Download Research on Circular Target Center Detection

Research on Circular Target Center Detection Algorithm
Based on Morphological Algorithm and Subpixel Method
Yu Lei1, Ma HuiZhu1, and Yang Weizhou1
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001,
China
1
Abstract - To satisfy the measuring precision requirement of
circular target center in the system of high precision
measuring and aiming tracing, a new algorithm， which is
used in mobile laser aiming system to detect the edge of the
circular target and orientate the center, was put forward.
First of all, the mathematics morphological algorithm of
variable structure which combines radial basis function (RBF)
neural network was used to orientate the edge position and
eliminate the noise, then employ the Zernike moments
algorithm to get the subpixel location of the object contour. At
last the least squares fitting method was employed to orientate
the circular target center. The results showed that this
algorithm not only could orientate the figure of the circular
target, but also has good anti-noise performance and high
precision as well as excellent stability.
recognition processing instead of pure matrix calculation),
and it can detect the edges rapidly and can effectively
suppress image noise at the same time, but when selecting the
inappropriate structure element, the testing is less effective.
The paper presents the method which combines the
mathematical morphology and the neural network is not only
change the traditional single structural element into multistructuring elements but also use the neural network trains the
element of the structure to complete the edge detection. Then,
the paper employs the Zernike moments to locate the circle
contour to subpixel level. At last, the least square fitting
method is used to locate the target circle center.
2
Circular target edge detection and
center location algorithm
Keywords: mobile laser-aiming system; mathematical
morphology; RBF neural network; subpixel location
2.1
1
Mathematical morphology is a mathematical tool which
is a shape-based image treatment.
Morphological changes can be divided into two kinds of
transformation, the binary and the gray-scale. Basic
morphological transformations include erosion, dilation,
opening operation and closing operation.
The traditional morphological approach is to use fixedstructure elements to process the image with opening and
closing an image operation, and then remove the noise.
However, the shortcoming of this approach is lack of
adaptability; different image processing effects are quite
diverse. The morphological processing method, which based
on RBF, is using the morphological neural network to train on
the structural elements to achieve the variable structuring
element. Realization processes are as follows:
1. Initializing structural elements, such as using 3 * 3 flat
square structuring element;
2. Select some sample points in the image (256 pixel value
of the main diagonal) as input of neural network and calculate
the corresponding neural network output;
3. Compare the output with the target output, and we can
use the method of sum of squares and subtractions as the
objective function, the aim of neural network training is to
make the objective function close to 0;
Introduction
The edge of the image can express the basic shape of the
image, including the main features information of the image.
The main task of edge detection is to identify and extract the
edge information for the preparation of image analysis, target
recognition and image coding. In general, to the detected
edges there are following requirements: (1) the accuracy of
edge location must be high, the edge shift does not occur; (2)
the edge of different scales should be have good response,
and to minimize missed inspections; (3) should be insensitive
to the noise, will not cause a false detection due to the noise;
(4) the detection sensitivity should be little impact on the edge
direction[1-4].
Image processing at the receiving antenna, the edge
detection directly affects the follow-up edges and center
location accuracy, according to the relevant information, the
traditional gradient detection operator[5] uses the differential
expression between the image pixel to extract the image edge,
although it is effective in extracting the edge, but it is
sensitive to the image noise and the calculation is slow.
Currently, the mathematical morphology[6] edge detection
which based on Characteristics of the image geometry has
been widely used because of it has little calculation (mode of
Edge detection algorithm which combines
mathematical morphology and RBF neural
network
4. Use some principles (generally, neural network uses the
gradient descent) according to the error function to modifies
the structural elements;
5. If it reaches maximum number of iterations or the error
is less than pre-set value, end the training and output the
current structure element as the optimal morphological
processing element of the image; otherwise go to Step 2,
continue to train the network.
RBF network is divided into three layers, the first layer
is input layer, the middle of it is a hidden layer, and finally a
layer to output layer, In the network, the mapping from input
to output is
m
f ( x)   wi ( x, ci )
(1)
i 1
In this formula， x  ( x1 , x2 , , xn )  R is the input
T
vector, W  ( w1 , w2 , , wm )  R
T
m
n
is the output weight
matrix, f (x) is the output vector, is the radial basis function,
ci is the radial basis function for the number i-th clustering
centers. The radial basis function uses Common Gaussian
function, that is,
  x  ci
 ( x, ci )  exp 
 2

