1.
International Journal For Research & Development in Technology
Volume: 2, Issue: 4, Oct -2014 ISSN (Online):- 2349-3585
21 Copyright 2014- IJRDT www.ijrdt.org
Segmentation Of Medical Images Using Metric
Topology – A Region Growing Approach
Gnanambal Ilango1,
B. Shanthi Gowri 2 ,
G. Rohith 3
1
Department of Mathematics,Government Arts College,Coimbatore, India.
2
Department of Mathematics,Sri Krishna College of Engineering and Technology
3
Department of Mechatronics Engineering,Sri Krishna College of Engineering and
Technology,Coimbatore,India.
Abstract— A metric topological approach to the region
growing based segmentation is presented in this article.
Region based growing techniques has gained a significant
importance in the medical image processing field for finest
of segregation of tumor detected part in the image.
Conventional algorithms were concentrated on segmentation
at the coarser level which failed to produce enough evidence
for the validity of the algorithm. In this article a novel
technique is proposed based on metric topological
neighbourhood also with the introduction of new objective
measure entropy, apart from the traditional validity
measures of Accuracy, PSNR and MSE. This measure is
introduced to prove the amount of information lost after
segmentation is reduced to greater extent which elucidates
the effectiveness of the algorithm. This algorithm is tested on
the well known benchmarking of testing in ground truth
images in par with the proposed region based growing
segmented images. The results validated show the validation
of effectiveness of the algorithm.
Index Terms— Metrics, Metric Topological
Neighbourhoods, Seeded Region Growing, Segmentation,
Accuracy, PSNR, MSE, Entropy.
I. INTRODUCTION
The interpretation of group of connected pixels with similar
properties into larger regions based on the predefined criteria
collectively termed as region growing based segmentation, has
gained a magnificent importance in medical image processing
field. The main goal of image segmentation is domain
independent partitioning of an image into a set of disjoint
regions that are visually different, homogeneous and
meaningful with respect to some characteristics or computed
property such as gray level, texture or colour to enable
interpretation of the image. Region growing based
segmentation has got the better tradeoff than the edge based
techniques owing to the fact that the edges are difficult to
detect in noisy conditions. The interpretation lies in the
isolation of the strongest region of interest with respect to the
highest gray scale value. This attempts to segregate the
particular group of pixels that are indifferent to the nominal
range of values in the histogram. The continuous path of gray
scale connected to the two points of the image requires a pre
requisite of predefined variance (threshold) failing to fulfill
will lead to the improper segregation. Based on this fact
several algorithms were experimented.
Yi-Ta Wu , Frank Y. Shih , Jiazheng Shi and Yih-Tyng Wu
[12] developed a top-down region dividing based approach
which combines the advantages of both histogram-based and
region-based approaches and showed that it is efficient for
segmentation without distorting the spatial structure. Geng-
Cheng Lin , Wen-June Wang , Chung-Chia Kang and Chuin-
Mu Wang [1] proposed a novel fuzzy knowledge based seeded
region growing for multispectral MR images and proved that
this method is more efficient than the functional MRI of the
brain automated segmentation tool, k-means and support
vector machine methods. Maria Kallergi , Kevin Woods,
Laurence P. Clarke , Wei Qian and Robert A . Clark [7]
developed local thresholding and region growing algorithms
and applied to digitized mammograms and proved that
computerized parenchymal classification of digitized
mammograms is possible and independent of exposure.
Pastore J., Bouchet A., Moler E. and Ballarin, V. [9]
described an automatic segmentation of computerized axial
tomography images with tumors by means of alternating
sequential filters of mathematical morphology and connected
components extraction based on continuous topology
concepts. Gnanambal Ilango and R. Marudhachalam [2]
presented an automatic method relevant to medical image
segmentation by means of fuzzy hybrid filters for denoising
and topological concepts to extract connected components and
edge detection. Tamas Sandor , David Metcalf and Young-Jo
Kim [11] described an automated procedure, based on the
principles of region growing and nearest neighbour
classification to extract continuous areas from brain CT scans
and showed that it is a fast and reasonably accurate method for
segmenting brain tissue. Sang Uk Lee and Seok Yoon Chung
[10] compared different thresholding techniques for
segmentation and concluded that image performances are
different. Junchai Gao , Zhiyong Lei , Zemin Wang and
Keding Yan [5] proposed a threshold and edge detection fused
segmentation algorithm which considers not only the adjacent
uniformity but also the local contrast.

