Image Anal Stereol 2018;37:127-137 doi:10.5566/ias.1914
Original Research Paper
FIBER SEGMENTATION IN CRACK REGIONS OF STEEL FIBER
REINFORCED CONCRETE USING PRINCIPAL CURVATURE
MARKUS KRONENBERGERB,1,2, KATJA SCHLADITZ1, BERND HAMANN3 AND
HANS HAGEN2
1Image Processing Group, Fraunhofer ITWM, 67663 Kaiserslautern, Germany; 2Computer Graphics and HCI
Group, University of Kaiserslautern, 67663 Kaiserslautern, Germany; 3Department of Computer Science,
University of California, Davis, CA 95616, U.S.A.
e-mail: markus.kronenberger@itwm.fraunhofer.de, katja.schladitz@itwm.fraunhofer.de,
hamann@cs.ucdavis.edu, hagen@cs.uni-kl.de
(Received February 22, 2018; revised May 5, 2018; accepted May 10, 2018)
ABSTRACT
This paper tackles the non-trivial image-processing task to segment hook-ended fibers in three-dimensional
images. For this purpose, a novel segmentation method is presented that relies on the following observation:
For a single fiber the configurations of principal curvatures that can occur on its surface are limited. Deviations
from these configurations indicate potential overlaps of fibers. The method that was developed based on this
observation is used to separate several simulated clusters of touching fibers as a proof-of-concept. Further, it is
applied to two images of cracked steel fiber reinforced concrete specimens arising from a 4-point bending test.
The method’s performance is compared to manual separation. Overall, we can state that the proposed method
yields satisfying results when data meets the following criteria: Low fiber volume density, circular fiber cross
section and sufficient spatial resolution of fiber-fiber contacts.
Keywords: fiber reinforced concrete, fiber segmentation, image analysis, X-ray micro-computed tomography.
INTRODUCTION
In civil engineering, fiber reinforced materials
enjoy growing popularity among scientists and
engineers. One important example is steel fiber
reinforced concrete (SFRC). This is a composition
of concrete and steel fibers that are added to
the cement matrix in liquid state. In contrast to
plain concrete the resulting material is able to bear
higher tensile stresses after crack formation (Thomas
and Ramaswamy, 2007). Application examples are
commercial buildings, industrial floors or tunnel
linings. In all of these applications, the post-crack
behavior is of particular interest. After a failure of the
concrete, the interplay between fibers and matrix is
essential for counteracting a further crack propagation.
For an in-depth review of the mechanics of crack
formation and propagation the reader is referred
to Afroughsabet et al. (2016).
Conventional approaches for investigating cracked
SFRC require the destruction of the crack region. In
contrast, X-ray micro-computed tomography (µCT)
allows to explore the material non-destructively based
on three-dimensional (3D) images. This technique
was used, e.g., by Schnell et al. (2011) in order to
quantitatively analyze cracked SFRC. They showed
that the alignment of fibers with respect to the crack
has a higher influence on the tensile stress that the
material can bear than the fiber volume fraction.
This demonstration was possible by a global analysis
using generalized projection lengths obtained as a
by-product when measuring the integral of mean
curvature (Ohser and Schladitz, 2009). This analysis
interprets the fiber system as a stationary random
closed set and thus does require only a separation
of the fiber system (foreground) from the concrete
(background).
The aforementioned investigation was possible
without a segmentation of individual fibers. However,
such a pre-processing would allow to analyze each
fiber separately. As a result, a variety of geometric
characteristics could be obtained directly. See Vecchio
et al. (2012) for object features. These, in turn,
describe the fiber geometry and therefore can be
used to quantify the deformation that a fiber
experienced during crack formation. Moreover, it
would also be possible to determine the contribution
of each fiber to the load transmission. Even height-
dependent investigations of the crack region would be
conceivable.
In the literature concerning fiber reinforced
composites, a wide range of methods have been
proposed in order to handle a segmentation
of individual fibers in 3D images. Basically,
these methods either perform a reasonable over-
segmentation and afterwards trace single fibers or
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they directly separate fibers at their contact areas.
Here, over-segmentation denotes a division into more
segments than intended.
A number of researchers have developed methods
for almost straight unidirectional (Requena et al.,
2009; Latil et al., 2011; Czabaj et al., 2014; Emerson
et al., 2017) or multidirectional fibers (Sencu et al.,
2016). All of them have in common that they
first detect center points of fibers in 2D slices and
afterwards trace them throughout the 3D image.
Herrmann et al. (2016) compute local orientation
vectors using the Hessian matrix and cluster them
in order to determine the preferred orientation of
overlapping fibers. Clusters consisting of uniquely
oriented and completely straight fibers could simply
be separated based on these orientations.
Apart from that, Altendorf (2011) combines
radius, orientation and probability measurements in
order to determine center points away from locations
where fibers are crossing. Afterwards center line
pieces are traced and simultaneously dilated using the
estimated fiber radius. The result are reconstructed
fiber parts that are disconnected at crossing locations.
