Edge Detection by Fuzzy Intensification
Vashno Dutt1, Raj Kumar2, Pardeep sinwar3
1,2
MRK Institute of Engg. & technology Saharanwas Rewari (Haryana)
[email protected], [email protected]
3
Principal GD Polytechnic Bhuna (Haryana)
[email protected]
ABSTRACT: A Gaussian membership function to
model image information in spatial domain has been
proposed in this paper. We enhance the contrast of
the image by intensification operator, in fuzzy
domain. The fuzzifier fh used for image modelling
can be changed interactively for diagnosis of medical
images. By minimizing the fuzzy entropy of the image
information, the parameter t is calculated globally.
To detect the edge of the image, a Gaussian type mask
in the fuzzy domain is used.
1. INTRODUCTION
The separation of a scene into object and
background is important step in image
interpretation. This process is carried out
effortlessly by the human visual system, but a
machine vision algorithm designed to mimic this
action requires object detection. The edge detection
in a given image is first step in computer vision for
the object segmentation. An edge represented image
reduces the amount the data to be processed,
retaining the information about the shape of the
objects in an image. In gray level image, edge
detection identifies the pixel located at the edge.
Many methods have been proposed for edge
detection. Earlier methods used gradient operator to
detect edges of particular orientation. They have
poor performance on blurred and noisy images.
Fuzzy set [1] offers a problem- solving tool between
precision of classical mathematics and inherent
imprecision of the real world. The imprecision in
the image is contained within gray value to be
handled using fuzzy set [2]. An image can be
considered as an array of fuzzy singletons having a
membership value that denotes the degree of some
image property.Fuzzy logic for the image contrast
enhancement was applied globally in [3] on an
image. Fuzzy rule based image intensity
enhancement and noise smoothing and edge
detection is discussed in [4]. Choi and
Krishnapuram [5] have suggested a robust filtering
technique for noise removal and edge enhancement.
Hanmandlu et al [6] has attempted contrast
stretching by fuzzy modelling of image information.
The main emphasis has been laid on the entropy
based
fuzzy
modelling
for
contrast
intensification.Edge detection is very useful in a
number of contexts. Edges characterize object
boundaries and are, therefore, useful for
segmentation, registration, and identification of
objects in scenes. Edge detection algorithm which
are quite useful in a broad set of application, have
already been developed [2] [3] and [4]. For edge
detection the methods commonly used are Gradient,
Laplacian and Sobel’s algorithm [7-9].
II.
MODELLING OF AN IMAGE AS A FUZZY SET
An image I of size MxN and intensity level (0 to L1), in fuzzy set notation, can be considered as
collection of fuzzy singletons,
X=
  xmn  =  mn xmn  ;
m  1,2,..., M ; n  1,2,..., N
(1)
 xmn 
where,
or
 mn x mn
represents the
 mn
of
intensity value
at
membership or grade of some property
xmn
,
xmn
m, n 
=0,1,...,L-1 is the
th
pixel.
To transform the gray intensity X in (0,255) to
fuzzy property plane in the interval (0,1), a
membership function is used. A Gaussian type
membership
function
is
given
as:
 xmn   e
 x
max
 xmn 2 / 2 f h2

L 1

C  
x 0
x   0.5 px 
2
'
X
; where
L 1
(2)
  p ( x)
x 0
is often selected, as it involves a single fuzzier, fh.
x  L  1 , is the maximum color value
Here, max
(5)
2.1 Determination of Fuzzifier:
present in the image. The membership values are
restricted
to
  e  x
the
2
max
/ 2 f h2
range

 ,1 ,
Optimization of the two measures leads to the value
of fh[6] as
with
.
For
computational
efficiency, histogram of color X is considered for
L 1
 x 
f h2
x
fuzzification. So,
represents the
membership function of color X for a gray value x,
with x = 1,2…L-1, defined by,
 X x   e x
2
2
max  x  / 2 f h
max
 x  px 
2
III.


INTENSIFICATION OPERATOR AS THE FIRST
OPERATION FOR EDGE EXTRACTION
Edge detection is a local operation performed on a
window. However, the detected image might not be
acceptable to the human for the desired application.
Therefore a re-look of edges information may be
desired. This requires the system to come back at
the original image. In the proposed approach, the
gray value distribution of the pixel needs to be kept
intact so that a re-look can be possible for the
desired result and re-iteration of algorithm can be
applied.
1 L1
'
'
  X' x  ln  X' x   1   X x  ln 1   X x  px  3.1. Derivation of Intensification Operator:

ln 2 x0
(4)
Where
 x
x 0
L 1
4
The Gaussian membership function of gray images
have typical values of fh = 135,112 and 95. It is
observed that higher fh corresponds to a brighter
image.
Definition- Fuzzy Entropy: H(X) of an image and
is defined as:
 
 x  px 
(6)
x

max

This function is the same as in (2), with mn
replaced by an index x, as the spatial information
(m,n) is lost in histogram. The fuzzifier fh, is a
parameter here which can have an assumed value
for the shape of the membership function. Though it
is desirable that fh should be based on the
information available from the image pixels. In this
regards, fuzzy entropy [3] and fuzzy contrast [6]
measures are used.

