Determination of Bifurcation Angles of the Retinal Vascular Tree
through Multiple Orientation Estimation based on Regularized
Morphological Openings
Sandra Morales
1,2
,
´
Alvar-Gin
´
es Legaz-Aparicio
3
, Valery Naranjo
1,2
and Rafael Verd
´
u-Monedero
3
1
Instituto Interuniversitario de Investigaci
´
on en Bioingenier
´
ıa y Tecnolog
´
ıa Orientada al Ser Humano, Universitat
Polit
`
ecnica de Val
`
encia, I3BH/LabHuman, Camino de Vera s/n, 46022 Valencia, Spain
2
Grupo Tecnolog
´
ıas de Inform
´
atica Aplicadas a la Oftalmolog
´
ıa, Unidad Conjunta UPV- FISABIO, Valencia, Spain
3
Universidad Polit
´
ecnica de Cartagena, Cartagena, Spain
Keywords:
Retinal Image, Bifurcation Angle, Multiple Orientation Estimation, Regularized Morphological Opening.
Abstract:
This paper describes a new approach to compute bifurcation angles in retinal images. This approach is based
on the estimation of multiple orientations at each pixel of a gray retinal image. The main orientations are
provided by directional openings whose outputs are regularized in order to extend the orientation information
to the whole image. The detection of vessel bifurcations is based on the coexistence of two or more than two
different main orientations at the same pixel. Once the bifurcations and crossovers has been identified, bifur-
cation angles are calculated. The proposed procedure of computing bifurcation angles by means of orientation
estimation at all pixels of the gray level image is much more stable than those methods which are based on the
skeleton of the vascular tree, since a slight variation of a pixel of the skeleton can produce a significant change
in the angle value.
1 INTRODUCTION
Retinal vasculature is able to indicate the status of
other vessels of the human body. Its study involves
a non-invasive or minimally invasive procedure and,
usually, it is included in the standard screening of
any patient with diseases in which the vessels may be
altered. Specifically, the identification and study of
bifurcations and crossovers has great significance in
cardiovascular diseases as well as in their early detec-
tion (Azzopardi and Petkov, 2011). Moreover, vas-
cular changes produced in systemic diseases usually
induce particular modifications in the vessels, such as
changes in bifurcation angles. Based on this fact, this
paper is focused on automatically determining the bi-
furcation angles calculated on vessel bifurcations pre-
viously detected. There are two approaches to de-
tect vessel bifurcations: methods based on geometri-
cal features and methods based on models (Abramoff
et al., 2010). The method proposed in this paper be-
longs to the first group. In particular, the detection of
vessel bifurcations is addressed by analyzing a mul-
tiple orientation vector field provided by the regular-
ization of directional morphological openings. The
orientation field is given by a directional signature
(Soille and Talbot, 2001) for each pixel using a set
of directional openings with a line segment. Then,
the orientation of a pixel is defined as the one asso-
ciated to the directional opening which produces the
maximum value of the signature of this pixel. Nev-
ertheless, the original approach from (Soille and Tal-
bot, 2001) does not deal with the multiple orientation
case and does not take into account that, locally, pix-
els in natural images can be associated to more than
one orientation, e.g., crossing lines, corners and junc-
tions (also known as X-, L- and Y-junctions, respec-
tively). To determine not only the main direction but
all the significant orientations, the directional signa-
ture is analyzed in the present work using multiple
peak detection on the curve interpolated by b-splines.
Thanks to the estimation of the main orientations of
each bifurcation point of the retinal images, the an-
gles generated by the vessels on all of them can be
analyzed.
In the literature, the most common approach to
detect retinal tree bifurcations and their correspond-
ing bifurcation angles is through the analysis of the
segmented vascular tree. In general, this segmenta-
tion can be performed by matched filters (Chaudhuri
et al., 1989), by region growing and scale-space anal-
126
Morales S., Legaz-Aparicio Á., Naranjo V. and Verdú-Monedero R..
Determination of Bifurcation Angles of the Retinal Vascular Tree through Multiple Orientation Estimation based on Regularized Morphological Openings.
DOI: 10.5220/0005170101260131
In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2015), pages 126-131
ISBN: 978-989-758-069-7
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
ysis (Mart
´
ınez-P
´
erez et al., 1999), by mathematical
morphology (Zana and Klein, 2001), etc. Regarding
the detection of vessel bifurcations and crossovers on
segmented retinal images it can be done, e.g., by us-
ing a set of trainable keypoint detectors and a bank of
Gabor filters (Azzopardi and Petkov, 2013), or with
matched filters (Ardizzone et al., 2008). The mea-
surement of the bifurcation angles is usually carried
out fitting the segmented vessels, or their skeleton, by
straight lines and then estimating the angle between
them (Gao et al., 2000; Martinez Perez et al., 2002;
MacGillivray et al., 2012). Segmentation-dependent
results are the major drawback of this approach: a
slight variation of a pixel in the skeleton results can
produce a significant change in the angle value, even
of several degrees.
