Blind Deconvolution

In image processing, blind deconvolution is a technique used to correct blurred images when the system Point Spread Function (PSF) is unknown or poorly known. The PSF is then estimated from the image. This technique has been used for decades (Ayers and Dainty 1988 Levin et al. 2009) and in this section a blind deconvolution iterative algorithm is applied to spectro-spatial data to study its applicability. 

In the case of one iteration of the blind deconvolution algorithm (Fig. 5.9) it can be observed that around the sources the background level has decreased slightly and hence the contrast has been increased. The detected PSF size is comparable to the theoretically expected, this is Xlbmax and the spatial size of the sources has slightly decreased compared to the dirty image (Fig. 5.6 top row). However, although the PSF includes some of the ripples expected from the dirty beam these ripples have not been removed in the deconvolution itself. 

For a 2-iteration blind deconvolution, the results do not improve. Although the PSF beam size is still consistent with the one expected from theory, the ripples have vanished from the PSF and are still present in the recovered datacube. The spatial size of the gaussian source approximates to a point source for the maximum wavenumber image (right) and does not correspond to the expected spatial size. 

When 5 blind deconvolution iterations are applied to the dirty data cube, the beam size does not correspond to the one expected from theory anymore, from which one can infer that the recovered datacube is unrealistic. 

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. 

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Figure 5. Example of blind deconvolution in action. Left tme object brightness and PSF. Middle simulation of corresponding observed image. Right the two components found by blind deconvolution. Source Thiebaut, 2002,...

In the limit Wpsp 0 or if no calibration data are available, myopic deconvolution becomes identical to blind deconvolution which involves to find the PSF and the brightness distribution of the object from only an image of the object. 

Stated like this, conventional and blind deconvolution appear to be just two extreme cases of the more general myopic deconvolution problem. We however have seen that conventional deconvolution is easier to perform than myopic deconvolution and we can anticipate that blind deconvolution must be far more difficult. 

Nevertheless a number of blind deconvolution algorithms have been devised which are able to notably improve the quality of real (i.e. noisy) astronomical images (e.g. Ayers and Dainty, 1988 Lane, 1992 Thiebaut and Conan, 1995). 

Figure 6b. Microscopic image of chromosomes improved by blind deconvolution.

For instance and following the MAP approach, blind deconvolution involves the minimization of the join criterion (Thiebaut and Conan, 1995 Thiebaut, 2002) ... 

Figure 5 shows an example of blind deconvolution by the resulting algorithm applied to simulated data. Of course the interest of blind deconvolution is not restricted to astronomy and it can be applied to other cases for which the instrumental response cannot be properly calibrated for instance in medical imaging (see Fig. 6a and Fig. 6b). 

Thiebaut, E., Conan, J.-M., 1995, Strict a priori constraints for maximum likelihood blind deconvolution, JOSA.A, 12, 485 Thiebaut, E., 2002, Optimization issues in blind deconvolution algorithms, SPIE 4847, 174... 

Fig. 5.9 Synthesised datacube layer after 1 iteration of the blind deconvolution algorithm for the minimum wavenumber (25cm , / ), central wavenumber (118cm , centre) and maximum wavenumber (212cm, right) (top). Restored PSF at each of the previous wavenumbers (bottom)...

Fig. 5.12 Spectral results of the Master simulation after 1 iteration of the blind deconvolution algorithm for the central pixel of the gaussian source (blue), the point source (green) tind the central pixel eUiptical source (red) (left). Detected spectra for three positions in the sky where no source...

G.R. Ayers, J. Christopher, Dainty. Iterative blind deconvolution method and its applications. Optics Lett. 13(7), 547-549 (1988)... 

A. Levin, Y. Weiss, F. Durand, W.T. Freeman. Understemding rmd evaluating blind deconvolution algorithms, in IEEE Conference on ComputerVision and Pattern Recognition, 2009. CVPR 2009... 

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. 

Kobayashi Y. Toyoda H. (1999). Development of an Optical Joint Transform Correlation System for Fingerprint Recognition, Opt Eng., Vol.38, p>p.l250-1210 McCaUum B. C. (1990). Blind Deconvolution by Simulated Annealing, Opt Commun., Vol.75, pp.101-105... 

In the blind deconvolution approach, the object and the PSF are assumed to be unknown and are estimated throughout the iterative process [34-36]. The algorithm uses the standard MLE algorithm described above, together with a PSF estimation for each iteration. The object is computed, using the MLE estimation, as follows ... 

Autoquant s adaptive blind deconvolution algorithm (Media Cybernetics, Rockville, MD). 

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