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Edge detection in noisy images based on the co-occurrence matrix
Pattern Recognition, Vol. 27, No. 6, pp. 765-775, 1994 Elsevier Science Ltd Copyright © 1994 Pattern Recognition Society Printed in Great Britain. All rights reserved 0031-3203/94 $7.00+.00
Pergamon
0031-3203(93) E0026-4
E D G E D E T E C T I O N IN NOISY IMAGES BASED O N THE C O - O C C U R R E N C E MATRIX DEOK J. PARK, KWON M. NAM and RAE-HONGPARKt Department of Electronic Engineering,Sogang University, C.P.O. Box 1142, Seoul 100-61l, Korea (Received 4 January 1993; in revised form 11 November 1993; received for publication 2 December 1993)
An edge detection method for noisy images is proposed based on the co-occurrence matrix. In the proposed technique based on the step-edge model, the gray level information is simply converted into a bit-map, i.e. the uniform and boundary regions of an image are transformed into a binary pattern by normalization using the local mean. In this binary bit-map pattern, 0 and 1 are densely distributed near the boundary region while they are randomly distributed in the uniform region. To detect the boundary region, the co-occurrence matrix on the bit-map is introduced. The effectivenessof the proposed technique is shown via a quantitative performance comparison to the conventional edge detection methods and the simulation results for noisy images are also presented. Abstract
Edge detection
Step-edge
Bit-map
Co-occurrence matrix
I. INTRODUCTION Edge detection is one of the most fundamental steps in image analysis. Edge points can be defined as pixels at which an abrupt discontinuity in gray level, color, texture, depth, or motion occurs. The intensity discontinuities occur due to the different surface reflectance of objects, illumination condition, or varying distance and orientation of objects from the viewer. 11'2) Since edge detection is a fundamental process in various applications such as image analysis, robot vision, and image data compression, much research has been devoted to the reliable detection of edges. The significance of a physical change in an image depends on the various applications. An intensity change that would be classified as an edge in some applications might not be considered as an edge in other applications. To detect the various edge elements existing in an image, therefore, the edge detector having different scales should be considered, t3'4~ Marr and Hildreth ~2) suggested that the original image be band-limited at several different cutoff frequencies, and the impulse response of the lowpass filter proposed to band-limit an image at different cutoff frequencies is Gaussianshaped. This implies that the Gaussian filter is optimal in terms of the smoothing and localization in both the spatial and frequency domains and the standard deviation ofa Gaussian filter is used as a scale parameter. Since the scale information for a given image is not known a priori, the selection of the proper scale for the edge detection varies with the spatial resolution and thus it is a very difficult problem. To avoid the difficulty
t Author to whom all correspondence should be addressed.
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of selecting the proper scale, therefore, it is necessary that image information be converted into a simpler form in order to be easily analyzed. In this paper, with the step-edge model corrupted by additive Gaussian noise, an edge detection method in noisy images is proposed based on the co-occurrence matrix. Since conventional gradient edge detection schemes take points having maximum gradient magnitude as an edge, it is difficult to detect edge points having small intensity changes. With the step-edge model, both sides along the boundary can be divided into two regions based on the local mean, thus boundary regions having small or large intensity changes are equally classified as edge points regardless of intensity changes. Thus we construct the binary pattern by decomposing an image into the relatively uniform regions and boundary regions, and define the co-occurrence matrix over the binary pattern to detect the edges. The proposed technique is observed to generate a thinner edge map than the conventional ones and is also shown to be robust to additive noise. In Section 2, the construction of the bit-map and the edge detection using the co-occurrence matrix obtained from the bit-map are presented. In Section 3, the performance of the proposed method for noisy images is compared to those of the conventional ones in terms of the quantitative measures, and finally the conclusion is given in Section 4. 2. PROPOSED EDGE DETECTION IN NOISY IMAGES BASED ON THE CO-OCCURRENCE MATRIX
The proposed edge detectiofi technique is based on the step-edge model corrupted by additive Gaussian noise and the features are extracted from the co-occur765
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rence matrix over the binary bit pattern. In this section, the construction of the bit-map from noisy images is discussed and the edge detection method based on the co-occurrence matrix is proposed. 2.1. Construction of the bit-map Consider a step-edge image corrupted by additive zero-mean Gaussian noise with a standard deviation a,. Then, the probability density function (pdf)po(g) of the gray level of the noisy image is given by
Pg(g) _
4(2=)1a, exp [,_ - (o-2~- m~)~]_]
(1)
where g denotes the gray level and mg signifies its mean. If the gray level at a given pixel is assumed to be independent of those at any other pixels, then a new pdf can be obtained for pixel values generated with a weighted local window,iS) As a special case, the mean m s a n d variance a s2 of a new pdf, whose gray level is normalized by the local mean mo, are given by ms = 1 and try-2 _ (a,/mg)2, respectively. In other words, a new pdf of a gray level normalized by the local mean also follows a Gaussian distribution function with the mean and standard deviation equal to 1 and try, respectively, and thus it is expressed by 1 exp[-(g--1)2] P" (g) = ~/(2n) a, L 2a] '
(2)
Now, consider a step-edge model, having constant gray levels gl and g2 (gl < g2), contaminated by additive zero-mean Gaussian noise with the variance a 2. In case that the local window is considered over the uniform region, the mean and standard deviation of the gray level of the pixels in a local window, normalized by the local mean m9 calculated from the gray level of pixels in the local window, are 1 and (a,/mg) 2, respectively, the pdf being given by equation (2). If the local window is considered at the boundary regions, then the mean of the gray level in each region
is not normalized to one. In this case, the mean and variance of the gl region are m,l = (gl/mg) and a21 = (tr,/m0)2, respectively. Similarly, the mean and variance of the g2 region are given by m~2 = (g2/m o) and a~22_(a,/mg) 2, respectively. Since the local mean mg satisfies the inequality ofg t 1 are satisfied for the gt and g2 regions, respectively. Therefore, the normalized distribution of the g~ region near boundaries has its mean less than 1 and that of the g2 region has its mean greater than 1. Based on the discussion mentioned above, the bitmap is constructed by thresholding the gray level with the local mean. Or equivalently, the normalized probability distribution is thresholded with 1. Over the uniform region the bit-map is distributed randomly since the probability distribution is symmetric along the axis ofg = 1. Whereas at the boundary regions, due to the shifted pdf with its mean smaller or larger than 1, it shows the tendency that 0 and 1 are densely distributed in the gl and 02 regions, respectively. Figure l(a) shows an ideal step-edge contaminated by Gaussian noise with tr, = 5 and Fig. l(b) illustrates its bit-map. It is observed that 0 and 1 are randomly distributed over the uniform regions whereas 0 and I are densely distributed along the boundary at the left and right sides having the lower and higher mean values, respectively. The construction of the bit-map mentioned above simplifies the information of an image by decomposing it into the uniform and discontinuous regions and then by representing it in terms of the simple bit pattern consisting of 0 and 1. Edge detection of noisy images is therefore equivalent to detecting the densely distributed regions with 0 and 1 in the bit-map. Thus the co-occurrence matrix or projection method can be used in detecting boundary regions in the bit-map. In case of using the projection information, the difference of the projection values obtained from both sides along the boundary can be used as a gradient value, but the computer simulation shows that the co-occurrence matrix gives better per-
(a)
Fig. 1. Construction of the bit-map: (a) noisy image with step edges (a,, = 5); (b) its bit-map.
Edge detection based on the co-occurrence matrix formance than the projection method in detecting the reliable boundaries in the bit-map. By increasing the number of quantization levels, we may construct a matrix having several levels rather than two levels, but the computational requirement becomes complex due to the increased number of levels and the dimension of the co-occurrence matrix. In the next subsection, the definition and property of the co-occurrence matrix are briefly discussed and the proposed edge detection method using the co-occurrence matrix is presented. 2.2. Edge detection based on the co-occurrence matrix The co-occurrence matrix is one of the most powerful tools in segmenting or classifying texture imagesJ 6) It implies the second-order statistics which manifest the distribution of gray level values at two neighboring pixels separated by the distance and direction predefined. In this paper, the co-occurrence matrix over the bit-map is introduced for the edge detection of noisy images. Let P be a direction operator and A be a k × k matrix whose element aij denotes the relative frequency with which two neighboring pixels separated by direction operator P occur, one with gray level zi and the other with gray level zi, 0 < i, j < k 1.(7)For instance, consider an image consisting of three gray levels, 0, 1, and 2. Here the 5 x 5 co-occurrence matrix window is assumed. -
0 0 0 1 2 1 1 0 1 1 2 2 1 0 0 . 1 1 0 2 0 0 0 1 0 1 If we define the direction operator P as "one pixel to the right", then we obtain the 3 x 3 matrix A given by 0 A=
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For example, aoo (top left element) denotes the number of times that a point having the gray level zo = 0 appears 1 pixel right to the pixel having the same gray level, while ao2 (top right element) signifies the number of times that a point having the gray level zo = 0 appears 1 pixel right to the point having the gray level z 2 = 2. It is important to note that the size of A is determined by the number of distinct gray levels defined in the input image. Since the bit-map consists of two levels with 0 and 1, the co-occurrence matrix A used for the proposed edge detection method is a 2 × 2 matrix. Matrix A has four elements where aol(alo ) is the number of times that the bit-pattern changes from 0(1) to 1(0), and aoo(a~l ) denotes the number of times that the bitpattern does not change, with two neighboring pixels
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separated by the direction operator P. Thus, with a step-edge model corrupted by additive zero-mean Gaussian noise, ao~ and alo elements relating to the number of changing pixels have large values over the uniform regions whereas aoo and a ~ elements have large values near the boundary regions. Our interesting elements of the matrix A, therefore, are aoo and a~ ~, and thus the edge detection relates to finding the direction operator which maximizes the gradient value defined based on the distribution of 0 and 1 on the bit-pattern. Generally, gradient edge operators such as Sobel and Prewitt operators use four-directional information. These operators are applied to an image along the vertical, horizontal, diagonal, and anti-diagonal directions and then the maximum value among them is taken as an edge magnitude with its direction information. Similarly, the direction operators for the cooccurrence matrix can be defined as follows: Pv: PH: PD: PA:
1 pixel below (vertical) 1 pixel right (horizontal) 1 pixel right and 1 pixel below (diagonal) 1 pixel left and 1 pixel below (anti-diagonal).
Considering the scanning direction, above direction operators Pv, Pn, PD, and PA represent the vertical, horizontal, diagonal, and anti-diagonal directions, respectively. Figure 2 indicates the pixel locations pertinent to four direction operators at the discrete coordinates system. For example, the elements aoo of the matrices Av, An, AD, and AA denote the numbers of times that, with the bit pattern at (i, j) being 0, the bit patterns at (i + 1, j), (i, j + 1), (i + 1, j + 1), and (i + 1, j - 1) are 0 for the direction operators Pv, PH, PD, and PA, respectively. We calculate the matrix A for each direction operator and the values ofaoo and a~l for each matrix A can be used to define the edge magnitudes in a bit-map. In case the sum of aoo and a ~ are simply used as an edge magnitude, thick edge points are obtained since boundary regions densely distributed with 0 or 1 are too wide, as shown in Fig. l(b). So, instead of finding points having large (aoo + a~ l), it would be more desirable to detect the regions in which the bit-pattern changes from 0 to 1, or vice versa. This scheme gives the thinner edges and their locations are closer to the actual edge locations.
