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It is possible to obtain an exact and unique solution of the source
function if and only if
is known for all
rays
that cut the region where
. That is for an
angle
we have to move the detector along
so that
measurements are made for all rays that go through the object. The
solution can be written as ( Gordon and Herman, 1974):
For the three-dimensional tomographic problem exact solutions can be
obtained if there is an observation point from which our image
projections are made on every plane
that cuts the region where the function
( Kudo and Saito, 1990; Smith, 1990). There exist a number of analytical
algorithms that determine
from a set of image projections which
closely satisfy the above condition ( Jacobsson, 1996; Grangeat, 1990).
However, in most cases these tomographic problems are ill-posed, meaning that they are sensitive to noise in the measurements or have either an infinite number of exact solutions or no exact solutions. With some regularization schemes, i.e. physical or mathematical constraints, the problem can, in a restricted sense, be transformed into a well-posed problem ( Ghosh Roy, 1991). Auroral and airglow tomography from ground-based multi-station measurements is ill-posed because only a few projections/images of the aurora are available and that the image data are all taken from below the object and thus are close to being linearly dependent.
The tomographic problem of ALIS is three-dimensional, and all performance,
resolution and error analysis on the actual geometry suffers badly from
the curse of dimensionality. That is, the total size of the
tomographic problem is proportional to the total number of pixels in
the images times the total number of volume elements (voxel = volume
element, small cube, c.f. pixel = picture element) in the retrieved
volume distribution. For a two-dimensional tomographic problem with
voxels per side, the number of unknowns is
projected onto
images with
pixels per one-dimensional image; the size of
the problem is
. For a three-dimensional tomographic system
the total number of pixels is
and the number of voxels
rises to
, giving a total size of the problem of
. For systems studying the ionosphere the
typical number of stations appears to be approximately 5 and typical
number of pixels is 1000 per image dimension. For the retrieved
volume distribution the number of voxels per side is of the
order of 100. When we go from a two-dimensional to a three-dimensional
tomographic problem the total size of the problem
increases by a factor
.
To simplify the analysis in this and the following two chapters, which describe the geometrical calibrations and the forward model, we will study a two-dimensional problem where emission distribution is to be determined in a vertical two-dimensional slice from one-dimensional images. This tomographic model system makes one-dimensional image measurements with the imaging characteristics of the ALIS cameras from four ground-based stations separated by 50 km from each other. The principles of tomographic inversion and all the characteristics of ground-based auroral tomography remain in this 2-D problem but the size is small enough to allow use of the most rigorous analysis of its resolution and retrieval characteristics.
Since ALIS currently only has six ground-based stations, and thus is very far from fulfilling the necessary requirements for obtaining exact reconstructions, we turn to approximate methods in order to obtain the best solution possible.
In order to be able to solve the tomographic problem it is necessary to have good knowledge about the forward problem or forward model. That is, we must know what fraction of the photons emitted in a volume element will be detected and where in the image the intensity will appear. In this model case we have to know how the auroral distribution in a slice is projected down onto the images, as depicted in Figure 3.3.
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If we solve the tomographic problem analytically, we will encouner the problem that the forward model is not a deterministic process, but rather a random process
Thus we see that the imaging process is a stochastic one for which the
probability of getting a set of measured images from a
source distribution
is:
The solution to the inverse problem is then to find the
source distribution
that is most probable given
the measurements
and considering
equation (3.9) and Bayes' rule the resulting problem is
The simple and robust iterative methods we have chosen are the
algebraic reconstruction technique (ART) and the simultaneous
iterative reconstruction technique (SIRT) ( Gordon and Herman, 1974). The
working principle of ART is then to take an initial guess
and project that guess down to an image by
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ART is a reconstruction scheme that converges fast but is
sensitive to noise in the images ( Aso et al., 1993). A modified version
is SIRT for which the current guess/solution is projected down
to all stations at once and the voxel intensity in the reconstruction
is updated with the average ratio of pixels from all stations.
It is possible to use a priori knowledge about the aurora, such as field-alignedness of the auroral structures ( Semeter, 1997; Aso et al., 1990), in the reconstruction process in order to bias the reconstruction and to avoid noise amplification thereby stabilising the solution. Several more general constraints can also be used to reduce the effect of noise in the images, e.g. spatial low pass filtering in 3-D as outlined in paper I, horizontal median filtering, horizontal smoothing of field aligned intensity variation Aso et al. (1993).
Several more advanced iterative schemes exist and the technically interested reader is directed to Veklerov and Llacer (1989b).