- The document discusses methods for computing moments of mouse-drawn shapes using only corner pixel locations. - It presents an algorithm that discards redundant vertices, retaining only points where the shape bends sufficiently. This improves time and space complexity over prior methods. - Moments are computed by dividing the shape into triangles using corner pixels and the origin, then summing the moments of the individual triangles.

1. International Association of Scientific Innovation and Research (IASIR)
(An Association Unifying the Sciences, Engineering, and Applied Research)
(An Association Unifying the Sciences, Engineering, and Applied Research)
Interna tional Journal of Emerging Technologies in Computational
and Applied Sciences (IJETCAS)
www.iasir.net
IJETCAS 14- 567; © 2014, IJETCAS All Rights Reserved Page 170
ISSN (Print): 2279-0047
ISSN (Online): 2279-0055
Improved Complexity of Area Sequence Moments for Mouse Drawn Shapes
Vinay Saxena
Department of Mathematics, Kisan P.G. College, Bahraich, U.P. 271801, INDIA
______________________________________________________________________________________
Abstract: A mouse drawn shape that is represented by a function of sequence of pixels can be completely specified by the locations of its corner pixels given in a correct sequence. The sequence can be specified according to either clockwise direction or counterclockwise direction tracing. Leu suggested a method which is much more efficient than the traditional occupancy array based method. This method computes the set of moments using only the comer pixels along a shape's boundary. The complexity of this method is linearly proportional to the number of comers along a shape's boundary, which in general is only a small fraction of the total number of pixels in the shape. In this work, we have been able to further improve the time as well as space complexity by introducing an intermediate optimization steps that discards most of the redundant vertices and retain only the points where the bend is sufficient to contribute towards the appearance of the shapes.
Keywords: Triangle Moment; Area Sequence Moment; Mouse Drawn Shapes; Optimal Shapes
________________________________________________________________________________________
I .INTRODUCTION
There are several algorithms known for evaluation of moments, central moments and their modifications. For two-dimensional shapes, most of them are based upon the entire shape including the interior points. In recent years, the time as well as the space complexity is reduced by Leu [1]; Li and Shen [2]; Chen [3];Jiang and Bunka [4] by evaluating the moments using only the boundary point. Leu [1] suggested a method which is much more efficient than the traditional occupancy array based method. He proposed a method, which computes the same set of moments using only the comer pixels along a shape's boundary. The basic approach is to construct a set of triangles using the shape's comers and the origin of the coordinate system. The moments of these triangles are computed first. The moments of the shape are then derived from the triangles moments.
In this work, we are concerned with a two dimensional mouse drawn shape that is represented by a function of sequence of pixels. These shapes can be completely specified by the locations of its corner pixels given in a correct sequence. The sequence can be specified according to either clockwise direction or counterclockwise direction tracing. In this approach we assume that corner sequences are specified in clockwise direction. The area sequence (mi[0][0] : i =1,2, ... , N) consists of an array of the area of triangular regions with pi, pi+l and the origin O(0,0) as the vertices of the ith triangle. Saxena and Kapoor [5] defined corresponding optimal sequences, when pixels are the optimal vertices left after the polygonal boundary is reduced. Without loss of generality, the shapes are drawn in the first quadrant of the x-y plane. The area sequences and corresponding optimal area sequence of the shapes in Figure 1 and Figure 2 are displayed in Figure 3 and Figure 4 respectively. The complexity of this method is linearly proportional to the number of comers along a shape's boundary, which in general is only a small fraction of the total number of pixels in the shape.
In this work, we have been able to further improve the time as well as space complexity by introducing an intermediate optimization steps that discards most of the redundant vertices and retain only the points where the bend is sufficient to contribute towards the appearance of the shapes.
Original Shape (a) Original Shape (b)
Figure 1: Mouse Drawn Shapes (a) - (b)

2. Vinay Saxena, International Journal of Emerging Technologies in Computational and Applied Sciences, 9(2), June-August, 2014, pp. 170-
175
IJETCAS 14- 567; © 2014, IJETCAS All Rights Reserved Page 171
Optimal Shape (c) Optimal Shape (d)
Figure 2: Optimal Shapes (c)- (d) corresponding to Original Shapes (a)- (b)
Figure 3: Area Sequence Moment i Vs mi [0][0] (Darker Shape [e] for Original shape (a) ; Lighter Shape [e’] for Optimal shape(c))
Figure 4: Area Sequence Moment i Vs mi [0][0] (Darker Shape [f] for Original shape (b) ; Lighter Shape [f’] for Optimal shape(d))
II. ALGORITHM FOR OPTIMAL POLYGON
Comment:
Input: The vertices pi = {(xi,yi),i=1,.....,n}. Denote u by the ith vertex of contour so that (xi, yi) = (u.x, u.y);

3. Vinay Saxena, International Journal of Emerging Technologies in Computational and Applied Sciences, 9(2), June-August, 2014, pp. 170-
175
IJETCAS 14- 567; © 2014, IJETCAS All Rights Reserved Page 172
Output: The outputs are the optimal vertices (xj*, yj*) and j*.(All arithmetic is in modulo n).
Step 1 Initiate i=0;
Step 2 Translate the origin of the coordinate system to the point (xi, yi) using
pj
1 .x= pj. .x –u.x; pj
1.y= pj .y-u.y
Step 3 Set j=i+1
Step 4 Compute Fj=[( pj
1 .x)2+( pj
1 .y)j
2)]1/2
Step 5 Change j to j+1
Step 6 Compute
Case (a) (when error norm is E2)
Case (b) (when error term is E1 )
Step7: If Fj  Fj-1 then go to step 5 else write j*=j-1 and (xj*,yj*) using
Set i = j* and go to step 2
Step8: Repeat this process until j* is repeated
Step9: Join ( p j .x, p j .y)successively to determine the optimal polygon.
Step10: End
III. MOMENTS FROM SHAPE CORNERS
In the case of mouse drawn shapes the boundary is extracted by the mouse. We have used the TArray class of
OWL to store the vertices in the boundary polygon. A boundary pixel is also a corner pixel when the pixel is
not collinear with the boundary pixel before it and the boundary pixel that follows it. In other words, the
boundary makes a turn at that location. A mouse drawn shape can be completely specified by the locations of its
corner pixels given in a correct sequence. The sequence can be specified according to either clockwise direction
or counter-clockwise direction tracing. In this approach we assume that corner sequences are specified in
clockwise direction.
Let the corners of the shape S be (x1,y1), (x2,y2), ... , (xn,yn), going around the shape in a clockwise direction.
Which corner point is taken, as the starting point of the list will have no effect on the outcome? Without loss of
generality, we also place the shape in the first quadrant of the x-y plane. Therefore, the origin (0,0) is located to
the lower-left side of the shape. To show how shape moments can be derived from the corner pixel list, we first
draw lines to connect every pixel to the origin. Since, every two neighboring corner pixels together with the
origin form a triangular region. Therefore, for a shape with n corner points, exactly n triangles, T1, T2, ….., Tn,
are defined. We now compute the moments of these triangles. Since the integration operation is a linear
operation, the moments of the entire shape can be derived easily from the moments of these triangles.
Computing the moments of a triangle:
Let T be a triangle which has corners (a,b), (c,d) and (0,0). Three regions may be formed when we draw a line to
link (a,b) and (a,0) and another line to link (c,d) and (c,0). Furthermore, we let a  c. Region 1 is the triangle
with corners (0,0), (a,b) and (a,0). Region 2 is a trapezoid whose corners are (a,0), (a,b), (c,d) and (c,0). Region
3 is the triangle with corners (0,0), (c,d) and (c,0). These regions are shown in Figure 5.The triangle T that, we
are interested in, can be reproduced by joining Region l and Region 2 and then subtracting Region 3. Since the
integration operation in Equation (1) is a linear operation, the (p,q)th moment of T can be found by adding the
(p,q) the moment of Region 1 to the (p,q)th moment of Region 2 and then subtracting the (p,q)th moments of
Region 3 from the sum.
The (p,q)th moments of Region 1, 2 and 3 can be computed according to the following three equations,
respectively.
F p x p y
p x p y p x p y
p x p y j j j
j j
k i
j
k k
j j
  