This paper is organized as follows: Section 2
firstly addresses the method to estimate multiple ori-
entations at each pixel, and then the approach fol-
lowed to detect the bifurcations and crossovers in reti-
nal images together with the calculation of the bifur-
cation angles is described. Section 3 shows the results
of the proposed method in a public database. Finally,
Section 4 closes the paper with the conclusions and
future work lines.
2 METHOD
The goal of this paper is to propose a new method
to compute bifurcation angles of the retinal tree from
a gray image without resorting to the skeletonization
of a segmented (binary) image. For that purpose,
three stages are necessary: estimation of the multi-
ple orientation vector field, detection of bifurcations
and crossovers and, finally, the measurement of the
desired angles. The key of the method is the way to
estimate the multiple orientations existing in a fundus
image. The algorithm proposed to carried out this es-
timation is depicted in Figure 1. This approach differs
from (Angulo et al., 2011), where the multiple main
orientations are estimated by analysing a block of the
image, whereas in this paper the multiple orientations
are estimated at each pixel.
-
f
kk
g
-
-
-
γ
L
θ
1
,l
γ
L
θ
2
,l
γ
L
θ
N
,l
-
-
-
g
θ
1
g
θ
2
g
θ
N
H
σ
H
σ
H
σ
-
-
-
˜g
θ
1
˜g
θ
2
˜g
θ
N
Multiple
peak
search
-
~
θ
.
.
.
.
.
.
Figure 1: Flowchart of the multiple orientation estimation
method.
2.1 Multiple Orientation Estimation
Let f (x) : E R be a gray-level image, where the
support space is E Z
2
and the pixel coordinates are
x = (x,y). Let us define g(x) as the absolute value of
the gradient of f (x), i.e.,
g(x) = k f (x)k =
r
f (x,y)
x
2
+
f (x,y)
y
2
. (1)
The directional opening of g(x) by a linear (sym-
metric) structuring element (SE) of length l and direc-
tion θ is defined as the directional erosion of g by L
θ,l
followed by the directional dilation with the same SE
(Verd
´
u-Monedero et al., 2011):
γ
L
θ,l
(g)(x) = δ
L
θ,l
[ε
L
θ,l
(g)](x), (2)
where the directional erosion and dilation are respec-
tively
ε
L
θ,l
( f )(x) =
^
hL
θ,l
(x)
{
f (x + h)
}
, (3)
δ
L
θ,l
( f )(x) =
_
hL
θ,l
(x)
{
f (x h)
}
. (4)
The proposed orientation model is based on a de-
composition of the gradient information by families
of linear openings, {γ
L
θ
i
,l
}
iI
, according to a particu-
lar discretization of the orientation space {θ
i
}
iI
.
In the next step of the proposed method, a filtering
is performed at each one of the directional openings
(depicted as H
σ
in Figure 1). The filtering diffuses the
orientation information and avoids angle mismatches
due to noise. The kernel H
σ
is the sampling of the
gaussian low-pass filter
H
σ
(ω
1
,ω
2
) = e
σ
2
(ω
2
1
+ω
2
2
)
2
, (5)
where σ is the spatial standard deviation of the filter.
More details of the method can be found in (Legaz-
Aparicio et al., 2014).
Once the directional openings have been filtered,
the directional signature at pixel x is defined as
s
x;l
(i) = ˜g
θ
i
(x). (6)
Then, s
x;l
(i) is interpolated using cubic b-splines and
its maxima correspond to the multiple orientations ex-
isting at pixel x. Finally, collecting all the orientations
estimated at all the pixels in the image provides the
multidimensional vector field
~
θ(x).
2.2 Detection of Bifurcations and
Crossovers
The method previously presented can be applied to
a gray or on a binary image. The resulting orienta-
tion vector field can be used then to detect the ex-
isting bifurcations and crossovers of the retinal tree
DeterminationofBifurcationAnglesoftheRetinalVascularTreethroughMultipleOrientationEstimationbasedon
RegularizedMorphologicalOpenings
127
(see Figure 2). In general, most of the methods which
detect significant points of a vascular tree are based
on a previously segmented, and therefore binary, im-
age. In this paper, the segmentation stage is skipped
and the method is applied directly to a gray image,
avoiding the segmentation result dependence. How-
ever, the method can also be applied to a segmented
image, providing a better orientation field due to the
binarization of the image.
