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