Manfred Weisensee 
  
in image space and, thus, yielding homologous image points, FastVision matches digital images directly to a 
radiometric model, i.e. an ortho image, of a surface in object space. The basic equation of FastVision is given with 
G(X.Y)zT'[G' G7, y )] 2 T"[G" (x, y*)] a... (1) 
An arbitrary number of two or more digital images G', G", ... is equally transformed by global or local, but usually 
linear transfer functions T', T", ... to the ortho image G(X,Y). In case of central perspective imaging the geometric 
relation between images and object space is given by the collinearity equations, other sensor models can be considered 
as well. There are several ways to derive a feasible procedure for surface reconstruction and ortho image production 
from (1). In the following, a method according to /Weisensee 1992/ is used. 
Expanding equation (1) into a Taylor series results in 
| o y? óG(x9 y? 
GU dX. Y edY ye Gc Y? 80 12 ay S0 ar 2) 
66(X^,Y?) é6(X^, y?) 
m and Ó—ÓÓOM—. 
SX óY 
with the gradients of the radiometric model of a surface in object space 
The collinearity equations account for a change dZ of the height of a point Z? in object space depending on a change 
dX, dY of its position X^, Y?. Being X'o, Y'o, Z'o the center of perspective of an image yields 
0 ' 
dX = Oe ele dZ-X'.dZ (3) 
az 25 
ay oy zzi vu dZ=Y',-dZ (4) 
az zm 
Joining equations (1) - (4) for one image G' gives 
0 0 0 0 
T'IG' (x. y))2 GCX", Y?) TIL) x, 4 HET) y, .dZ (5) 
While the image function G'(x',y') is given by discrete samples of a signal (intensity, optical density or gray value), the 
ortho image G(X°,Y°) is represented by an interpolating function G(X,Y) of order m,n as well as the height Z(X,Y) of 
the surface in object space with coefficients oy; and aj; and function values G;j and Z;; in separate grids respectively. 
G(X. = Wo X.Y)6, (6) 
i=0 j=0 
ZX. 0)=YY a (X.7)Z, (7) 
i=0 j=0 
Using the function G(X,Y) again should not be too confusing. Before G(X,Y) denoted the ortho image in a continuous 
manner, in the following G(X,Y) means an interpolating function from which also the gradients in equation (5) are to be 
computed. 
Splitting up the ortho image and surface geometry in equations (6) and (7) into approximations G(X",Y") and Z(X?, Y?) 
and changes dG(X5, Y?) and dZ(X, Y?) leads to 
Ga Y» 2 Ya o o", e 
i=0 j=0 
  
966 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.