Figure 2: Left: real retinal image. Right: close-up of the
image with its estimated multiple orientation vector field.
The detection procedure takes into account only
the two main orientations at each pixel. The method
considers that a bifurcation or crossover exists if two
conditions happen. The first condition is that the mag-
nitude of the regularized opening has to be greater
than a threshold, | ˜g
θ
i
| > th. The second condition to
be fulfilled is that the two main orientations must dif-
fer more than 2∆θ, where ∆θ is the angular resolution
of the multiple orientation estimation method. After
processing these two conditions, a refinement stage
is necessary to reduce false positives and then to dis-
criminate between bifurcations or crossings and high
curvature areas.
2.3 Classification of Detected Points
In order to determine which interest points are bifur-
cations and crossovers and which ones are high cur-
vature areas, a control area around each point of inter-
est is established. The control area is a window with
circular shape centred in the point of interest. The ra-
dius of the window must be large enough to contain
the vessels belonging to the point of interest. At this
stage, a coarse segmentation based on a top-hat mor-
phological filter followed by a threshold is performed.
The aim of this segmentation is to speed up the clas-
sification stage.
The classification is carried out by placing a cir-
cular window at the interest point and analyzing the
number of branches that cross the circle defined by
the mask. According to the number of branches that
cross the circle, the points of interest are classified
(a) (b) (c) (d)
Figure 3: Circular windows placed at different detected in-
terest points with two main orientations.
into four types:
If there is only one branch, the point of interest is
a vessel termination (see Figure 3(a)).
If there are two branches, the point of interest is a
vessel with high curvature (see Figure 3(b)).
If there are three branches (in general, an odd
number greater than one), the point of interest will
be a bifurcation (see Figure 3(c)).
Finally, if there are four branches (in general, an
even number greater than two), the point of inter-
est will be a crossover (see Figure 3(d)).
Since we are interested in detecting bifurcations and
crossovers, only the points of interest whose mask
is crossed by more than two branches will be con-
sidered. Figure 4(a)-(b) show the detected interest
points and Figure 4(c)-(d) show the bifurcations and
crossover after the classification stage.
(a) (b)
(c) (d)
Figure 4: (a) Detection of interest points: bifurcations,
crossings and high curvature areas. (b) Close-up of (a). (c)
Bifurcations after the classification. (d) Close-up of (c).
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2.4 Measurement of Bifurcation Angles
Once the bifurcations have been classified, it is pos-
sible to measure the bifurcation angles. The angle is
calculated by performing the difference between the
orientation vectors of each branch. From the intersec-
tion of the circular window and the branch, an inte-
rior point is chosen as representative of this branch,
and its orientation vector will define the orientation
of the branch at this pixel. Note that the choice of
this pixel is not critical since the orientation vector
field varies slowly inside the vessel. Due to the dis-
cretization of the orientation space of the multiple ori-
entation method, the orientation is estimated using the
ASGVF method (Verd
´
u-Monedero et al., 2011) which
achieves a higher angular resolution. Since the AS-
GVF method only estimates the orientation and it is
defined between 90
and 90
degrees, the direction
of each vector can be obtained taking into account the
location of the representative pixel of each branch in
relation to the centre of the bifurcation. Considering
the quadrant of the representative pixel, the direction
of the vector can be obtained, providing a vector field
which is defined between 0
and 360
degrees.
Finally, after the conversion of the orientation
space, it is possible to calculate the angular difference
between the vessels contained in the window and thus
to obtain a measure of the bifurcation angle.
3 RESULTS
The performance of the proposed method has been
evaluated on images belonging to DRIVE database
(Staal et al., 2004). This database contains 40 reti-
nal images of 565 × 584 pixels as well as their man-
ual segmentation performed by experts. DRIVE is
often used to evaluate segmentation algorithms de-
signed specifically for the retinal vascular tree. In
this paper, two methods for the computation of bi-
furcation angles have been compared: the proposed
method based on the orientation vector field and the
method described in (Morales et al., 2014).
Regarding the proposed method, it has been ap-
plied directly to the green component of the origi-
nal RGB image. The directional openings were per-
formed using an oriented linear structuring element
of 7 pixels. This length comes from the trade-off be-
tween the curvature of the vessels and the angular res-
olution of the structuring element (it provides ∆θ =
15
and produces a filter bank with 12 branches). The
low-pass filtering of the directional openings is per-
formed using σ=1. The threshold th used to consider
that a pixel belongs to a bifurcation or crossover is the
25% of the maximum intensity value. In Figure 5, the
angles measured on two images of DRIVE database
can be observed. Only a region of interest of these
images has been shown for better visualization.
