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International Archives of the Photogrammetry, Remote Sensing and Spatial [Information Sciences, Vol XXXV, Part B2. Istanbul 2004
The corresponding bi-dimensional formulation of the generic g
order spline can be obtained simply by:
piso LL = EY 0) (6)
Figure 4 shows the behaviour of the first order bi-dimensional
spline known also as bilinear spline.
Figure 4. Bi-dimensional first order spline or bilinear spline
If we suppose that d(t) can be modelled as:
N
dit) 9 Ag" (rr) (7)
A
k=
the spline coefficients {A} can be estimated from the
corresponding observation equations:
d.t.) = NS REC. Ej + Vi (8)
kzl
by using the classic least square estimation method.
This ordinary spline interpolation approach suffers a rank
deficiency problem when the spatial distribution of the data is
not homogeneous. To make evident this concept, in figure 5.a a
sample of 7 observations and the first order splines, whose
coefficients we want to estimate, are shown. With this data
configuration the third spline can not be determined because its
coefficient never appear in the observation equations: the
unacceptable interpolation results is shown in figure 5.b.
The simplest way to avoid this problem is to decrease the spline
resolution with the consequent decreasing of the interpolation
accuracy, especially where the original field d(t) shows high
variability.
Since homologous points detected on geographical maps are
usually not regularly distributed in space, the use of single
resolution spline functions leads to two different scenarios.
In the first one, with low resolution spline functions, the
interpolating surface is stiff also in zones where a great amount
of points is available.
d(t)
\ A ^ ^N A
/ \ ZEN \ V CUN
z \ / \ 4 x \
4 x \ / \ ‘ 4 \
‘ / / ~ i =
4 X K x « s
^ s \ 4X / x
/ / 4 s / \ \ s
z \ / \ 4 \ 4 \
4 / NU N s \
z M M. x x b
(a)
(b)
Figure 5. Examples of mono-resolution spline interpolation:
data (a) and interpolation (b)
On the opposite, in the second case, corresponding to high
resolution spline functions, a more adaptive surface is obtained
but the lack of points in some area can give rise to local
phenomena of rank deficiency, making the interpolation
unfeasible. The multiresolution approach removes this problem.
2.2.2 The multi-resolution spline interpolation approach
The main idea is to combine splines with different domain
dimension in order to guarantee in every region of the field a
resolution adequate to the data density, that is to exploit all the
available information implicitly stored in the sample data.
To show the advantage of this approach we suppose to
interpolate the mono-dimensional data set shown in figure 6.a.
The classic spline interpolation approach requires to use a grid
resolution in such a way that every spline coefficient appears at
least in one observation equation. Figure 6.b shows the
maximum resolution interpolation function which is consistent
with the data set. The constraint on the grid resolution avoid the
interpolation function to conform to the field data in high
variability locations. Moreover, the use of the smallest
allowable resolution can make the estimations sensible to the
single observations in regions. where data are sparse; the
consequence is the generation of unrealistic oscillations, due to
the fact that the noise is insufficiently filtered.
(a) (b)
Sample data (a) and result of spline interpolation
using mono-resolution approach (b)
Figure 6.
In one dimension the multi-resolution can be obtained by
modelling the interpolation function d(t) as: