573 
Figure 2. (a) Sampling from the CCD array in spatial 
mode, (b) Sampling from the CCD array in 
spectral mode. These are as is done by the FLI 
and CASI instruments. 
array of the FLI and CASI. The rectangular area represents 
the 2-dimensional CCD array and the masks cover the 
elements of the array included in the summations that produce 
the bands. Conversely, the spectral mode data is recorded 
with full spectral resolution and is reduced spatially. Figure 2 
(b) shows the method used by the FLI and CASI. Robust 
models of the expected spectral signal's shape can be used to 
compensate for information lost during sampling, but the 
spatial signal does not yield to such models readily. 
A common method of spatial mode sampling, which has been 
used in multispectral scanners since their inception, is to 
average over discrete, strategically placed, spectral bands. 
This is done on the premise that the spectra will be well- 
behaved between the bands, and even more importantly, 
within the bands, so that the shape of the entire spectra can be 
correctly inferred. 
To reconstruct the spectrum a model is required, which can be 
as simple as linear interpolation or as complex as an inverted 
Gaussian (Miller et al., 1990). The model can provide the 
missing component of the signal that was not measured, yet is 
well understood, to aid in the reconstruction. For example 
absorption features can be avoided based their well known 
spectral position and width. This is done because the 
absorption feature contributes high frequency of, known 
phase, whose content is undesirable. Conversely, the inverted 
Gaussian model rejects the high frequencies of the signal to 
precisely parameterize the phase and the dominant frequency 
that characterizes the subtle changes in slope and position of 
the vegetation reflectance red edge. The drawback of under 
sampling comes when the assumptions of the model break 
down, or when the expected spectrum is not known, so no 
model is applicable. The aliasing that occurs as a result of 
undersampling makes it impossible to distinguish between 
high and low spectral frequencies. 
Alternatively, the spectral range can broken into uniform 
contiguous spectral bands that sample the signal optimally 
according to linear system theory. No assumptions are made 
about the well-behavedness of the signal, but no use is made 
of the spectra's characteristic structure either. If only a small 
number of bands are allowed and the spectral range is great 
then the high frequency portion of the signal will be filtered 
out and the resulting spectrum will be featureless and uninte 
resting. However, in terms of sampling theory, this is the best 
approach because the power in the aliased frequencies is 
minimized and therefore the signal can be accurately recon 
structed, albeit at low resolution. 
Spectral mode only became possible with the advent of 
Unsampled 
Spectral 
Spatial 
Overlap 
- max 0 max 
Spatial Frequency 
(cycles/element) 
Figure 3. Representation of the spectral and spatial modes 
in the Fouier domain. Note the Unsampled 
region contains the high-spectral/high-spatial 
frequencies. 
imaging spectrometers, but the concept of spatial averaging to 
change the resolution of an image is not new. The spatial 
signal is not well behaved and does not yield to models that 
would help with interpolation of an undersampled signal. The 
only tenable model of the signal is given by linear systems 
theory which requires that the sampling be done in accordance 
with the Nyquist criterion. 
Spectral mode undersampling is quite different than for spatial 
mode because the digital levels of the pixels in a scene, 
although strongly correlated, are essentially random events. 
The spatial location of any arbitrary feature cannot be 
predicted, since no model of how the signal will vary across 
the scan-line is possible. When the signal is undersampled 
then the only valid way that the signal can be analyzed is using 
probability theory. 
FOURIER DOMAIN REPRESENTATION 
The two data sets can be overlaid in the Fourier domain to 
elegantly demonstrate what information the data samples do, 
and do not, contain. Consider the discrete 2-dimensional 
Fourier transform of the 2-D signal measured by an imaging 
spectrometer for a single image line. The name given to the 
Fourier spectrum of the spectral signal will be the spectral 
frequency spectrum and is analogous to the Fourier spatial 
frequency spectrum. The use of the term spectrum in 
reference to the Fourier spectrum must be distinguished from 
the spectrum used to describe intensity of light within a range 
of electromagnetic wavelengths. 
The spatial mode data has high spatial resolution and low 
spectral resolution and so represents full coverage of the 
spatial frequency spectrum but only covers the lower range of 
spectral frequencies (see figure 3, "Spatial"). The spectral 
mode data is the converse with high spectral resolution and 
low spatial resolution, so it covers the full spectral frequency 
spectrum but only the lower range of spatial frequencies 
(figure 3, "Spectral"). As is evident in Figure 3, overlaying 
the two spatial/spectral frequency spectrum plots results in two 
regions which are uniquely sampled by the individual modes, 
a region that is sampled by both modes and a region that is not 
sampled at all. The region which is labeled “Overlap” repre 
sents the low spatial and low spectral resolution image that 
could be formed either by averaging the spatial data in the 
spectral direction or the spectral data in the spatial direction. 
Thus the overlap region represents truly redundant data. The 
area labeled “Unsampled” represents the portion of the signal 
that contains high frequency both spatially and spectrally.