Finally, angle differences between the orientation
vectors at the ends of fiber parts and the connection
vectors between the ends are compared in order to
unite the best matching parts.
The separation of curved fibers is also subject
of current research. Gaiselmann et al. (2013), for
example, assume fibers to have a previously known
radius and an approximately constant circular cross
section. By combining these two assumptions with
the Euclidean distance transformation measuring the
distance to the fiber surface they extract center
lines by a radius-dependent thresholding. However,
when this approach is applied to CT images, an
over-segmentation of such lines can occur due to
discretization effects and noise. Therefore, center line
pieces are traced in a subsequent step. Randomly
pieces are joined that fulfill user selected distance
and angle criteria. Afterwards, the candidate with
the highest mean value, computed over an enlarged
overlapping subset of the initial CT image, is
considered to be a true fiber. This procedure is repeated
until all fiber pieces are processed.
Pinter et al. (2016) use the same assumptions as
the method described before, but suggest to use a
circular voting filter, which they define as weighted
geometric mean of two measurements: The first part
is a surface normal overlap measure, which exploits
that for a fixed radius and a continuously circular
cross section one can find a high response close to the
center line. The second part is a coherence measure
that considers the eigenvalues of the structure tensor
to decide if a structure is cylindrically shaped. This
shape-related interpretation of the eigenvalues goes
back to Frangi et al. (1998). Center lines can then be
obtained by a global thresholding that is controlled by
the user. In this case, too, an over-segmentation can
occur when the method is applied to CT images. Local
orientation information are used to perform a suitable
reconnection.
Teßmann et al. (2010), on the other hand, use
the approach by Frangi et al. (1998) directly to
identify center points. Afterwards, they use local
orientation vectors, which are a by-product of the
applied approach in order to trace the points to
complete center lines.
Another popular concept for single fiber
segmentation is template matching (Eberhardt and
Clarke, 2002; Martin-Herrero and Germain, 2007;
Weber et al., 2012). A 3D image is processed by
comparing its local structure with a pre-defined
template (line, ellipse, cylinder). The consensus
between the local structure and the template provides
information about the most probable positions of the
center line of fibers as well as their orientations. Both
information are exploited later to trace individual
fibers.
In other applications, researchers had not only
to deal with curved, but also deformed fibers. In
order to tackle this complicated issue, Lux (2013)
proposed an approach that first computes a skeleton
and disassembles it to meaningful parts. Afterwards,
the parts are properly assembled to complete fibers,
which is controlled by several parameters. In order to
make this approach more user-friendly, Huang et al.
(2016) reduced the number of required parameters
using the idea of punishing terms.
Up to now, the only method that directly segments
fibers without the need of a subsequent tracing step
was proposed by Viguié et al. (2013). This approach
is able to separate curved and deformed fibers that
slightly overlap. For this purpose, it estimates the
orientation field inside of the fiber component using
the main axis of inertia like in the case of Altendorf
(2011). Afterwards, the method interprets high values
of the gradient of this field as potential locations where
fibers are overlapping and removes them.
All described approaches face problems when they
are applied to overlapping fibers with hooked ends.
Fig. 2 shows renderings of such fibers contained in
cracked SFRC specimens. Here, we have to deal with
straight, almost parallel fibers as well as strongly
curved ends. For that reason, methods geared to almost
straight fibers are not practical. However, also the
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Image Anal Stereol 2018;37:127-137
other mentioned approaches do not lead to the desired
result. All approaches, except the one by Viguié et al.
(2013) partially or completely rely on local orientation
information while tracing fibers. While this does not
pose a problem when having fibers that are bent on a
large scale, it can lead to problems when considering
fibers with strong local bends like the hooked fiber
ends in the present case. See Fig. 6g for an example.
In such a situation orientation information can be
misleading when tracing fibers.
On the contrary, the method suggested by Viguié
et al. (2013) does not require any of the mentioned
assumptions and is capable to perform the
fiber separation without subsequent tracing step.
Unfortunately, it is possible that fibers in SFRC are
almost parallel and do overlap in the 3D image (Fig. 2).
For parallel overlapping fibers the gradient of the
orientation field is low as the orientations are similar
or even identical. Therefore, the method is not able to
separate fibers in such situations satisfactorily.
Thus, there is a need for a method that can separate
fibers with the characteristics that can be found in
cracked SFRC. This paper contributes by
– giving a thorough overview over existing
approaches for single fiber segmentation and
– presenting a novel segmentation method for hook-
ended fibers in 3D images that can be controlled
using only four input parameters.
MATERIALS AND METHODS
CRACKED SFRC
The cracked SFRC specimens that are considered
within this work arise from a 4-point bending test.
