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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: