Study on Improved Algorithm for Image Edge Detection
Jiang Lixia
School of Electric Engineering Bohai Shipbuilding Vocational College Huludao, China bhcyjlx@163.com
Zhou Wenjun, Wang Yu
School of Computer and Information Engineering Heilongjiang Institute of Science and Technology Harbin, China zwj196836@163.com selectivity and resolution of variable time and frequency domains. And Fourier Transform has better ability on frequency domain processing, but it is inferior to wavelet transform on the time domain analysis ability. So the wavelet transform has been obtained widely development in the fields such as signal and image processing, pattern recognition and electromagnetic fields and so on [3]. Although the wavelet transform has well theoretical foundation for image edge detection, it is not sensitive to direction properties. It is absolutely necessary and significant to improve on wavelet transform algorithm for image edge detection that the sensitive to direction properties could be enhanced and the detected result of image become clearly. II. TRADITIONAL WAVELET TRANSFORM ON IMAGE
EDGE DETECTION
2 2
Abstract—It has enormous development with the wavelet theory applied to image edge detection for its well properties of multi-scale edge detection. The traditional wavelet algorithm has bad sensitive to direction properties that applied to analysis image edge detection, and it is the major disadvantage. So the traditional wavelet about this is improved and puts forward a kind of new wavelet transform algorithm used for image edge detection. Compared improved wavelet algorithm with traditional wavelet for edge detection, it shows that new wavelet transform is more suitable for image edge detection and the clearer detection result is obtained. Complete image edge as well as accurate positioning and can reserve better detail information. Keywords- Image edge detection; Algorithm; Wavelet; Canny operator
I.
INTRODUCTION
The edge of image is the most fundamental characteristic of the image. The edge detection for images is the important research content in the field of image processing, so its detection algorithm is the research hotspot in this field. For a long time, it is an objective all along to obtain a kind of detection algorithm with such advantages as great noiseproof ability as well as accurate positioning feature and without the problems of un-detection or false detection [1]. Up to now there are many kinds of edge detection algorithm for images, and the classical Sobel algorithm is one of them. The classical Sobel algorithm has such advantages as small amount of calculation and high calculation speed, so it has been got extensive application in many fields. Because of the limited edge direction that the classical Sobel algorithm could only detect image with low noise-proof character and the appliance has many limitations too. In addition, the detection methods based on Canny algorithm and its varieties have also been used because of the “best” edge detection wave filter in respect of high precision index. For the advantages such as lower false rate and accurate positioning of image edge detection, the practical appliance purpose is very outstanding among all kinds of “best” edge detection wave filters [2]. The wavelet transform is called as magnifying glass on signal processing for its good limitation in the respects of both time domain and frequency domain, in addition, because of the advantages on its orthogonality, direction
For arbitrary function f ( x, y ) ? L ( R ) , its Fourier transform is?
? (Z , [ ) f
? ? f ( x, y ) e
RR
(Zx [y ) dxdy
(1)
2
For two functions f ( x, y ), g ( x, y ) ? L ( R ) , their convolution is?
2
f
g ( x, y )
? ? f (u, v) g ( x u, y v)dudv
RR
2 2
2
(2)
Transform them into one function \ ? L ( R ) and that is a wavelet. If it meets the allowance conditions?
C\
1
? (Z [ ) \ dZd[ f (2S ) ? ? Z2 [ 2 RR
2 2 2 2
(3)
If\ ? L ( R ) L ( R ) , the preceding formula means
? (0,0) that\
0 , that is ? ?\ ( x, y)dxdy
R R
2 2
0.
Suppose function f ( x, y ) ? L ( R ) , its continuous wavelet transform concerning wavelet function \ could be expressed as:
W\ 1 f ( x, y )
( f
\ s )( x, y ) 1 s2
?? f (u, v)\ (
R
(4) xu y v , )dudv s s
978-1-4244-5586-7/10/$26.00
C
2010 IEEE
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u v 1 \ ( , ). 2 s s s 2 2 Just as we know, in a grey image f ( x, y ) ? L ( R ) ?the
In which
S is scaling function?\ s (u , v)
W22j f ( x, y ) means the same things at the vertical direction.