The Institute of Construction Material Technology
manufactured a large beam of SFRC in cooperation
Fig. 1. Sketch of a 4-point bending test. Dimensions
are given in millimetres.
with a local ready-mixed concrete plant. The exact
composition of the concrete (cement, aggregates, etc.)
can be found in Bund (2011). For the reinforcement of
the concrete steel fibers of type Dramix RC-80/60-BN
with a diameter of 0.75 mm and an overall length of
60 mm were added. The mixing of both components
was performed in liquid state at a dosage of 25 kg of
steel fibers per cubic meter concrete. See Bund (2011)
for further details. The prepared beam is positioned on
two supporting pins, while two loading pins are used
in order to apply force to the beam. See Fig. 1 for
a sketch of the test setup. After crack formation, the
interesting part shown in green is cropped out and cut
in two halves, A and B, in order to get a better aspect
ratio for the subsequent CT scan.
3D images of specimens A and B are acquired
using µCT. See the volume renderings in Fig. 2. They
have a size of 737×573×1510 and 746×559×1530
voxel. This approximately corresponds to 6.9× 5.4×
14.2 and 7.03×5.27×14.42 cm3 for an isotropic voxel
side length of 94.27 µm.
(a) (b)
Fig. 2. 3D renderings of two cracked SFRC specimens.
Concrete mass is indicated in green, crack in
red, and fibers in white color. Specimens: Institute
of Construction Material Technology, University of
Kaiserslautern. Imaging: Fraunhofer ITWM.
(a) (b)
Fig. 3. Blurred contacts of two (a) or three (b) fibers in
the 3D image of specimen A.
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The reinforcing fibers are made of steel and thus
do not physically overlap. However, in the 3D image,
virtual overlap does occur (Fig. 3), mainly due to
the so-called partial volume effect (PVE): Computed
tomography is based on the discrete projection images
that the flat bed detector generates in each angular
position of the specimen. Thus, each voxel represents
a small subvolume of the specimen. Voxels at the
boundary of two material components carry a weighted
mean of the gray values corresponding to each of
the two components. Thus, the boundaries between
components of strongly differing X-ray absorption
contrasts are blurred. Clearly, this effect can be
reduced by increasing the resolution. However, this
measure is limited by the requirement to image a
representative volume.
In the present case, the experimental setup dictates
the size and shape of the two specimens. In order to
obtain meaningful results, the complete crack and all
crossing fibers have to be covered. This is favorable
for the subsequent evaluation, but the corresponding
specimens are quite large. While µCT devices allow for
spatial resolutions down to the submicrometer range,
the achievable resolution always depends on the size of
the specimen which is to be imaged. Thus, the obtained
voxel side length of 94.27 µm has to be considered
as trade-off in order to handle the large size of the
specimens. In addition, the prismatic shape can lead
to problems when acquiring 3D images. Then X-rays
have to pass through significantly different amounts of
material. This can cause various CT artifacts. In the
case of specimens A and B slight cupping artifacts
(a)
(b)
(c)
Fig. 4. Gray-value distribution in an xy-slice of the 3D
image of specimen A. The cross sections of fibers close
to image corners show pronounced disturbances by
streaks due to beam hardening artifacts (b) compared
to their equivalents in the middle of the image (c).
in the concrete matrix are notable. Further, effects
similar to beam hardening within the included fibers
emerge. Here, the cupping occurs along the structure
and differs with alignment to the X-ray direction.
The result are gray-value fluctuations within the fibers
and streak effects around their surface. These cause
a change of the otherwise heterogeneous fiber cross
section shape (Fig. 4). Moreover, they worsen the
blurring effect between different fibers. For a more
detailed description of such physics-based X-ray CT
artifacts the reader is referred to Buzug (2008).
PVE as well as CT artifacts influence how exact
the steel fibers can be captured by the acquired 3D
images. While artifacts can be reduced, the PVE is
µCT inherent and thus determines the lower bound
of how fine grained the fibers can be captured. Due
to the PVE at least one layer of voxels can not be
unambiguously assigned to steel fibers or concrete. As
a consequence, an inaccuracy of one voxel (around
0.09 mm) at the interface between the materials needs
to be taken into consideration when evaluating the
fibers. Note that the PVE can be reduced by increasing
the resolution. However, this is not practicable in the
present case due to the trade-off between specimen size
and resolution.
SINGLE FIBER SEGMENTATION
METHOD
Our method for segmenting single fibers is
based on the analysis of principal curvatures on
the surface of fibers. In the following, we first
repeat the corresponding mathematical background
and afterwards we explain how we use the concept
within the proposed method.
Principal Curvatures
Principal curvatures are a known concept from
differential geometry that is useful for describing
the shape of surfaces locally. For the subsequent
explanation of the proposed segmentation method we
make use of the definitions by do Carmo (1976) and
Kreyszig (1991).