The parameters of the orientation estimation have
been chosen to deal with this particular resolution.
However, the orientation estimation method can be
improved by considering a multiscale approach as de-
scribed in (Angulo et al., 2011). There, the estimation
of the orientation properties is provided by directional
openings by line segments of variable length, which
produce directional signatures for various scales.
With respect to the algorithm presented in
(Morales et al., 2014), the main difference is that it
only can be applied to binary images. Moreover, once
the vessels have been segmented with this approach,
the retinal vascular tree must be labelled in order to
perform the angle measurements. The vascular tree
labelling is focused on obtaining the vascular skele-
ton, detecting its bifurcation points through hit-or-
miss transformations and a tracking process. Then,
the angle formed by the daughter branches of each bi-
furcation point is calculated. The branches are fitted
by straight lines using least-squares in a circular win-
dow centred on these points. As it was mentioned in
Section 1, the main drawback of the methods based
on the vascular skeleton is its excessive dependence
between the skeleton pixels and the measured angle.
Figure 6 shows the measured angles provided by the
method described in (Morales et al., 2014) on differ-
ent expert hand-made segmentations.
Comparing the results shown in Figure 6, it can
be appreciated that the measured angles can be quite
different, reaching a maximum angle variation on the
same image of around 7% in the Figure 6a and around
6% in the Figure 6b. Note that the variation is due to
the fact that the fitted line of the branches depends
directly on the skeleton pixels considered, and the
skeleton shape depends on the segmentation. Only
the change in one pixel of the skeleton can modify the
fitted line and therefore the calculated angle. The an-
gles shown in the Figure 6a and 6b correspond with
the existing bifurcations within the red rectangles of
the Figure 5a and 5b respectively. Since the method
proposed in this paper is based on the grey image and
therefore, it avoids the segmentation and skeletoniza-
tion stages, the provided measurements are much
more accurate, stable and faithful to reality.
4 CONCLUSIONS
This paper has addressed a novel method to measure
bifurcation angles in retinal images. The method is
DeterminationofBifurcationAnglesoftheRetinalVascularTreethroughMultipleOrientationEstimationbasedon
RegularizedMorphologicalOpenings
129
(a) (b)
(c) (d) (e) (f) (g) (h)
Figure 5: Angles of the bifurcations with the ASGVF orientation vector field on two images from DRIVE database. (a) A
region of interest of the 19 test image. (b) A region of interest of the 40 training image. (c)-(e) Close up of selected
bifurcations marked in red in (a). (f)-(h) Close up of selected bifurcations marked in red in (b). The bright pixels inside the
vessels indicate at what points the value of the vector field has been taken.
(a) (b)
Figure 6: Bifurcation angles provided by the method described in (Morales et al., 2014) on the selected bifurcations of the
Figure 5. This method is based on the skeletonization of a segmented (binary) image. (a) Skeletonization of two different
expert segmentations of the 19 test’ image. (b) Skeletonization of of two different expert segmentations of the 40 training
image. Slight differences in the skeleton cause large differences in the measured angles.
based on a multiple orientation vector field which is
obtained from gray images. The coexistence at some
locations of two or more orientations provides the in-
terest points where a bifurcation can exist, but also
high curvature vessels are wrongly provided. After
a classification stage, the false positives are reduced
and then the measurement is performed using the ori-
entation estimation provided by the ASGVF of the
gray image. Since the orientation method is based
on the gradient of the gray image, no segmentation
is required. The proposed method can measure ac-
curately the bifurcation angles and it is more robust
BIOSIGNALS2015-InternationalConferenceonBio-inspiredSystemsandSignalProcessing
130
than other methods which use the skeleton of the seg-
mented image. The measurements based on the skele-
ton depends highly on this process. The change of a
single pixel of the skeleton can modify the line which
fits the branch and therefore it involves variations in
the angle measurement.
As future work, we will evaluate the performance
of the proposed method on all retinal images from
DRIVE (Staal et al., 2004) and STARE (Hoover et al.,
2000) databases. Comparisons with other state-of-art
methods will also be done.
ACKNOWLEDGEMENTS
This work was supported by Ministerio de Econom
´
ıa
y Competitividad of Spain, Project ACRIMA
(TIN2013-46751-R).
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DeterminationofBifurcationAnglesoftheRetinalVascularTreethroughMultipleOrientationEstimationbasedon
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