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Let us consider the bit-map with the n x n (n = 5) co-occurrence matrix window near the boundary region, centered at ®, as shown in a solid line in Fig. 3. The window contains the region with a significant intensity change. For each direction, the absolute difference between aoo and a 11 is (n - 1) for an ideal example in Fig. 3. Since each element of a matrix A is considered within an n x n local window, the calculation of each element is done on an (n - I) x (n - 1) local window for each direction. Whereas in the case that the co-occurrence matrix window is overlaid toward the uniformly distributed region with 0, centered at ®, as shown as a dotted line in Fig. 3, the absolute difference between aoo and a l l is greater than (n - 1). It is due to the fact that aoo is greater than al 1 in the densely distributed region with 0, whereas a l l is greater than aoo in the densely distributed region with 1. Therefore, if we define the sum of a0o and a l l as an edge magnitude only for the region satisfying the condition laoo - a 111 < (n - 1), then these conditions can prevent the thick edges near the boundary region, and more thinned edge points can be obtained. In the relatively uniform region, the magnitudes of aoo and a 1~ are similar due to the random distribution of the bit-pattern. Thus the condition [aoo - a~ 11< (n - 1) is satisfied, but since the sum of a0o and a 11 is relatively smaller than that for boundary regions, these regions may be easily eliminated by taking the proper threshold value. In summary, the whole steps of the proposed edge detection in noisy images based on the co-occurrence matrix are as follows: (1) With the pdfofthe gray level normalized by the local mean, the bit-map is constructed by comparing the normalized gray level to unity. (2) 2 x 2 co-occurrence matrix A is obtained for each direction operator. (3) If the co-occurrence matrix A for each direction satisfies the condition of laoo - a111 < (n - 1), then (aoo + a l l ) is defined as an edge magnitude for that direction. (4) Take the maximum value of (aoo + a11) among values derived for 4 directions as an edge magnitude. (5) Edge map is obtained by thresholding.
3. SIMULATIONRESULTSAND DISCUSSIONS In the performance analysis of edge detection, the analytic and experimental analysis for the correct and false detection of edges are useful performance indicators. In this paper, the performance of the proposed edge detection method is compared to those of the several conventional ones based on the well-known performance criterion. The performance comparison is performed on the real 256 x 256 images which are uniformly quantized to 8 bits. The performance of edge detection operators can be compared in various ways. The gradient image may be compared visually or quantitatively. In this paper, the quantitative performance measures for noisy images are considered. Let N I be the number of ideal edge points and N A denote the number of detected edge points which do not coincide with the ideal edge points. Then the error rate Pe of edge detection is defined by 18) Po --
NA • Nl
(3)
Another figure of merit F introduced by Abdou and Pratt 19} for the noise performance of edge detection operators is defined by F
1
IA X~
1
max {ll, IA} i--~l 1 + ~d2(i)
(4)
where I~ and I A are the numbers of ideal and actual edge points, respectively, d(i) denotes the pixel distance of the ith edge point detected incorrectly to the correct edge location, and ~ is a scaling constant. In our experiment • is set to 1. These performance measures Pe and F are observed to emphasize the existence of edges and locality of detected edges, respectively. Synthetic images used for the performance comparison of edge detection contain ideal step edges in the vertical, horizontal, diagonal, and anti-diagonal directions, as shown in Fig. 4. These synthetic images, with the minimum gray level of 40 and the identical step size of 40, are contaminated by zero-mean Gaussian noise having the standard deviation a,. The conventional methods such as the Canny's gradient method, (1o) Marr and Hildreth's LoG method, (2) and Jain's stochastic method "1) are considered for the performance comparison of the proposed method. The first and second derivatives of a Gaussian function are used for the Canny's method, and Marr and Hildreth's method, respectively, and the standard deviation of a Gaussian function is used as a scale parameter. The stochastic gradient method proposed by Jain is the best linear mean-square semi-causal finite impulse response (FIR) filter and a 7 x 7 mask is designed for the noisy image with SNR = 9.t11) Since Pe and F used as the edge performance measures depend on the parameters of each method, the resultant numbers of detected edge points for each method are made to be almost the same by adjusting the parameters of each method. Also the mask size of
Edge detection based on the co-occurrence matrix
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Fig. 4. Synthetic test images (tr, = 10): (a) vertical edge; (b) horizontal edge; (c) diagonal edge; (d) antidiagonal edge. each method is set to 7 x 7. Figures 5 and 6 show the results of Pe and F, respectively, for each method, as a function of the noise level. Figures 5 and 6 show the average results for 5 noisy images with the same tr,, generated by different random number seeds. Due to the similar performance between the edge masks for the vertical and horizontal directions and between those for the diagonal and anti-diagonal directions, only the results for vertical and diagonal directions are shown. Examining the error rate P~ shown in Fig. 5 demonstrates that the LoG method using the second derivative of a Gaussian function is more sensitive to noise than any other methods. The more the noise level in an image is, the more false edges are detected. On the contrary, it is observed that other edge detectors except the LoG are relatively insensitive to noise due to the smoothing effect inherent to each edge mask. In case of the vertical direction, as the noise level increases, the proposed method and Jain's method give relatively small error rate which is comparable to Canny's method. Whereas, in the case of the diagonal direction, Canny's method shows the best performance among any other methods. The figure of merit F also shows
the similar tendency as the error rate Pe. From the above results, it is observed that the LoG method is sensitive to noise and thus its performance is severely degraded. From the computer simulation results mentioned above, for the vertical direction, the proposed method shows better performance if the noise level increases. But, for the diagonal direction, Canny's method gives better results than the proposed one, which is due to the fact that the Canny's gradient method gives the biased edge magnitude depending on the edge orientation. According to the experimental results of an amplitude response to the edge orientation,191 gradient methods such as Sobel or Prewitt operators demonstrate the biased results as the edge orientation changes from the horizontal direction to the diagonal one. Figure 7 shows the ideal step-edge model 19) passing through the center of an edge mask and, with these models, the sensitivity of the proposed edge operator to the edge orientation is shown in Fig. 8 along with those of Sobel operators, Canny's method, and Jain's method. Note that the edge gradient response of a vertical edge is normalized to unity. These curves indicate