2.
International Journal For Research & Development in Technology
Paper Title:- Segmentation of Medical Images Using Metric Topology-
A Region Growing Approach (Vol.2, Issue-4) ISSN(O):- 2349-3585
22 Copyright 2014- IJRDT www.ijrdt.org
In this article, a new seeded region growing segmentation
algorithm is proposed based on metric topological –
neighbourhoods of different metrics and grouping
criterion.This work is organized as follows: Seeded region
growing method of segmentation, Basic definitions, Proposed
segmentation Algorithm, Method of evaluation of
segmentation, Experimental work, Result analysis, Conclusion
and the Figures and Tables of the experimental results.
II. SEEDED REGION GROWING METHOD OF
SEGMENTATION
Region based segmentation is a classical method. This method
tries to extract the object that is connected based on some
predefined criterion. This criterion may be based on the
intensity information and edges in the image. One of the
region based segmentation methods is the seeded region
growing method. This is a procedure that groups pixels in the
whole image into sub regions based on predefined criterion
and processed in four steps. (i) Select a seed pixel in the
original image. (ii) Select a set of similarity criterion such as
gray level or colour and set up a stopping rule. (iii) Grow
region by appending to the seed, those neighbouring pixels
that have predefined properties similar to seed pixel. (iv) Stop
region growing when no more pixels meet the criterion for
inclusion in that region.
Here from the histogram of the image, the gray level
of the region of interest is selected. Considering the gray level
of the region of interest, a seed point is selected as a point
having maximum number of – neighbourhoods of different
metrics. For the similarity criterion, a grouping criterion is
defined based on metric and gray level difference. The
connected region of interest is grown by including the –
neighbours of the seed point satisfying the grouping criterion
with the seed point. The quality of the segmented region of
interest from the medical images for the proposed algorithm is
compared with the ground truth images and the results are
analysed.
III.BASIC DEFINITIONS
Definition 1[8]
A metric on a set is a function having the
following properties:
( )
( ) ( )
( ) ( ) ( )
Given a metric on , the number ( ) is called as the
distance between and in the metric . For a given ,
consider the set ( ) * ( ) + ( ) of all
points y whose distance from is less than . Here ( ) is
called the - ball centered at If is a metric on the set ,
then the collection of all - balls ( ) for and
, is a basis for a topology on , called metric topology
induced by . A set is open in the metric topology induced
by iff for each , there is a such that ( )
. If is a topological space, is said to be metrizable if
there exists a metric on the set that induces the topology
of . A metric space is a metrizable space together with a
specific metric that gives the topology of .
Definition 2[4]
An image may be defined as a two-dimensional function
( ), where and are spatial (plane) coordinates, and
the amplitude of at any pair of co-ordinates ( ) is called
the intensity or gray level of the image at that point. If
and the intensity values of are all finite, discrete quantities,
we call the image a digital image. A digital image is
composed of a finite number of elements ( ) each of which
has a particular location and value. These elements are called
picture elements or pixels.
Definition 3[3]
Let , where *( )
+ be the set of spatial co-ordinates
of an image. For any metric space ( ) any
and any , consider the set ( ) * ( )
+ . ( ) is called the (open) - neighbourhood of
in . ( ) is an open ball centered at .
Definition 4[6]
Let ( ) ( ) . Consider the
functions
i. , defined by ( )
| | | |. Then ( ) is a metric space.
The metric is called the City – Block metric or Manhattan
metric.
ii. , defined by ( )
* | | | | + . Then ( ) is a metric
space. The metric is called the Chessboard metric.
Definition 5[3]
For any point , the 4 - neighbourhood of denoted by
( ) is defined as
( ) * ( ) + ( ) is also
denoted by ( ) and the 8 - neighbourhood of denoted by
( ) is defined as
( ) * ( ) + ( ) is also
denoted by ( ).
Definition 6[3]
For any point , the 4 - neighbours of are ( ) * +
and the 8 - neighbours of are ( ) * +.
Definition 7[3]
Let where *(
) +. Let ( )
( ) . Consider the function
defined by ( ) | | | | . ( )
is a metric space. For any point , the neighbourhood
of is defined as ( ) * ( ) +.
Hence the neighbourhood of is
( ) * ( ) + and the neighbourhood
of is ( ) * ( ) +. The

3.
International Journal For Research & Development in Technology
Paper Title:- Segmentation of Medical Images Using Metric Topology-
A Region Growing Approach (Vol.2, Issue-4) ISSN(O):- 2349-3585
23 Copyright 2014- IJRDT www.ijrdt.org
neighbours of are ( ) * + and the neighbours
of are ( ) * +.
Definition 8[3]
Let where *( )
+ . Let ( ) ( ) .
Consider the function
defined by ( ) | | | | .
( ) is a metric space. For any point p, the -
neighbourhood of p is defined as
( ) * ( ) +. Hence the -
neighbourhood of p is defined as
( ) * ( ) + and the -
neighbourhood of p is defined as
( ) * ( ) + . The - neighbours of
p are ( ) * + and the - neighbours of p are
( ) * +.
Definition 9[3]
Let where {*( )+
*( )+ } .
Let ( ) ( ) . Consider the
function defined by ( )
,| | | | - . ( ) is a metric space.
For any point , the neighbourhood of is defined as
( ) * ( ) +. Hence the
neighbourhood of is defined as
( ) * ( ) + and the
neighbourhood of is defined as
( ) * ( ) +. ( ) is also
denoted by ( ) . The neighbours of are ( )
* + and the neighbours of p are ( ) * +.
Definition 10
Grouping criterion:
Let be an image in levels of gray and a topology
associated with and let Let be a metric on .
Define such that (( )) where is
the maximum of the gray level difference between and the
elements of . Let Given a fixed , an
element is said to belong to if ( ) for
some and (( ))
IV.PROPOSED SEGMENTATION ALGORITHM
VHLT ALGORITHM
Let be an image in levels of gray.
Step I:
From the histogram of the image , select the gray
level for the region of interest to be segmented.
Step II:
Find the seed point of the region of interest. Here the
seed point is a point with maximum number of vertical
neighbours with gray level difference less than which is very
small. If more than one point has the maximum number of
vertical neighbours then choose any one of those points as the
seed point.
Let * + where is the seed point of the region
of interest.
Step III:
Choose If a point where
such that ( ) (( )) -----(1)
then include in and rename it as .
Repeat Step III again and again for the elements of till no
point satisfying the condition (1). That is all the
vertical neighbours of are obtained.
Step IV:
If a point where such that ( )
where (( )) -----(2)
then include in and rename it as .
Repeat Step IV again and again for the elements of till no
point satisfying the condition (2). That is all the
horizontal neighbours of are obtained.
Step V:
If a point where such that ( )
where (( )) ------(3)
then include in and rename it as .
Repeat Step V again and again for the elements of till no
point satisfying the condition (3). That is all the LT
neighbours of are obtained.
The set is the segmented region of interest.
V.EVALUATION MEASURES
(i) Let be the set of pixels in the image. Define the
ground truth as the set of pixels that were labeled as
tumor by the expert. Similarly define the segmented tumor
as the set of pixels that were labeled as tumor by the
algorithm. ̅and ̅ the set of pixels that were labeled as non-
tumor by the experts and algorithm respectively. The true
positive ( ) set is defined as , i.e., the set of
pixels common to and . i.e., the set of pixels that were
labeled as tumor by the expert and algorithm. The true
negative ( ) set is defined as ̅ ̅, i.e., the set of
pixels common to ̅ and ̅. i.e., the set of pixels that were
labeled as non-tumor by the expert and algorithm. The false
negative ( ) set is defined as ̅, i.e., the set of
pixels common to and ̅. i.e., the set of pixels that were
labeled as tumor by the expert and non-tumor by the
algorithm. The false positive ( ) set is defined as ̅

4.
International Journal For Research & Development in Technology
Paper Title:- Segmentation of Medical Images Using Metric Topology-
A Region Growing Approach (Vol.2, Issue-4) ISSN(O):- 2349-3585
24 Copyright 2014- IJRDT www.ijrdt.org
, i.e., the set of pixels common to ̅and . i.e., the set of
pixels that were labeled as non-tumor by the expert and tumor
by the algorithm.
The segmentation evaluation measure “ Accuracy” is defined
as
Accuracy =
( ) ( )
( ) ( ) ( ) ( )
.
(ii) The Mean Square Error (MSE) is defined as
∑ ∑ ‖ ( ) ( )‖
Where I(i,j) = Input image and K(i,j) = Segmented image.
The Peak Signal to Noise Ratio (PSNR) is defined as the ratio
of peak signal power to noise power in logarithmic scale.
Since the MSE represents the noise power and peak signal
power, it is unity in case of normalized image signal.
PSNR = [ ] where MaxI = Maximum possible
pixel value of the image. The typical values of PSNR in a
lossy image is between 20 and 50 db.
(iii) For the validation in terms of amount of information
retained after segmentation subjected to accurate recovery of
information, a new parameter which can be introduced is the
Entropy. Entropy (E) which is the amount of information per
pixel in the source is formulated as
E = ∑ where = Number of image pixels
expressed in bits.
VI. EXPERIMENTAL WORK
In this work, magnetic resonance images of brain affected
by tumor are taken. The seed point of the region of interest is
selected using the histogram of the image and metric
topological - neighbourhoods . The region of interest is
grown from the seed point using the metric topological -
neighbourhoods ( ) ( ) ( ) and the
grouping criterion. The proposed region growing segmentation
algorithm is implemented using MATLAB 7.0. The
performances of the proposed algorithm are evaluated using
the segmentation evaluation measures Accuracy, MSE, PSNR
and Entropy.
VII.RESULT ANALYSIS AND DISCUSSION
The magnetic resonance images of brain affected by tumor
are segmented using the proposed region growing
segmentation algorithm based on metric topological -
neighbourhoods and the grouping criterion. The experimental
results show that the quality of segmentation by the VHLT
algorithm is 98.6% or more. PSNR values are in the nominal
range 20 – 50 db. The difference between the entropy values
of the original and segmented images is negligibly small i.e.
less than 0.01. Table 1 shows the entropy values of the
original and segmented images. Table 2 shows the
accuracy ,quality of segmentation, PSNR and MSE values. Fig
1 shows the ground truth images. Fig 2 shows the original
magnetic resonance images of brain affected by tumor, the
corresponding segmented region and the segmented region
with boundary using the proposed VHLT region growing
segmentation algorithm respectively.
VIII. CONCLUSION
In this work, a new region growing segmentation algorithm
based on metric topological - neighbourhoods and grouping
criterion is introduced. To demonstrate the performance of the
proposed region growing segmentation algorithm,
experiments have been conducted on magnetic resonance
images of brain affected by tumor. The performance of the
proposed segmentation algorithm is measured using the
segmentation evaluation measure „Accuracy‟. The
experimental results indicate that the quality of segmentation
by the proposed VHLT algorithm is 98.6% or above. The new
parameter Entropy is validated between the original and
segmented images. The amount of information present in the
original image approaches the segmented portion which
proves the effectiveness of algorithm by retaining the essential
information without loss of data. This retains maximum
fidelity of image. Further, the observed objective measure
PSNR meet the nominal range of 20-50 db. This work is a
new metric topological approach for region growing
segmentation of magnetic resonance images.
FIGURES
Fig 1 – Ground Truth Images
Fig 2 – Segmentation by VHLT Algorithm

5.
International Journal For Research & Development in Technology
Paper Title:- Segmentation of Medical Images Using Metric Topology-
A Region Growing Approach (Vol.2, Issue-4) ISSN(O):- 2349-3585
25 Copyright 2014- IJRDT www.ijrdt.org
Table 1 – Entropy Values
Table 2 - Accuracy , Quality, PSNR and MSE values.
ACKNOWLEDGEMENT
The first author thank The University Grants Commission,
India for its financial support to carry out this work.
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