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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004
2.1.3 Modulation Transfer Function Calculation: Once
the edge spread function has been determined, the MTF can be
calculated as follows. The line spread function of the system is
calculated by taking the numerical derivative of the equally
spaced ESF samples. A Fast Fourier Transform is performed on
the resulting LSF, the normalized magnitude of which yields
MTF. Care must be taken in selecting the number of points
calculated along the ESF with respect to the sampling rate in
order to obtain the desired number of points in the resulting
MTF. Appropriate scaling of the frequency axis of the MTF
must also be performed to represent the calculated MTF in
terms of the Nyquist frequency of the imaging system. The
Nyquist frequency is defined as the highest sinusoidal
frequency that can be represented by a sampled signal and is
equal to one half the sampling rate of the system.
2.2 Algorithm Calibration
2.2.1 Calibration Approach: Once the algorithm has been
developed, it is important to determine the accuracy and
precision of the measurement tool. Identification of systematic
errors can improve the overall accuracy of the measurement and
quantification of precision allows error estimates to be
associated with individual measurements.
In order to characterize performance, images of synthetic edges
were generated with varying edge characteristics such as edge
magnitude, length, noise, angle and sharpness. As the edges are
computer generated, the actual value of each parameter is
known by construction. The MTF measurement algorithm is
then run on each edge and results compared to the actual values.
Surfaces are then fit to the measured bias and standard
deviation as a function of edge characteristics.
Subsequently, when an edge is analyzed from an operational
image, the edge characteristics are also measured. Prior to
reporting results from the algorithm, the measurement values
are corrected for bias and an error estimate is assigned based on
the fits obtained from the synthetic edges.
2.2.2 Test Suite: In generating the test set, a preliminary
experiment was performed to determine the edge characteristics
that impact the bias and standard deviation of the results from
the synthetic edges. The values of the parameters spanned the
range expected to be obtained operationally from 1 meter GSD
imagery. From the list of characteristics above, it was
determined that the measurement bias and standard deviation
was invariant to edge angle and edge sharpness. Note that while
the edge measurement technique is invariant to edge angle,
operationally, the edge angle is limited to small angles around
the along scan and cross scan principal directions.
A full scale experiment was then performed using eight values
of edge magnitude, five values of noise standard deviation and
eight levels of edge length. This results in 320 test cases. One
hundred edges for each test case were then generated and
analyzed, bringing the total experiment sample size to 32,000
edges. For each edge constructed, the edge angle and sub-pixel
edge location were randomized over small ranges. These
variations, along with the actual Gaussian noise sequence,
characterized by the noise standard deviation parameter,
provided the various instances of edges used in the calibration
experiment.
2.2.3 Calibration Results: For each test case, consisting of
a common edge magnitude, noise standard deviation and edge
length, the mean and standard deviation of the MTF
measurement at eight spatial frequency points were tabulated. A
multiple parameter least squares regression was performed on
the standard deviations with 93 degrees of freedom (Mandel,
1964). The correlation coefficient of the regression for the
Nyquist frequency is 0.92. Surface plots for the fit of the
standard deviation of MTF measurements at the Nyquist
frequency as a function of edge magnitude and noise standard
deviation for two edge lengths are shown in Figure 3.
Fit to Standard Deviation of Nyquist MTF
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Figure 3a. Surface fit for standard deviation of Nyquist MTF for
edge length = 30
Fit to Stondard Deviation of Nyquist MTF
p.06: Edge Lengin = 60
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Std. Dev. ot Nyquist MTF
Figure 3b. Surface fit for standard deviation of Nyquist MTF
for edge length = 60
Observe that the results of the regression follow intuition. For
example, as the noise increases, the standard deviation of MTF
increases or alternatively, confidence decreases. As edge
magnitude increases, confidence increases and as edge length
increases, confidence increases.
The bias for each point is determined by subtracting the true
value used to construct the edge from the mean value measured
by the algorithm. Again, a multiple parameter least squares
regression was performed on the measurement bias with 93
degrees of freedom. The correlation coefficient of the
regression for the Nyquist frequency is 0.63. A plot of the MTF
bias at the Nyquist frequency as a function of test case number
and the residual error after bias correction is shown in Figure 4.