Using osculating circles: Let us consider a point
p on a regular plane curve. Then, a circle can be found
that optimally approximates the curve infinitesimally
close to p. This circle is called osculating circle. Using
its radius r, the curvature at p is defined as:
κ =
1
r
. (1)
In the surface case this idea is applied too. There, for
any point one can extract a plane curve by intersecting
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Algorithm 1 Single Fiber Segmentation
Input: A binary image I, the size of a filter kernel σ and three thresholds ξ1,ξ2,ζ
1: Estimation of κminr (x) using spherical granulometry
2: Estimation of κmin f (x) and κmax f (x) using fundamental forms controlled by σ
3: Separation of surfaces of different fibers using κminr (x) ,κmin f (x) ,κmax f (x) controlled by ξ1,ξ2
4: Labeling of surfaces of individual fibers
5: Removal of small labels with less than ζ voxels
6: Assignment of all unlabeled parts to nearest labeled surface parts
Output: A label image containing individual fibers only
the surface with a plane, described by the normal
vector and a vector lying within the tangent plane.
By running through all possible tangent vectors and
computing the curvatures of the resulting plane curves,
one maximum κmaxr and one minimum κminr can be
identified, which are called principal curvatures.
Using fundamental forms: Let S be a
differentiable surface in 3D that is defined by an
explicit function f : R2 → R3. This function assigns
every pair (u,v) ∈ R2 to a point of S ⊂ R3. fi with
i ∈ {u,v} denote the first partial derivatives w.r.t. i and
fi j with i, j ∈ {u,v} the partial derivatives of second
order of f . Then, at each point of S, the normal vector
is given by
N =
[ fu, fv]
‖[ fu, fv]‖2
,
where [·, ·] denotes the cross product. Using these
notations we define the first and second fundamental
form as
F1 =
(
〈 fu, fu〉 〈 fu, fv〉
〈 fv, fu〉 〈 fv, fv〉
)
,
F2 =
(
〈 fuu,N〉 〈 fuv,N〉
〈 fvu,N〉 〈 fvv,N〉
)
,
where 〈·, ·〉 denotes the scalar product. In simple terms
F1 allows to calculate lengths of curves or areas of
regions on deformed surfaces. F2, on the other hand,
is an operator that describes the deformation of a
surface by measuring the changing of N. The principal
curvatures κmin f and κmax f can then be defined using
F1 and F2 as
κmin f = H −
√
H 2−K ,
κmax f = H +
√
H 2−K ,
(2)
with
H =
1
2
trace
(
F2
F1
)
, K =
det(F2)
det(F1)
.
Proposed algorithm
The input for our method is a binary 3D image
I : Ω→ {0,1}, where Ω is a cuboidal subset of Z3.
Let 1 indicate the fiber component X ⊂ Ω and let 0
indicate the background X = Ω\X . We will call x ∈Ω
a voxel for the rest of this work. Further, we consider
the 26-adjacency system as neighborhood of a voxel,
i.e., x is a neighbor of y iff ‖x− y‖∞ ≤ 1 with y ∈ Ω.
Here, ‖·‖∞ denotes the maximum norm.
Algorithm 1 presents a high-level description of
the suggested single fiber separation method. Next, we
are going to explain each of its steps in detail.
Step 1 – Minimum principal curvature via
granulometry: In the first step of the proposed
algorithm the spherical granulometry distribution g(x)
of I is computed (Soille, 2013). g(x) gives the radius1
of the largest ball that covers x and is completely
contained in X . For all voxel x of the background X
the function g(x) returns zero. See Soille (2013) for
details how to efficiently implement such a measure
using morphological opening.
Unfortunately, noise and discretization artifacts are
captured close to the surface. For this reason, the
following adaptation is applied: We use the internal
gradient (Rivest et al., 1992) with a 3× 3× 3 cube as
structuring element in order to identify the set Xs of all
voxels representing the surface of the fiber component
X . Afterwards we replace their values with the values
of the nearest neighbors in X \Xs using the Euclidean
distance d(·, ·):
g′(x) =
g
(
argmin
y∈X\Xs
{d(x,y)}
)
, if x ∈ Xs
g(x) , otherwise.
(3)
It is possible that several voxels y have the same
minimal distance d(x,y) in Eq. 3 and therefore argmin
does not have to be unique. We solve this in our
1Usually, the granulometry computes the diameter that is twice the radius.
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BENT FIBER
(a) ξ1 = 0.15, ξ2 = 0.05
(b) ξ1 = 0.15, ξ2 = 0.15
FIBER WITH ELLIPTICAL CROSS
SECTION
(c) ξ1 = 0.15, ξ2 = 0.15
(d) ξ1 = 0.5, ξ2 = 0.15
OVERLAPPING FIBERS
(e) ξ1 = 0.15, ξ2 = 0.15
(f) ξ1 = 0.5, ξ2 = 0.15
Fig. 5. Parameter selection for different fiber shapes. In each case, the input for the proposed method, the
separated surfaces, and the final result are visualized. The value of σ was 2, and the value of ζ 100. Different
colors indicate different connected components.
implementation by taking the first y that fulfilled the
condition.
