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IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 3
Plastic rotation and tension stiffening effect analysis in beams using photogrammetry
where
ω
is a scale factor.
Since only the ratio of the homography parameters is relevant, the
number of unknowns to be solved is eight. Therefore, an exact so-
lution is obtained using four targets. Usually, there are many more
targets available and the resulting system is over determined. In this
case, the solution is obtained by minimizing the norm
Ah
of Equa-
tion 5 for ‘n’ points. The eigenvector corresponding to the least eigen-
value of
T
A A
allows to directly obtaining the solution.
It should be noted that the homography define a map between
two planes, independently of their position and orientation. The ho-
mography parameters can then be used to compute the real plane
coordinates for all the targets and at any stage evaluated.
The differences in the coordinates in relation to the reference stage
allow to directly computing the corresponding displacement field.
Then, an auxiliary mesh is assembled by means of a Delaunay
triangulation and using the reference targets [13]. Lastly, the strain
field is directly computed by a strain-nodal displacement matrix as-
sociated to the auxiliary mesh (see [3] for more details).
It should be mention that all acquired images were orientated and
scaled to 1:5 using the homography parameters. This value is used
since the mean resolution of the original image frames was circa
0.2mm/pixel. Later, the value is also used to measure the crack
width (see Section 5.3).
3.2 Image processing
Digital image processing is a technique which allows detecting dis-
continuities in the image, i.e, points where sudden changes in the in-
tensity level of the pixels occur. Therefore, detecting and measuring
cracks on concrete surfaces is enabled. In the most cases, the detec-
tion of discontinuities in an image was performed by applying edge
detectors [6-8]. This allows obtaining a binary image which enhances
the crack pattern to be characterized. However, the method presents
results strongly dependent on the surface conditions (other source
of discontinuities). Thus, surfaces need to be carefully prepared and
to adequate lighting conditions have to be assured. Therefore, com-
bined approaches have emerged, in which the strain field is used to
define critical regions where image processing is performed [15].
In this section, a brief description of the digital image processing op-
erations required to enhance and measure cracking is presented. The
surface of the specimen was painted white in order to obtain a homoge-
neous background, thus further enhancing cracking appearing during
the experimental test. This method was developed for monitoring crack-
ing by combining digital image processing and mathematical morphol-
ogy operations, and a complete description of the procedure, includ-
ing experimental examples, can be found in [7]. After orientating and
scaling all the images according to the previous Section, the procedure
comprises the following main steps: i) binarization of the images using
the Otsu’s method; ii) mathematical morphology operations (cleaning,
linking and filling) in order to prevent other sources of discontinuities
(e.g. surface imperfections) to be misleadingly taken as cracks; and fi-
nally iii) measuring any selected crack on the scaled image.
4. Experimental program
4.1 Overview
The method was applied to monitoring an experimental test un-
w
i
- i-th crack width
d - effective depth of the cross-section
xi - neutral axis depth
Ɵ - rotation
3. Photogrammetry and image processing
3.1 Photogrammetry
Photogrammetry allowsmeasuring thedisplacement fieldat targets placed
at the surface of the specimen [1-4]. Alternatively, advanced image corre-
lation techniques can be applied to avoid the use of pre-defined targets. In
both cases, after obtaining the displacement field, the strain field is comput-
ed applying standard finite element method (MEF) procedures [5].
All operations performed after image acquisition to obtain the strain field,
are described in this section, i.e.: i) target detection, where the positions of
all targets are identified in all images; ii) homography, which allows scaling
and orientating all images and thus obtaining the coordinates of the targets
in the surface of the specimen; and iii) mesh generation, where aDelaunay
triangulation is applied to define the post-processing mesh [13].
The first step, target detection, is performed using the Hough trans-
form [14] to identify the geometrical center of the targets, at all stag-
es. In summary, the average pixel radius allows computing a para-
metric transform, which results in a map of peaks coincident with the
geometrical center of the targets [14, 15]. Secondly, after detecting
all targets, their position is obtained in the world coordinate system.
Since all displacements are expected to occur within a plane, a
simple homography is established to match image coordinates with
real plane coordinates. The concept is accomplished by solving the
following system of equations for the reference stage:
(5)
where X
i
and Y
i
are the real plane coordinates for each target ‘i’
provided by the reference grid size painted at the surface of the
specimen (in this case 20 x 20 mm
2
), x
i
and y
i
are the correspond-
ing coordinates in the image, and h
1
to h
9
are the homography pa-
rameters. Thus, the following relation can be written, for any point
belonging to the surface of the specimen [16]:
(6)
و
2 3
4 5 6
7 8 9
1
1
X h h h x
Y
h h h y
h h h
=
