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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
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 A(t) = h(x, y) has been sampled at m
locations t,,t,,...,t
stm»
t; 2 (x,, yj) . The interpolation domain is
[Emin>Enax ]- The field observations y, are modelled by means
of a suitable combination of bilinear spline functions and noise
V;: yy =h(t,)+v, .
Let define the following
1-4 0sg«l
9(q) - 10 q>1
p--q) q<0
INO UA
The height field is given by the
M-A | N41 N,-1
h(t)= S > > Ai iy Pax, {ox Yor (ay) (1)
h=0 | iy=0 iy=0
where qx = X= Axi, i Xmin , qy = Vi = Ay,l, = min >
AX, Tmax ET mn Ay, = J max = s. Mis the number of
2 or
levels, Ais is the coefficient of the spline at the grid node
(..), N,is the number of nodes at the A level,
N za" rp,
In the estimation, all the field observations are tiled in a vector
y, -y*Yv-Ak*v, E[v| 20 C, 2 C,, - o7l
where AA are respectively the vector containing all the
À
hjix siy
coefficients to be estimated and the design matrix
obtained by applying (1) to the observations. The estimation of
A=(ATA)'Ay, 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 f,f 21,
observations exist in at leastk (k=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 f=1k =1
correspond to no redundancy in the estimation, while bigger
values smooth the interpolating ficld. 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, f 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 (A20,L..,M —-1) levels have been
already activated, for a total number of N,, splines estimated
coefficients. The criterion to activate or not the M + 1 -th level
is based on a significance analysis. Let suppose that N,,., is
the number of splines activated with the new level and use
A, to indicate the vector containing all the splines
coefficients of the new level. We want to evaluate the
following hypothesis
H, s 2EÍS ls 0} Q)
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
2 2
Yu = Yo Yu Tu 7 Vy /(m- Ny)
ut ^ A2 Ed a =
Vua P Yo Yu Gun = Yun fm —Ny 7 Nu)
where y, -[A...,^,] is the vector containing all the field
observations, $,,,y,,., are the a posterior estimates provided
by LS. From a geometrical point of view the situation is
depicted in Fig. 1.
Yo
9S(w)/ | 0 me)
i
y CRT
rra.
Yim)
Vim)
VtMa)
Figure 1. Geometric interpretation of the significance analysis
ofa new level.
If (2) holds, yz E {yo} €Vu » and the usual significance
analysis on the a posterior variances can be applied
22 A2
(m em Ny JO TR (m = Ny = Ny, )O 4 A
a2
Nya Ow
2
Tr Nd
s
Vus =F
7 M+1 N Nr Nr
A ne Nu Nu /(m-N,-Ny,4)
(3)
where 7;,F,, 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, 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
