International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B4, 2012 
XXII ISPRS Congress, 25 August - 01 September 2012, Melbourne, Australia 
9 
corresponding to different halving steps, are considered. A new 
level halves the width of the support of the previous level. 
We suppose that a field h{t) = h(x,y) has been sampled at m 
locations t,,t 2 ,...,t m , t, = (x i ,y i ). The interpolation domain is 
[tram’tmax ] • The field observations y 0i are modelled by means 
of a suitable combination of bilinear spline functions and noise 
K : To, =h(t,) + v,. 
Let define the following 
(p(q) = 
\-q 
0 < q < 
0 
q > i 
(p(-q) 
q < 0 
( pM) = ( pi q / / y) 
k w+l to indicate the vector containing all the splines 
coefficients of the new level. We want to evaluate the 
following hypothesis 
// 0 :{X M+1 =£{^ +1 } = 0} (2) 
If HO is true, the coefficients of the new level are not 
significant, the relevant estimates can be discarded and the 
iterative process can be stopped. Otherwise, the coefficients 
should be kept and a new higher level should be tested. Let 
define the following quantities 
v m = y 0 -= V L ~ N m ) 
V A/+1 ~ yo — ÿ M+1 ’ &M+1 = V A/ + 1 ^ m — N M — N M + 1 ) 
The height field is given by the 
Ml [" N h -1 N h -1 
h( t) = I IIA ^ (Va )<Pay„ (Vr) 
h=0 i x =0 i Y =0 
(1) 
where y 0 = [h l ,...,h m ] T is the vector containing all the field 
observations, ÿ M ,ÿ M+l are the a posterior estimates provided 
by LS. From a geometrical point of view the situation is 
depicted in Fig. 1. 
where q x -x- Ax h i x - ;c min , q Y = y - Ay h i - y min , 
Ax, = X — , , Av k = . ? m '" . M is the number of 
n 1 
levels, A h ( . is the coefficient of the spline at the grid node 
(i x ,i Y ), N h is the number of nodes at the h level, 
N h = 2 h+l +1 . 
In the estimation, all the field observations are tiled in a vector 
y„ = y + V = AX + V, E{ v} = 0, C vv = C vv = all 
where L,A are respectively the vector containing all the 
A h ( . . coefficients to be estimated and the design matrix 
obtained by applying (1) to the observations. The estimation of 
H = (A / A) ‘A r y 0 is based on the well known LS principle. 
Two innovative aspects characterize our interpolation 
approach. 
Given a level, each local spline is individually activated if no 
spline of some lower level has the same application point. 
Moreover the spline is activated if at least /,/>1, 
observations exist in at leasts (£ = 1,2,3,4) quarters of the 
spline support: f,k are input by the user. They must be 
choosen according to two criteria. Clearly / -l,k =1 
correspond to no redundancy in the estimation, while bigger 
values smooth the interpolating field. Moreover, particular 
spatial configurations of the observations can produce a LS 
system that, although redundant, is either rank deficient or ill 
conditioned: in these cases, / and k should be increased 
independently of redundancy considerations. The individual 
activation of the splines guarantees a real multi-resolution 
interpolation. 
The levels are activated iteratively from level 0 to level M. A 
new level is activated if and only if its splines significantly 
improve the accuracy of the interpolation. 
Let suppose that M (/? = 0,1,...,M-1) levels have been 
already activated, for a total number of N M splines estimated 
coefficients. The criterion to activate or not the M + 1 -th level 
is based on a significance analysis. Let suppose that N M+X is 
the number of splines activated with the new level and use 
Figure 1. Geometric interpretation of the significance analysis 
of a new level. 
If (2) holds, y = £{y 0 } e V M , and the usual significance 
analysis on the a posterior variances can be applied 
O - )àl, - (m - N. 
N. 
t)*i 
^m + . 
(3) 
where x]»F i j indicate respectively a chi square variable with 
i degrees of freedom and a Fisher variable with (i,j) degrees 
of freedom. A threshold value F a with significance a can be 
set and the zero hypothesis (2) can be tested by (3). 
2.1 Storage requirements 
To evaluate the DTM storage requirement of the different 
models, at first a numerical comparison between grids, TINs 
and our multi-resolution approach is here presented. 
Particularly, an occupation of 64 bits (8 bytes) is hypothesized 
for the horizontal coordinates and the height of a point. In the