[IEEE 2008 International Conference on Machine Learning and Cybernetics (ICMLC) - Kunming, China...

Proceedings of the Seventh International Conference on Machine Learning and Cybernetics, Kunming, 12-15 July 2008

A PARAMETRIC METHOD FOR EDGE DETECTION BASED ON RECURSIVE MEAN-SEPARATE IMAGE DECOMPOSITION

SHAHAN NERCESSIAN1, KAREN PANETTA1, SOS AGAIAN2

1Department of Electrical and Computer Engineering, Tufts University, Medford, MA, 02155, USA 2Department of Electrical and Computer Engineering, University of Texas at San Antonio,

San Antonio, TX, 78249, USA E-MAIL: [email protected], [email protected], [email protected]

Abstract: Edge detection has played an important role in the field

of computer vision. A parametric edge detection method based on recursive mean-separate image decomposition is introduced. A method for automatic parameter selection and two methods for thresholding are also suggested. Experimental results show that the proposed method outperforms many popular edge detection methods, including Sobel, Prewitt, Frei-Chen, and Canny both visually and by quantitative edge map evaluation. Proper parameter selection can also provide segmentation of materials such as potential threat objects in x-ray luggage scan images.

Keywords: Edge detection; feature extraction; image segmentation;

image decomposition; object detection

1. Introduction

Edge detection is a central part of determining structure in images. For this reason, it has served as a basis for feature extraction, object detection, object recognition, image enhancement, and image segmentation. Edges in an image are points at which intensity values vary sharply. Many popular edge detection methods, such as Prewitt and Sobel, use gradient approximations to determine this variation [1]. The Frei-Chen edge detector generalizes this process by projecting an image onto an edge subspace to determine points that correspond to edges [2]. The Canny edge detector suppresses noise effectively by Gaussian smoothing an image as a preprocessing step and using the derivative of a Gaussian to determine variation in the image [3].

We introduce a generalized bi-orthogonal set of kernels and illustrate its application for edge detection. We show that Prewitt, Sobel, and Frei-Chen kernels for edge detection are essentially instances of the general set. The kernels are integrated into a parametric edge detection system which uses recursive mean-separate image

decomposition. An input image is decomposed into subimages, and edge detection is performed on each subimage with its own parameter set. These results are fused to form the final edge map. Two methods for thresholding edge maps are also suggested. Ultimately, parameters are trained to generate an optimal edge map that can be used to generate feature vectors for detection and recognition. The parameters can be selected by reconstructing the original image from its edge map [4] and using Bovik’s structural similarity index (SSIM) [5] between the original and reconstructed image as an objective edge evaluation measure. Pratt’s Figure of Merit (FoM) [6] is used as a quantitative evaluation method for synthetic images. The aim of the system is to provide edge maps which are superior visually as well as edge maps which will be more suitable for specific object detection purposes, such as the detection of potential threat objects in x-ray luggage scan images. The experimental results show that the proposed method outperforms standard edge detection methods, and that the right parameter set can also segment materials that may correspond to threat objects in x-ray luggage scan images.

The remainder of this paper is organized as follows: Section 2 describes recursive mean-separate image decomposition, Section 3 briefly describes methods for edge detection, Section 4 introduces the parametric method for edge detection, and Section 5 shows experimental results of the system.

2. Recursive mean-separate image decomposition

Recursive mean-separate image decomposition is a technique which has been used in extensions of histogram equalization [7,8]. Figure 1 shows a block diagram outlining the steps of recursive mean-separate image decomposition.

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Split subimage based on mean

Split subimage based on mean

Split image based on meanInput image

Subimage 1

Subimage 2

Subimage 3

Subimage 4 Figure 1. Recursive mean-separate image decomposition with n=2

An input image is recursively split n times based on the mean of the image. Pixels less than or equal to the mean of the image form one subimage while pixels greater than the mean of the image form the other subimage. The result of the image decomposition is k=2n

subimages, as illustrated in Figure 2 for n=2. This paper considers the use of either k=2 (n=1) or k=4 (n=2) subimages.

