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Stopping criteria
It has long been known that iterative solutions to tomographic
problems improve initially but deteriorate after some point
( Gordon and Herman, 1974). This highlights the importance of knowing
when it is time to stop.
What we want to do is to test whether the reconstruction gives
projections that are statistically compatible with the observed
images. If we had a large number of samples from one random
process, it is possible to test whether the random process has a
hypothetical probability distribution . The method to do this
is the test of fit for which the range of the random
variable is divided into intervals, with theoretical
probabilities for a random sample to fall into the
interval if the hypothesis
is true.
With a total of samples
and samples in
interval , the test function

(3.15) 
has an asymptotic distribution with degrees of
freedom. is a measure of the deviation between the true
distribution and the hypothetical one. The hypothesis is accepted if
the value calculated from equation (3.15) is less than a
critical value corresponding to a chosen significance level
( Bronstein and Semendyayev, 1997).
In the case of tomography, the pixels in the projection images of the
reconstructed threedimensional source distribution are the expected
values of the probability distributions from which the measured pixel
values are the random samples. Here we only have one sample from each
probability distribution; it should be noted that the
probability distribution for a pixel value is known and assumed to be
determined only from the expected value.
For auroral imaging with ALIS, the pixel intensity has a probability
distribution that is
as
derived in section 3.1. With the
above relation between expected pixel intensity and the
random distribution of the measured pixel intensity, it is possible
to use the above test after the modification that the
random process for each pixel is divided into intervals with equal
probabilities and that the measured pixel intensity is put
into its corresponding interval.
The hypothesis that the measured images are random samples from
a set of random processes with expected values
is
accepted if the value calculated from
equation (3.15) is less than is required at the corresponding
significance level. (The values of
are calculated from
the source distribution.) For processes that have continuous probability distributions, it is
no problem to divide the range into equal intervals for different
expected values and widths but for discrete probability
distributions this can be a problem; for Poissonian processes a
method to overcome this problem is described by Veklerov and Llacer (1987).
An example that can illustrate the working procedure outlined above is
to look at the outcome of the roll of four nonstandard dice. What
separates these dice from ordinary dice is that they have
unknown numbers of dots; that
is, they might have dots from 1 to 6 or from 3 to 8 or any other
sequence of
dots
. If we make a measurement of the dot numbers
and then get two models which estimate the expected value of each
die, as presented in table 3.4, the algorithm for this
stopping criteria is as follows.
Table 3.1:
The number range denotes that the die have faces
with
eyes.

First we divide the probability distribution into a number of
intervals, for this case two intervals with
and with . For model 1 all events
fall into interval 1 and the value according to
equation (3.15) is:

(3.16) 
and for model 2 die falls into interval and dice
and fall into interval giving a value, according to
equation (3.15), of:

(3.17) 
For a significance level of and one degree of freedom, the
critical value of is
3.84, i.e. if is larger than 3.84 we must reject the
hypothesis that the model is the cause of the observations. Model 1,
with
, is not likely given the observations and has to be
rejected, and for model 2, with
, we cannot reject the
hypothesis and should be satisfied and stop the iteration.
If we have come this far the reconstruction gives projections
that fit the measured images in a statistical sense. However, there
are not yet any guarantees that the spatial distribution of the errors
is well behaved, i.e., the large positive errors might be grouped
together in one region of the images and small errors
might be grouped in another region. One further condition for
optimal reconstructions is that the spatial distribution of the
errors is also random. This can be tested with the same method as
above: the image location of errors in an intensity interval
is calculated and then the images are divided into 10 by 10 regions
and the number of errors from the interval is
calculated region by region. If the test function

(3.18) 
where is the total number of errors in the intensity error
interval and is the total number of regions, is
also less than the corresponding significance value, there is no
reason to continue with the iteration.
By the way, about the cartoon animation on the even pages  it starts
of at page 2. The images in the top right corner are ALIS data of HF
enhanced airglow from Silkimuotka and the images in the lower left
corner is the projected images of the retrieved volume emission. The
animation starts at 17:40:15 UT with 10 s between each frame and
covers about one HFpump onoff cycle.
Next: Summary
Up: Tomography
Previous: Error sensitivity
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copyright Björn Gustavsson 20001024
