Figure 4. (a) FLI spatial image of the Bruce Peninsula and Lake Huron, (b) FLI spectral image of the same scene as in (a).
(c) Overlay of the spatial and spectral images (a) and (b). Geometric correction was done using the AIR II system at
Moniteq Ltd.
Truncation Window Artifacts
Any rectangular feature in an image will transform into the
Fourier domain to have a strong cross in its power spectrum.
Since the discrete two-dimensional Fourier transform (DFT)
of an image is effectively the transform of the periodic
extension of the image, which is tiled in the plane to infinity in
each direction, it has an inherent rectangular feature. The
sharp edges at the boundaries where the tiles meet, increases
the image’s power spectral density in the horizontal and
vertical directions. This manifests itself as the cross-shape in
the power spectrum. By apodizing the image, so that there is
a smooth transition to zero at the boundaries, the sharp edge
discontinuities due to an image’s periodic extension disappear,
effectively removing the cross-shaped artifact from the power
spectrum.
If this is done to spectral/spatial frames from a CCD array of
an imaging spectrometer, the cross-shape pattern is reduced
but is not removed. The power in the signal is still concen
trated along the frequency axes because of the strong
correlation between the spectral elements and between the
spatial elements, but not between the spectral and spatial
elements. Most of the features of the CCD image are linear
and align themselves with either the spatial or the spectral
directions but do not appear to cross the image at other inter
mediary angles; that is features are localized in the spatial
direction or the spectral direction but rarely both
simultaneously.
Figure 4(a) shows an FLI spatial mode image of the coastline
of the Bruce Peninsula on the Lake Huron side. The cross
shaped artifact can be seen in the power spectrum of the image
which is shown in figure 5(a). Note that the origin (zero
frequency) is at the centre. This artifact can be removed by
apodizing the window so that the gray-levels go gradually to
zero at the bounds of the window. This has been done in the
figure 5(d) by using a circular Gaussian apodization with a
width that is 5% of the image. Finally, figure 5(g) shows the
power spectrum of the apodized image and the cross has
disappeared. The cross-shape in the unapodized power
spectrum was purely an artifact of the rectangular window.
Figure 4(b) shows the spectral mode image of the same scene
with the pixels placed in their correct geometric position and
size. Notice the rake effect that results from the gaps between
the recorded spatial elements. Figure 5(b) shows the power
spectrum of the spectral mode image with multiple peaks
which due to the gaps between the the tines of the rake. Each
of the peaks bears the distinctive cross-shape. After apodi
zation (figure 5(e)) the cross-shape which is aligned with
truncation window disappears and the power spectrum
becomes much more like the spatial mode image particularly in
the low frequencies.
Compare the results of performing the same sequence of
operations on an image of the CAS I CCD array with that of the
FLI images. Figure 1 (b) shows the CCD array image where
the horizontal and vertical directions correspond to the spatial
and spectral directions respectively. Figure 5(c) is its power
spectrum where the horizontal and vertical directions corre
spond to the spatial frequency and the spectral frequency
directions respectively. The cross pattern is, again, very
pronounced in the power spectrum. After apodizing the image
(shown in figure 5(f)) and taking the power spectrum (figure
5(i)), the cross pattern is still very strong. Here the cross is
inherent to the data rather than purely an artifact of the
truncation window, illustrating a significant difference
between a spectral-spatial image and a spatial-spatial image.
DOING THE SAMPLING
This marked difference can be exploited by a new sampling
strategy which only uses the signal from within the cross,
rather than uniformly across the entire tile in the frequency
domain. This is based on the assumption that the dominant
power in the signal will fall into one of three categories; low-
spatial with high-spectral frequency, high-spatial with low-
spectral frequency or low-spatial with low-spectral frequency.
The signal component which is high-spatial with high-spectral
frequency will have Fourier coefficients will not be recorded
and are taken to be zero.
The advantage of the method is that it can dramatically reduce
the data rates required for recording and/or downlinking. The
signal can be more accurately reconstructed using this type of
sampling than from either spectral of spatial mode data
samples alone. However, this alters the nature of the imaging
spectrometer into one more like a synthetic aperture radar in
that the signal must be transformed after recording before it
can be viewed as an image. The data reduction can only be
realized by recording/downlinking in the transformed state.
Sampling Examples
Consider the power spectrum shown in figure 1(b). If only
the 16 lowest spectral frequencies are passed (figure 6(a)),
then the inverse transform (figure 6(d)) has all of the fine
structure in the spectrum smoothed out. This is a type of
spatial mode sampling which captures the low-spectral/low-
spatial and the low-spectral/high-spatial frequencies. The
difference between the original CCD image frame and the
inverse transform is shown in figure 6(g). Notice that there is
considerable ringing. The same is be done in the spatial
direction in figure 6(b) with the inverse transform in figure 6
(e) and the difference image is in figure 6(h). This is a type of
spectral mode sampling which captures the low-spectral/low-
spatial and the high-spectral/low-spatial frequencies. Again,
ringing is very prominent. In both cases resolution has been
lost and ringing has corrupted the imagery because of the
sharp cut-offs used in the Fourier domain.
If the filtering is done so that only the high-spectral/high-
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