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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
Table 1: The Results from Split-Sample Routine Designed to
Assess the Suitability of Each Interpolation Technique
Error (2) Descriptive Statistics
Max Min Mean Std Dev Rmse
Ya A a
INTERPOLATION METHOD
195% Split sample
Bilinear 538085 6.8551 -0.1444° 19168 1.8999
IBicubic 51996: -3.0855. -0.0085. 1.5887. 1.5499
INearest Neighbour 13.34 1163 01398 3.5087 3.4848
IBiharmonic Spline 7.1825 127806; -0.3186° 2.6703 2.6697
|
[450 Split sample
IBilinear 10.1808. 14.8968 01296. 29564. 29367
IBicubic . 12.3605! -15709| 0.1721: 30472 3049
Nearest Neighbour 15.37 -15.4.. 0.2402, 3.4617, 3.4674
IBiharmonic Spline 15.1701, -15.7385| 0216. 3.0555. 3.0608
|
25% Split Sample
Igiinear 133931. -15.1774| 01575. 29994 3.0018
|Bicubic . 113701 -16.7194: 0.1026. 29863 2.9861
INearest Neighbour 1599 -1614 0.1597; 3.6569! 3.6585
[Biharmonic Spline 117705! -182333 0.1919; 30508 3.0553
Table 1 above shows the results from the interpolation
method comparison on a 1m grid. It can clearly be seen that
there is a difference in the statistics of the calculated errors.
A discussion of these results follows in section 3.1.
Table 2: The Results from the Split-Sample Routine
Designed to Quantify the Differences in Errors Introduced by
the Methods at a Variety of Grid Resolutions.
Error (:) Descriptive Statistics
Max error (m) {Min error(m) Mean (m) :Std Dev (m) RMSE (m)
INTERPOLATION METHOD
Bilinear Am 5.56 444. 902 1.98 1.96
2m 524 7.26 -0.16 1.93 1.92
4m 4.29 -7.44 -0.73 2.12 2.22
Bicubic 1m 6.13 6.30 0.02 2.64 2.01
Im’ 5.83! 657 015. 1.96 194
4m 2.94 -7.50 -0.49 2.02 2.05
Nearest Neighbour Am 8.40 714; 0.29 2.58, 2.58
2m 8.40] £94 03% 241 242
4m 8.33 -1472 -0.06 2.67 2.65
Biharmonic Spline 1m 7.301 627 0.13, 2.18 2.7
2m, 92 515 0.16 240, 239
Am 13.09 2273 -0.28 3.93 3.91
3.1 Discussion
The results presented in Table 1 show that bilinear and
bicubic algorithms were found to produce the lowest RMSE
of all the interpolators. It was surprising that the biharmonic
spline method did not produce lower errors. This was thought
to be caused by the tendency for this algorithm to produce
strong artificial oscillations in the surface reconstruction in
unconstrained regions as indicated by the large maximum
and minimum errors. The results also showed that, in
general, the nearest neighbour interpolator produced the
highest RMSE value. Despite preserving discontinuities
across the surface, the nearest neighbour algorithm was
found to introduce a large amounts of error. This was
probably due to its inability to model oblique surfaces - as
there are no slopes the changes between groups of values are
Very steep and create discontinuities which are not
necessarily present in the raw data.
In terms of differences in errors created at different
resolutions, the bilinear, bicubic, and biharmonic splining
Interpolators produced relatively stable range and mean
errors, and there appeared to be only a minimal difference
between the surfaces produced at different resolutions. It
was, however, noted that the resolution which produced the
999
lowest error was that which was as close as possible to the
original point spacing, which in this instance was ~2m. The
nearest neighbour interpolator produced higher errors at
larger grid spacings than any of the other methods. In all
cases interpolation onto the 4m grid resulted in higher errors,
due to the loss of information in this approach. The increase
in error was in the order of 50-80cm. In such cases it remains
the decision of the end-user as to whether the decreases in
accuracy are outweighed by faster computation and smaller
file sizes.
3.2 The Spatial Pattern of Error
The pattern of individual errors was examined by plotting the
locations of the interpolated points and assigning them a size
in proportion to the error calculated for that point (Figure 3).
It was observed that there was a strong spatial dependence of
the highest magnitude errors in the biharmonic splined
surface, and that many of the highest magnitude errors
occurred at the edges of the dataset. It was considered that
these edge errors were skewing the statistical analysis. As
such it is suggested that the biharmonic spline should be used
with an edge buffer to eliminate some of the largest errors
(see Smith et al, 2003b). It was also noted that there was a
general patterns of larger errors which coincided with the
occurrence of breaklines in the dataset (in this instance
breaklines included building and vegetation edges).
However, there was a clear difference in the amounts of
errors over different breaklines, with the errors caused by
building breaklines tending to be smaller than those caused
by vegetation.
- + - Men, Soi P EE + —— À n codem
44287 — 44281 44294 44299 44268 44267 44098 44258 4.07
y 10
Figure 3 Errors created in the interpolation of DSM using
Biharmonic Spline method on a 2m grid. Size of the disks
indicates amount of error at that location. Contours underlain
for context. Note the occurrence of some large errors at the
edges of the dataset. This edge effect for the biharmonic
splining algorithm is explored further in Smith et al (2002b).
There was also a difference noted in the pattern of errors
produced by different methods at different resolutions.
Figure 4 shows the differences in the distribution of errors
for both the bilinear and the nearest neighbour methods at the
3 different scales. It can be seen that the errors produced by
the nearest neighbour method are much more dense and of
higher magnitude over vegetated areas than the bilinear
errors. The increase in errors over breaklines at higher grid
resolutions can also be observed. Refer to Figure 1 for
context.