Figure 2. An input image decomposed into k=2n

subimages based on its mean with n=2

3. Methods for edge detection

Edge detection methods such as Prewitt and Sobel use two masks to approximate the gradient of an input image I to determine edge points. The magnitude and orientation of the gradient are given as

22 )*()*( yx GIGIG += (1)

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

x

yG GI

GI**

tan 1θ (2)

Frei-Chen masks are nine 3x3 orthonormal kernels which attempt to locally represent images as a weighted sum of 9 basis functions. This decomposition is useful as the first four masks (F1-F4) form an edge subspace, while

the next four masks (F5-F8) form a line subspace. The final mask is used to determine the average value information of the local image. The Frei-Chen edge detector works by projecting an input image onto the edge subspace. The magnitude of the projection of the input image I onto the edge subspace is given as

( )∑=

=4

1

2*n

nFIG (3)

The greater the magnitude of the gradient |G| is at a point, the higher the intensity variation is at that point and the more likely the point is to being an edge. Therefore, points at which the magnitude of the gradient is above a certain threshold T are considered to be edge pixels.

The Canny edge detector begins by first Gaussian smoothing an input image as a preprocessing step. It then uses the derivative of a Gaussian to approximate the gradient of the input image, given as

2

22

22

),( σ

σ

yx

x exyxG+

−−= (4)

2

22

22

),( σ

σ

yx

y eyyxG+

−−= (5)

where σ2 is the variance of the Gaussian filter used to smooth the input image in the preprocessing step. After calculating the magnitude and orientation of G, non-maximum suppression is applied to G to thin edges. Edges are thinned by suppressing pixels that are not local maxima along their gradient direction. Lastly, hysteresis thresholding is performed to determine edge fragments. A fragment is considered to be an edge fragment if every pixel in the fragment is greater than a low threshold T2 and if at least one of the pixels in the fragment is greater than a high threshold T1. This process eliminates the edge streaking evident in thresholding with a single global threshold T as it considers edge fragments instead of individual edge pixels.

4. Parametric method for edge detection

Given the edge detection methods described, we introduce a generalized set of kernels, given as

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Sobel is an instance of the generalized set using F1 and F2 with a = 2. Prewitt is an instance of the generalized set using F1 and F2 with a = 1. The Frei-Chen edge subspace is an instance of the generalized set using F1-F4 and 2=a . The generalized set of kernels is bi-orthogonal, as {F1, F2, F3, F4, F5, F8} and {F6, F7} are both orthogonal subsets. F1-F4 are standard edge detection kernels forms, F5 is a discrete Laplacian operator which enhances edges, F6 and F7 can be used for line detection, and F8 is an averaging filter. Effectively, edge maps can be generated using any combination of F1-F7 producing edge maps of varying quality and detail based on the value of the parameters. The magnitude of the gradient can be calculated by

∑∈

=}{

2)*(en

nn FIG ρ (6)

where e is the set of indices corresponding to kernels used for edge detection. For the normal use of edge detectors, ρ1= ρ2,= …= ρk-1= ρk=1. Alternatively, the gradient magnitude can also be calculated by

)*(max}{

nen

FIG∈

= (7)

Figure 3 shows a block diagram outlining the steps of the parametric method for edge detection. The input image is smoothed as a preprocessing step, and decomposed into subimages I1, I2, … Ik-1, Ik. The image is enhanced and the gradient is approximated using different parameter sets for each corresponding subimage. Each result is multiplied by its respective binary mask Bm, where

⎭⎬⎫

⎩⎨⎧ ∈

=otherwise

IyxIyxB m

m 0),(1

),( (8)

The results are fused as a weighted sum given as

∑=

=k

mmm GG

1

α (9)

where |Gm| is effectively the contribution of corresponding nonzero pixels in Bm to the magnitude of the gradient |G|. The fused result is thinned and/or thresholded to output a binary edge map.