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actual edge orientation ~ (rad) Fig. 8. Edge gradient amplitude ratio response as a function of actual edge orientation. that the degree of biased responses to edge orientation is large for Jain's method, Canny's method, and Sobel operators, in decreasing order. But the proposed technique shows the unbiased response to edge orientation. This result explains the dependency of the performance measure on edge orientation as shown in Figs 5 and 6. Edge detection results for the syntactic images with a n = 10 are shown in Fig. 9. The proposed method detects some false edges in the uniform region, but it is observed to be robust to noise in the boundary region. The Canny's method and Jain's method give the large smoothing effect in the uniform region but thick edges appear in boundary regions. The LoG method shows many false edges in the uniform regions
and a lot of disconnected edges due to the influence of noise. In this paper, with the step-edge corrupted by additive Gaussian noise, the performance comparison of edge detection for noisy images is discussed. Figures 11 and 13 show the edge maps for each method obtained by thresholding the gradient magnitude obtained from noisy images shown in Fig. 10. These results may give rise to the different edge maps depending on the threshold value. A global threshold value is chosen experimentally so that the amount of false edges at the uniform region and the loss of significant edges are minimized. Since the LoG method is sensitive to noise, false edges detected in the uniform region are greatly reduced by taking the absolute sum of the differences of second derivatives, as a threshold, between at a zero-crossing point and its four neighboring pixels. But some false edges still appear near the boundary region. Gradient methods such as the Canny's method and Jain's method are relatively insensitive to noise and detect thick edges. Thus, though the number of edge points obtained by thresholding is more than that of the proposed technique, each method shows similar edge detection results. The proposed method obtains a thinner edge map than any other methods, and is relatively insensitive to intensity changes. Thus the edge detection results do not greatly depend on the noise level. Thinned results of Figs 11 and 13 are shown in Figs 12 and 14, respectively, and the parameters are adjusted to give the similar numbers of edge points. The numbers of edge points obtained in the Toy image are 4132 for the proposed method, 4168 for the Canny's
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method, 4142 for the Jain's method, and 4172 for the LoG method. In the same manner, for the House image as shown in Fig. 14, 2549 for the proposed method, 2522 for the Canny's method, 2515 for the Jain's method, and 2506 for the LoG m~thod. In the Canny's and Jain's methods, the thinning process is accomplished based on the non-maximal suppression scheme by using the gradient magnitude and direction information. But these methods have a drawback: not detecting the edge points where the local maximum point does not exist. Since the thinned edge map of the proposed method can be obtained from a binary pattern,
the loss of edge information is minimized since the edge map obtained by threshoiding is relatively thinner than those by any other methods. In comparison of the amount of computation, the proposed method has similar computational complexity to the gradient-based methods, most of the computation being due to the calculation of the co-occurrence matrix. In the proposed method, if a bit-map is not randomly constructed in the uniform region, then it often detects false edges and, in the smoothly changing region, it may have a defect in not detecting the edge points accurately.
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Edge detection based on the co-occurrence matrix 4. CONCLUSION In this paper, with the step-edge model corrupted by Gaussian noise, an edge detection technique for noisy images based on the co-occurrence matrix is proposed. In the proposed method, the uniform and boundary regions in an image are simply converted into the bit-map having only 0 and 1, by using the local mean. 0 and 1 are densely distributed near the boundary regions while randomly distributed in the uniform regions. Thus the edge detection in noisy images is equivalent to finding the densely distributed region on the bit-pattern. Thus the co-occurrence matrix is introduced to detect the boundary region. The effectiveness of the proposed technique is shown via a quantitative comparison with the conventional edge detection methods, and the simulation results for noisy images are also shown. It is observed that the proposed method is effective in noisy images. Further research will be focused on the accurate edge detection in a complex image where the gradual intensity change occurs and on the extension of the proposed technique to the multi-level co-occurrence matrix case.
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2. D. Marr and E. Hildreth, Theory of edge detection, Proc. R. Soc. Lond. B 207, 187-217 (1980). 3. F. Bergholm, Edge focusing, 1EEE Trans. Pattern Analysis Mach. lntell. PAMI-9, 726-741 (1987). 4. A. Schrift, Y. Y. Zeevi and M. Porat, Pyramidal edge detection and image representation, SPIE Proc. Visual Comnmnications and Image Processina '88 1001, Cambridge, Massachusetts, pp. 529-536 SPIE, Bellingham, Washington (1988). 5. J. F. Haddon, Generalised threshold selection for edge detection, Pattern Recognition 21, 195-203 (1988). 6. P.C. Chen and T. Pavlidis, Segmentation by texture using a co-occurrence matrix and a split-and-merge algorithm, Comput. Graphics Image Process. 10, 172-182 (1979). 7. R.C. Gonzalez and P. Wintz, Digital Image Processing, 2nd Edn, pp. 414 419. Addison Wesley, Reading, Massachusetts (1987). 8. T. Peli and D. Malah, A study of edge detection algorithms, Comput. Graphics Image Process. 20, 1-21 (1982). 9. I. E. Abdou and W. K. Pratt, Quantitative design and evaluation of enhancement/thresholding edge detectors, Proc. IEEE 67, 753-763 (1979). 10. J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Analysis Math. Intell. PAMI-8, 679-698 (1986). 11. A. K. Jain, Fundamentals of Digital Image Processing, pp. 347-357. Prentice-Hall, Englewood Cliffs, New Jersey (1989).
REFERENCES
1. J. S. Lim, Two-Dimensional Sional and Image Processing, pp. 476-494. Prentice-Hall, Englewood Cliffs, New Jersey (1990).
About the Author--DEoK J. PARKwas born in Kyung-gi, Korea, on 21 May 1967. He received the B.S. and M.S. degrees in electronic engineering from Sogang University, Seoul, Korea, in 1991 and 1993, respectively. He is currently with Samsung Electronics. His research interests are computer vision and pattern recognition.