 


[( . ) ( . ) ]
[( . ) *( . ) ( . ) *( . ) ]
[( . ) ( . ) ]
    /
   


    
   
F p x p y
p x p y p x p y
p x p y j j j
j j
k i
j
k k
j j
  


 


[( . ) ( . ) ]
|( . ) *( . ) ( . ) *( . ) |
[( . ) ( . ) ]
/
/
     
   


   
     
p x p x u x
p y p y u y
j j
j j
*
*
*
*
*
*
. . .
. . .
 
 



4. Vinay Saxena, International Journal of Emerging Technologies in Computational and Applied Sciences, 9(2), June-August, 2014, pp. 170-
175
IJETCAS 14- 567; © 2014, IJETCAS All Rights Reserved Page 173
  
a bx a
p q
pq m x y dydx
0
/
0
1 , (1)
 
 



c
a
x a b
c a
d b
p q
pq m x y dydx
( )
0
,2 (2)
  
c dx
p q
pq m x y dydx
0
/ 2
0
,3 (3)
On simplification;
For region 1, we have
dx
q
y
m x bx
a q
p
pq
/ 2
0
0
1
1 , ]
1
 [ 


(4)
( 1)( 2)
. 1 1
  

 
q p q
a bp q
For region 2, we have two cases;
Case 1: when d  b

 

 


c
a
x a b
c a
q d b
p
pq dx
q
y
m x
( )
0
1
2 . ]
1
[
   




c
a
p q x a b dx
c a
d b
q
x 1 [ ( ) ]
1
1
   


 




c
a
q r
q
r
q
r
p x a dx
c a
d b
x C
q
[ ( )] .
( 1)
1 1
1
0
1
( ) . [ { . ( ) } ]
( 1)
1 1
1
0
1
0
1 1 1 b x C x a dx
c a
d a
C
q
q r j j
q
r
c
a
q r
j
q r
j
q q r r p
r 



   


 

       
]
2
( )
( ) . ( ) .
( 1)
1 1
0
1
0
2 2
1 1 1 dx
p q r j
c a
b C a
c a
d a
C
q
q
r
q r
j
p q r j p q r
q r j
j
q q r r
 r 


 

      
    
   





 ( 5)
Case 2 when d = b;
  
c
a
b
p q
pq m x y dydx
0
2 .
( 1)( 1)
( )
.
1 1
1
 


 

p q
c a
b
p p
q (6)
For region 3, we have
( 1)( 2)
.
]
1
.[
1 1
/
0
1
.2
  



 


q p q
c d
dx
q
y
m x
p q
c
a
dx c
q
p
pq
(7)
pq.T pq.1 pq.2 pq.3 m  m  m m (8)

5. Vinay Saxena, International Journal of Emerging Technologies in Computational and Applied Sciences, 9(2), June-August, 2014, pp. 170-
175
IJETCAS 14- 567; © 2014, IJETCAS All Rights Reserved Page 174
y (a,b)
Figure 5: Regions R1  {(0,0), (a,b), (a,0)}, R2  {(a,0), (a,b), (c,d), (c,0)}, R3  {(0,0), (c,d),(c,0)} for
T  {(0,0), (a,b), (c,d)}
In the second step, we now associate with each triangle a sign value. The way in which a sign in assigned to a
triangle is very straightforward. For triangle Ti which is defined by (xi,yi), (xi+l, yi+l) and (0,0) as the boundary
trace goes from (xi,yi) to (xi+1, yi+l) and (0,0), as the boundary trace goes from (xi, yi) to (xi+l, yi+l), if the point
(0,0) is on the right-hand side of this boundary segment, we will assign the + 1 sign to Ti. On the other hand, if
the point (0, 0) is on the left-hand side of the boundary segment, the sign of Ti will be -1 . Whether the point
(0,0) is on the left or the right side of the boundary segment can be determined easily. If, the angle from the x-axis
to the line which is defined by (xi, yi) and (0,0) is greater than the angle from the x-axis to the line which is
defined by (xi+1 yi+l) and (0,0), then the point (0,0) is on the right-hand side of the boundary segment. On the
other hand, if the first angle is less than the second angle, the point is on the left-hand side of the boundary
segment. If the two angles are the same (0,0) is right on the extension of the line segment. In this case, the sign
value, which we associate with the triangle, is not important, since the size (area) of the triangle is O.
That is,
1 -1 1
1
1, tan ( ) tan ( )
( )
1,
i i
i i
y y
when
Sign i x x
otherwise
 


 
 
 
(9)
The way in which sign are assigned to the triangle is in accordance with the fact that the shape boundary is
traced in a clockwise direction. When we travel along the shape boundary in a clockwise direction, the interior
of the shape is to our right and the exterior is to our left.
The (p,q)th moment of the shape S is now can be computed according to the following equation:
. ( )
1
. , 1
m m Sign i
n
i
T pq s pq 
 (10)
where Sign (i) is the sign associated with triangle Ti . Equation (10) shows that moments of S can be found
through adding and subtracting the moments of the triangle according to their polarity values. This is because
the shape S itself can be reconstructed through adding and subtracting the triangles.
Using this technique, a C++ program has been developed that evaluates the moments using vertices pi and we
apply this program to the same shapes by descriptor as discussed in [5],[6]. Numerical results obtained show
that in all the cases we obtain the same set of features that proves the correctness of the computer program. In
addition to this, we also develop a VC ++ program, which plots pixel Vs triangle's area corresponding to that
pixel. Without loss of generality we draw sequences {(i, mi(0][0] ) ,i=l, .... ,n} in the first quadrant of x-y plane.
We call these sequences as Area sequences. Figure 3 - 4 displays these sequences as well as their corresponding
optimal area sequence.
(c,d)
x
O (0,0) (a,0) (c,0)

6. Vinay Saxena, International Journal of Emerging Technologies in Computational and Applied Sciences, 9(2), June-August, 2014, pp. 170-
175
IJETCAS 14- 567; © 2014, IJETCAS All Rights Reserved Page 175
IV. CONCLUSION
We show that only a small portion of the vertices plays an important role and most of the vertices could be eliminated by an optimization technique, thus reducing the time complexity significantly with minimal effect on the accuracy. We emphasize that the optimization technique will not compromise on the wisdom of the machine.
Acknowledgment
I am grateful to Dr. V.V. Kapoor who has developed the GUI of the program using the Multiple Document- view architecture of Borland VC ++ as per the requirements of the problem taken up in this work.
REFERENCES
[1] Leu J.G. ,1991, Computing a shape's moments from its boundary. Pattern Recognition. 24: 949-957.
[2] Li B.C. and Shen J. ,1991, Fast computation of moment invariants. Pattern Recognition. 24: 807-813.
[3] Chen C.C. , 1993, Improved moment invariants for shape discrimination. Pattern Recognition. 26 : 683-686.
[4] Jiang X.Y. and Bunke H., 1991, Simple and fast computation of moments. Pattern Recognition. 24: 801-806.
[5] Saxena V. and Kapoor V.V., 2011,. Behavior of Normalized Moments under Distortion and Optimization, Recent Research in Science and Technology, 3(7): 73-76. [6] Saxena V. , 2011, On Low Order Moment Based Features for Hand Drawn Shapes, Interface between Statistics, Mathematics and Allied Sciences, Excel India, N Delhi. 149-156.