2




(2)
The issue of RBF network designation is to determine
the number of hidden layer nodes and the corresponding
position and width of the central nodes. When the number of
hidden layer nodes and the position and width of the central
nodes in the RBF network is confirmed, RBF network will
form a linear equation which from input to output , and then
the output weighting vector can be get by using least squares
method .
As the noise which is varied can be superimposed on the
image, so the single structure element for the morphological
filter have poor adaptability, good filtering effect can not be
played on all kinds of noise, so it is necessary to use different
structure elements to different types of noise adaptively. In
this paper, we use the structural elements which are trained by
RBF neural network to filter the image which is contaminated
by the noise, and hope for the best filter effect.
The original image with noise is set to F, and target
expected is set to images D, and the structure element is set to
B. We take the sub-matrix Fij (i  1, 2,, m; j  1, 2,, n)
which has the same size with the structure element B from F
as input sample, take the target image D corresponds to the
center elements of sub-matrix d as the network desired output
with structural elements as a template, then the network
training samples are (Fij, d). Among them,
 fij

fi 1, j
Fij  


 fi  r 1, j
fi , j  s 1 

fi 1, j 1  fi 1, j  s 1 
,

fi  r 1, j 1  fi  r 1, j  s 1 
fi , j 1 
(i  1, 2, , m; j  1, 2, , n)
(3)
In short, the process of morphological filtering algorithm
based on RBF neural network is as follows:
(1) Initialize the structure elements B = [0,0, ... ..., 0], and
error limit  is given;
(2) Input
P
samples
in
turn
to
calculate
yk  { f k1  b1 , f k 2  b2 , , f kM  bM } ;
(3) Calculate error E 
1 P
 ( yk  d k ) 2
2 P k 1
otherwise, continue;
(4) Adjust
the
structure
bi (t  1)  bi (t )  bi  bi (t )  i ;
, if E<  ;
element
(5) Return (2), repeat until to all the training samples mode,
the network output can meet the requirements.
2.2
Zernike Moments
The results of edge detection are smoothened by the
morphology method. The noises in the images could be
removed. However, the accuracy of edge could not approach
to the subpixel extent. Here, we just employ the existing
Zernike moments operator.
The reason of using Zernike moments is the special
property of circular polynomials of Zernike moments. Only
three masks are used to calculate four parameters of every
edge point, as shown in Fig. 1. k is the step height, h is the
background gray level, l is the perpendicular distance from
the center of the circular kernel and the edge makes an angle
off with respect to the x -axis.
2.2.1
Fig.1 subpixel step edge model
Theory about Zernike moments
Zernike moments for an image f ( x, y ) is defined as [5],
ANzMz 
Nz  1


x 2  y 2 1

f ( x, y )VNzMz
(  ,  )dxdy
(4)
where Nz  1 /  is a normalization factor and it is ignored in
future discussion. In discrete form, ANzMz can be expressed as
So

ANzMz   f ( x, y )VNzMz
(  , ), x 2  y 2  1
x
  tan 1 (
y
(5)
It can be seen from (5) that in a discrete image, the
neighborhood of that point should be mapped onto the interior
Solving (10) and (11), the edge parameter
where
RNzMz (  )
l
RNzMz (  )  
(1) ( Nz  s)！
Nz  Mz
Nz  Mz
(
(
s！
 s )！
 s )！
2
2
(15)
That means, the three Zernike moments A00 , A11 , A20 [6]
could locate the edge to subpixel accuracy.
Nz  2 s
(7)
If an image is rotated by an angle  , the Zernike moments
2.3
2.2.2
 jv
(8)
The least square fitting center algorithm
In general, the circular equation can be expressed as
of the original image Anm and the Zernike moments of the
rotated image has the following relationship:
  ANzMz (  )e
ANzMz
( x  x0 ) 2  ( y  y0 ) 2  r 2
n
For calculating edge parameters l and  , three
masks A00 , A11 , A20 , should be deduced. According to (6), the
orthogonal complex polynomials can be written as:
2
2
V00  1 , V11  x  jy , V20  2 x  2 y  1 . In this work we the unit
circle is divided into 7×7 homogeneous grids, masks are



calculated when making integral for V00 , V11 , V20 on the
dashed area of every grid. Herein, assuming f ( x, y ) to be
constant over every pixel, convolving these masks with the
image points can get Zernike moments[5].
According to (8), the relationship between Zernike
moments of original image A00 , A11 , A20 and rotated image
  A11e j , A20  A20 .
 , A11 , A20
 can be given as A00  A00 , A11
A00
Furthermore, the following equations can be deduc-ed
based on theory of Zernike moments,
k
 k sin 1 l  kl 1  l 2
2
A11 
2k
(10)
1  l 
2 3
3
(11)
When the edge is rotated an angle  , it will be aligned
parallel to y-axis so that

x 2  y 2 1
C   ( ( x  x0 )2  ( y  y0 )2  r )2
i 1
(17)
Where: ( xi  yi ) is the coordinate of feature points for the
arc; n is the number of the feature points which involved in
the fitting calculation.
Least square fitting is used to achieve the minimum
objective function to solve some unknown parameter value
within the certain region. When the circle radius involve in
fitting as a constraint, according to the Lagrange multiplier
method, the least squares objective function can be written as
n
C   ( ( x  x0 ) 2  ( y  y0 ) 2  r ) 2   (r  r )
(18)
i 1
It will be turning constrained least squares method into
the unconstrained least square method. Using the Gauss Newton or Levenberg-Marquardt method can solve the
relevant parameters.
3
Experimental results
2 3
3
A11 
(9)
1  l 
2kl
(16)
For nonlinear least-squares circle fitting, the optimization
objective function is[9]
Zernike moments operator edge detection
  h 
A00
(14)
 xs   x  cos  
 y    y   l  sin  