About the Autbor--Kwo~ M. NAM was born in Kangnung, Korea, on 24 July 1968. He received the B.S. and M. S. degrees in electronic engineering from Sogang University, Seoul, Korea, in 1991 and 1993, respectively. He is currently with Samsung Electronics. His research interests include computer vision and image coding.
About the Autbor--RAE-HOsG PARK received the B.S. and M.S. degrees in electronics engineering from Seoul National University, S¢oul, Korea, in 1976 and 1979, respectively. He received the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, California, in 1981 and 1984, respectively. He .joined the faculty of the Department of Electronic Engineering at Sogang University, Seoul, Korea, in 1984, where he is currently a professor. In 1990, he spent his sabbatical year at the Computer Vision Laboratory of the Center for Automation Research, University of Maryland, College Park, Maryland, as a visiting associate professor. His current research interests are computer vision, pattern recognition, and video communication. He is a member of the IEEE and KITE.
Pergamon
0031-3203(93) E0026-4
E D G E D E T E C T I O N IN NOISY IMAGES BASED O N THE C O - O C C U R R E N C E MATRIX DEOK J. PARK, KWON M. NAM and RAE-HONGPARKt Department of Electronic Engineering,Sogang University, C.P.O. Box 1142, Seoul 100-61l, Korea (Received 4 January 1993; in revised form 11 November 1993; received for publication 2 December 1993)
An edge detection method for noisy images is proposed based on the co-occurrence matrix. In the proposed technique based on the step-edge model, the gray level information is simply converted into a bit-map, i.e. the uniform and boundary regions of an image are transformed into a binary pattern by normalization using the local mean. In this binary bit-map pattern, 0 and 1 are densely distributed near the boundary region while they are randomly distributed in the uniform region. To detect the boundary region, the co-occurrence matrix on the bit-map is introduced. The effectivenessof the proposed technique is shown via a quantitative performance comparison to the conventional edge detection methods and the simulation results for noisy images are also presented. Abstract
Edge detection
Step-edge
Bit-map
Co-occurrence matrix
I. INTRODUCTION Edge detection is one of the most fundamental steps in image analysis. Edge points can be defined as pixels at which an abrupt discontinuity in gray level, color, texture, depth, or motion occurs. The intensity discontinuities occur due to the different surface reflectance of objects, illumination condition, or varying distance and orientation of objects from the viewer. 11'2) Since edge detection is a fundamental process in various applications such as image analysis, robot vision, and image data compression, much research has been devoted to the reliable detection of edges. The significance of a physical change in an image depends on the various applications. An intensity change that would be classified as an edge in some applications might not be considered as an edge in other applications. To detect the various edge elements existing in an image, therefore, the edge detector having different scales should be considered, t3'4~ Marr and Hildreth ~2) suggested that the original image be band-limited at several different cutoff frequencies, and the impulse response of the lowpass filter proposed to band-limit an image at different cutoff frequencies is Gaussianshaped. This implies that the Gaussian filter is optimal in terms of the smoothing and localization in both the spatial and frequency domains and the standard deviation ofa Gaussian filter is used as a scale parameter. Since the scale information for a given image is not known a priori, the selection of the proper scale for the edge detection varies with the spatial resolution and thus it is a very difficult problem. To avoid the difficulty
t Author to whom all correspondence should be addressed.
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of selecting the proper scale, therefore, it is necessary that image information be converted into a simpler form in order to be easily analyzed. In this paper, with the step-edge model corrupted by additive Gaussian noise, an edge detection method in noisy images is proposed based on the co-occurrence matrix. Since conventional gradient edge detection schemes take points having maximum gradient magnitude as an edge, it is difficult to detect edge points having small intensity changes. With the step-edge model, both sides along the boundary can be divided into two regions based on the local mean, thus boundary regions having small or large intensity changes are equally classified as edge points regardless of intensity changes. Thus we construct the binary pattern by decomposing an image into the relatively uniform regions and boundary regions, and define the co-occurrence matrix over the binary pattern to detect the edges. The proposed technique is observed to generate a thinner edge map than the conventional ones and is also shown to be robust to additive noise. In Section 2, the construction of the bit-map and the edge detection using the co-occurrence matrix obtained from the bit-map are presented. In Section 3, the performance of the proposed method for noisy images is compared to those of the conventional ones in terms of the quantitative measures, and finally the conclusion is given in Section 4. 2. PROPOSED EDGE DETECTION IN NOISY IMAGES BASED ON THE CO-OCCURRENCE MATRIX
The proposed edge detectiofi technique is based on the step-edge model corrupted by additive Gaussian noise and the features are extracted from the co-occur765
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rence matrix over the binary bit pattern. In this section, the construction of the bit-map from noisy images is discussed and the edge detection method based on the co-occurrence matrix is proposed. 2.1. Construction of the bit-map Consider a step-edge image corrupted by additive zero-mean Gaussian noise with a standard deviation a,. Then, the probability density function (pdf)po(g) of the gray level of the noisy image is given by
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(1)
where g denotes the gray level and mg signifies its mean. If the gray level at a given pixel is assumed to be independent of those at any other pixels, then a new pdf can be obtained for pixel values generated with a weighted local window,iS) As a special case, the mean m s a n d variance a s2 of a new pdf, whose gray level is normalized by the local mean mo, are given by ms = 1 and try-2 _ (a,/mg)2, respectively. In other words, a new pdf of a gray level normalized by the local mean also follows a Gaussian distribution function with the mean and standard deviation equal to 1 and try, respectively, and thus it is expressed by 1 exp[-(g--1)2] P" (g) = ~/(2n) a, L 2a] '
(2)
Now, consider a step-edge model, having constant gray levels gl and g2 (gl < g2), contaminated by additive zero-mean Gaussian noise with the variance a 2. In case that the local window is considered over the uniform region, the mean and standard deviation of the gray level of the pixels in a local window, normalized by the local mean m9 calculated from the gray level of pixels in the local window, are 1 and (a,/mg) 2, respectively, the pdf being given by equation (2). If the local window is considered at the boundary regions, then the mean of the gray level in each region
is not normalized to one. In this case, the mean and variance of the gl region are m,l = (gl/mg) and a21 = (tr,/m0)2, respectively. Similarly, the mean and variance of the g2 region are given by m~2 = (g2/m o) and a~22_(a,/mg) 2, respectively. Since the local mean mg satisfies the inequality ofg t
(a)
Fig. 1. Construction of the bit-map: (a) noisy image with step edges (a,, = 5); (b) its bit-map.
