mber of
(11)
(11)
(9)
(6)
ras
red accor-
parameters.
an equal
re 2. Side
eck point
eight one
jinting to
and 3 uses
The came-
22 Solution of DLT parameters
As mentioned earlier, the DLT parameters can be
calculated either in a direct way where the corrections
to the observations are assumed to be small and can be
ignored or by an iterative non-linear solution. The
first type with a direct solution will be called linear
DLT and the second type will be called iterative DLT.
Direct solution: The equations for the direct solu-
tion can be written as:
F, = L,X + LaY + L3Z + L, -
LgxX - LyoxY - L41xZ - (x-v,) = 0 (8)
F, - L5X + LeY + LyZ + La -
LoyX - LygyY - LyyyZ - (y-vy) = 0 (9)
where v,, v
points.
y are residuals of the measured image
Iterative solution In the iterative solution, all
observations have corrections except the control
points, which are assumed to be without error. The
equations for the iterative solution of the parameters
can be written as:
Fa = LyX + LaY + L,Z + 1, * La (X-49 X -
Lo(x-v) Y - Lq(x-v) Z - (x-v,) = 0 (10)
F, = L5X + La Y + L7Z + Lg = Lo (y-vy)X =
Lio (y-v,) Y = Lu (y-vy)Z m (y - vy) -0 (11)
Since equations (10) and (11) are non-linear they must
be linearized before they can be used in a least squares
solution.
23 Solution of parameters in the collinearity
equations
As in the case with the iterative DLT, all observations
have corrections. The collinearity equations for the
iterative solution can be written as:
Fs = cT, - Nx, + N(x-v,) = 0 (12)
Fo=¢T, = Ny, + NG -vy) = 0 (13)
In case D, see table 2, equation (12) and (13) are lineari-
zed with respect to the unknown parameters Xo, Yo, Zo,
Q, 6, K. For case C they are also linearized with respect
to parameters x,, yy, and c.
24 Solution of object coordinates
Both the DLT and the collinearity equations are treated
37
as non-linear. Both the comparator readings and the
calculated parameters are treated as observations.
2.5 Evaluation
The methods have been evaluated with respect to pre-
cision and internal reliability. Precision refers to the
statistical variability of the result and the internal reli-
ability indicates to what extent gross errors in an
observation can be detected.
Precision values have been calculated for all control
points in the 11 by 11 by 11 grid. An average value is
calculated for the X, Y and Z component of the check
point coordinates.
Internal reliability is evaluated by using the cofactor
matrix ob, and the correlation matrix of the residuals
resulting from the parameter calculations. An average
of the off-diagonal elements in the correlation matrix
has been calculated to get a value for comparison of
how correlated the parameters are in the four cases.
Each diagonal element of the covariance matrix or.
gives an indication of to what extent a gross error in
the observations can be detected into a specific obser-
vation. A value of above 0.5 has been selected to indi-
cate that an error can be detected in that position with
a high degree of confidence.
3. ADJUSTMENT PROCESS
In a first step, the transform parameters for each of the
four cases A, B, C, and D in table 2, are calculated using
a least square adjustment method. In the same step,
cofactor matrix o, and the corresponding correlation
matrix of the measurement residuals are computed. In
a second step the calculated parameters and their cal-
culated cofactor matrix Q". is used to calculate the
standard deviation o of the check points in order to
estimate the precision of the coordinates of each point.
For all test set-ups the adjustment is overdetermined,
i.e. the number of observations exceeds the number of
parameters carried in the adjustment process, see
table 3.
case |test |parameters | observations | redundancy
(u) (n) (r)
A 1,2 11 12 1
B 1,2 11 12 1
B 1,2 9 12 3
c 1,2 6 12 6
A 3,4 11 26 15
B 3,4 11 26 15
C 3,4 9 26 17
D 3,4 6 26 20
Table 3: The four tested methods A, B, C, and D and redundancy in
the adjustment process. See table 1 and 2 for cases and tests.