Afterwards, an approximation for the minimum
principal curvature is calculated using g′(x). For each
voxel x of the fiber surface g′(x) can be considered as
estimation of the fiber radius. In the special case of a
fiber, this radius extracted of a fitted ball comes close
to the idea of the osculating circle and therefore we use
g′(x) as radius r in Eq. 1:
κminr(x) =

1
g′(x)
, if x ∈ Xs ,
0 , otherwise.
Please note that κminr(x) ≥ 0 due to the fact that the
radius estimated by the granulometry is also always
larger than zero.
Step 2 – Principal curvatures via fundamental
forms: In the second step the principal curvatures
κmin f (x) and κmax f (x) based on the fundamental
forms are estimated for I. The discretization of
the fundamental forms to the discrete setting of
binary images has already been described in previous
work (Kronenberger et al., 2015). The subsequent
computation of the principal curvatures, i.e., Eq. 2,
is then straightforward. In contrast to the previous
work we here switched the approach that we use for
calculating normal vectors. Instead of using the center
of mass approach like in Kronenberger et al. (2015),
we here convolve 3D images with a Gauss filter of
size σ and afterwards use the idea of finite differences,
as suggested, e.g., by Thirion and Gourdon (1995)
or Wernersson et al. (2011) for deriving the needed
normal vectors.
Step 3 – Surface separation: The surface of
an almost straight fiber is described by a strongly
negative minimum principal curvature (κmin  0)
in combination with a maximum principal curvature
close to zero (|κmax| ≈ 0). This is true except for
fiber ends and for hooked parts where |κmax| increases.
Using this knowledge, we can identify overlapping
areas by investigating the compliance of κmax f (x) and
κmin f (x) with those assumptions. In our experiment,
some adaptations turned out to be helpful:
First, pronounced gray-value fluctuations cause the
radius of the discrete fibers to vary and thus it is
complicated to determine a value range of κmin only
allowing for single fibers. For this reason, we add
κminr (x) to κmin f (x) and take the absolute value. This
makes sense, as both measures nearly reflect the same
quantity, but with different sign. Hence, the resulting
term should be close to zero.
Second, the terms |κmin f (x) + κminr (x)| and
|κmax (x)| should be close to zero, when evaluated for
the lateral surface of a single fiber. Since “close to
zero” is always a relative statement, we scale both
terms with κminr (x) so that the terms can be interpreted
more intuitively. κminr (x) is defined using the idea of
osculating circles and therefore illustrative.
Using the previously discussed aspects we came up
with two conditions∣∣∣κmin f (x)+κminr (x)∣∣∣< ξ1 ·κminr (x) , (4)∣∣∣κmax f (x)∣∣∣< ξ2 ·κminr (x) , (5)
to separate surfaces of different fibers by removing
non-fiber-like parts:
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Image Anal Stereol 2018;37:127-137
(a) (b) (c)
(d) (e) (f) (g) (h)
Fig. 6. Simulated fiber clusters and their separated versions obtained with our method. Different colors represent
various connected components.
X̃s(x) =
{
1 , if Eqs. 4 and 5 are satisfied,
0 , otherwise.
Here, ξ1 and ξ2 control the separation process
and can be used to compensate for bent fibers as
well as small deformations in the cross sections. Their
values should be selected between 0 and around 1.
Note that due to the nature of the granulometry and
its adjustment in Step 1, the difference between the
approximation κminr (x) and the estimation κmin f (x)
increases with deviation from the ideal fiber shape.
Step 4 – Labeling: After removing surface voxels
that connect different fibers in Step 3, we label the
connected components (Soille, 2013) on X̃s (x) w.r.t. a
26-adjacency.
Step 5 – Clean up: Due to discretization and
noise it is possible that in Step 4 some labels do
not represent fiber surfaces and consist of comparably
less voxels. That is why we remove all connected
components with less than ζ voxels.
Step 6 – Voxel to label assignment: In the
last step, we run through every voxel of the fiber
component that is not yet labeled and search for its
nearest neighbor that already has a label. Afterwards
we assign the label found to the starting voxel.
Parameter selection
The proposed method is controlled by four
parameters. The first parameter σ is used to set the
degree of smoothing that is applied within the principal
curvature estimation using fundamental forms. It has
to be selected according to the structure within the
3D image as well as the image quality. The reader
is referred to the works of Thirion and Gourdon
(1995) as well as Wernersson et al. (2011) for details
concerning a good choice of this parameter. In all of
our experiments it was sufficient to select σ as 2.
The actual separation of the surfaces of different
fibers is controlled by ξ1 and ξ2. Fig. 5 illustrates
the effect of different choice of these two parameters.
If the value for ξ2 is chosen too small an over-
segmentation along a bent fiber can occur (Fig. 5a).
We can therefore think of ξ2 as controlling the allowed
bending of a fiber. A higher value has to be selected in
order to achieve the desired result (Fig. 5b).
In cracked SFRC changes from a circular to an
elliptic cross section are possible as, e.g., visible
in Fig. 4. A suitable choice of ξ1 can solve over-
segmentations occurring due to such elliptical cross
sections (Fig. 5c and 5d).
