y to a 
(1) 
usually 
metric 
sidered 
]uction 
change 
(3) 
(4) 
(5) 
ie), the 
Y) of 
l. 
(6) 
(7) 
inuous 
e to be 
X9. Y?) 
(8) 
Manfred Weisensee 
  
m n 
UGX YN a SY a6. (9) 
i=0 j=0 
TRS NOK ZN (10) 
i=0 j=0 
dX y y NN UY sz, (11) 
i=0 j=0 
Inserting equations (8), (9) and (11) into (5) gives 
m. n m n 
T'IG' x, y')] E SO AI) eS TY SO XSYO dG 
i=0 j=0 i=0 j=0 
óG(X? y?) óG(X? y23 m n 
EE Munro REY XXS.Y az. 
= 4 S zZ Exe )4Z, 
Error equations of a parameter estimation in the Gaufj-Markov model result from equation (12) by introducing e.g. the 
bilinear interpolation. 
Z(XpYp)= > 
1 1 
i=0 j=0 
tX SYM Z, (13) 
with coefficients 
ap=l -X, -Y, +X,-Y, 
aom tX, XY, 
a, = TY, 7 Xp *Ÿp 
a, = Xp. 
Other interpolating functions have also been introduced for ortho images, e.g. wavelets in /Tsay 1996/, and object 
surfaces, e.g. Bezier splines in /Weisensee 1992/. A true 3D model for the geometry of objects has been derived by 
/Schlüter 1999/. 
Some parts of a surface may have almost constant reflectance, leading to difficulties in solving surface reconstruction 
due to small gradients in equation (12), especially, when choosing a small grid spacing. One way to overcome this is to 
introduce some kind of regularization into surface reconstruction, cf. /Franek and Müller 1990/ and /Korten et al. 1988/, 
e.g. by adding conditions for the curvature of a surface model formed by a regular grid and an interpolating function. 
C (Z;)= (Z1 — 27; + Zin ) + dZ | — 2dZ , + dz; 
Cr (Zi )= Zi -22;t Zin; ) + d2; 1; > 2dZ; +dZ 
Co(2,)2 (2-2 Z +Zu tdZ,-dZ 
(14) 
i+1j 
ju inj ij4l -dZi; *tdZ,ja 
In equations (14) the curvatures are the second derivatives of the discrete surface model. In a surface reconstruction 
procedure they can be treated as additional direct observations with value 0 and weights depending locally on the 
gradients of the image signal, /Schreyer 1998/. 
Comparable to other tasks of geometric rectification, a pixel transfer can be carried out either directly, i.e. by projecting 
a pixel onto the object surface, or indirectly by projecting a regular grid from object space into the digital images and 
resampling the image signal. These values are then treated as pseudo measurements in the reconstruction process. The 
advantage of the regular grid becomes apparent from equation (12). The coefficients o;; and aj; are constant for all facets 
of the ortho image and the surface model and can be computed in advance for any given grid of pseudo measurements. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 967