International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
  
MAN 
do =} S A, kB) (9) 
h=0 k=0 
where alt) is the first order spline with h resolution on the 
domain [-A,, Aj]; M is the number of different resolutions used 
for the interpolation (levels); Ay, is the k™ spline coefficient at 
resolution h; Ny, 1s the number of spline with resolution h; Ay is 
the half-domain of the spline at the resolution h. In order to 
uniformly distribute the spline into the whole domain 
D=[tminstmax] the (9) can be rewrite as follow: 
; M-1 NN, 2 PRO 
dije S A, 2 edi (10) 
h=0 k=0 ; À Loin ) 
The model requires the imposition of constraints on the À 
coefficients to guarantee the occurrence of a spline only in 
locations where data are enough to its coefficient estimation 
and to avoid the contemporaneous presence of two or more 
splines, obviously at different resolution, at the same grid 
position. The second event in fact causes the singularity of the 
normal matrix in the least square estimation. In order to avoid 
it, the i spline with resolution h; is activated in point t;, and 
therefore the A; coefficient is not zero if: 
e at least f observations are located inside its definition domain; 
* no j spline exists with resolution h; such as t; = t; and h;<h; 
The f parameter acts as filtering factor to be used in the 
interpolation to avoid singularity. 
The results of two multi-resolution spline interpolations are 
shown in figure 7. 
  
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(b) 
Figure 7. Results of multi-resolution spline interpolation with 4 
(a) and 5 (b) levels 
The bi-dimensional formulation. can be directly obtained 
generalizing the mono-dimensional case. 
We suppose that d(t) 7 d(t;,t;) can be modelled as: 
M-M[N4- Nan! 
d(t) = > > SG 0, t - Ac.) 
h=0 1-0 k=0 
(11) 
where: 
210 
Ay 7 X grid resolution; 
À, = y grid resolution; 
9. (t) o. (t) o.) 
M - number of different resolutions used in the model; 
tu 7 [I kJ" = node indexes (l.k) of the bi-dimensional grid; 
Ay 7 coefficient of the spline at the grid node ty; 
Nın = number of x grid nodes at the h resolution; 
N,, = number of y grid nodes at the h resolution. 
As in the mono-dimensional model, a spline only starts up 
where data are enough to its coefficients estimation. Moreover, 
as usual, for each grid node only one spline is defined. It is 
important to notice that even though in a multi-resolution 
approach there is one grid for each level of resolution, the grid 
at resolution i+1 is built starting from the grid at resolution i by 
halving the grid step in both, directions. The i-resolution grid 
nodes are therefore a subset of the ones at i+1 resolution (see 
figure 8). 
  
  
  
  
  
  
  
  
  
  
  
(a) (b) 
Figure 8. Interpolation grid with mono (a) and multi-resolution 
(b) approach 
2.2.5 The automatic resolution choice 
By passing from N to N+1 interpolation levels we introduce a 
certain number of splines whose coefficients, computed by least 
square estimation, are not null because of the stochastic 
deviation due to the noise. It is necessary to consider if the 
contribution of these new splines is significant, that is if they 
add new information to the field modelling or they only “chase” 
the noise. Given: 
ni = number of spline used with N levels; 
n;+n, = number of spline used with N+1 levels; 
Save — set of coefficients of the new n» splines; 
we find the N level such as the statistical hypothesis: 
Hosis Liz Ns eSouals 
operator, is accepted. Without detailing the test, from the 
deterministic and stochastic model of the least squares approach 
we compute a variate F, which can be compared, with a fixed 
significance level o, with the critical value F, of a Fisher 
distribution of (n,,N-(n,+n,)) degrees of freedom. The test to 
accept the hypothesis Hy: {N is the resolution to choice, that is 
the increase of the spline number with n, new splines does not 
improve the fitting model and therefore their coefficients are 
null! can be formulated as follow: if Ho is true then Fy must be 
smaller than F,, with probability (1-a)), otherwise Hg is false and 
we have to iterate the test with N+1 resolution levels. 
being E! the expectation 
ce as 
2.3 Application example 
The procedure presented in the previous paragraphs was tested 
both on artificial scenarios appropriately designed for estimate 
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