2 The method

Given an image I_i,j (corrected for flat-field and dark-current), we say that it is of good quality if it contains numerous structures of small dimension. The relationship between the number of structures at the different scales defines the optimal analysis scale for the image. The fundamental hypothesis of the whole method is that such a scale, which we will call $p_{\rm o}$, is a good estimator of image quality. We observe that, given an image, it is easy to decide if it is blank (uniformly illuminated) or contains structures (detailed). I define an image to be blank if:

$$\overline n - \sqrt{\overline n} < n_{i,j} <\overline n + \sqrt{\overline n}$$ (1)

for a fraction $\alpha$ of the pairs (i,j); here n_i,j is the number of counts recorded in the pixel i,j and $\overline n$ is the mean number of counts in the image. If all the image is detailed, after dividing it into $N_{\rm p}$ windows of dimensions $p\times p$, we expect that a good number of them continue to contain structures. I define $r_{\rm p}\equiv n_{\rm p}/N_{\rm p}$ as the ratio between the number of detailed windows and the total number of windows. Information theory will provide (as we will see shortly) a criterion for determining the optimum sampling dimension, $p_{\rm o}$.

2.1 The criterion of maximum specific information

Dividing the image into $N_{\rm p}$ windows, we can ask what the probability, P, is that a given window is detailed: to a first approximation $P_{\rm p}=r_{\rm p}$.

In other terms, we have $N_{\rm p}$ measurements, each of which can give a positive or negative result with probability $P_{\rm p}$ or $(1-P_{\rm p})$, respectively. Following information theory (Bijaoui 1981), the specific information we obtain for one of such a series of measurements is given by the expression:

$$S_{\rm p} = -r_{\rm p}\cdot \ln_2(r_{\rm p}) - (1-r_{\rm p})\cdot \ln_2(1-r_{\rm p}).$$ (2)

The "best'' ratio, $r_{p_{\rm o}}$, can now be defined as the ratio that provides the largest specific information; then $p_{\rm o}$(i.e. the step which provides the best ratio) is the best sampling scale. Therefore, $p_{\rm o}$ directly provides an estimate of the image quality: the smaller its value, the better the image.

Figure 1: Specific information as a function of the scale, p, for an image of solar granulation and a numerical simulation of a uniformly illuminated image (the lowest amplitude curve). Although the latter presents a peak, it is much smaller than unity; therefore the measurement is not reliable

Figure 2: Image of solar granulation ( $223\times 223$ pixels) obtained on 1999 July 1 with the Themis IPM. Image scale: $29\hbox {$^{\prime \prime }$ }\times 29\hbox {$^{\prime \prime }$ }$

The reliability of the estimate is measured by the peak value itself, which stays between 0 ( $r_{\rm p}=0$, $r_{\rm p}=1$) and 1 ( $r_{\rm p}=1\!/2$). In general, reliable estimates are obtained for $S_{p_{\rm o}}\simeq 1$. This is illustrated by Fig. 1, in which $S_{\rm p}$ is plotted as a function of p for the image of solar granulation shown in Fig. 2 and for a numerical simulation of a blank image. The value adopted for $\alpha$ is $68\%$. The results, for the image of the solar granulation and the blank one, are $S_{p_{\rm o}}\simeq 1$ and $S_{p_{\rm o}}<0.7$, respectively.

These values support the choice made for $\alpha$ and, at the same time, also provide an answer to a possible criticism to the method, based on the fact that statistics in Eq. (1) ave not valid for small values of p. In fact, when errors due to a wrong choice of the statistics are large, as in the blank image, then the estimate of $p_{\rm o}$ loses its significance, which allows one to work with the same statistical function for any p, provided that $S_{p_{\rm o}}>0.7$.