Master Thesis - Fachbereich Informatik

5.1. EXPERIMENTAL DESIGN 101
where lj(i) indicates the length of the jth single measurement of tube i, l(i) themean
over all single measurements of this tube, and mi is the total number of single measurements
of tube i. σtube is expressed in terms of pixels.
A large per tube standard deviation represents a uncertainness in the results. In this
case, the mean describes the data only roughly. If the uncertainness is too large, it may
be better to blow out the particular tube, since the probability of a false positive decision
increases proportional with the standard deviation.
Sequence Standard Deviation The standard deviation of a sequence σseq is computed
analogue to σtube, but not with respect to the single measurements of one tube, but to the
computed total length ltotal of N tubes:
�
�
�
σseq = � 1
N − 1
N�
i=1
� �2 ltotal(i) − ltotal
(5.9)
where ltotal is the mean over all total measurements. Finally, all measurements can be
represented by a Gaussian distribution function G(x) as:
G(x) =
�
1 (x − µseq)
√ exp
2π 2 �
(5.10)
σseq
2σ2 seq
where µseq = ltotal. The production is most accurate if the distance between the given
target length and the mean of this distribution is small.
Ground Truth Distance The difference between the vision-based length measurement
results and the manually acquired ground truth data can be seen as relative error assuming
the ground truth data is correct. Interesting are the minimum and maximum ground truth
distance (GTD) of a sequence of tubes defined as:
GT Dmin = min {(ltotal(i) − lgt(i)) | 1 ≤ i ≤ N} (5.11)
GT Dmax = max {(ltotal(i) − lgt(i)) | 1 ≤ i ≤ N} (5.12)
where ltotal(i) is the computed total length of tube i, lgt(i) the corresponding ground
truth length, and N the number of tubes considered. If the mean ground truth distance
GT D is approximately zero, the deviation is distributed equally. Otherwise, if GT D > 0,
the measured length is predominantly larger than the ground truth measurement. Accordingly
if GT D < 0, the opposite is valid. In both cases, the systematic error indicates
the system is probably not calibrated correctly.
Root Mean Square Error (RMSE) The root mean square error measure is used to
compare the measurements of the visual inspection system to manually acquired ground
truth data over a sequence as follows:
�
�
�
RMSE = � 1
N�
(ltotal(i) − lgt(i))
N
2
(5.13)
i=1