A Novel Method to Extract Vasculature from the
Optical Imaging of Intrinsic signals using Fuzzy
Morphological Approach
Sreekala K.
Assistant Professor, Christ University Faculty of Engineering, Bangalore
Abstract: Here a method is proposed for the
extraction of Vasculature from the cerebral cortex
image, that was created using Optical imaging of
intrinsic signals. Automatic separation of vascular
structure in optical cerebral cortex images is
important in clinical practice and preclinical study.
Image and structuring elements are considered as
fuzzy sets, so the fast implementation of Mathematical
Morphology is possible. The basic idea is to replace the
fixed structuring elements by variable structuring
elements and the segmentation is done using fuzzy
morphological techniques.
I. INTRODUCTION:
OPTICAL imaging of intrinsic signals (OIS) is an
extremely powerful method that can be used to obtain
high resolution spatial maps of functional properties from
cerebral cortex [1]. This technique maps the brain by
measuring intrinsic activity-related changes in tissue
reflectance. Functional physiological changes, such as
increases in blood volume, hemoglobin oxymetry
changes, and light scattering changes, result in intrinsic
tissue reflectance changes that are exploited to map
functional brain activity [2]. Extraction of Vasculature
from the resulting gray optical images is extremely
important to basic researches, clinical diagnostics, and
intraoperative procedures.
Mathematical Morphology, the foundation of
Morphological image processing, is based on set theory.
Initially it was developed for binary images, and later got
extended to gray scale images. It relies on a set of pixel
locations called structuring elements which is convolved
over the image. It is a special mask filter which is used to
enhance the input image. In Fuzzy Mathematical
Morphology the image and structuring elements are gray
scale. Gray scale images can be represented as fuzzy sets.
So these fuzzy concepts can be used in Traditional
Mathematical Morphology which leads to Fuzzy
Mathematical Morphology. This work uses Fuzzy
Mathematical Morphology as the basic root to extract
vasculature.
In Fuzzy Mathematical Morphology the image and
structuring elements are gray scale. Gray scale images can
be represented as fuzzy sets. So these fuzzy concepts can
be used in Traditional Mathematical Morphology which
leads to Fuzzy Mathematical Morphology. Another
important point that has to be considered here is the
structuring element. Unlike the Traditional Mathematical
Morphology where the structuring elements are taken as
images, the Fuzzy Mathematical Morphology takes
structuring element as fuzzy sets which is obtained from
the image itself. In the proposed method the image and
structuring element are converted to fuzzy sets and the
basic morphological operations erosion, dilation, and
opening are performed to extract the vasculature structure.
II. BACKGROUND
A. Fuzzy Mathematical Morphology
If two predicates[3] P and Q are given, then the basic
operators used in fuzzy set theory are
Conjunction ( P  Q ) ,
Disjunction ( P  Q ) ,
Negation (~ P ) and
Implication (~ P  Q ) .
The most Commonly used Conjunction
Implication formulae are given below.
Godel Brouwer:
C ( a, t )  a  t
sa
sa
s,
I ( a, s )  
1,
Kleene Dienes:
0,
C ( a, t )  
t
t  1 a
t  1 a
I ( a, s )  1  a  s
and
In a gray scale image, Each gray level is associated
with a value between 0 and 1. Fuzzy Morphological
operators can be defined by means of fuzzy logic[4].
Let A and B belongs to the set of image parts, then
From the lattice theory[6] adaptive morphological
filtering operations i.e; closing and opening are
respectively defined as
A  B  y  f ,
 y  f ,
 y  f ,
 hm ( f )
:


x



 mh   mh ( f )( x)
(11)
 mh ( f )
:


x



 mh   mh ( f )( x)
(12)
y A yB
A( y )  B( y )
I ( A( y ), B( y ))  1
Where I denote the binary Implication.
The fuzzy erosion of an image f by a structuring element
B at a point x is given by
 F ( f , B)( x)  inf I ( B( y)), f ( y)
y f
(1)
Similarly, being A and B part of image f, then
A  B    y  f
y  A, y  B
 y  f , C ( A( y ), B( y ))  1
where C is binary conjunction.
Then the fuzzy dilation of an image f by a structuring
element at a point x is defined as
 ( f , B)( x)  sup C ( B( y)), f ( y)
F
(m, h, f )     C  I
C. Adaptive Sequential Morphological Filters
The adaptive morphological filters described by
equations (11) and (12) are generally neither size
distribution nor anti-size distribution. Sequential filters
are built by naturally reiterate adaptive dilation or erosion.
So the adaptive sequential dilation, erosion, closing and
opening are respectively defined as[7]
(m, p, h)     N  C

h
m, p
(2)
y f
From morphological theory, fuzzy opening is described as
 ( f , B)   ( ( f , B), B)
F
F
F
(3)

h
m, p
and fuzzy closing as
 ( f , B)   ( ( f , B), B)
F
F
F
(4)
Therefore the fuzzy inner edge is defined as
 FInt f  f   F ( f , B)
(5)
 mh , p
and the fuzzy outer edge is defined as

F
Ext
f   ( f , B)  f
F
(6)
Top  Hat F ( f )  f   F ( f , B)
Top  Hat ( f )   ( f , B)  f
F
:
 mh ( ( f )
:
I
 mh     mh ( f )

P
sup f ( w)
h
wR m ( x )



infh f ( w)
wR m ( x )
times

I

 mh , p   mh , p ( f )

I

 mh , p   mh , p ( f )
 m,h p
and
(15)
(16)
 m,h p
provides
h
 hm, p and  m,
p.
ASFOC mh ,n ( f ) :
II

h
h
 f   m, pn     m, p1 ( f )( x)
(17)
ASFCOmh ,n ( f ) :


(14)
(8)
(m, h, f )     C  I
 (f)
I
: 
f
I
: 
f
times
Thus the extension of well known alternating sequential
filters can be defined as
The elementary dual operators of adaptive dilation
and erosion are defined as[5]



P
(13)
the two sequential morphological filters
B. Adaptive Morphological Operators and Filters


x


 
x

I

: f

I
 mh     mh ( f )

(7)
and Fuzzy Top-Hat by closing is
h(
m


The morphological duality of
Fuzzy Top-Hat by opening is
F

 hm, p
I

: f

(9)
(10)
 II

h
h
 f   m, pn     m, p1 ( f )( x)
III. PROPOSED ALGORITHM
(18)
Original image
structuring element. The image was eroded and then
dilated by means of Kleene-Dienes conjunction. The
structuring element used is 5X5 in size and cone
shaped[3]. Its values range from 0 to 1, and it is obtained
from the following formula:
Fuzzification
Filtering and
reconstruction
operations
Fuzzy image opening
Fuzzy
mathematical
morphology
Top-Hat transform
Defuzzification
Fig.1. flowchart of the proposed method
A flowchart of the proposed method is shown in figure(1).
The proposed method follows these steps:
Step1: Image fuzzification.
Step2: Image filtering and reconstruction by closing.
Step3: Image fuzzy opening.
Step4: Calculation of top-hat transform.
Step5: Image purification.
Step6: Image Defuzzification.
Step 1: Image fuzzification.
To be able to operate with fuzzy logic in gray
level images, the first step is to fuzzify the image. This
consists of taking the values of each pixel from the
original image to values between 0 and 1. There are
several techniques that allow so. The one employed in this
work is the sigmoid function[8]:
1 1
 arctan( t )
2 
f(x, y) - k
f(x, y) - j
f(x, y) - i
f(x, y) - j
f(x, y) - k
f(x, y) - 2i f(x, y) - k f(x, y) - 2j 
f(x, y) - i f(x, y) - j f(x, y) - k 
f(x, y)
f(x, y) - i f(x, y) - 2i 

f(x, y) - i f(x, y) - j f(x, y) - k 
f(x, y) - 2i f(x, y) - k f(x, y) - 2j 
If any value is below zero, it is assigned a zero value.
Purification
 (t ) 
 f(x, y) - 2j
 f(x, y) - k
1 
 f(x, y) - 2i
f(x, y) 
 f(x, y) - k
f(x, y) - 2j
Step4: Calculation of fuzzy top-hat transform.
The Fuzzy Top-Hat transform was calculated.
The objective is to extract the locally brilliant elements
from the image. To do so, fuzzy Top-Hat by opening was
used.
The purpose is to obtain the graph peak
representing the object to be segmented (brilliant object).
The strategy is to create a new image with the irrelevant
information, that is to say, to eliminate the peak applying
an opening to the original image. Then, by subtracting it
from the original image, a new one obtained is built from
the information of interest.
Step5: Image purification.
Image purification is nothing but the removal of noise.
Even after Top-Hat transform, there will be some amount
of noise present in the extracted output. This can be
removed by adaptive sequential filtering operations.
Step6: Image Defuzzification.
Defuzzification process takes place in order to
convert the fuzzy partition matrix to a crisp partition.
Here, to achieve “defuzzification”, the inverse function of
the sigmoid function is used.
IV. RESULTS AND DISCUSSION
(7)
Step2: Image filtering and reconstruction by closing.
Firstly, in order to remove noise, a pre-filtering
process is employed. It consists of a filtering operation
followed by a closing by reconstruction.
Step3: Image fuzzy opening.
Morphological opening operator can wipe off the
light object with smaller dimension than the used
The proposed method was applied to many
cerebral cortex images. This type of images accounts for a
great deal of noise which leads to notorious difficulty
when it comes to applying other traditional segmentation
techniques. When processing these images with the
proposed algorithm, the lines of interest were well
segmented. The results are shown below(figure(1)) and
the efficiency of the proposed method can be clearly
appreciated.
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Figure(2): (a). Original image (b). Fuzzified
image (c). Tophat transform (d). Defuzzified image
V. CONCLUSIONS
Here, method for the extraction of vasculature
structure from cerebral cortex image is presented. The
method is developed by using mathematical
morphology combined with the fuzzy techniques. These
techniques are first employed to smooth and strengthen
the images as well as to suppress the background
information. Then, it enhances the image. Finally, a
purification procedure is introduced using adaptive
sequential filters and defuzzification is done. The
vasculature are then extracted.
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