578 
spatial frequencies are blocked then the result is the cross 
shaped mask shown in figure 6(c). The inverse transform is 
shown in figure 6(0 with the difference image in figure 6(i). 
On the whole the differences between the original and the 
inverse transform are extremely small. The differences lie 
mostly along a single row. In the original image that row is a 
an atmospheric absorption feature which implies a high 
frequency component to the spectral direction of the signal. 
The small scale variations in the signal along that row imply 
high frequency in the spatial domain also. Thus the differ 
ences are the result of blocking the high-spectral/high-spatial 
frequencies, exactly as expected. 
Figures 7(a)-(i) show the exact same sequence of transforma 
tions using a 10 element wide Gaussian shaped transition 
rather than a sharp cut-off. In fact this is a much more realistic 
representation of how spectral or spatial mode sampling is 
actually be done. This reduces the ringing considerably in the 
spectral and spatial modes (figures 7(g) & (h)). When applied 
to the cross-shaped mask there appears to be shift in the 
ringing to lower frequencies and there is also a reduction in the 
RMS error from 10.5 digital-levels to 9.4. 
DISCUSSION 
An important consideration when using any sampling 
technique is how much of the signal is lost irretrievably. This 
can be computed from the full CCD array and expressed by 
the information density function. This calculation provides an 
accurate quantitative estimate of the size of the error expected 
in the reconstructed image (Huck, 1985). The total power in 
the signal from all elements can be found by integrating the 
power spectrum. The power in the signal under the cross can 
also be computed. The ratio of the power under the cross to 
the total power quantifies how much of the original CCD full 
frame signal is contained in the sample. If the fraction is very 
high then the sample can be reliably used to reconstruct the full 
CCD frame. Otherwise care must be taken in interpretation. 
The cross pattern sampling is the simplest method that makes 
sampling in Fourier domain worthwhile. The number of 
samples could be varied across the frequency domain so that 
areas of strong signal power could be sampled more densely 
than areas of of weak power, regardless of whether they fall 
along the axes. What is required is some model of where the 
power in the signal will fall within the power spectrum. 
CONCLUSIONS 
What implications does this have for imaging spectrometers 
and how their data should be sampled? The spatial domain 
should not be undersampled, in general, because the spatial 
frequency content of the image cannot be predicted and 
therefore the resulting image can have spurious values which 
can lead to incorrect image analysis using linear systems 
theory. In the spectral domain the image can be undersampled 
because the spectral correlations are much better behaved and 
are much more predictable. Undersampling should only be 
done when the behaviour of the intervening samples is known 
or predictable. This is often the case with the spectral signal 
and is the assumption that has been used (perhaps unwittingly) 
by analysts to interpret multispectral remote sensing imagery 
for years. 
It is apparent from Fourier transforming a typical imaging 
spectrometer CCD array that the spatial/spectral frequency 
content of the imaging spectrometer data is mainly one of low- 
spatial/low-spectral, low-spatial/high-spectral or high spatial/ 
low-spectral. There is very little power in the high-spatial/ 
high-spectral frequencies. If we are going to sample from the 
CCD array in the most efficient manner, then we must sample 
where we expect to find the power in the signal. In the 
imaging spectrometer the power in the signal is in the cross 
pattern that lies along the spatial and spectral frequency axes. 
The cross pattern can be most efficiently sampled in the 
Fourier domain and must be recorded there to realize the data 
reduction. 
Reconstructing the full spatial and spectral resolution is not 
possible in the most rigorous sense. The low-spatial and low- 
spectral frequencies will be faithfully represented. The low- 
spatial/high-spectral or high-spatial/low-spectral frequencies 
will be reconstructed with good results but aliasing from the 
high-spectral/high-spatial frequencies can occur. If high- 
spectral/high-spatial frequencies are present in the signal they 
will be aliased. This reconstruction method will give good 
results if the energy in the signal in the high-spectral/high- 
spatial frequencies is a small fraction of the total power. In 
general the accuracy of the reconstruction will depend on the 
fraction of the power, of the total signal, which falls into the 
high-spatial/high-spectral range. 
ACKNOWLEDGEMENTS 
We would like to thank the people at Moniteq Ltd. for 
allowing us to work with the FLI which lead to this paper and 
for letting us include the FLI imagery. We are indebted to 
Itres Research Ltd. for allowing us to use the CASI to collect 
the unique full frame spectrometer imagery. Thanks to Wayne 
Rasband of the National Institute of Health and Arlo Reeves of 
Dartmouth College who developed NIH Image and the FFT 
enhancements to it. Thanks also to Bill Connor who let us use 
a prelease version of Imagine. Both of these programs made 
the data processing so much easier on the Macintosh. We also 
wish to gratefully acknowledge the support of the Earth 
Observations Laboratory, personnel and its Director Ellsworth 
LeDrew. This research is supported, in part, by a Centre of 
Excellence grant from the Province of Ontario to the Institute 
for Space and Terrestrial Science and an NSERC operating 
grant to John R. Miller. 
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