CLEAN algorithm visualized on a simulation of three signals of relative amplitudes 1 5 10 and equal decay rates. The sparsely sampled signal (a) is Fourier transformed (b), then the mask (c) is subtracted to yield the residual spectrum. Reconstruction after the first iteration is shown (d). The final result of the CLEAN procedure (e) can be used to obtain reconstruction of the time-domain signal (f)...

Finally, the residual spectrum may be added to replica in order to retain smaller features that might have been omitted by the CLEAN algorithms, or to reintroduce the usual noise a(/) The latter might be useful to judge which peaks selected during the iterations are false [87]. It was emphasized that displaying the replica without the addition of residual spectrum is merely a cosmetic operation and does not improve the sensitivity at all [88]. 

It is noteworthy that the uncertainty of peak amplimdes caused by the presence of noise e(f) limits the capability of CLEAN algorithm to improve the quality of spectrum [89]. This, however, should apply for the most of reconstruction algorithms, e.g., similar conclusions were drawn for the maximum entropy method [90]. 

It has also been agreed that a fixed number of iterations is difficult to apply in practice for NMR spectra and could lead to misinterpretations of the results of CLEAN algorithm [89]. Therefore, one should rather use the intensity threshold as the stopping condition. 

It should be noted that the use of an appropriate analytic function instead of discrete replica may be beneficial as this (1) is less influenced by noise and (2) allows one to reproduce the wings of the resonances, which are neglected in the original CLEAN algorithm due to intensity threshold. On the other hand, if the line widths in the Fourier domain are mainly due to signal truncation, the fitted parameters poorly reflect the tme signal properties, and this may affect the performance of described procedure. 

The original CLEAN algorithm was invented to deconvolve effectively the Fourier spectrum from the PSF. In radio astronomy it was either impossible or impractical to arrange detectors on a regularly spaced grid due to malfunctioning of the part of equipment, occultations caused by the Moon, or if telescopes were operating over a... 

According to these observations, the power of CLEAN algorithm was utilized in high-dimensional (3D and 4D) NMR spectroscopy of proteins (see Fig. 11), where sparse sampling has to be employed due to practical limitation on experiment time (see Sect. 9.4). 

In conclusion, the application of CLEAN algorithm to sparsely sampled data is especially beneficial if (1) the technique features good thermal sensitivity and (2) a high dynamic range of peak amplitudes is expected. Otherwise, artifact suppression is hampered or irrelevant in view of the general noise level. 

In this thesis the CLEAN algorithm is used for the data synthesis of Double Fourier Modulation data and is described in detail in Chap. 5. In general terms, it is basically a numerical deconvolving process applied in the 0x, 0y) domain. It is an iterative process, which consist of breaking down the intensity distribution into point source responses, and then replacing each one with the corresponding response to a clean beam, this is, a beam free of side lobes. 

In conclusion, a blind deconvolution algorithm is not suitable for DFM dirty datacubes because for a proper restoration, more information regarding the dirty beam has to be applied. As the information of the dirty beam can be extracted from the known v-map, previous knowledge of the dirty beam can be used. One algorithm that makes use of a known dirty beam is the interferometric CLEAN algorithm. 

The CLEAN algorithm (Hogbom 1974) is essentially a brute force deconvolution. The starting point is the fact that the measured dirty image Id 0x. y) is the convolution of the true intensity or sky map I(6x, 9y) with the dirty beam B(0x, 9y), this is... 

The CLEAN algorithm is vastly used and is one of the most successM iterative deconvolution algorithms which attempts to obtain an image compatible with the data assuming it is made of point-like sources. 

The input to AIPS is the dirty image (in this case the dirty data cube) and the dirty beam. After applying the CLEAN algorithm for each spectral channel, the CLEAN data cube is obtained. 

Fig. 5.16 Synthesised datacube layer after restoration with the CLEAN algorithm for the minimum wavenumber (25cm , left), central wavenumber (llScm, centre) and maximum wavenumber (212 cm , right) (top). Corresponding interferometric dirty beam at each of the previous wavenumbers (bottom)...

Pig. 5.18 Spectral results of the Master simulation after restoration with the CLEAN algorithm for the central pixel of the gaussian source (blue), the point source (green) and the central pixel elliptical source (red) Qeft). Detected spectra for three positions in the sky where no source intensity is expected (right)... 

In conclusion, these results suggest that the CLEAN algorithm is valid for Double Fourier Modulation data. Applying some FTS algorithms (i.e. phase correction algorithms) in localised areas where there is only one source or the sources are unresolved could be included to further improve the results in the spectral domain. 

The synthesis of Double Fourier Modulation data has been performed with a blind deconvolution algorithm and with the CLEAN algorithm via AIPS (Greisen et al. 2003). Only the second one has been proved robust enough for DFM data. 

The next step will be to perform some modifications on the CLEAN algorithm by coding a DFM specific one, in order to include some FTS algorithms such as phase correction and apodization. Apodization requires a unique zero path difference (ZPD), and research would need to be performed to design tailored apodization filters for DFM data. 

C. Buccella, M. Feliziani, and G. Manzi, Detection and localization of defects in shielded cables by time-domain measurements with UWB pulse injection and clean algorithm postprocessing, IEEE Transactions on Electromagnetic Compatibility, vol. 46, no. 4, pp. 597-605, 2004. 

Deconvolution algorithms rely on a priori information about the sky brightness to interpolate into portions of the u-v plane that were not sampled. The two most common methods are clean and maximum entropy method (MEM). The former assumes that the sky is mostly dark and that sources can be described as a combination of point sources, while the latter assumes positivity for the sky brightness of sources. Figure 3 illustrates deconvolution with the CLEAN algorithm. 

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