For the binary function, the function value changes most quickly along gradient direction, so the gradient module at the gradient direction can reach maximum value. The smoothed gradient function for image f ( x, y ) could be expressed as?
grey change speed rate at the edge point can reach the maximum. As a result, the edge point corresponds to the maximum point of first derivative or zero cross point of second derivative for the smoothed function. Because the zero cross point of second derivative includes both the maximum point of first derivative and the minimum point of fist derivative, whereas only the maximum point of first derivative corresponds to edge point, thus the false edge appears. On the other hand, because the second derivative is sensitive to noise, we adopt the first order differential operator. As for the binary function, the function value changes most quickly along the gradient direction, so the gradient module of image edge element points at the gradient direction could reach maximum value. Support T ( x, y ) is smooth function and it meets?
grad ( f
T 2 j )
? T ( x, y )dxdy 1 °?? ?R ° lim T ( x, y ) 0 ? x , y orf
Here, it is defined that
W 1j f ( x, y ) arctan( 22 ) , and W2 j f ( x, y ) so the gradient direction for ( f
T 2 j )( x, y ) can be
It is defined that D ( x, y ) determined. The below formula could be achieved.
?W21j f ( x, y )? ? ? 2 ? ? ?W2 j f ( x, y )? ? ?w ? wx ( f
T 2 j )( x, y ) ? ? 2j? ? w ( f
T j )( x, y )? 2 ? ? ? ? wy
(9)
(5)
\ 1 ( x, y )
wT ( x, y ) 1 ? then both functions \ ( x, y ) wx 2 j and \ ( x, y ) are two-dimension wavelets. With s 2 ?
wT ( x, y ) ? wx
grad ( f
T 2 j )
?
\ 2 ( x, y )
the below formula can be obtained?
? 1 \ j ( x, y ) ° ° 2 ? °\ 2j ( x, y ) ° ? 2 W21j f ( x, y )
1 1 x y \ ( j , j) 22 j 2 2 1 2 x y \ ( j , j) 22 j 2 2 1 f
\ 2 j ( x, y )
( f
2j 2j w T j ( x, y )) wx 2
(6)
The edge points can be determined and oriented through two wavelet transforms. As a result, with the edge detection of every image element point, its gradient direction is determined first of all. And then the point can be found out whose modulus is maximum value along gradient direction. In order to achieve better visual effects for the edge images, it is necessary to select one or more threshold values. The final edge image for original image through dual threshold suppression could be obtained by the Canny algorithm after getting the maximum point. III.
THE IMPROVED WAVELET ALGORITHM ON IMAGE DETECTION
1 2j 1 2j
1 2 ( f
\ 2 ( f
\ 2 j )( x , y ) j )( x , y )
2
2
(10)
W21j ( x, y ) W22j ( x, y )
2
2
(7)
W22j f ( x, y )
w ( f
T 2 j )( x, y ) wx 2 f
\ 2 j ( x, y )
( f
2j 2j
w ( f
T 2 j )( x, y ) wx 1 It indicates that the essence of W2 j f ( x, y ) is a partial
derivative of smoothed function at horizontal direction. Hence, its partial maximum corresponds to mutation place of smoothed image at the horizontal direction. Similarly,
w T j ( x, y )) wx 2
(8)
According to the definition of traditional wavelet transform, the formula (1), (2), (3) can be proved tenably. If 2 2 the wavelet function \ ? L ( R ) meets the allowance condition as in formula (3), and where 1 2 2 2 \ ? L ( R ) L ( R ) , the preceding formula means
? (0,0) 0 and ? ?\ ( x, y )dxdy 0 . It is supposed that \
RR
f ( x, y ) ? L2 ( R 2 ) , its continuous wavelet transform concerning wavelet function \ could be expressed as?