 x
x 0
(3)
H(X ) 
1

2
 X  (x) is the modified fuzzy value of pixel
x after some operation on  X (x) . The p(x) stands
for the frequency of occurrence of the gray intensity
x. The lesser the entropy, the lesser is the fuzziness
of the image, which in other words, the image is
well enhanced.
Definition - Fuzzy Contrast: Then the fuzzy
contrast is written as:
We are confined to image enhancement using fuzzy
contrast intensification operator by using a modified
intensification operator [3], INT given as a sigmoid
function:
  ( x) 
1  e
1
t (  X ( x )0.5)

(7)
Where t is termed as the intensification parameter
which lies between 5 to 11.
3.1. Defuzzification of Image pixels:
The membership values are transformed back to the
spatial domain after the desired operations are
applied in fuzzy property domain. The
corresponding inverse operator from the fuzzy
domain to spatial domain is given as:

x '  xmax   2 ln  ' X x  f h2
p(x)
0.03
0.09
0.22
0.25
0.19
0.09
0.00
0.0
0.75
0.0
1.0
 X xmn  x 

1/ 2
(8)
 ' x 
where,
and x are the modified
membership
function
and
spatial
values
respectively.It may be noted that the intensification
operator does not change the frequency of
occurrence of a membership function. However,
after transforming back to the spatial plane, the
distribution might change due to enhancement.
IV.
0.13
0.25
1.0
0.5
1.0
The fuzzification of image using histogram reduces
calculation time.
V.
FUZZY EDGE DETECTION ALGORITHM
The proposed edge detection filter given by the
following equation.
AN EXAMPLE OF IMAGE INTENSIFICATION
(9)
An example is given to demonstrate the above
process of fuzzification of image data followed by
intensification and restoring back from fuzzy to
spatial domain. Assume a 32 X 32 image having 8
gray levels ranging from 0 to 7 with the following
occurrences of gray levels to represent the
histogram data for limited size image:
Gray level
0
4
1
32
256
96
192
2
6
3
7
124
96
128
0
5
Number of pixels
If we subjectively determine the membership of the
fuzzy set as follows:
 X xmn 
= 0.0/0 + 0.0/1 + 0.25/2 + 0.5/3 +
0.75/4 + 1.0/5 + 1.0/6 + 1.0/7
Then the given images can be modelled by this
fuzzy set bright as the fuzzy set
X = { 0,1,2,3,4,5,6,7 } and .p(x) and
can also be found as x ϵ fuzzy set
0
1
2
3
4
5
6
7
 X xmn 
Where f(x,y) is the gray value of an image pixel at
location (x,y) and g(x,y)i s the modified value. α
and β varies from -1 to 1 such that total 8neighbourhood or 4-neighbourhood of central pixel
can be covered around (x,y). The edge filter
operates 3X3 window around the central pixel. The
resultant gray value of the pixel lies in (0,1). By
experiments, it was found that the edge details can
be interactively changed by changing the value of
fh. To increase the edge details in the resulting
image the value fh has to be lesser, and vice versa.
Also, the 8-neighbourhood filter leads to lesser
unconnected edges or the impulse noise in the final
image than the 4-neighbourhood filter. We have
also used our intensification operator as discussed in
eqn. (7), to sharpen the edges of the image and
removing less promising and unconnected spots of
an image.
VI.
RESULT AND DISCUSSION
The proposed algorithm is implemented on MatLab.
First, the intensification operator is calculated.
Using different values of intensification parameter,
t, and the set of intensified images are generated.
The edge information for each of the intensified
image is observed for its edge information quality.
Since this is a subjective measure by human
operator, yet standard edge detectors are used for
the comparison to choose the desired quality.
Further, the iterations are repeated for the other
value of the fuzzifier fh. This enables the operator
to get a variety of edge information, as the threshold
value of the edge pixel can be tuned around any
desired pixel value due to the various values of fh
and t.
REFERENCES
[1] Bezdek, J.C., and Pal, S.K.,: Fuzzy Models for Pattern
Recognition : Methods that search for structure in data, A
selected reprint ume, IEEE Press 1992.
We can easily observe that variety of edge image
with different thickness can be obtained by simply
changing the shape of fuzzy membership functions.
This also strengthens our belief that fuzzy algorithm
serves better edge performance and provides more
flexibility to edge information depending upon the
need. We have used the algorithm on various test
images. The fuzzifier and intensifications were
calculated for fh and t respectively. The detailed
method for doing this in given in [6].The original
image with edges at different value of fh and t, for
two test images as shown in fig 1 and in fig. 2.
VII
CONCLUSION
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The fuzzy edge detector proposed in this paper
extracts the edge using local edge operator over a
small window. However, the image is intensified
globally with the intensification operator a priory.
This operation can bring out the edge around the
desired ray value. This automatic handling of the
intensification followed by edge detection is carried
out by computer. The human operator can interact
in defining the desired intensification to get the
edge feature of the object. Medical professional
may desire to interact of the images to observe the
pathological information at various gray levels. By
varying the fh value and cross-over point different
classes of edge can be extracted.
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Fig. 1: Edges extracted after intensification on original with
fh=112, 167 and 95(top left to bottom right) and t=7 parameters
calculated in sub-images(top left); Canny edge(bottom
left)
Fig. 2: Edges with fh=122, t=5 (top right) and t=7 (bottom left)