6 Results on restoration of polarization images

 

In order to measure polarization structure in the vicinity of a bright point source, it is necessary to deconvolve the point source response from the data frames taken at each position angle of the polarizer and then to form the polarization maps from the deconvolved images. The aim here is to detect polarization structure within an offset distance of a few times the diffraction limit from the point source. Several different approaches to restoration have been attempted in order to obtain detailed information on the fine structure of the Homunculus nebula close to the central source $\eta$ Carinae. This was motivated by the need to detect and measure the polarization of the three knots found in the 0.4'' vicinity of $\eta$ Car by speckle imaging in the optical (Weigelt & Ebersberger 1986 and Falcke et al. 1996). The polarization data for $\eta$ Car will be used to exemplify these experiments; the scientific conclusions will be reported in Walsh & Ageorges (1999). A preliminary discussion of restoration of these images, without considering the polarization, has been given by Ageorges & Walsh (1998).

6.1 Image restoration trial

Two deconvolution techniques have been applied to the data: Richardson-Lucy (R-L) iterative deconvolution (Lucy 1974; Richardson 1972) and blind deconvolution ("IDAC'', Jefferies & Christou 1993; Christou et al. 1997). The major difference between these methods is related to the treatment of the point spread function (PSF). With the Richardson-Lucy method, a PSF is required a priori to deconvolve the data, while for blind deconvolution, the PSF is determined from variations in the target object data. The blind deconvolution method uses an initial estimate, which can be a Gaussian for example. Since the adaptive optics PSF changes with time and is not spatially invariant (see e.g. Christou et al. 1998), blind deconvolution should be better suited than the Richardson-Lucy method, which assumes a PSF constant in time. The exact spatial variation of the AO PSF is not known. However in the present case, this is a minor problem since the source itself ($\eta$ Car) has been used as wavefront sensor reference star. Moreover with the pixel scale chosen, all the valuable information in the short band data is enclosed in the isoplanatic angle; the spatial variation of the PSF is thus negligible over the area of the $\eta$ Carinae images, which is not the case for the time variation.

  

Figure 7: Comparison of the results of two deconvolution techniques for $K_{\rm c}$ data of $\eta$ Carinae and the Homunculus nebula. On the top row are shown data from the second polarizer sequence, at 0 (left) and 180$^\circ$ (right), both deconvolved with "plucy'' and convolved with a Gaussian of 3 pixels FWHM. The bottom left image is identical to the top left one but the data restored is from the first polarizer sequence (0$^\circ$ image). The result of blind deconvolution is presented in the bottom right hand image for the first data set - $K_{\rm c}^1$ - at 0$^\circ$ polarizer angle

  

Figure 8: PSF estimated (left) and obtained (right) from an IDAC blind deconvolution of the $K_{\rm c}$ image of $\eta$ Car (see Table 2 for observational details). The PSF estimate is the shift-and-add of the point source (HD 94510) observed shortly before the target data with the $K_{\rm c}$ filter. It has been used as first guess for blind deconvolution

  

Figure 9: $K_{\rm c}$ data of $\eta$ Car. at 0$^\circ$ for the two different data sets acquired. These images have been obtained after convergence of the blind deconvolution algorithm and reconvolution with a Gaussian of 3 pixels FWHM. This has to be compared with the right hand pictures of Fig.  7 top and bottom respectively, which are the equivalent results for "plucy'' deconvolution

A comparison of the Richardson-Lucy method and IDAC - "Iterative Deconvolution Algorithm in C'', i.e. the blind deconvolution algorithm used, was made using the $K_{\rm c}$ data on $\eta$ Car (Table 2). The aim was to test the reality of structures revealed in the near environment of the central star of this reflection nebula. For the R-L restoration, the Lucy-Hook algorithm (Hook & Lucy 1994), in its software implementation under IRAF ("plucy''), was employed. The principle is the same as for the Richardson-Lucy method, except that it restores in two channels, one for the point source and the other for the background (considered smooth at some spatial scale). The estimated position of the point source is provided and the initial guess for the background is flat. $K_{\rm c}$ data taken at polarizer angles of 0 and 180$^\circ$ were restored (called $K_{\rm c}$0 and $K_{\rm c}$180). For the $K_{\rm c}$0 image, blind deconvolution was also performed. It should be noted that although the polarizer angles are effectively identical, the Strehl ratio is not identical between the two data sets ($K_{\rm c}^1$ & $K_{\rm c}^2$) and is higher for $K_{\rm c}^1$ (27.9% against 22.1%). Although this could be considered an advantage, it has a drawback since the four bumps around the PSF (see Fig. 8 for the appearance of the PSF) are more pronounced. These bumps ("waffle pattern'') correspond to a null mode of the wavefront sensor as a result of an inadequacy in the control loop. The problem of the four bumps distributed symetrically around the source is that although they are in the PSF they do not vary; they are fixed in time and position and therefore not removed from the image as part of the PSF. There is however a way to overcome this problem, and that is by forcing them to be in the PSF.

Figure 7 presents the deconvolution results obtained with both methods on the two separate data sets (Table 1) and Fig. 8 shows the PSF derived from blind deconvolution. The "plucy'' deconvolved data have been restored to convergence and then convolved with a Gaussian of 3 pixels FWHM. The blind deconvolved data were not restored to convergence but limited to 1000 iterations to be comparable, in terms of number of iterations, with the Lucy deconvolution. The resulting image seems thus more noisy than the Lucy deconvolved ones. Note that neither of the methods used succeeded in removing the 4 bumps from the $K_{\rm c}$ images of the first observational sequence ($K_{\rm c}^1$0).

