Master Thesis - Fachbereich Informatik

90 CHAPTER 4. LENGTH MEASUREMENT APPROACH
length [pixel]
418
417.5
417
416.5
416
415.5
Measurements
Polynomial Fit
415
0 50 100 150 200
x
250 300 350 400
(a)
Correction [pixel]
1.4
1.2
1
0.8
0.6
0.4
0.2
Perspective Correction Function
0
0 50 100 150 200
x
250 300 350 400
(b)
Length [pixel]
418
417.5
417
416.5
416
415.5
Corrected Measurements
Mean
415
0 50 100 150 200
x
250 300 350 400
Figure 4.25: Perspective correction. (a) The measured length varies depending on the
image position in terms of the left measuring point. Due to perspective the length of one tube
appears larger at the image center than at the image boundaries. The effect of perspective
can be approximated by a 2nd order polynomial. (b) The correction function computed from
the polynomial coefficients. (c) The result of the perspective correction.
4.6.1. Distance Measure
The distance between the two measuring points pL and pR (see Section 4.2) is computed
over the Euclidean distance. Thus, the pixel length l ofatubeisdefinedasfollows:
l = � (pR − pL) 2 (4.19)
where l is expressed in terms of pixels. In the following, l(x) denotes the pixel length of
a tube at position x where x = xpL , i.e. the position of a measurement is defined by the
x-coordinate of the left measuring point.
4.6.2. Perspective Correction
Figure 4.25(a) shows the measured pixel length l(x) of a metal reference tube (gage) at
different image positions. The sequence was acquired at the slowest conveyor velocity.
In the ideal case l should be equal independent of the measuring position. However, the
measured length is smaller at the boundaries and maximal at the image center due to
perspective. This property is consistent between tubes. To approximate the ideal case, a
perspective correction can be applied to the real measurements. Mathematically this can
be expressed as:
lcor(x) =l(x)+fcor(x) (4.20)
where lcor is the perspective corrected pixel length, and fcor a correction function. The
perspective variation in the measurements can be approximated by a 2nd order polynomial
of the form:
f(x) =c1x 2 + c2x + c3
(c)
(4.21)
where the coefficients of the polynomial ci have to be determined in the teach-in step
by fitting the function f(x) to measured length values l(x) in least-squares sense. Then,
the correction function fcor canbecomputedas: