transformations. Following a Delauney triangulation between 
the given GCPs, quintic polynomials are fitted locally, form- 
ing a piecewisely defined but smooth interpolation surface 
(Akima, 1978, Wiemker, 1996). 
Ground control point registration remains important when- 
ever continuous image flight data is missing. Elastic image 
registration shows promising results on airborne scanner im- 
age data. We see the following advantages of elastic regis- 
tration over conventional global polynomial methods: 
e |t allows for local corrections which are necessary due 
to the non-instantaneous image formation process of 
airborne scanner data. 
e |t gives the operator improved interactive control. Sin- 
gle features can be 'pinned down' locally to a forced 
fit while only negligibly changing the global match. 
e Sometimes proper elevation data is missing, e.g. for 
man-made objects and other objects smaller than the 
resolution of the digital terrain model (DTM). Dis- 
tortions resulting from these effects can be 'hand- 
corrected' locally, again without severely changing the 
global match. 
Registration by control point matching will remain important 
even though attempts to reconstruct flight path and sensor 
attitude are made. On the one hand, comprehensive flight 
data will not always be available, particularly not for already 
archived imagery. On the other hand, even with reconstructed 
flight path and sensor attitude, a forced fit of certain control 
points — particularly for significant features of uncertain ele- 
vation — may still be desirable as pointed out in the above 
items. Therefore, elastic registration as investigated in this 
paper for airborne line scanner imagery could then be the 
method of choice. 
2. ELASTIC IMAGE REGISTRATION USING 
THIN-PLATE SPLINES 
Within the field of medical image analysis, Bookstein (1989) 
has introduced an approach for elastic registration of 2D im- 
ages which is based on thin-plate splines. This approach 
allows to represent local deformations between two datasets. 
Originally, thin-plate splines have been introduced by Duchon 
(1976) in the context of surface interpolation. These splines 
uniquely minimize the bending energy of a thin plate 
][ &- + 2g2, + g5,) dz dy — min (1) 
R2 
and thus have a sound physical interpretation. 
The input of the registration algorithm of Bookstein (1989) is 
a set of N corresponding point landmarks q; — (z;, y;) and 
q; — (zi,yi) (analogue to the afore mentioned ground con- 
trol points) that have been located in both datasets. Given 
this data an interpolating transformation f : R? — R? is 
determined which maps one image to another: 
x = f(x) = ( A iy x= (2,9), (2) 
while forcing the corresponding point landmarks to exactly 
match each other. The transformation model consists of a 
global affine part and a pure elastic part, where the latter one 
is a superposition of certain radial basis functions: 
N 
ao + a1x + ay + S wiiU(ri) 
i=1 
f(x) = N (3) 
bo +bız + bay - Y, woiU(ri) 
i=1 
where 
U(r) = rilnri, ri = |x — qi (4) 
is the fundamental solution of the biharmonic equation in 
2D. Thus, with (3) we have an analytic expression of the 
transformation between the two images. In the follow- 
ing we assume that N corresponding points q; = f(q;) 
with q; — (zj,yi) and qi — (zi,yi) in an irregular spac- 
ing have been specified. Then, the parameters of the 
transformation pi — (ao,01,02,U11,...,U1N) and p» = 
(bo, 01, b2, w21,..., w2N) can easily be computed by solving 
a linear system of equations (using v1 — (x1, ..., 2y,0,0,0) 
and va — (yi, ..., yv, 0,0, 0)): 
A 
«8 p x)» [= 1256 10) 
v = Lpi, (6) 
where 
1 Ir yi 
xu roosont () 
1 £N JN 
and 
0 U(ri2) abi U(rin) 
U(r21) 0 Ne. U(ran) 
Us : : (8) 
U(rvi) ^ U(rnn-1) 0 
with r;; — |qi — q;| as the distance between the points q; 
and qj. Note, that the coefficients of the global affine and 
the pure elastic part are determined simultaneously, i.e., both 
parts are computed in one step. For this approach a unique 
solution exists (i.e., we have no local minima), it is in general 
numerically well-conditioned (i.e., robust), and also computa- 
tionally efficient. The approach is invariant under translation, 
rotation, or scaling of either set of landmarks and it is well- 
suited for user-interaction, e.g., there are no free parameters 
that have to be tuned by a user. 
With the same scheme as set out above, also transformations 
using other radial basis functions U(r) can be determined. 
E.g., analogue to the thin-plate spline function U(r) — r?Inr 
for the 2D case, the function U(r) — r is the solution for three 
dimensions. 
3. APPLICATION TO EXPERIMENTAL AIRBORNE 
SCANNER IMAGERY 
The image data was recorded by a DAEDALUS AADS 1268 
multispectral line scanner during campaigns in 1991 and 1995 
over the city of Nürnberg in cooperation with the German 
Aerospace Research Establishment (DLR), at flight altitudes 
of 1800 m with a nadir ground resolution of 4.2 m (Fig. 2, 
950 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996