The data acquired at the polarizer position angles of 0 and 180$^\circ$, deconvolved with the same algorithm ("plucy'') both show identical structures (upper row of Fig. 7). This example serves to illustrate the stability of the "plucy'' method when applied to AO data while using a reasonable PSF estimate. The image from the first polarizer sequence, polarizer angle 0$^\circ$, deconvolved using IDAC is shown as the lower right image in Fig. 7 and is to be compared with the upper left image deconvolved with "plucy''. It is clear that similar structures appear in both restorations and that there are no significant features in one restoration which do not appear in the other. The differences in the images are mainly due to the fact that the blind deconvolution has been stopped before fully resolving the data and the final image is thus more noisy. Moreover the presence of the four bumps is enhanced in this image. The major difficulty in this deconvolution is that these noise structures are convolved with extended emission from the Homunculus nebula. Being in the middle of the nebula, the flux identified on these bumps is then a convolved product of the waffle pattern and the extended structure of the nebula. It is thus very difficult for the program to isolate these four "point sources'' and recover properly the true shape of the nebula at these positions. In order to fully compare the different deconvolution techniques, blind deconvolution has been pushed to convergence for $K_{\rm c}$0 (data set 1 & 2). The results (Fig. 9) are to be compared with the right hand side of Fig. 7. The structures close to $\eta$ Car emphasized by the two deconvolution processes, excluding the four bumps, confer a degree of confidence in the scientific results which will be presented in Walsh & Ageorges (1999).

6.2 Polarimetry restoration trial

In the case of polarimetric data, the deconvolution problem is more severe since the photometry must be preserved in the restored images in order to derive a polarization map. The Richardson-Lucy algorithm is superior to blind deconvolution in that it should preserve flux. Experiments were performed on the $K_{\rm c}$ $\eta$ Car data set, restoring each of the nine polarizer images with the PSF derived from the unpolarized standard at the same polarizer angle. The results were poor even when the restored image was convolved with a Gaussian of 3 pixels FWHM. They illustrate the effect of the variable PSF and thus the difficulty to recover polarization data at high angular resolution so close to the star. Huge fluctuations in the value of the polarization were seen in the vicinity of $\eta$ Car. The differing PSF of the unpolarized star and of $\eta$ Car (the AO correction was much better for the $\eta$ Car images than for the standard star) produced restored images with large differences in flux at a given pixel in the different polarization images. At present there is no known method to recover the true PSF from the data and conserve the flux through restoration. A possible (although computer intensive) solution is to determine the PSF from blind deconvolution and use the result for the PSF in another algorithm known to preserve the flux. This has been performed here: the PSF determined by blind deconvolution has been used both with the Richardson-Lucy and Lucy-Hook algorithms. Since the IDAC blind deconvolution algorithm normalised the input image at the beginning of the iterations, the final image was rescaled back to the original total count to allow error estimation of the polarization image. Polarization maps for the three methods ("IDAC'' alone and combined with R-L and "plucy'' methods) have been created and compared after reconvolution with a 3 pixel Gaussian.

  

Figure 10: Polarization map of $K_{\rm c}$ deconvolved data of $\eta$ Car. overplotted on the high resolution intensity map. For clarity, the polarization vectors have been calculated on pixels binned 4 by 4 (0.2 $\times$ 0.2''). The intensity map is at the nominal resolution of 0.1''

  

Figure 11: Differential polarization map of $K_{\rm c}$ deconvolved data of $\eta$ Carinae. It represents the difference between the polarization map derived after pure blind deconvolution and R-L deconvolution using the PSF derived by blind deconvolution. The errors are clearly under 5% except at the border and thus low signal_to_noise level of the nebula and at the position of the telescope spider ($\Delta \alpha$ and $\Delta \delta$ = 0). Compared to the rest of the map some bigger differences can also be found at $\approx$ 0.5'' from $\eta$ Car and is attributed to the effect of the PSF wings in the deconvolution process

From the high resolution restored images an attempt has been made to derive the polarization map. Figure 10 illustrates the result obtained while using the PSF determined by the blind deconvolution with the R-L algorithm (30 iterations with the accelerated version), after reconvolution with a Gaussian of 3 pixel FWHM. The overall centro-symmetric pattern of polarization observed at larger scale and resolution is recognisable here as well. The major deviation from this pattern at $\Delta \alpha$ and $\Delta \delta$ zero (i.e. east-west and north-south through the image of $\eta$ Car) is due to the spider of the telescope. The presence of this feature is hard to identify on the intensity map underplotted but clearly present at this position in the original (undeconvolved) data. Figure 11 is a vectorial difference between results obtained with Lucy deconvolution and blind deconvolution. Special care has been taken to avoid the vector difference to add when the position angles were separated by close to 2$\pi$. Some vectors at the border of the noise cut-off (e.g. at $\Delta \alpha \approx -3.0''$) detected in the Lucy map but not in the other are not represented here to avoid confusion with the differential vectors plotted. Major differences can be found at approximately 0.5'' from the center and correspond to differences in the deconvolution due to the wings of $\eta$ Carinae. At $\Delta \alpha$ = 0 and $\Delta \delta$ = 0 $\pm$ 0.3'', the important difference between the two reconstructed polarization maps is meaningless since these positions correspond to the spider of the telescope and the data are poorly restored here.