Edge detection based on the co-occurrence matrix formance than the projection method in detecting the reliable boundaries in the bit-map. By increasing the number of quantization levels, we may construct a matrix having several levels rather than two levels, but the computational requirement becomes complex due to the increased number of levels and the dimension of the co-occurrence matrix. In the next subsection, the definition and property of the co-occurrence matrix are briefly discussed and the proposed edge detection method using the co-occurrence matrix is presented. 2.2. Edge detection based on the co-occurrence matrix The co-occurrence matrix is one of the most powerful tools in segmenting or classifying texture imagesJ 6) It implies the second-order statistics which manifest the distribution of gray level values at two neighboring pixels separated by the distance and direction predefined. In this paper, the co-occurrence matrix over the bit-map is introduced for the edge detection of noisy images. Let P be a direction operator and A be a k × k matrix whose element aij denotes the relative frequency with which two neighboring pixels separated by direction operator P occur, one with gray level zi and the other with gray level zi, 0 < i, j < k 1.(7)For instance, consider an image consisting of three gray levels, 0, 1, and 2. Here the 5 x 5 co-occurrence matrix window is assumed. -
0 0 0 1 2 1 1 0 1 1 2 2 1 0 0 . 1 1 0 2 0 0 0 1 0 1 If we define the direction operator P as "one pixel to the right", then we obtain the 3 x 3 matrix A given by 0 A=
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For example, aoo (top left element) denotes the number of times that a point having the gray level zo = 0 appears 1 pixel right to the pixel having the same gray level, while ao2 (top right element) signifies the number of times that a point having the gray level zo = 0 appears 1 pixel right to the point having the gray level z 2 = 2. It is important to note that the size of A is determined by the number of distinct gray levels defined in the input image. Since the bit-map consists of two levels with 0 and 1, the co-occurrence matrix A used for the proposed edge detection method is a 2 × 2 matrix. Matrix A has four elements where aol(alo ) is the number of times that the bit-pattern changes from 0(1) to 1(0), and aoo(a~l ) denotes the number of times that the bitpattern does not change, with two neighboring pixels
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separated by the direction operator P. Thus, with a step-edge model corrupted by additive zero-mean Gaussian noise, ao~ and alo elements relating to the number of changing pixels have large values over the uniform regions whereas aoo and a ~ elements have large values near the boundary regions. Our interesting elements of the matrix A, therefore, are aoo and a~ ~, and thus the edge detection relates to finding the direction operator which maximizes the gradient value defined based on the distribution of 0 and 1 on the bit-pattern. Generally, gradient edge operators such as Sobel and Prewitt operators use four-directional information. These operators are applied to an image along the vertical, horizontal, diagonal, and anti-diagonal directions and then the maximum value among them is taken as an edge magnitude with its direction information. Similarly, the direction operators for the cooccurrence matrix can be defined as follows: Pv: PH: PD: PA:
1 pixel below (vertical) 1 pixel right (horizontal) 1 pixel right and 1 pixel below (diagonal) 1 pixel left and 1 pixel below (anti-diagonal).
Considering the scanning direction, above direction operators Pv, Pn, PD, and PA represent the vertical, horizontal, diagonal, and anti-diagonal directions, respectively. Figure 2 indicates the pixel locations pertinent to four direction operators at the discrete coordinates system. For example, the elements aoo of the matrices Av, An, AD, and AA denote the numbers of times that, with the bit pattern at (i, j) being 0, the bit patterns at (i + 1, j), (i, j + 1), (i + 1, j + 1), and (i + 1, j - 1) are 0 for the direction operators Pv, PH, PD, and PA, respectively. We calculate the matrix A for each direction operator and the values ofaoo and a~l for each matrix A can be used to define the edge magnitudes in a bit-map. In case the sum of aoo and a ~ are simply used as an edge magnitude, thick edge points are obtained since boundary regions densely distributed with 0 or 1 are too wide, as shown in Fig. l(b). So, instead of finding points having large (aoo + a~ l), it would be more desirable to detect the regions in which the bit-pattern changes from 0 to 1, or vice versa. This scheme gives the thinner edges and their locations are closer to the actual edge locations.
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Let us consider the bit-map with the n x n (n = 5) co-occurrence matrix window near the boundary region, centered at ®, as shown in a solid line in Fig. 3. The window contains the region with a significant intensity change. For each direction, the absolute difference between aoo and a 11 is (n - 1) for an ideal example in Fig. 3. Since each element of a matrix A is considered within an n x n local window, the calculation of each element is done on an (n - I) x (n - 1) local window for each direction. Whereas in the case that the co-occurrence matrix window is overlaid toward the uniformly distributed region with 0, centered at ®, as shown as a dotted line in Fig. 3, the absolute difference between aoo and a l l is greater than (n - 1). It is due to the fact that aoo is greater than al 1 in the densely distributed region with 0, whereas a l l is greater than aoo in the densely distributed region with 1. Therefore, if we define the sum of a0o and a l l as an edge magnitude only for the region satisfying the condition laoo - a 111 < (n - 1), then these conditions can prevent the thick edges near the boundary region, and more thinned edge points can be obtained. In the relatively uniform region, the magnitudes of aoo and a 1~ are similar due to the random distribution of the bit-pattern. Thus the condition [aoo - a~ 11< (n - 1) is satisfied, but since the sum of a0o and a 11 is relatively smaller than that for boundary regions, these regions may be easily eliminated by taking the proper threshold value. In summary, the whole steps of the proposed edge detection in noisy images based on the co-occurrence matrix are as follows: (1) With the pdfofthe gray level normalized by the local mean, the bit-map is constructed by comparing the normalized gray level to unity. (2) 2 x 2 co-occurrence matrix A is obtained for each direction operator. (3) If the co-occurrence matrix A for each direction satisfies the condition of laoo - a111 < (n - 1), then (aoo + a l l ) is defined as an edge magnitude for that direction. (4) Take the maximum value of (aoo + a11) among values derived for 4 directions as an edge magnitude. (5) Edge map is obtained by thresholding.