In the case of overlapping fibers, as shown in
Fig. 5e and 5f, the parameters ξ1 and ξ2 have to be
selected according to the findings described before.
However, one could now draw the conclusion that the
parameter values should always be selected high, but
this is not the case. It is rather a compromise between
choosing ξ1 and ξ2 as high as possible, but not too high
since then the fibers are not separated any more.
The last parameter ζ is a threshold for removing
separated surfaces parts that are too small to be
considered as real fibers. In all of our experiments we
selected ζ as 100 voxel.
RESULTS
PROOF-OF-CONCEPT
Several configurations with overlapping fibers
were simulated in order to perform a proof-of-concept.
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All configurations are shown in Fig. 6. We selected
configurations being similar to those typically found
in 3D images of cracked SFRC (Fig. 2). Since fibers
are initially straight, except for their hooked ends, it is
possible that straight parts touch each other, requiring
us to handle such configurations. We simulated three
pairs of straight fibers with a diameter of eight voxels
with overlapping parts crossing at angles 0◦, 45◦ and
90◦.
Configurations involving hooked ends are of
special interest due to their more complicated
geometry. Thus, we simulated several relevant cases,
shown in the second row of Fig. 6.
Our method can separate all simulated clusters
illustrated in Fig. 6. For all eight fiber pairs we
set the value of the smoothing parameter σ to 2.
The parameter values of ξ1 and ξ2, controlling the
separation step, were set to 0.3 and 0.1, respectively.
The threshold value of ζ was set to 100.
APPLICATION
We used the proposed method to separate fiber
clusters in two 3D images of cracked SFRC, shown in
Fig. 2. In order to apply the single fiber segmentation
approach to these gray-scale images we performed
the following pre-processing: First, a morphological
closure filter was employed, using a ball with
an approximate diameter of 2.31 voxels, to act
as a structuring element. This step removed small
variations in gray values, while avoiding an excessive
blurring between different fibers.
For the further processing a segmentation of the
fiber component in both images is necessary. This
is done by a binarization using global gray-value
(a) (b)
Fig. 7. Example binarization result of global
thresholding (b) for one xy-slice of the 3D image
of specimen A (a). Here, white color indicates the
segmented fiber component and black color the
background.
thresholding (Fig. 7). For every voxel it is decided
whether it belongs to the fiber component or not
depending on a threshold t. This makes sense, as
usually the gray-values representing steel have a higher
value than the ones representing air or concrete. The
value of t was chosen high enough to remove blurring,
but low enough to keep fibers or parts of them. A
high value of t had to be selected in order to reduce
pronounced blurring effects between fibers in regions
very close to image corners, see Fig. 8a to 8c. At the
same time, the value of t must not be too high as this
can cause fibers to disappear, as can be seen in Fig. 8d
to 8f. For the image of specimen A we set t to 15 750,
and for the image of specimen B we set it to 21 000.
(a) t = 18000 (b) t = 21000 (c) t = 24000
(d) t = 18000 (e) t = 21000 (f) t = 24000
Fig. 8. Binarized fibers for specimen B using different
threshold values.
(a) (b)
Fig. 9. Fiber of interest selection. 3D renderings of the
fiber component (indicated in light gray color) and
the segmented crack (indicated in dark gray color)
of specimen A. (a) shows all fibers and clusters that
do not overlap with the crack and therefore will no
longer be considered. Whereas (b) presents all fibers
and clusters, which are separated into individual fibers
in this experiment.
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Image Anal Stereol 2018;37:127-137
(a) n = 2 (b) n = 2 (c) n = 2 (d) n = 2 (e) n = 2
(f) n = 4 (g) n = 4 (h) n = 4 (i) n = 5 (j) n = 5
(k) n = 6 (l) n = 6 (m) n = 14 (n) n = 15
Fig. 10. Single fiber segmentation of all clusters contained in the 3D images of specimens A and B touching the
segmented crack. Individual fibers are shown in different colors, with n being the number of fibers.
Thresholding did segment fibers, in addition to
segmenting very small objects due to noise. Therefore,
we removed all objects consisting of less than 132
voxels. This choice seemed reasonable, as it is roughly
the volume of a voxelized half-ball with a diameter
of 7.96 voxels, corresponding to the diameter known
from the production process of the fibers.
For the analysis of cracked SFRC only fibers
crossing or touching the crack are of interest.
We extracted the crack using global thresholding,
manually removed sticking-out parts, and filled holes
produced by the fibers using a closure filter with a
ball of diameter 45.61 voxels as structuring element.
Afterwards, we identified all fibers and clusters
touching the crack via simple superimposition (Fig. 9).
The pre-processing steps produce two binary
images only containing fibers and fiber clusters
touching the segmented crack. We applied our single
fiber segmentation method to both images. Again, we
set σ to 2. Due to bent fibers and elliptical cross
sections we had to increase the values of ξ1 and ξ2
compared to the proof-of-concept case. For the binary
image of specimen B we defined the values of ξ1 as
0.4 and of ξ2 as 0.2. However, the change of the cross
section was even more pronounced in the binary image
of specimen A. As a consequence, we had to increase
the value of ξ1 to 0.45. The value of ζ was 100 voxels
in both cases.
