International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
Surface Creeted Usng Nearest Nelohbour interpolation on a 1m Grid
no
Pest ores
x Coordnate
Surface Crested Using Ecubic Interpolation on a tm Gad
x coordinate
Sutsce Crealed Using Bi^armmonic Spios interpolation on a tm God
Height imedres)
x coacdeute
y coordnw e
(d)
Figure 2: Showing the (a)bilinear, (b) nearest neighbour, (c)
bicubic, and (d) biharmonic spline surfaces created using the
interpolation methods at 1m grid spacing. For information
regarding the specifics of the four algorithms see Sandwell
(1987), Watson (1992), and Smith et al (2003b).
The differences in surface form between the DSMs shown in
Figure 2 can clearly be seen. The nearest neighbour surface
is blocky and 'stepped' in appearance, whilst the spline
surface is much smoother, with many of the building and
vegetation edges appearing curved. The quantified
differences between the surface height predictions and the
raw data are presented below.
2.2 Comparing the Surfaces
For the purposes of this investigation, the error ( € ) at each
investigated point within the surfaces was considered to be
998
the difference between the raw data point, Zx)., and the
interpolated value, Zi(x), for that location (see eq.
below).
&(x) 2 Z(x) - Zi(x) (1)
where &(x) = measured error at location (X), Z height
value, Zi = interpolated height value.
This calculation was repeated across the surface to assess the
success of the four interpolation algorithms at the
investigated grid spacings. Model suitability was assessed in
relation to how much error was introduced to the surface by
each of the techniques - the most suitable model being the
one which introduced the least error ( € ).
In order to assess model suitability, some of the raw data
points had to be omitted from the surfacing process. If all of
the data points were used to interpolate a surface, then the
goodness-of-fit of the surface could not be assessed with
these same data points as this would yield an overly
optimistic (low) prediction error. For this reason a standard
validation procedure was employed which involved omitting
some raw data from the interpolation procedure and assessing
the success of the procedure to model in the absence of these
known values. The method chosen is called the split-sample
validation routine which is advocated by Declercq (1996). In
this procedure part of the sample values are omitted, the
interpolation is performed, and the difference between the
predicted and the raw data values at these locations are
calculated. This difference is then used as a measure of the
success of the algorithm. The usefulness of this technique is
increased when it is used iteratively, where the number of
omitted points is progressively increased and the differences
calculated. This can return useful information regarding the
stability of the algorithm, and its ability to cope with
differences in input point density. The split-sample
methodology was used in this way in Lloyd and Atkinson
(2002). The authors used sample sizes of 95%, 50% and 25%
of original points to measure the effects of interpolation for
rural DEMs. The methodology outlined in Lloyd and
Atkinson (2002) is adhered to in this study, and the same
proportions of omitted data are used here for the comparison
of interpolation methods. For the grid spacing investigation a
random selection of 5% of the raw points were omitted and
the surfaces produced at different resolutions. The success of
each resolution was then assessed by calculating the
difference between the omitted data and the surface
predictions at these locations. The amount of omitted data
points in each sample was not varied for this part of the
investigation.
In each investigation the omitted data points were chosen ina
random selection process, and the tests run multiple times to
ensure different points were selected each time and similar
results obtained. This ensured the reliability of the
investigation.
3. RESULTS
The model errors were calculated in accordance with
Equation (1), and are recorded in Table 1 below.
[nternatic
Eds
Table 1: "
Assess th
|
'INTERPOLA
195% Split sz
{Bilinear
iBicubic
[Nearest Nei
IBiharmonic :
L————
L————
[450 Split s:
|Bilinear
iBicubic
Nearest Nei
|Biharmonic
|
i
Bilinear
[ficubic
{Nearest Nei
IBiharmonic
Fam
Table |
method «
there 1s :
A discus
Table 2
Designec
the Meth
INTERPOLAT
Bilinear
Bicubic
Nearest Neigl
Biharmonic Sj
31 Dis
The res
bicubic
of all th
spline n
to be c:
strong €
unconst
and mi
general,
highest
across
found |
probabl
there ar
very s
necessa
In terr
resoluti
interpo
errors,
betwee
was, he