Microsoft Word - 26-2866_s_ETASR_V9_N4_pp4457-4462


Engineering, Technology & Applied Science Research Vol. 9, No. 4, 2019, 4457-4462 4457  
  

www.etasr.com Bhatti et al.: Active Contours Using Harmonic Global Division Function 

 

Active Contours Using Harmonic Global Division 

Function 
 

Muhammad Tarique Bhatti 

Department of Mechanical Engineering, 
Quaid-e-Awam University of 

Engineering, Science & Technology,  

Nawabshah, Pakistan  
trqbhatti@quest.edu.pk 

Ahsin Murtaza Bughio 

Department of Electronics Engineering, 
Quaid-e-Awam University of 

Engineering Science & Technology, 

Nawabshah, Pakistan 
ahsan.murtaza@quest.edu.pk 

Ali Anwar 

Department of Electronics Engineering, 
Quaid-e-Awam University of 

Engineering Science & Technology, 

Nawabshah, Pakistan 
alianwer@quest.edu.pk 

Toufique Ahmed Soomro 

Department of Electronics Engineering, 
Quaid-e-Awam University of 

Engineering Science & Technology,  

Nawabshah, Pakistan 
etoufique@yahoo.com 

Muhammad Afzal Soomro 

Department of Mathematics and 
Statistics, Quaid-e-Awam University of 

Engineering Science & Technology, 

Nawabshah, Pakistan 
m.a.soomro@quest.edu.pk 

Shafiullah Soomro 

Department of Basic Science and Related 
Studies, Quaid-e-Awam University of 

Engineering Science & Technology, 

Nawabshah, Pakistan 
s.soomro@quest.edu.pk 

 

 
Abstract—This paper presents the region-based active contours 

method based on the harmonic global signed pressure force 
(HGSPF) function. The proposed formulation improves the 

performance of the level set method by utilizing intensity 

information based on the global division function, which has the 

ability to segment out regions with higher intensity differences. 

The new energy utilizes harmonic intensity, which can better 

preserve the low contrast details and can segment complicated 

areas easily. A Gaussian kernel is adjusted to regularize level set 

and to escape an expensive reinitialization. Finally, a set of real 

and synthetic images are used for validation of the proposed 

method. Results demonstrate the performance of the proposed 

method, the accuracy values are compared to previous state-of-
the-art methods. 

Keywords-image segmentation; active contours; HGSPF 

function 

I. INTRODUCTION  

Segmentation is a technique to isolate particular image 
regions having a variety of uses in image processing and 
computer vision [1]. Images with feeble edges, noise, and 
obscured boundaries have some regarding this aspect. In 
manual segmentation, human mistakes can deliver more false 
results, so, a computer-based system is required, which can 
bolster experts and radiologists to partition accurate tumor 
muscles or true regions [2]. A few techniques have been 
proposed regarging the image segmentation issue. Authors in 
[3] proposed an underlying contour based procedure for 
separating objects in an image. Active contour image 
segmentation is a strategy, in which an implicit curve is 
deformed using some minimization procedure until it achieves 
the ideal boundary. There are two significant groupings of 
active contour techniques, which are edge based strategies [4-
6] and region based techniques [7-14]. Edge based procedures 

set up an edge force by using the edge data of an image, which 
turn towards the required boundaries. These models do not 
accomplish right segmentation on images with weak and 
obscured boundaries. Region based methods use image pixel 
information and develop region based force terms, which 
divide the pixels inside an image. The work in [9] is viewed as 
the most utilized model for global region based segmentation. 
This strategy is not suitable for unclear boundaries and images 
which have intense pixel contrast and complicated background 
variations. To minimize such problems, authors in [8] proposed 
an altered global region based technique, which utilizes four 
intensity means for pixel regions across the image. Due to the 
impact of global division algorithim, this technique enhances 
segmentation. Authors in [7] proposed a strategy which utilizes 
global pixel data from [9] and builds up a SPF (Signed Pressure 
Force) function. Also, it utilizes a Gaussian kernel to regularize 
a level set function in each cycle. A modified SPF method was 
formulated in [15] using harmonic means instead of arithmetic. 
This method reduces the approximation error of the previous 
methods. Motivated by [7, 8, 15], we propose a novel harmonic 
global signed pressure force (HGSPF) function. HGSPF 
function utilizes global division and harmonic intensity 
information in its design. It yields better results over images 
having complicated intensity information and/or low contrast. 
Moreover, a Gaussian filter has been induced to regularize 
level set and to overcome the need of level set reinitialization. 
The proposed strategy is validated over synthetic and real 
images. The results are compared in terms of accuracy, which 
indicates the effectiveness of the proposed method compared to 
previous state of the art methods. 

II. RELATED WORK 

Authors in [9] proposed a mainstream global intensity 
technique in the light of [16]. This technique relies upon 

Corresponding author: Shafiullah Soomro



Engineering, Technology & Applied Science Research Vol. 9, No. 4, 2019, 4457-4462 4458  
  

www.etasr.com Bhatti et al.: Active Contours Using Harmonic Global Division Function 

 

gradient based segmentation techniques. Lets suppose an image 

:I RΩ ⊂  with a level set : Rφ Ω ⊂  and C is a curve for level 
set: the following formulations are proposed: 

2

2 1 1

2

2 2

1

2

( , , ) | ( ) |

                        | ( ) |

Length( ) Area( ( ))

CV
E c C I x u

I x u M dx

C v in C

u M dxλ

λ

µ

Ω

Ω

= −

+ −

+ +

∫

∫  (1) 

where µ, v, λ1 and λ2 > 0 are constants. Length term regularizes 
the contour C. M1 and M2 are defined as: 

1 ( ( ))M H xφ= ε     (2) 

2
(1 ( ( )))M H xφ= −

ε

    (3) 

where Hε(ϕ) is a Heaviside function is defined as: 

1 2
( ) 1 arctan

2
Hε

φ
φ

π ε
= +

  
  
  

   (4) 

ε accomplishes the smoothness of the Heaviside function. u1 
and u2 are statistical means of an image inside and outside of 
curve C respectively, which are explained as: 

1

1

1

( )I x M dx

u
M dx

Ω

Ω

=
∫

∫
          (5) 

2

2

2

( )I x M dx

u
M dx

Ω

Ω

=
∫

∫
     (6) 

Minimizing (1) by gradient descent [17], we achieve the 
following variational formulation: 

( 2 21 1 2 2( ) ( ) div ( )
| |

I u I u v
t

ε

φ φ
λ λ µ δ φ

φ

∂ ∇
= − − + − + −

∂ ∇

 
 

  
 (7) 

where δε(ϕ) is a Dirac function defined as: 

2 2
( )

( )
ε

ε
δ φ

π φ ε
=

+
     (8) 

where µ controls the smoothness of the level set function, v is 
utilized to pace contour evolution and λ1, λ2 are constants for 
the inner and outer force of the contour. It is considered as a 
sound technique for segmenting images with noisy and 
obscured regions. However, this strategy is not fit for images 
which have higher intensity distinction or images with 
convoluted intensity information in the background. 

To overcome this obstacle, authors in [8] proposed an 
enhanced global division intensity term, which can segment 
complex intensity regions inside homogenous images. Global 
division function is joined in their energy definition in a new 

region based term. This strategy can segment objects with 
enormous distinction of intensity regions. They proposed the 
following formulations: 

1 2 11 12 21 22

2

1 11

2

1 12

2

2 12

2

2 22

2

2

1

1

( , , , , , , )

( ( ) )( ( ) )

(1 ( ( ) ))( ( ) )

( ( ) )( ( ) )

(1 ( ( ) ))( ( ) )

MIN
E u u w w w w

H I x u I x w dx

H I x u I x w dx

H I x u I x w dx

H I x u I x w M dx

M

M

M

φ

Ω

Ω

Ω

Ω

=

− −

+ − − −

+ − −

+ − − −

∫

∫

∫

∫

   (9) 

where ( ( ) ),( 1,2))iH I x u i− =  is a global division intensity 
function. Instead of considering two intensity terms, this 
procedure considers four intensity terms, two inside and two 
outside the contour.  

The technique developed in [9] cannot authentically capture 
an object with bigger intensity distinction. Figure 1 displays the 
segmentation of one complex image for both methods. 

 

 
(a) 

 
(b) 

Fig. 1.  Image segmentation on images having intensity difference (a) 

Chan-Vese [9], (b) [8] 

An active contour strategy with respect to ACSLG (active 
contour with selective local and global) technique was 
proposed in [7] as a blend of GAC [18] and [9] methods. In this 
model, a region based SPF term was proposed, which can 
perfectly stop the curve at weak or fuzzy edges. This model can 
capture outside regions if initialized inside and capture inside 
regions if initilzed outside the object. In addition, this model 
uses the Gaussian kernel to regularize the level set dimension 
set. They propose the energy term as: 

( )( ( )) | | div
| |

( ( ))

spf I x
t

spf I x

φ φ
φ α

φ

φ

∂ ∇
= ∇ +

∂ ∇

+ ∇ ∇

   
   
      (10) 

A novel SPF force term is introduced: 

( )
1 2

1 2

( ) ( ) / 2
( ( ))

max | ( ) ( ) / 2 |

I x u u
spf I x

I x u u
=

− +

− +
    (11) 

where u1 and u2 are global intensities computed from the given 
image defined in (5) and (6) correspondingly. 

More recently, authors in [15], summed up the Chan-Vese 
[9] energy formulation and proposed another exceptional case. 



Engineering, Technology & Applied Science Research Vol. 9, No. 4, 2019, 4457-4462 4459  
  

www.etasr.com Bhatti et al.: Active Contours Using Harmonic Global Division Function 

 

The new energy in this method reduces the estimate error and 
can more effectively safeguard the regions with minute 
difference between ROIs and background, hence this method 
can viably segment images, particularly with low contrast. 
Authors came up with a two-phase harmonic approximation in 
their energy formulation instead of the traditional arithmetic 
intensity approximations in [9]. They proposed the following 
general form of energy function: 

2

1 1

2

2 2

1

2

( , ) | f( ( )) ( |

                | f( ( )) ( |

Length( ) Area( ( ))

)

)

E C I x f u

I x f u M dx

C v in C

u M dxλ

λ

µ

Ω

Ω

= −

+ −

+ +

∫

∫  (12) 

where f is reciprocal harmonic mean average i.e., f(x)=1/x. The 
harmonic mean intensities are usually less than one, while the 
traditional means u1 and u2 in (1) are greater than one. So, in 
this case, this method penalizes the approximation error in a 
lighter way than Chan-Vese method, which does this in a 
stronger way. For low contrast images u1 and u2 remain close to 
each other, which makes the contour stationary and does not 
capture the required boundaries. Therefore, this method can get 
image boundaries in low contrast images. 

III. PROPOSED METHOD 

The following harmonic energy function is proposed: 

1 2 11 12 21 22

2

1 11

2

1 12

2

2 22

2

2 12

1

1

2

2

( , , , , , , )

( ( ( )) ( )( ( ( )) (( )))

(1 ( ( )) ( ))( ( ( )) ( ))

(1 ( ( ( )) ( )))( ( ( )) ( ))

( ( )) ( )( ( ( )) ( ))

)

)

)

(

(

MIN
E u u w w w w

H f I x f u f I x f w dx

H I x f u f I x f w dx

H f I x f u f I x f w

H I x f u f I x f w dx

d

M

f M

M

f M

x

φ

Ω

Ω

Ω

Ω

=

− −

+ − − −

+ − − −

+ − −

∫

∫

∫

∫

 
(13) 

where ( ( ( )) ( )),( 1,2)iH f I x f u i− =  is a harmonic global division 

function. In (13), we take four harmonic means, two for each 
side of the contour. After minimizing the above problem, we 
get the following new harmonic solutions: 

1 1

11

1 1

( ( ( ( )) ( ))

( ( ( ( ))) ( )) (

)

( )))

H f I x f u M dx

w
H f I x f u f I x M dx

Ω

Ω

−

=
−

∫

∫
  (14) 

1 1

12

1 1

(1 (( ( )) ( )))

(1 ( (( ( ))) ( ))) ( ( ))

Hf I x f u M dx

w
H f I x f u f I x M dx

Ω

Ω

− −

=
− −

∫

∫
  (15) 

2 2

21

2 2
)

)( (( ( )) ( ))

( ( (( ( )) ( )) ( ( )) )

Hf I x f u M dx

w
H f I x f u f I x M dx

Ω

Ω

−

=
−

∫

∫
 

 (16) 

2 2

22

2 2

(1 ( (( ( )) ( )))

(1 ( (( ( ) )

)

) ( ) ( ( )))

H f I x f u M dx

w
H f I x f u f I x M dx

Ω

Ω

− −

=
− −

∫

∫
   (17) 

Lets take an image :I RΩ ⊂ , a level set : Rφ Ω ⊂ , and a 
closed contour C equivalent to a zero-level set: C = {x ϵ Ω|ϕ(x) 
= 0}. The following energy term is proposed: 

( ) ( ) ( ) ( )
HGSPF HGSPF HGSPF
E L vA Pφ λ φ φ α φ= + +

  
(18) 

where Lhspf (ϕ) is the length and Ahspf (ϕ) is the area terms 
respectively. These terms are calculated as: 

( ) ( ) ( ) | |
hgspf
L hgspf I dx

ε
φ δ φ φ

Ω

= ∇∫   (19) 

( ) ( ) ( )
hgspf
A hgspf I H dxεφ φ

Ω

= −∫    (20) 

where hgspf (I) is a HGSPF function which is defined as:  

( ( ) )
( )

max(| ( ) |)

I x HG
HGSPF I

I x HG

−
=

−
  (21) 

where HG is a newly developed two-phase harmonic global 
fitted image model which is characterized as: 

11 12 21 22

12 21 22 11 21 22 11 12 22 11 12 21

4w w w w
HG

w w w w w w w w w w w w
=

+ + +
 (22) 

where w11, w12, w21 and w22 are the intensity means described in 
(14)-(17). By the calculus of variations [17], the final level set 
of an energy functional EHGSPF  in (27) can be defined as: 

.div ( ) ( ) ( ) ( )
| |

hgspf I vhgspf I
t

ε ε

φ φ
λ δ φ δ φ

φ

∂ ∇
= +

∂ ∇

 
 
 

  (23) 

The HGSPF term has the incentive to modulate its sign 
within the [-1,1] region depending on the initialization of the 
curve. If it is placed outside of an object, the curve deforms 
inwards and if placed inside, the curve expands in the outward 
direction. Traditional level set methods use some initialization 
techniques to maintain the level set ϕ as signed distance 
function (SDF). The proposed method totally discards the 
costly re-initialization strategy by utilizing the penalization 
term from LSEWR method [9]. The initialization function for 
our method is defined as: 

0 0

0

0

( , 0) 0

x

x t x

x

ρ

φ

ρ

− ∈ Ω − ∂Ω

= = ∈∂Ω

∈Ω − Ω







   (24) 

where ρ>0 is constant. The steps of the proposed method are: 

1. Initialize ϕ by ϕ0 using (24) at t=0. 

2. Compute w11(ϕ), w12(ϕ), w21(ϕ) and w22(ϕ) using (14), 
(15), (16) and (17) respectively. 

3. Compute HGSPF(I) from (21). 

4. Solve the PDE of ϕ using (23).  



Engineering, Technology & Applied Science Research Vol. 9, No. 4, 2019, 4457-4462 4460  
  

www.etasr.com Bhatti et al.: Active Contours Using Harmonic Global Division Function 

 

5. If the result does not change then terminate, otherwise, 
move to step 2 and repeat. 

IV. EXPERIMENTAL RESULTS 

The experiments were performed on MATLAB 8.0 
installed in a PC running Windows 10, with Quad Core CPU 
2.4GHZ and 8GB RAM. In this paper, these parameters were 
selected manually: λ=1, µ=200/2552, v=15, α=0.2/∆t, ∆t=1.0, 
K=5, ρ=4, ε=1.5 and s=1.5. Initially, the method was tested on 
some simple images, which are demonstrated in Figure 2.  

Figure 3 delineates segmentation utilizing the proposed 
method and some previous methods. Results demonstrate that 
the proposed technique accomplishes the required 
segmentation results correctly. Chan-Vese [9] had also 
segmented some noise, while LBF neglected to accomplish the 
ideal segmentation results. 

Figures 4-5 exhibit the results of the Chan-Vese [9], LBF 
[21, 22] and the proposed strategy on mammograms taken from 
the mini MIAS database [20]. The results demonstrate that 
tumor segmentation has been efficiently segmented by the 
proposed technique while the other methods delivered 
inadmissible outcomes. 

  

  

 

 
Fig. 2.  Proposed method segmentation results and their initialization 

 

     

     
(a) (b) (c) (d) (e) 

Fig. 3.  Noisy image segmentation results: (a) Images and initial contours, (b) Chan–Vese, (c) LBF, (d) LCV and (e) proposed  

 

 
(a) (b) (c) (d) (e) 

Fig. 4.  Segmentations result on a real set of images, (a): original images, (b) ground truth (c) Chan-Vese, (d) LBF, (e) proposed 



Engineering, Technology & Applied Science Research Vol. 9, No. 4, 2019, 4457-4462 4461  
  

www.etasr.com Bhatti et al.: Active Contours Using Harmonic Global Division Function 

 

 
(a) (b) (c) (d) (e) (f) 

Fig. 5.  Quantitative comparison on mammogram images: (a) Original images, (b) ground truth. (c) [9], (d) [7], (e) [19], (f) proposed 

V. QUANTITATIVE COMPARISON 

For quantitative results, we used images taken from a 
publically accessible dataset [20] (Figure 5). The results exhibit 
that the proposed strategy improved the results of previous 
methods. We have utilized an accuracy metric to validate 
nature of our outcomes. The formulation of the metric is 
defined as: 

TP TN
Accuracy

TP TN FP FN

+
=

+ + +
   (34) 

where TP (true positive) stands for the segmented region, TN 
(true negative) for the accurately unsegmented regions, FP 
(false positive) stands for regions erroneously considered as 
true and FN (false negative) are the undetected tumor areas. 
Table I demonstrates the accuracy metric study of proposed 
technique and previous methods. We can see that the accuracy 
of the proposed methodology is essentially improved in 
comparison with the previous methods. 

TABLE I.  ACCURACY COMPARISON 

 Method 

Image [9] [7] [19] Proposed 

Row 1 0.5368 0.9587 0.7247 0.9628 

Row 2 0.6856 0.7456
 

0.7124
 

0.9698 

Row 3 0.5789 0.8321 0.2354 0.9321 

 

VI. CONCLUSION 

A novel HGSPF (harmonic global signed pressure force 
function) was presented in this paper. A global division 
function was used to approximate intensity means inside image 
regions. With the incursion of global intensity means and 
harmonic function, the proposed technique brought better 
outcomes in terms of accuracy, than the previous methods. 
Gaussian filter was adapted to eliminate an expensive 
reinitialization and to regularize level set. Finally, quantitative 
analysis demonstrated the effectiveness of the proposed 
method. 

 

 

REFERENCES 

[1] S. Soomro, K. N. Choi, “Active Contours Based on An Anisotropic 
Diffusion”, Digital Image Computing: Techniques and Applications, 

Canberra, Australia, December 10-13, 2018 

[2] B. W. Hong, B. S. Sohn, “Segmentation of regions of interest in 
mammograms in a topographic approach”, IEEE Transactions on 

Information Technology in Biomedicine, Vol. 14, No. 1, pp. 129-139, 
2010 

[3] M. Kass, A. Witkin, D. Terzopoulos, “Snakes: Active contour models”, 

International Journal of Computer Vision, Vol. 1, No. 4, pp. 321-331, 
1988 

[4] C. Li, C. Xu, C. Gui, M. D. Fox, “Level Set Evolution Without Re-

Initialization: A New Variational Formulation”, IEEE Computer Society 
Conference on Computer Vision and Pattern Recognition, San Diego, 

USA, June 20-25, 2005 

[5] V. Caselles, R. Kimmel, G. Sapiro, “Geodesic Active Contours”, 

International Conference on Computer Vision, Cambridge, USA, June 
20-23, 1995 

[6] Y. Liu, G. Captur, J. C. Moon, S. Guo, X. Yang, S. Zhang, C. Li, 

“Distance regularized two level sets for segmentation of left and right 
ventricles from cine-MRI”, Magnetic Resonance Imaging, Vol. 34, No. 

5, pp. 699-706, 2016 

[7] K. Zhang, L. Zhang, H. Song, W. Zhou, “Active contours with selective 
local or global segmentation: A new formulation and level set method”, 

Image and Vision Computing, Vol. 28, No. 4, pp. 668-676, 2010 

[8] H. Min, W. Jia, X. F. Wang, Y. Zhao, R. X. Hu, Y. T. Luo, F. Xue, J. T. 
Lu, “An intensity-texture model based level set method for image 

segmentation”, Pattern Recognition, Vol. 48, No. 4, pp. 1547-1562, 
2015 

[9] T. F. Chan, L. A. Vese, “Active contours without edges”, IEEE 

Transactions on Image Processing, Vol. 10, No. 2, pp. 266-277, 2001 

[10] X. F. Wang, D. S. Huang, H. Xu, “An efficient local Chan-Vese model 
for image segmentation”, Pattern Recognition, Vol. 43, No. 3, pp. 603-

618, 2010 

[11] C. Li, C. Y. Kao, J. C. Gore, Z. Ding, “Minimization of region-scalable 

fitting energy for image segmentation”, IEEE Transactions on Image 
Processing, Vol. 17, No. 10, pp. 1940-1949, 2008 

[12] S. Lankton, A. Tannenbaum, “Localizing region-based active contours”, 

IEEE Transactions on Image Processing, Vol. 17, No. 11, pp. 2029-
2039, 2008 

[13] S. Soomro, F. Akram, A. Munir, C. H. Lee, K. N. Choi, “Segmentation 

of Left and Right Ventricles in Cardiac MRI Using Active Contours”, 
Computational and Mathematical Methods in Medicine, Vol. 2017, 

Article ID 8350680, 2017 

[14] S. Soomro, F. Akram, J. H. Kim, T. A. Soomro, K. N. Choi, “Active 
Contours Using Additive Local and Global Intensity Fitting Models for 

Intensity Inhomogeneous Image Segmentation”, Vol. 2016, 2016 



Engineering, Technology & Applied Science Research Vol. 9, No. 4, 2019, 4457-4462 4462  
  

www.etasr.com Bhatti et al.: Active Contours Using Harmonic Global Division Function 

 

[15] A. Razi, W. W. Wang, X. C. Feng, “An Active Contour Method Using 
Harmonic Mean”, IEEE International Conference on Signal and Image 

Processing, Beijing, China, August 13-15, 2016 

[16] D. Mumford, J. Shah, “Optimal approximations by piecewise smooth 
functions and associated variational problems”, Communications on 

Pure and Applied Mathematics, Vol. 42, No. 5, pp. 577-685, 1989 

[17] G. Aubert, P. Kornprobst, Mathematical Problems in Image Processing: 
Partial Differential Equations and the Calculus of Variations, Springer, 

2006 

[18] B. Appleton, H. Talbot, “Globally optimal geodesic active contours”, 

Journal of Mathematical Imaging and Vision, Vol. 23, No. 1, pp. 67-86, 
2005 

[19] S. Soomro, K. N. Choi, “Robust Active Contours for Mammogram 

Image Segmentation”, International Conference on Image Processing, 
Beijing, China, September 17-20, 2017 

[20] The mini-MIAS database of mammograms, available at: 

http://peipa.essex.ac.uk/info/mias.html 

[21] C. M. Li, C. Kao, J. Gore, Z. Ding, “Implicit active contours driven by 
local binary fitting energy”, 2007 IEEE Conference on Computer Vision 

and Pattern Recognition, Minneapolis, USA, June 17-22, 2007 

[22] C. M. Li, C. Kao, J. Gore, Z. Ding, “Minimization of region-scalable 
fitting energy for image segmentation”, IEEE Transactions on Image 

Processing, Vol. 17, No. 10, pp. 1940–1949, 2008