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Figure 1. Model of slope and exposure configuration 
3. PRECISION OF TARGET COORDINATES 
Target number and object coordinates are expressed as 
P1(X,,Y,,Z1),P2(X2,Y2,22),,,. Observation equations of bundle 
adjustment for the vision metrology at the two epochs of time 
(hereinafter expressed by suffix I and IT) are shown below. 
vd4rvA4Aro-P (1) 
where V is error vector; A, and A, are design matrices for size 
(m, q) and (m, n); X is the vector for internal and external 
orientation, respectively; X is the space vector of the target; e 
is a discrepancy vector; and P is a weight matrix. This gives 
the weight of the observation for the image coordinate with unit. 
Observed values of image coordinates are assumed to be 
independent. 
By applying the least square method to Equation (1), and 
eliminating X , and equivalent observation equations are 
obtained as follows: 
V=AX+e:P 
PzP-PA(PA)! ATP 
(1°) 
Because therc are no absolute datum points, the rank of the 
observation equations decreases by 7. Thus, constraints are 
added so that the mean variance of the space coordinates 
becomes a minimum. The most probable value of space 
coordinates of the targets X and variance-covariance matrix 
X pare given as: 
X=-0"A Be 
i^ % | Q) 
6 =V"PV/f,f =m-n+7 
Q = ATP a, 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BS. Istanbul 2004 
. . . . . a” . 
where oi is an observation variance of the unit weight. 6% is 
a posteriori estimate. The most probable value of object space 
coordinates should be at least smaller than the deformation 
criterion to be detected. 
4. SENSITIVITY OF DEFORMATION 
MEASUREMENT 
This paper adopts the simultaneous adjustment of object space 
coordinates of targets for epoch I and epoch II. Another 
alternative is to compare the coordinates of the targets by 
superposing the coordinates after independently adjusting the 
observations (Benzao). Because different cameras were used for 
the two epochs, internal orientation elements were set to be 
independent for each epoch. The targets in Block A were 
common for the two epochs. The targets in Block B are treated 
to different for each epoch and were tagged differently. X 
denotes the common target coordinates, while X, and 
X ;; denote the target coordinates numbered as different targets. 
The equation of the adjustment calculation is as follows: 
X 
XI 
veld, Ay] |* 4 X; |*e :P (3) 
Il 
Xy 
A variance covariance matrix >, of y=|x" x} x, is 
obtained in the same way as Equation (2) as follows: 
S UK 
S CU Er (4) 
Xon Xs 
To detect the deformation, the coordinate difference d is tested. 
SV À (5) 
The variance covariance matrix Y for d is obtained as 
follows: 
= = = S 
2, =>, X» X +2, (6) 
The relationship between the displacement vector d and the 
parameter used for the testing C is given by: 
V, * Mc» d (7) 
If the number of points included in Block B is p and they have 
common deformation property, M becomes a matrix of 3p x 3 
derived by gathering the p unit matrices.