Up: How to determine the
Subsections
2 The method
Given an image I_{i,j} (corrected for flatfield and darkcurrent), 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 ,
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:

(1) 
for a fraction
of the pairs (i,j); here n_{i,j} is the number of counts recorded in the pixel i,j and
is the mean number of counts in the image.
If all the image is detailed, after dividing it into
windows of dimensions
,
we expect that a good number of them continue to contain structures.
I define
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, .
Dividing the image into
windows, we can ask what the probability, P, is
that a given window is detailed: to a first approximation
.
In other terms, we have
measurements, each of which can give a positive
or negative result with probability
or
,
respectively.
Following information theory (Bijaoui 1981), the specific
information we obtain for one of such a series of measurements is given by the
expression:

(2) 
The "best'' ratio,
,
can now be defined as the ratio that provides the largest specific information; then (i.e. the step which provides the best ratio) is the best sampling scale.
Therefore,
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 (
pixels) obtained
on 1999 July 1 with the Themis IPM. Image scale:

The reliability of the estimate is measured by the peak value itself, which
stays between 0 (
,
)
and 1 (
). In general,
reliable estimates are obtained for
.
This is illustrated
by Fig. 1, in which
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
is .
The results, for the image of the solar granulation and the
blank one, are
and
,
respectively.
These values support the choice made for
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
loses its
significance, which allows one to work with the same statistical function for
any p, provided that
.
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