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The image being obscure is the common problem that can result in the picture quality reducing. It may lead to images obscure in the course of image shooting, image R transmission, image processing and many factors such as Where S is scaling function and\ s (u , v) \ ( su , sv) . diffraction of lights, inaccurate focusing problems as well as Support T ( x, y ) is a smooth function and it can meet the the relative movement of scenery and viewfinder. In addition, the bad high-frequency performance of electronic system can formulas (5), (6). Similarly, it is defined that also damage high-frequency component of image and lead to w T ( x , y ) T x y w ( , ) 2 the image blur. And the sharpening technology could be used ? \ ( x, y ) , the \ 1 ( x, y ) wx wx to strengthen the target boundary in the images and image 1 2 details [5]. The common methods include gradient module functions \ ( x, y ) and \ ( x, y ) are two-dimensional operator, Roberts operator, Sobel operator, Lagrangian j wavelets. With s 2 , the below formula can be obtained? operator, etc. 1 1 j j Actual image often become blurred against processing ? °\ 2 j ( x, y ) \ (2 x,2 y ) because of containing noise or image information (12) ? 2 2 j j transmission lost. Gradient sharpening is a good method of ° ?\ 2 j ( x, y ) \ (2 x,2 y ) image enhancement and can make the fuzzy image clear. The clarity of sharpen image was obviously improved. And then the standard wavelet transforms \ 1 ( x , y ) and The simulation test of image edge detection with wavelet 2 2 2 \ ( x , y ) with f ( x , y ) ? L ( R ) has two components as transform is carried out. The image of Woman.tif is below: sharpened firstly and then the edge of image is extracted 1 based on traditional wavelet transform and improved wavelet ( , ) W21j f ( x, y ) f
\ 2 x y j algorithm proposed by using MATLAB. And the simulation result is shown in figure 1 to figure 3. w ( f
T 2 j ( x, y )) (13)
W\ 1 f ( x, y )
( f
\ s )( x, y )
IV. (11)
SIMULATION ON IMAGE EDGE DETECTION
??
f (u, v)\ ( sx u, sy v)dudv
W22j f ( x, y )
w ( f
T 2 j )( x, y ) wx 2 f
\ 2 j ( x, y ) (f
w T j ( x, y )) wx 2
(14)
Figure 1. The original image of women.tif
wx
The smoothed gradient functions for image f ( x, y ) could be expressed as?
w ( f
T 2 j )( x, y ) wx
grad ( f
T 2 j )
grad ( f
T 2 j )
?W21j f ( x, y )? ? 2 ? W ? j f ( x, y ) ? ? 2 ? ? ?w ? wx ( f
T 2 j )( x, y ) ? ? ? ? w ( f
T j )( x, y )? 2 ? ? ? ? wy
2 2
(15)
Figure 2. The picture of image edge detection based on traditional wavelet transform with Canny operator
1 2 ( f
\ 2 ( f
\ 2 j )( x, y ) j )( x, y )
(16)
W21j ( x, y ) W22j ( x, y )
2
2
Figure 3. The picture of image edge detection based on improved wavelet algorithm
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As shown in figure 2 and figure 3, image clarity handled with improved wavelet edge detection algorithm is superior to that of the best currently with Canny operator. It indicates that the improved wavelet algorithm is a effective method for image edge detection. V.
CONCLUSION
wavelet transform can extract complete edge with accurate positioning and keep better detail information. REFERENCES
[1] Liu Cai, “a kind of advanced Sobel image edge detection algorithm”, Guizhou Industrial College Transaction (Natural Science Edition), 2004, 33(5):77-79. Zhang Hongqun, Zhang Xue, “B-spline wavelets edge detection for image based on Cinny Criteria”, Information Technology, 2003, 27 (10):28-30. Written by Albert Boggess, Francis J. Narcowich, translated by Rui Guosheng, Kang Jian, “Wavelet and Fourier Analysis Basis”, Electronic Industry Publishing Company, 2003, 27(10):28-30. Stephane Mallat, Zhong shifen. Characterization of Signals from Multi-sale Edges [J]. IEEE Trans on Pattern Analysis and Machine Intelligence, 1992, 14 (7): 710-732. He Bin, Ma Tianyu. Digital Image Processing [M]. Beijing, China: People Postal Press, 1992. Li Bicheng, Luo jianshu. Wavelet Analysis and Application [M]. Beijing, China: Electronic Industry Press, 2003.
Wavelet transform has good time-frequency localization and multi-resolution advantages, and particularly suitable for local analysis and edge detection. Gradient sharpening process enhanced fuzzy image and can make its clarity better [6]. The improved algorithm of image edge detection based on wavelet transform combines the above two methods and the experiment results show that the method is very effective for the image edge detection. From the above results, we can know that for the same image, the improved algorithm can obtain the best detection effect compared with the Canny operator detection method. The edge detection based on new
[2]
[3]
[4]
[5] [6]
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