The pre-processed 3D images of specimens A
and B included overall 47 single fibers and 14 fiber
clusters. Our method was able to separate all clusters
properly, while leaving single fibers unchanged. Small
clusters of 2 to 4 fibers as well as large clusters of 14
to 15 fibers had to be handled, see Fig. 10 showing
segmentation results.
In order to validate our results, we visually
compared them with a manual segmentation. We
observed that the last step of our method – assignment
of all unlabeled voxels to the nearest labels – does not
always lead to a desirable or meaningful separation,
see Fig. 11. In Figs. 11a and 11b the cross section
of the fiber is not circular, being one assumption
of our method; therefore, inaccuracies can result.
However, such effects can also occur, less pronounced,
in locations where the cross sections are circular, see
Fig. 11c. In our targeted application – computing
features of the fibers – such small inaccuracies of the
fiber surface are statistically not important.
We performed our experiment on a computer
running the Red Hat Enterprise Linux Workstation
(a) (b) (c)
Fig. 11. Possible inaccurate segmentations.
Release 7.3 (64-bit), equipped with an Intel Core
i7-6700 CPU and 64GB of memory. Starting with
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KRONENBERGER M ET AL: Fiber segmentation in crack regions of SFRC
binary images of segmented fibers, our method had
a runtime of around 47 minutes for specimen A and
around 55 minutes for specimen B. The main reason
for these relatively long runtimes is our current naive
implementation. Curvature estimation and adaptation
of granulometry in the surface can be accelerated
substantially via parallelized computations. However,
our goal was to demonstrate the viability of our general
algorithmic approach, and we will improve runtime
behavior in the future.
CONCLUSION
A novel segmentation method for hook-ended
fibers has been presented. Its performance has been
demonstrated using simulated fiber clusters as a proof-
of-concept. Further, the method has been applied
to fibers within 3D images of two cracked SFRC
specimens arising from a 4-point bending test. In
both cases it produced satisfying results even in the
presence of parallel fibers, hooked ends and slightly
deformed cross sections.
The suggested method is applicable when the
following criteria are met:
Low fiber volume density: The fiber separation is
realized by removing non-fiber-like surface parts. If
too many fiber-fiber contacts are very close together
the remaining surface parts in the overlap region can
completely disappear. The resulting over-segmentation
would then have to be solved manually.
Almost circular fiber cross section: Our
experiments show that the suggested method is able to
handle slightly deformed cross sections. Nevertheless,
the underlying idea builds on the homogeneity of the
principal curvatures for a cylindrical shape. Deviations
from the circular cross section violate this assumption
and therefore should be viewed with caution.
Sufficient spatial resolution of fiber-fiber contacts:
The proposed segmentation method is based on local
investigations of the surface shape. In order to be able
to distinguish locally between a blurred overlap and a
deformed fibre cross section the fiber-fiber contact has
to appear at least slightly constricted. Otherwise the
proposed method is not able to separate the fibers.
In the previous section, we have applied our
method to 3D images of cracked SFRC specimens. It
is planned to evaluate a complete series of such images
in order to gain a deeper understanding of the post-
crack behavior of such material. One advantage of the
presented method is that the complete segmentation is
controlled by four well interpretable parameters.
ACKNOWLEDGEMENT
The authors thank Frank Schuler and Bianca Bund
for preparing the two SFRC specimens and fruitful
discussions. This research has been supported by the
German research foundation (DFG) within the IRTG
2057 “Physical Modeling for Virtual Manufacturing
Systems and Processes” and the Fraunhofer High
Performance Center Simulation and Software based
Innovation.
REFERENCES
Afroughsabet V, Biolzi L, Ozbakkaloglu T (2016). High-
performance fiber-reinforced concrete: a review. J
Mater Sci 51:6517–51.
Altendorf H (2011). 3D Morphological Analysis and
Modeling of Random Fiber Networks. PhD Thesis.
École Nationale Supérieure des Mines de Paris;
Technische Universität Kaiserlautern.
Bund B (2011). Untersuchung des Nachrissverhaltens von
Stahlfaserbeton mithilfe der Computer-Tomographie.
Diplomarbeit. Technische Universität Kaiserslautern.
Buzug TM (2008). Computed Tomography: From
Photon Statistics to Modern Cone-Beam CT. Berlin,
Heidelberg: Springer.
Czabaj MW, Riccio ML, Whitacre WW (2014).
Numerical reconstruction of graphite/epoxy composite
microstructure based on sub-micron resolution X-
ray computed tomography. Compos Sci Technol
105:174–82.
do Carmo M (1976). Differential geometry of curves and
surfaces. Englewood Cliffs: Prentice-Hall.