3. SIMULATIONRESULTSAND DISCUSSIONS In the performance analysis of edge detection, the analytic and experimental analysis for the correct and false detection of edges are useful performance indicators. In this paper, the performance of the proposed edge detection method is compared to those of the several conventional ones based on the well-known performance criterion. The performance comparison is performed on the real 256 x 256 images which are uniformly quantized to 8 bits. The performance of edge detection operators can be compared in various ways. The gradient image may be compared visually or quantitatively. In this paper, the quantitative performance measures for noisy images are considered. Let N I be the number of ideal edge points and N A denote the number of detected edge points which do not coincide with the ideal edge points. Then the error rate Pe of edge detection is defined by 18) Po --
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(3)
Another figure of merit F introduced by Abdou and Pratt 19} for the noise performance of edge detection operators is defined by F
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(4)
where I~ and I A are the numbers of ideal and actual edge points, respectively, d(i) denotes the pixel distance of the ith edge point detected incorrectly to the correct edge location, and ~ is a scaling constant. In our experiment • is set to 1. These performance measures Pe and F are observed to emphasize the existence of edges and locality of detected edges, respectively. Synthetic images used for the performance comparison of edge detection contain ideal step edges in the vertical, horizontal, diagonal, and anti-diagonal directions, as shown in Fig. 4. These synthetic images, with the minimum gray level of 40 and the identical step size of 40, are contaminated by zero-mean Gaussian noise having the standard deviation a,. The conventional methods such as the Canny's gradient method, (1o) Marr and Hildreth's LoG method, (2) and Jain's stochastic method "1) are considered for the performance comparison of the proposed method. The first and second derivatives of a Gaussian function are used for the Canny's method, and Marr and Hildreth's method, respectively, and the standard deviation of a Gaussian function is used as a scale parameter. The stochastic gradient method proposed by Jain is the best linear mean-square semi-causal finite impulse response (FIR) filter and a 7 x 7 mask is designed for the noisy image with SNR = 9.t11) Since Pe and F used as the edge performance measures depend on the parameters of each method, the resultant numbers of detected edge points for each method are made to be almost the same by adjusting the parameters of each method. Also the mask size of
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Fig. 4. Synthetic test images (tr, = 10): (a) vertical edge; (b) horizontal edge; (c) diagonal edge; (d) antidiagonal edge. each method is set to 7 x 7. Figures 5 and 6 show the results of Pe and F, respectively, for each method, as a function of the noise level. Figures 5 and 6 show the average results for 5 noisy images with the same tr,, generated by different random number seeds. Due to the similar performance between the edge masks for the vertical and horizontal directions and between those for the diagonal and anti-diagonal directions, only the results for vertical and diagonal directions are shown. Examining the error rate P~ shown in Fig. 5 demonstrates that the LoG method using the second derivative of a Gaussian function is more sensitive to noise than any other methods. The more the noise level in an image is, the more false edges are detected. On the contrary, it is observed that other edge detectors except the LoG are relatively insensitive to noise due to the smoothing effect inherent to each edge mask. In case of the vertical direction, as the noise level increases, the proposed method and Jain's method give relatively small error rate which is comparable to Canny's method. Whereas, in the case of the diagonal direction, Canny's method shows the best performance among any other methods. The figure of merit F also shows
the similar tendency as the error rate Pe. From the above results, it is observed that the LoG method is sensitive to noise and thus its performance is severely degraded. From the computer simulation results mentioned above, for the vertical direction, the proposed method shows better performance if the noise level increases. But, for the diagonal direction, Canny's method gives better results than the proposed one, which is due to the fact that the Canny's gradient method gives the biased edge magnitude depending on the edge orientation. According to the experimental results of an amplitude response to the edge orientation,191 gradient methods such as Sobel or Prewitt operators demonstrate the biased results as the edge orientation changes from the horizontal direction to the diagonal one. Figure 7 shows the ideal step-edge model 19) passing through the center of an edge mask and, with these models, the sensitivity of the proposed edge operator to the edge orientation is shown in Fig. 8 along with those of Sobel operators, Canny's method, and Jain's method. Note that the edge gradient response of a vertical edge is normalized to unity. These curves indicate
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actual edge orientation ~ (rad) Fig. 8. Edge gradient amplitude ratio response as a function of actual edge orientation. that the degree of biased responses to edge orientation is large for Jain's method, Canny's method, and Sobel operators, in decreasing order. But the proposed technique shows the unbiased response to edge orientation. This result explains the dependency of the performance measure on edge orientation as shown in Figs 5 and 6. Edge detection results for the syntactic images with a n = 10 are shown in Fig. 9. The proposed method detects some false edges in the uniform region, but it is observed to be robust to noise in the boundary region. The Canny's method and Jain's method give the large smoothing effect in the uniform region but thick edges appear in boundary regions. The LoG method shows many false edges in the uniform regions
