
Next: Pixel fieldofview
Up: The Forward model 
Previous: The Forward model 
Contents
Voxel  Pixel fieldofview intersection
In order to calculate the contribution from a voxel to a pixel in an
image, the volume of the intersection between the pixel fieldofview
and the voxel has to be calculated. The intersection between a voxel
and the pixel fieldofview is a body with between four and ten
faces. In order to avoid having to calculate the volume of the
complex body with all its bounding planes, the intersection volume
is approximated by the area spanned by the pixel fieldofview,
, at distance, , to the voxel
multiplied by the pixel lineofsight intersection length :

(5.3) 
In order to study
the error in the approximation, we turn to the corresponding
twodimensional example illustrated in
figure 5.2. The manageable problem is a narrow circle
sector intersecting three voxels with intensities
and, respectively. For a voxel representation of a
function in three dimensions to model the function well the function
should have slow spatial variations compared to the size of the
voxels, i.e. neighbouring voxels should have
intensities that do not differ much:
Where and are small compared to .
Figure 5.2:
Intersection between a ``2D voxel'' and a ``1D fieldofview''

The true contribution from voxel 1 is
. The
proposed approximation is that the contribution from voxel 1 is
given by
. Here is the area of region
. In order to calculate the difference between the exact and the
approximate contributions, we calculate the true and approximate
contributions from the region :
Calculating the difference between the true and the approximate
contribution, and using equations (5.4 and 5.5), gives:

(5.8) 
After doing some geometry and some algebra the areas of the different
regions can be obtained:


(5.9) 


(5.10) 


(5.11) 


(5.12) 
Substituting these equations into equation (5.8) and expanding
to the third order in , gives:

(5.13) 
Here it is seen that the approximate contribution, , is
correct to within the third order in and the
second order in . With values typical for ALIS and auroral
tomography this gives relative errors that are of the order
.
As proven the approximation for voxelpixel intersection is good
enough. A significantly more interesting topic is whether or not the
voxel approximation of the continuous volume emission distribution is
good enough. As a first test a simple model with only 1 voxel in
altitude and a 2D Gaussian intensity distribution is used with
various sizes of the voxels. These voxel representations are projected
down onto images with 256 by 256 pixels as shown in
figure 5.3.
Figure 5.3:
Projection of a twodimensional Gaussian with varying size
of the voxels. From left, 256 by 256 voxels with 0.5 km
horizontal side length, then respectively 128 by 128 voxels with
1 km side, 64 by 64 voxels with 2 km side and 32 by 32 voxels
with 4 km side.

It can be seen that for voxel sizes that are comparable to the
typical dimensions of the structures in the distribution the images
become coarse. Looking at the differences between images projected from
voxels with different resolutions, as presented in
figure 5.4,
Figure 5.4:
Differences between the projection from 256 by 256 voxels
and, from left, respectively, projection from 128 by 128 voxels,
from 64 by 64 voxels, and from 32 by 32 voxels.

it is seen that the voxel
representation needs to be a posteriori justified, or that the
number of voxels must be increased when the result of the
tomographic inversion shows structures in the volume emission
distribution which have widths of only a few voxels.
Next: Pixel fieldofview
Up: The Forward model 
Previous: The Forward model 
Contents
copyright Björn Gustavsson 20001024
