G (X9«dx Y9 «d = G (X© YO) + dG (XO YO) + 
3 O yO 3 O yO ap 
ER à SEE} 
ox oY | dY 
Together with (10) this leads to: 
G(x.y) - T7! (G cx9.Y9) - aa cx9. Yo) + 
dG x9. Y9) aG x9. Y9) (12) 
+ We Ty : ar) 
The following equations, expressing the dependence of 
changes dZ in the Z-coordinates on changes dX, dY in 
X- and Y-coordinates, are known from analytical photo- 
grammetry: 
O 
dX oZ dz 79.7 dz x dz (13) 
a y9_y ; 
dY s dZ*-——9*d721.-d7. (14) 
oZ 70. > z 
Substituting (13) and (14) in (12) leads to: 
Gy) » T7! ( Gax9.Y9) -aecx9 39) 
XC x9 YO) AGCXO YO) (13) 
G 4, , 
* ox X2 dz * oY 2 % Y az) 
Now, the functions describing object surface and object 
grey values are introduced (see fig. 1.1). Up to now 
bilinear functions are used for each facet: for a surface or 
Z-facet and for a grey value or G-facet. The complete 
representation of object surface and the complete object 
grey value function by piecewise polynomial functions, 
depending on the unknown grid values Z,, and G,4 can 
be written as: 
GX EN =F Xa CXY) Spy (16) 
1 
ZOUD VZD - 
rs 
ni *M 
Ya, 000 Z an 
s 
with known functions œ,, (X,Y) and a,, (X,Y). Splitting of 
(16) into approximate values of the object grey values 
GX9.19) and their changes dG(XO,YO) leads to: 
Se vO yO O vo O 
Ga uv) u Z oq(X 33:98 (18) 
vO vO OQ vO 
dG(X-,Y >= 2 ol X Xd 
Splitting of Z(X,Y) can be done in an analogous way: 
26 y0 vO O y0,,70 
20 A Lau Y Mz 
az En To (PP YO 2 
r s 
In equation (15) the partial derivatives of the grey value 
function are computed from (18): 
5,050 ty 7940; “du CO Y9) 
2co 13. X SGOT) hat ao 
aX = T auc 30 ^ X 
- 21,40 40 Y yO y0p da, X X9. YO) 
> . 
oY ET 2000 — 
826 
From the above given relationships the fundamental 
differential equation of digital photogrammetry is obtai- 
ned: 
G(x.y) - 
/ 
Lu 2 E ay (XOX) Gy 3 Z a (XOX) "dO, 
  
~ QO oO f O vo O yO \ 
Ey EC am XY e Sta OY, 
k 1 de, (X9. YO) 9X z oY z 
m 
à La 121 102 | (2D 
r 
w M3 
In order to reconstruct an object with the derived method 
at least a second picture is needed. The values associated 
with picture P' in (2D have to be replaced with those 
associated with the pictures PP etc. The evaluation of 
(21) for all pixels from all pictures leads to the following 
linear Gauf-Markov model: 
E(D = A, x (22) 
(noni 
n, is the number of observation equations of type (2D 
and n,sn, is the number of unknowns: dG,,, dZ,, and 
the parameters of T, T etc. 1 is the vector of differences 
between the measured grey values G and those grey 
values, which result from the approximate grey values of 
the object model: 
Gy) - Ta($ Yaa 019-69) (2) 
k 1 
This overdetermined problem can be solved by a least 
squares approach, which minimizes vB, v, and which 
leads to the following system of normal equations: 
T = AT 
ATP Ax = ATPL (24) 
P, is the weight matrix associated with the observed 
image grey values. 
As can be realized from equation (21), FAST Vision is a 
non-linear problem and the solution of (24) may be 
computed by Newton-Gauf iteration. For latest details 
concerning FAST Vision see (WEISENSEE, 1991). 
From now on, but only in this paper, not in practice, we 
suppose, that the parameters of the transfer functions 7 
T .. are already given. Then, the matrix A, consists of 
two submatrices: the coefficient matrix A, for the 
unknowns x, of the object surface and the coefficient 
matrix A for the grey value unknowns xg: 
A, -(Ag Ag. 
As the elements of A, depend on the partial derivatives 
of the grey values, see (12) and (21), linear depen- 
dences of some columns of this submatrix can occur, if 
the measured grey values are constant in larger regions 
of the pictures, or if they increase or decrease linearly in 
X-Y-space. This will cause the product