It is advantageous to use a simple local adaptive thresholding method rather than a single global threshold T. The matrix T assigns a threshold for each pixel G(x,y), given as

βµ += ),(),( yxyxT local (10)

Edge map

Split subimage based on mean(binary masks)

Split subimage based on mean(binary masks)

Split image based on mean

Fusion

Image enhancement

Image enhancement

Image enhancement

Image enhancement

Gradient approximation




X

X

X

X

Edge thinningand/or

thresholding

Input image Gaussian smoothing

Figure 3. Block diagram for the parameterized method for edge detection

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where µlocal(x,y) is the mean of pixels within a local area of G(x,y) and β is an additive constant. Figure 4 illustrates the difference between global and local adaptive thresholding method with about the same tolerance for noise.

(a)

(b) (c) Figure 4. (a) Input image and Sobel edge map using (b) global thresholding and (c) local adaptive thresholding

Alternatively, the hysteresis thresholding used in

Canny can also be extended to the parametric edge detection system. With that said, although hysteresis thresholding eliminates edge streaking, it does not determine edge fragments in the most appropriate manner. As long as a single pixel comprising the fragment is above the high threshold, all other pixels in the fragment can be below the high threshold and the fragment can still be considered an edge fragment. A more effective method would be to threshold the mean or median of an edge fragment. Namely, a fragment is considered an edge fragment if the mean or median of the fragment is greater than a certain threshold T. This is a more desirable and appropriate method because it eliminates one of the thresholding parameters and the thresholding decision incorporates statistics of the pixels comprising the edge fragments. Figure 5 illustrates the difference between hysteresis thresholding and the suggested revision.

(a)

(b) (c) Figure 5 . (a) Input image and Canny edge map using (b) hysteresis thresholding and (c) suggested revision

Figure 6 shows a block diagram of the reconstruction-based objective edge evaluation method.

X Reconstruction SSIM index Edge map evaluation

Edge map

Input image Figure 6. Reconstruction-based objective edge evaluation method

The original input image is multiplied by the edge map, and the result is reconstructed by linear interpolation. Bovik’s SSIM is given as

))((

)2)(2(),( 2222

yxyx

xyyxyxSSIMσσµµ

σµµ++

= (11)

where x and y are the two images to be compared, (µx, µy) are the means of (x, y), (σx, σy) are the standard deviations of (x, y), and σxy is the covariance of x and y. It can be used to measure the structural similarity between the original and reconstructed image, and this in turn can be used as an objective edge evaluation measure. The edge map which yields the maximum SSIM can be taken to be best edge map by the objective evaluation method.

5. Experimental results

Figure 7 compares the results of our system with the Sobel, Prewitt, Frei-Chen, and Canny edge detectors. The images used for testing are x-ray luggage scan images which contain threat objects such as guns, bombs, or other explosives.

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(a) (b) SSIM = .2807

(c) SSIM = .2806 (d) SSIM = .2806

(e) SSIM = .2808 (f) SSIM = .2815 Figure 7. (a) Input image and edge map using (b) Sobel, (c) Prewitt, (d) Frei-Chen, (e) Canny, and (f) the parametric method for edge detection along with their respective SSIM reconstruction values

Compared to Sobel, Prewitt, Frei-Chen, and Canny,

the parametric edge detection method detects more true edges and neglects more extraneous edges. Specifically, the parametric edge detection method result in Figure 7(f) detects the detonator cords connected to the bottle of explosives as well as removes some of the more superfluous edge details on the gun. This is of paramount importance for an edge-based detection or recognition system, which relies on obtaining coarse edge detail while still detecting edges corresponding to all of the objects in an image. Note that the objective evaluation method corresponds to visual results, with the parametric edge detection method outperforming standard edge detection methods.

