Scale multiplication in odd Gabor transform domain for edge detection

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Abstract

In this paper, we propose an adaptive edge detection technique based on scale multiplication in odd Gabor transform domain (ESMG). With adjacent scale multiplication in odd Gabor transform domain, a sharpened edge response output is obtained, which can more effectively resist the inverse influence from noise contamination on the performance of edge detector. Based on odd Gabor filter with single scale, it is shown that Rayleigh distribution can be feasibly adopted to model the real pdf of edge response. Thus the pdf of sharpened edge response output can be approximately modeled by an exponential distribution since there exists strong correlation between two edge response outputs with two adjacent scale factors. In determining the threshold for the sharpened edge response, an adaptive strategy is applied, in which the nonlinear relation of the threshold with the mean and variance of exponential distribution is exploited. Moreover, an optimization problem is finally formulated to find the adaptive adjustment factor. The experimental results on both synthetic and real world natural images show that our scheme is robust and takes on good edge detection performance.

Introduction

Edge detection, which has attracted the attentions of many researchers, is one of the most important areas in lower level computer vision because the success of higher level processing such as object detection, object recognition, and scene interpretation relies heavily on good edge detection. Generally, edges are referred to those pixels around which there is a large gray variation. Most existing edge detection methods are gradient-based, such as some simple edge detection operators Roberts and Sobel as well as some sophisticated operators Canny [1] and Susan et al. [2]. An overview of edge detection techniques was given in [3]. Some new developed edge detection algorithms can be found in [4], [5]. In this paper, we focus on the multi-scale analysis of odd Gabor transform to develop a new edge detection technique.

Gabor transform, or named by short time Fourier transform (STFT), can obtain the lower limit of joint time-frequency resolution under Heisenberg Uncertainty Principle. Daugman [6] has taken a deeply investigation on it from the view of visual nerve perceptibility and found that the receptive profile of simple cells in mammalian visual systems can be closely fitted by two-dimension Gabor function. Now two-dimension Gabor transform has obtained comprehensive applications in computer vision field such as image segmentation, object recognition, and so on.

Under Canny’s three optimal criterions for edge detection, R. mehrotral [7] pointed out that odd Gabor filter has good edge detection performance. In real applications, as suggested in [7] a simple edge operator should be used to estimate the gradient orientation. Thus the estimation error based on the above simple edge operator, which generally is high especially in the noise case, will largely influence the accurate construction of the odd Gabor filter templates. In addition, we cannot implement the convolution between odd Gabor filter and the input image by using FFT since each pixel corresponds to different odd Gabor filter template.

Multi-scale analysis is a popularly adopted technique in computer vision. A systematic analysis for multi -scale space theory has been given by Tony Lindeberg [8]. The applications of multi scale in edge detection can be dated back to Marr and Hildreth’s work [9]. Based on the framework of Canny [1], some techniques for edge detection based on multi scale wavelet transform have been developed [10], [11], [12]. In addition, Park et al. [13] have proposed a region based multi scale edge detection method. Recently Zhang [14] extended the ideas of Xu [15] and Sadler [16] and proposed a scale multiplication based edge detection scheme, in which adjacent scales are multiplied in wavelet domain to sharpen the edge responses. In our work, the similar scale multiplication idea was applied to the odd Gabor transform domain, but a more favorable edge response output energy function was presented.

Voorhees et al. [17] pointed that the derivative based edge response can be fitted by Rayleigh distribution, which is also shown in our work to be applicable for odd Gabor filter based edge response. Thus the pdf of sharpened edge response output can be approximately modeled by an exponential distribution since there exist strong correlation between two edge response outputs with two adjacent scale factors. Moreover, an adaptive threshold determining scheme is proposed based on the analysis of the pdf of sharpened edge response output.

This paper is organized as follows: Section 2 gives an introduction to odd Gabor based edge detection and proposes an improvement on it. To sharpen the edge response, a scale multiplication in odd Gabor transform domain is proposed and a more favorable edge response output energy function is given in Section 3. In Section 4, we theoretically analyze the sharpened edge response output energy distribution property and propose an adaptive threshold selection scheme. Some experimental results of synthetic and natural images are presented in Section 5. Finally we give our conclusions in Section 6.

Section snippets

Odd Gabor filter based edge detection

Gabor filters, which have been shown to fit well the receptive fields of the majority of simple cell in the primary visual cortex [6], are modulation products of Gaussian and complex sinusoidal signals. A 2D Gabor filter oriented at angle θ is given by:G(x,y,σg,ω,θ)=g(x,y,σg)·exp[jω(xcosθ+ysinθ)]where g(x,y,σg)=12πσg2exp[-12σg2(x2+y2)] is a Gaussian function, σg is the standard deviation of the circle Gaussian along x and y, and ω denotes the spatial frequency.

Fig. 1 shows the even real and odd

Scale multiplication in odd Gabor transform domain

Multi-scale method is a commonly adopted technique in computer vision. Here, we form the multi-scale odd Gabor filter asGoik(x,y,σg,ω,θi)=2-2kGoi(2-kx,2-ky,σg,ω,θi)where k  Z is the scale factor and i  [1,2] denotes orientation index.

Following the idea of Zhang [14], Xu [15] and Sadler [16], an adjacent scale multiplication in odd Gabor transform domain is performed to obtain a sharpened edge response output Esj(x,y), and we have:Esj(x,y)=k=jj+JEk(x,y)Ek(x,y)=i[Tik(x,y)]2where Ek(x,y) is the

Rayleigh distribution of edge response based on odd Gabor filter

In [17] Voorhees et al., have pointed that the derivative based edge response can be fitted by Rayleigh distribution. For the first derivative property of odd Gabor filter, the odd Gabor filter based edge response E (x,y) can also be fitted well by Rayleigh distribution. To evaluate the degree of fitness between the real pdf p(x) and the estimated distribution q(x) of odd Gabor based edge response, the Bhattacharya distance is adopted as similarity measure, and we have:sp(x),q(x)=-+p(x)q(x)dx

Experimental results and analysis

Because of the subjectivity of edge detection, it is difficult to compare the performance of two edge detectors on most real-world natural images [19]. Here, we use some synthetic [20] and real world images to test the effectiveness of ESMG.

Conclusion

We proposed an adaptive edge detection technique based on the multi scale analysis in odd Gabor transform domain (ESMG). To overcome the shortcomings of original odd Gabor-based edge response obtaining method, an improved scheme is suggested. With adjacent scales multiplication in odd Gabor transform domain, a sharpened edge response output, which can more effectively resist the inverse influence from noise contamination on the performance of edge detector, can be obtained. By analysis of edge

Acknowledgments

This work was funded in part by National Natural Science Foundation of China (Nos. 60373028, 60475010 and 90604032), Program for New Century Excellent Talents in University, Specialized Research Fund for the Doctoral Program of Higher Education (No. 20030004016), Specialized Research Foundation of BJTU (No. 2005SZ005), Research Foundation of Beijing Jiaotong University (No. 2005SM013).

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