and a lot of disconnected edges due to the influence of noise. In this paper, with the step-edge corrupted by additive Gaussian noise, the performance comparison of edge detection for noisy images is discussed. Figures 11 and 13 show the edge maps for each method obtained by thresholding the gradient magnitude obtained from noisy images shown in Fig. 10. These results may give rise to the different edge maps depending on the threshold value. A global threshold value is chosen experimentally so that the amount of false edges at the uniform region and the loss of significant edges are minimized. Since the LoG method is sensitive to noise, false edges detected in the uniform region are greatly reduced by taking the absolute sum of the differences of second derivatives, as a threshold, between at a zero-crossing point and its four neighboring pixels. But some false edges still appear near the boundary region. Gradient methods such as the Canny's method and Jain's method are relatively insensitive to noise and detect thick edges. Thus, though the number of edge points obtained by thresholding is more than that of the proposed technique, each method shows similar edge detection results. The proposed method obtains a thinner edge map than any other methods, and is relatively insensitive to intensity changes. Thus the edge detection results do not greatly depend on the noise level. Thinned results of Figs 11 and 13 are shown in Figs 12 and 14, respectively, and the parameters are adjusted to give the similar numbers of edge points. The numbers of edge points obtained in the Toy image are 4132 for the proposed method, 4168 for the Canny's
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method, 4142 for the Jain's method, and 4172 for the LoG method. In the same manner, for the House image as shown in Fig. 14, 2549 for the proposed method, 2522 for the Canny's method, 2515 for the Jain's method, and 2506 for the LoG m~thod. In the Canny's and Jain's methods, the thinning process is accomplished based on the non-maximal suppression scheme by using the gradient magnitude and direction information. But these methods have a drawback: not detecting the edge points where the local maximum point does not exist. Since the thinned edge map of the proposed method can be obtained from a binary pattern,
the loss of edge information is minimized since the edge map obtained by threshoiding is relatively thinner than those by any other methods. In comparison of the amount of computation, the proposed method has similar computational complexity to the gradient-based methods, most of the computation being due to the calculation of the co-occurrence matrix. In the proposed method, if a bit-map is not randomly constructed in the uniform region, then it often detects false edges and, in the smoothly changing region, it may have a defect in not detecting the edge points accurately.
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Edge detection based on the co-occurrence matrix 4. CONCLUSION In this paper, with the step-edge model corrupted by Gaussian noise, an edge detection technique for noisy images based on the co-occurrence matrix is proposed. In the proposed method, the uniform and boundary regions in an image are simply converted into the bit-map having only 0 and 1, by using the local mean. 0 and 1 are densely distributed near the boundary regions while randomly distributed in the uniform regions. Thus the edge detection in noisy images is equivalent to finding the densely distributed region on the bit-pattern. Thus the co-occurrence matrix is introduced to detect the boundary region. The effectiveness of the proposed technique is shown via a quantitative comparison with the conventional edge detection methods, and the simulation results for noisy images are also shown. It is observed that the proposed method is effective in noisy images. Further research will be focused on the accurate edge detection in a complex image where the gradual intensity change occurs and on the extension of the proposed technique to the multi-level co-occurrence matrix case.
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2. D. Marr and E. Hildreth, Theory of edge detection, Proc. R. Soc. Lond. B 207, 187-217 (1980). 3. F. Bergholm, Edge focusing, 1EEE Trans. Pattern Analysis Mach. lntell. PAMI-9, 726-741 (1987). 4. A. Schrift, Y. Y. Zeevi and M. Porat, Pyramidal edge detection and image representation, SPIE Proc. Visual Comnmnications and Image Processina '88 1001, Cambridge, Massachusetts, pp. 529-536 SPIE, Bellingham, Washington (1988). 5. J. F. Haddon, Generalised threshold selection for edge detection, Pattern Recognition 21, 195-203 (1988). 6. P.C. Chen and T. Pavlidis, Segmentation by texture using a co-occurrence matrix and a split-and-merge algorithm, Comput. Graphics Image Process. 10, 172-182 (1979). 7. R.C. Gonzalez and P. Wintz, Digital Image Processing, 2nd Edn, pp. 414 419. Addison Wesley, Reading, Massachusetts (1987). 8. T. Peli and D. Malah, A study of edge detection algorithms, Comput. Graphics Image Process. 20, 1-21 (1982). 9. I. E. Abdou and W. K. Pratt, Quantitative design and evaluation of enhancement/thresholding edge detectors, Proc. IEEE 67, 753-763 (1979). 10. J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Analysis Math. Intell. PAMI-8, 679-698 (1986). 11. A. K. Jain, Fundamentals of Digital Image Processing, pp. 347-357. Prentice-Hall, Englewood Cliffs, New Jersey (1989).
REFERENCES
1. J. S. Lim, Two-Dimensional Sional and Image Processing, pp. 476-494. Prentice-Hall, Englewood Cliffs, New Jersey (1990).
About the Author--DEoK J. PARKwas born in Kyung-gi, Korea, on 21 May 1967. He received the B.S. and M.S. degrees in electronic engineering from Sogang University, Seoul, Korea, in 1991 and 1993, respectively. He is currently with Samsung Electronics. His research interests are computer vision and pattern recognition.
About the Autbor--Kwo~ M. NAM was born in Kangnung, Korea, on 24 July 1968. He received the B.S. and M. S. degrees in electronic engineering from Sogang University, Seoul, Korea, in 1991 and 1993, respectively. He is currently with Samsung Electronics. His research interests include computer vision and image coding.
About the Autbor--RAE-HOsG PARK received the B.S. and M.S. degrees in electronics engineering from Seoul National University, S¢oul, Korea, in 1976 and 1979, respectively. He received the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, California, in 1981 and 1984, respectively. He .joined the faculty of the Department of Electronic Engineering at Sogang University, Seoul, Korea, in 1984, where he is currently a professor. In 1990, he spent his sabbatical year at the Computer Vision Laboratory of the Center for Automation Research, University of Maryland, College Park, Maryland, as a visiting associate professor. His current research interests are computer vision, pattern recognition, and video communication. He is a member of the IEEE and KITE.