 s   
(6)
s
A20
A11
The subpixel location of image edge is
jMz
is a radial polynomial defined as
(13)
l can be given
as:
of the unit circle for evaluating Zernike moments ANzMz of an
image point. The complex polynomials VNzMz (  , ) can be
expressed in polar coordinates as
VNzMz (  , )  RNzMz (  )e
Im[ A11 ]
)
Re[ A11 ]
f '( x, y ) ydxdy  0
(12)
Experiment 1.
Based on RBF neural network morphological filtering
method, in order to maximize the elimination of noise, we
should make a edge detection on the 256 × 256 lena image
with Salt and pepper noise density 0.2. The threshold variance
is 1.2. The error curve of RBF is shown as FIG 2. The
simulation result shows that the algorithm converge to the
predictive error when iteration times go to 487.
160
140
120
均方误差
100
80
60
40
20
0
0
50
100
150
200
250
300
350
400
450
500
迭代次数
Fig.2 Error curve
Experiment 2
Fig.3 and Fig.4 show the images processed
morphological edge detection methods based on fixedstructure element and RBF neural network respectively.
Image noise was significantly inhibited by neural network
morphological filter which based on RBF neural network, and
the goals are clearer, only some partial edge burr.
(a) lena image
(b) noisy lena image
(c) fixed-structure
element morphological method
Fig.3 Edge detection of noisy image with fixed-structure
element morphological method
Experiment 4
Table1 describes circle center coordinates, which is
located after edge detection by Morphological method based
on RBF, subpixel location by Zernike moments and fitting by
least square method, continuesly。Based on 10 photos taken,
this algorithm reduced the average error and increases the
measurement accuracy.
Table 1: The comparison of circular target center coordinate
orientated by fixed structural element mathmatics morphology
and the arithmetic proposed in this paper
True value (x,y)
Detected value (x,y)
95
(b) noisy lena image
(c) Morphological
method on RBF
Fig 4 Edge detection of noisy image with morphological
method based on RBF
Experiment 3
Figure 5 is the circle target image which collected by CCD
in the experiment and detected the edge of the image by
the proposed method.
95.3607
106.6744
95
109
95.1533
109.1312
95.5
103.5
95.3637
103.5568
95.5
99.5
95.3456
99.5903
94.5
90.5
94.5624
90.5265
94
106
93.6302
105.6657
94.5
107.5
94.4348
107.6682
95.5
110.5
95.4272
110.5469
95.5
107.5
95.5017
107.6032
94.5
100.5
94.3976
100.5594
Average error
0.1479
0.1342
4
(a) lena image
107
Conclusion
For the circular target centre and edge detection
algorithm for a mobile laser aiming system, this paper
proposes an algorithm which is the combination of RBF
neural network and variable structure element of mathematical
morphology, Zernike moments and least squares fitting
algorithm, then get the target center sub-pixel location.
Experimental results show that method is very effective to the
edge detection, the center positioning accuracy is high. The
method improves the mobile measurement accuracy of laser
targeting system effectively, and expands the mobile
application field of laser aiming system. It can effectively
meet the needs of the mobile laser tracking system in the areas
of precision edge measurement.
5
References
[1] Serra J. Introduction to mathematical morphology[M].
New York: Academic Press, 1982.
[2] Plaza A. Dimensionality reduction and classification of
(a) experiment taken to the circular target (b) obtained
using the proposed circular target edge image
Fig 5 the circle target image
hyperspectral image data using sequences of extended
morphological transformations[J]. IEEE Transactions on
Geoscience and Remote Sensing, 2005, 43(3): 466-479.
[3] Benediktsson J A, Palmason J A. Classification of
hyperspectral data from urban areas based on extended
morphological profiles[J]. IEEE Transactions on
Geoscience and Remote Sensing, 2005, 43(3): 480-491.
[4] Rao Haitao, Wong Guirong. The image edge-detection
[5]
[6]
[7]
[8]
[9]
based on mathematics morphology [J]. Journal of Suzhou
University(Natural Science),, 2004, 20(2): 42-45.
Chen T, Wu Q H, Rahmani-Torkaman R, et al. Apseudo
top-hat mathematical morphological approach to edge
detection in dark regions[J]. Pattern Recognition, 2002,
35(1): 199-210.
Jiang Mingyan, Yuan Dongfeng. A multi-grade mean
morphologic edge detection[A]. 2002 6th International
Conference on Signal Processing,Beijing, China, 2002.
F. Zernike, Physical (1934) 689.
QU Ying-dong. A Fast Subpixel edge measurement
method based on Sobel-Zernike moment operator. OptoElectronic Engineering, 30(5):59-61,Oct. 2003
Craig M Shakarji. Least-squares fitting algorithms of the
NIST algorithm testing system[J]. Journal of Research of
the National Institute of Standards and Technology, 1998,
103(6): 633-641.