Additionally, the right parameter set can isolate the threat objects in the image, as shown in Figures 8 and 9. Even for a noisier, more obscured x-ray luggage scan such as the one in Figure 9(a), the parametric edge detection method is able to isolate the bomb, handgun, and laptop computer in the image with few other edges.

(a)

(b) (c) Figure 8.(a) Input image (b) segmentation of potential threat objects and (c) segmentation of bottle containing explosives using the parametric method for edge detection

(a) (b) Figure 9. (a) Input image and (b) segmentation of potential threat objects using the parametric method for edge detection Lastly, Figures 10 and 11 compare the results of our system with the Sobel, Prewitt, Frei-Chen, and Canny edge detectors for synthetic input images. Pratt’s Figure of Merit is used as a quantitative edge evaluation measure. Its value ranges from 0 to 1 with 1 corresponding to an identical match with the ideal edge map. The Pratt’s Figure of Merit shows that the parametric method for edge detection outperforms the other edge detection methods.

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(a) (b) FoM = .3102

(c) FoM = .2186 (d) FoM = .4029

(e) FoM = .9375 (f) FoM = .9439 Figure 10. (a) Synthetic input image and edge map using (b) Sobel, (c) Prewitt, (d) Frei-Chen, (e) Canny, and (f) the parametric method for edge detection along with their respective Pratt’s Figure of Merit

(a) (b) FoM = .5080

(c) FoM = .4912 (d) FoM = .8691

(c) FoM = .8908 (d) FoM = .8919

Figure 11. (a) Synthetic input image and edge map using (b) Sobel, (c) Prewitt, (d) Frei-Chen, (e) Canny, and (f) the

parametric method for edge detection along with their respective Pratt’s Figure of Merit

6. Conclusions

A parametric method for edge detection has been introduced. Two edge thresholding methods were also suggested. The performance of the proposed method has been compared with that of other standard edge detectors, and results show that the proposed system can outperform standard edge detection methods both by subjective visual results and by objective edge evaluation methods. The basis of the system is the use of mean-separate image decomposition, with each subimage being processed its own way. Future research will be dedicated to improving the parameter selection process, both to provide better general detail edge maps as well as improving segmentation results, particularly of threat objects in x-ray luggage scan images.

7. References

[1] R. Gonzalez and R. Woods, “Digital Image Processing: Second Edition”, Addison-Wesley publishing company, 2002.

[2] W. Frei and C. Chen, “Fast Boundary Detection: A Generalization and a New Algorithm,” IEEE Trans. Computers, Vol. C-26, No. 10, pp. 988-998, 1977.

[3] J.Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell., pp. 679–714, 1986.

[4] S. Carlsson, “Sketch based coding of gray level images,” IEEE Trans. Image Proc., Vol. 15, No. 1, pp. 57-83, 1988.

[5] Z. Wang, E. P. Simoncelli and A. C. Bovik, “Multi-scale structural similarity for image quality assessment,” IEEE Asilomar Conf. Signals, Systems and Computers, 2003.

[6] I. E. Abdou and W. K. Pratt, “Quantitative Evaluation of Enhancement/Thresholding Edge Detectors”, Proc. IEEE, Vol. 67, No. 5, pp. 753-763, 1979.

[7] Y. Kim, “Contrast Enhancement Using Brightness Preserving Bi-Histogram Equalization,” IEEE Trans. Cons. Electronics, Vol. 43, No. 1, pp. 1-8, 1997.

[8] S. Chen and A. R. Ramli, “Contrast Enhancement using Recursive Mean-Separate Histogram Equalization for Scalable Brightness Preservation,” IEEE Trans. Cons. Electronics, Vol 49, No. 4, pp. 1301-1309,2003.

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[IEEE 2008 International Conference on Machine Learning and Cybernetics (ICMLC) - Kunming, China...

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