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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
l. Determination of required, ideal frequency response
H(w) (see Fig. 1) — determination of cut — off frequency
We
2. Calculation of infinite impulse response hı(n) via
inverse transformation of ideal frequency response
3. Truncating impulse response using a window function
to receive finite and causal impulse response. The size of
window is equal to the filter order.
The impulse responses of a FIR filter are also the filter
coefficients.
In the first phase a one-dimensional filter was elaborated, and
then it was transformed to the 2-dimensional filter for the 3-
dimensional space. The determined filter coefficients h(n/,n2)
were used to filter the analysed topographical surface
FFT testing and determination of the cut-off frequency
Using FFT the amplitude spectrum and power spectrum is
determined. From those graphs a concrete information
characterizing the topographical surface in the frequency
domain can be obtained. The example of periodogram is
presented on Fig 2.
Fourier Frequency Spectrum
$27
a
t2
Amplitude
a ot [M as 04 os
Frequency
Figure 2. The example of a periodogram
The cut-off frequency can be determined by the graph analysis.
That parameter posses the following properties (Hassan, 1988):
eAt a low frequencies, below the cut-off frequency, the
periodogram values come out mainly from the true
elevations of the topographical surface
eAt the frequencies close to the cut-off frequency the
influence of noise (field objects) and the true terrain
elevations is identical.
e At the frequencies higher than the cut-off frequency, the
periodogram values come out mainly from the noise.
In this area of the periodogram mainly the small
values appear, and the curve shape is rather irregular.
Determination of the filter order using the geostatistical
theory
The innovative value of geostatistics, invented by George
Matheron in Fontainebleau (France) in 1960-ies, consists in
treating an analysed parameter, in our case it is the elevation, as
a "regionalized variable" (Matheron, 1962, 1963). The values of
that "regionalized variable" are a function of locations of
measured points. The structure of variability is described in a
synthetic form by the so-called semivariogram, which
determines dependence between the average variance of the
1149
differences of elevation values (semivariance) and distances
between the surveyed points.
For a regular grid of measured points the semivariance values
y(d) are given by equation:
2.[z(x;) = zx; m dr
y(d)- ^ (9)
2n
n
where z(xj, z(x;td) — the elevations values in the points
separated by the distance d
n — the number pairs of surveyed points separated by
distance d
The semivariogram is a plot of semivariance as a function of
distance between measured points (Fig. 3).
A ©
y(d) «—— —»
C +Co [77 m7
Co
»
d
Figure 3. A hypothetical semivariogram. C, - a local variability
in the scale of a single sample, so-called the nugget
effect; C - the sill value; @ - the influence range; d
— distance (lag); (d) - semivariance
To enable a quantitative assessment of variability of elevations
on the tested surfaces, it is necessary to estimate the empirical
semivariograms by the simple analytical functions, which can
be treated as the geostatistical variability models. In
geostatistics occur a number of semivariogram models, among
others: linear, spherical, logarithmic, Gaussian, exponential, etc
(David, 1998). The analysis of theoretical semivariograms
provides the so-called range of influence c (see Fig. 3). The
range can be used as a measure of spatial dependency. This
range it is a greatest distance from the analysed point to the
surrounding points, which have real influence on the value of
the interpolated parameter assigned to the analysed point. In a
more general sense, range of influence is used to predict
optimal filter order for use in filtering.
3. THE EMPIRICAL RESEARCH
3.1 The test surfaces
A laser data, concerning two test fields (Vaihingen/Enz, and
Stuttgart), provided by the OEEPE project, were used (Pfeifer
et al., 2001). The topographical surfaces of those two test fields
are very rough, of diversified shape and field objects.
In the both cases the first and last laser impulse reflection was
registered together with the returned laser impulse intensity.