Eberhardt C, Clarke A (2002). Automated reconstruction of
curvilinear fibres from 3D datasets acquired by X-ray
microtomography. J Microsc 206:41–53.
Emerson MJ, Jespersen KM, Dahl AB, Conradsen K,
Mikkelsen LP (2017). Individual fibre segmentation
from 3D X-ray computed tomography for characterising
the fibre orientation in unidirectional composite
materials. Compos Part A Appl Sci 97:83–92.
Frangi AF, Niessen WJ, Vincken KL, Viergever MA (1998).
Muliscale vessel enhancement filtering. In: Wells WM,
Colchester A, Delp S, eds. Proc 1st Int Conf Med Image
Computing and Computer-Assis Interven, MICCAI ’98.
Lect Not Comput Sci 1496:130–7.
Gaiselmann G, Manke I, Lehnert W, Schmidt V (2013).
Extraction of curved fibers from 3D data. Image Anal
Stereol 32:57–63.
Herrmann H, Pastorelli E, Kallonen A, Suuronen JP
(2016). Methods for fibre orientation analysis of X-ray
tomography images of steel fibre reinforced concrete
(SFRC). J Mater Sci 51:3772–83.
136
Image Anal Stereol 2018;37:127-137
Huang X, Wen D, Zhao Y, Wang Q, Zhou W, Deng D (2016).
Skeleton-based tracing of curved fibers from 3D X-ray
microtomographic imaging. Results Phys 6:170–7.
Kreyszig E (1991). Differential geometry. Mineola: Dover.
Kronenberger M, Wirjadi O, Freitag J, Hagen H (2015).
Gaussian curvature using fundamental forms for binary
voxel data. Graph Models 82:123–36.
Latil P, Orgéas L, Geindreau C, Dumont P, Du Roscoat
SR (2011). Towards the 3D in situ characterisation of
deformation micro-mechanisms within a compressed
bundle of fibres. Compos Sci Technol 71:480–8.
Lux J (2013). Automatic segmentation and structural
characterization of low density fibreboards. Image Anal
Stereol 32:13–25.
Martin-Herrero J, Germain C (2007). Microstructure
reconstruction of fibrous c/c composites from X-ray
microtomography. Carbon 45:1242–53.
Ohser J, Schladitz K (2009). 3D images of materials
structures: Processing and analysis. Chichester: Wiley.
Pinter P, Bertram B, Weidenmann KA (2016). Novel method
for the determination of fibre length distributions from
µCT-data. Proc Conf Indust Comput Tomo iCT.
Requena G, Fiedler G, Seiser B, Degischer P, Di Michiel M,
Buslaps T (2009). 3D-quantification of the distribution
of continuous fibres in unidirectionally reinforced
composites. Compos Part A Appl Sci 40:152–63.
Rivest JF, Soille P, Beucher S (1992). Morphological
gradients. Nonlinear Image Processing III SPIE/SPSE
1658:139.
Schnell J, Breit W, Schuler F (2011). Use of computer-
tomography for the analysis of fibre reinforced concrete.
In: Sruma V, ed. Proc fib Symp Prague 2011. 583-6.
Sencu R, Yang Z, Wang Y, Withers P, Rau C, Parson
A, Soutis C (2016). Generation of micro-scale finite
element models from synchrotron X-ray CT images
for multidirectional carbon fibre reinforced composites.
Compos Part A Appl Sci 91:85–95.
Soille P (2013). Morphological image analysis: Principles
and applications. Berlin, Heidelberg: Springer.
Teßmann M, Mohr S, Gayetskyy S, Haßler U, Hanke R,
Greiner G (2010). Automatic determination of fiber-
length distribution in composite material using 3D CT
data. EURASIP J Adv Sign Proc 2010:545030.
Thirion JP, Gourdon A (1995). Computing the differential
characteristics of isointensity surfaces. Comput Vision
Image Und 61:190–202.
Thomas J, Ramaswamy A (2007). Mechanical properties
of steel fiber-reinforced concrete. J Mater Civil Eng
19:385–92.
Vecchio I, Schladitz K, Godehardt M, Heneka M (2012).
3D geometric characterization of particles applied to
technical cleanliness. Image Anal Stereol 31:163–74.
Viguié J, Latil P, Orgas L, Dumont P, du Roscoat SR, Bloch
JF, Marulier C, Guiraud O (2013). Finding fibres and
their contacts within 3D images of disordered fibrous
media. Compos Sci Technol 89:202–10.
Weber B, Greenan G, Prohaska S, Baum D, Hege
HC, Mller-Reichert T, Hyman AA, Verbavatz JM
(2012). Automated tracing of microtubules in
electron tomograms of plastic embedded samples
of Caenorhabditis elegans embryos. J Struct Biol
178:129–38.
Wernersson ELG, Hendriks CLL, Brun A (2011). Accurate
estimation of gaussian and mean curvature in
volumetric images. In: Proc 2011 Int Conf 3D
Imaging Modeling Process Visual Transmis. 312–7.
137