3 Discussion

First of all I show the principal characteristic that differentiates the OWM from the contrast method. As already mentioned in the introduction, the contrast is sensitive, above all, to the variations of intensity on a large scale. This is made clear by writing the contrast as:

$$\sigma = \sqrt{\sum_i P_i},$$ (3)

where P_i is the power of i-th Fourier frequency and remembering that, in the case of granulation, the power falls drastically for spatial frequencies of the order of $8\ {\rm Mm}^{-1}$ (Schmidt et al. 1981). Such a limit makes the method effective when we compare two similar images captured with a short time difference from each other, but the method is less reliable when we want to choose the best images in a long series.

Figure 3: The images of different quality class, from the same series as that in Fig. 2: a) contrast = 4.4%, $p_{\rm o}=6$, b) contrast = 5%, $p_{\rm o}=8$. The indication given by the two quality parameters are opposite. The contrast give image b) as the best one, while it is apparent by eye that image b) as a better resolution

Figure 4: Quality maps of the images shown in Fig. 3. The images were divided in $6\times 6$ windows, corresponding to the optimum width for image a). The detailed windows are givn in white, "flat'' windows in black. Note the presence in b) of a large black region which, nevertheless, contributes significantly to the higher contrast of b) with respect to a)

To quantify the above, I show, in Fig. 3, two images of granulation recorded with $\sim 7$ min difference. The contrast shows image b) to be better than a) contradicting simple visual analysis. The OWM, on the contrary, correctly classifies the two images, providing for a) the parameter $p_{\rm o}=6$ and for b) $p_{\rm o}=8$ (remember that good images correspond to small parameters). Besides, the method gives the quality map of the aforesaid images, in which the more degraded zones can be identified. In other words, the OWM is sensitive directly to the size of the structures present in the image, while the contrast method is determined only by the percentage of white or black pixels present in it. To clarify this idea, we consider two chessboards of black and white squares of $4\times 4$ and 8 $\times$ 8 pixels: evidently both chessboards have the same contrast, while my method provides as quality parameter 3 for the first one and 6 for the second.

One of the main properties of the method is that it provides reliable results even for small images. To clarify the meaning of the term "small'', consider Fig. 5, which shows an artificial image with uniform quality over its entire area. We estimate $p_{\rm o}$ considering at first the whole image, then only $1\!/4$, $1\!/8$ of it, and so on. The estimates are essentially constant as a function of the size of the sub-image which is considered ( $p_{\rm o}=4$ in the interval size = 256, size = 12). The OWM gives coherent results down to a minimum size of $12\times 12$ pixel, i.e. for images as small as 3 times the estimated $p_{\rm o}$ (equal to 4 in this case).

Figure 5: Artificial image with uniform quality. White spots represent the peak of Gaussians with a full width at half maximum (FWHM) of 6 pix; the maximum is 1000 ADU. Nearby peaks are separated by 5 pix

Figure 6: $p_{\rm o}$ as a function of the FWHM of the Gaussian which has been used to degrade the image in Fig. 2

Consider now the convolution of the image of Fig. 2 with Gaussians of FWHM = 2 pixel, 3 pixel, etc. The values of $p_{\rm o}$ as a function of the adopted FWHM are plotted in Fig. 6, which shows that the method is sensitive even to the smallest degradation factors. Finally I have analysed a series of 608 images taken at Themis IPM during about one hour of observation on 1999 July 1. In Fig. 7 the results obtained with the OWM are compared with the contrast method. The plot shows that there are many intervals of contrast which do not identify a unique class of quality. In other words, there is a danger of the situation described in Figs. 3 and 4 occurring.