On Hierarchical Combination of Discrete Wavelet Transform with Canny Edge Detector

Transcription

1 On Hierarchical Combination of Discrete Wavelet Transform with Canny Edge Detector Tiehang Duan Department of Computer Science and Engineering The State University of New York at Buffalo Abstract Discrete Wavelet Transform and Canny Edge Detector have been state of the art edge detection methods, each has its own unique strength. In this article, we provided three combination methods and developed two implementation algorithms to utilize the unique strength of both methods under the unified framework. The three methods we developed in the article have different bias on DWT and Canny Detector and can be applied to different practical situations. The comprehensive experiment result shows the algorithms have decent performance in edge detection and are computational efficient. I. INTRODUCTION The two state of art edge detection methods involved in this article, namely the Discrete Wavelet Transform and Canny Edge Detector, both serves as important and matured methods in edge detection and they each have their own strengths. For the wavelet transform, its strengths lies in its flexibility of choosing different wavelet type and scales and can get edge information on different detailed levels. For canny edge detector, it has superior performance in post processing of the edge information with methods including non maximal suppression and hysteresis threshold to eliminate spurious edges. In this article, we come up with three combined methods to synthesize the strength of both and form robust algorithms on edge detection. In section 2, we have a brief review on the theory establishment of DWT and Canny Detector, in Section 3, we discuss the possible combination mechanisms of the two methods, in section 4, we describe the experimental setup and analyze the results of the comprehensive experiments we performed on these combined methods, and conclusion is reached in Section 5. A. Wavelet Transform II. THEORY ESTABLISHMENT The inspiration of Discrete Wavelet Transform comes from the need of getting both temporal information and frequency information simultaneously while maintain computation efficiency at the same time. Traditional Fourrier transform take the viewpoint that the whole axis of the time domain is one period and it has the drawback of not able to show the frequency band information and temporal position information simultaneously. To solve this problem, people developed the method of Windowed Fourier Transform [2], and its main methodology is to pick an interval of finite length (the period of interest) on the time domain and make the assumption that the signal intensity other than this period is all zero, which is the same as adding a mask to the original signal. And Fourier Transform is applied to this specified length of signal, which provides information on both temporal and frequency. In terms of applying the mask, people find out that the simple u(t m) u(t n) function will bring problems that the transform will take the edges of the mask to be the original signal thus introduced high frequency noise to the signal. To prevent this, people designed masks with the margin smoothly goes to zero. Still, Windowed Fourier Transform suffers from computational complex issues as under most circumstances, people may just want to know the information on some specific frequency, while the WFT computes result thoroughly on the whole frequency domain. There are several ideas to simplify the computation for WFT on the time domain such as applying discrete masks g(x n s0). And when people turn to the idea of directly calculate the information on specific locations of interest on the frequency domain instead of the whole information on the axis, they turn to Discrete Wavelet Transformation [1]. Discrete Wavelet transformation is the convolution of the original signal with predefined functions on the time domain [11], the purpose of which is to turn the original signal into the decomposed form [7] n 1 f(t) = C 00 φ(t) + C j,k Ψ j,k (t) (1) j=0 Where φ(t) is the scaling function and Ψ(t) is the wavelet function. The wavelet function is achieved from the dilation and translation of the mother wavelets with the form [4] ψ j,k (t) = 2 j/2 ψ( t k2j 2 j ); j, k Z; (2) And scaling function is achieved in the same way φ j,k (t) = 2 j/2 φ( t k2j 2 j ); j, k Z; (3) After each iteration of the decomposition, we can get the detailed edge information from the high pass component W H (n, j) = m W L (m, j 1)h(m 2n) (4)

2 And the low pass component is used for further decomposition of the next level W L (n, j) = m W L (m, j 1)g(m 2n) (5) Where g(n) and h(n) are the filter coefficients corresponding to the scaling functions φ(n) and wavelet functions ψ(n) [4]. One of the major advantages of Discrete Wavelet Transform is computational effective in getting specified frequency information (or resolution) added with the temporal location of this information. It accomplish this by scaling its short function to be flatter to get low frequency information, and sharper(or finer) to get high frequency information. In practice, people usually perform the finest wavelet transform on the original signal, then scale the wavelet function by 2 every time to get the edge information of lower frequency on the original signal, just like stripping off the leaves of a cabbage [1]. People can choose freely on the specified frequency level they care(also on the temporal axis), without calculating redundant result on other frequency bands, and this computation efficiency attribute makes it to have wide application in image processing and compression [8] [5]. B. Canny Edge Detector The canny edge detector is built on three criteria with edge detection [3]: (1) Low Error Rate: There should be a low probability of failing to mark real edge points, and low probability of falsely marking nonedge points. (2) Good localization: The points marked as edge points by the operator should be as close as possible to the center of the true edge. (3) Only one response to a single edge. This is implicitly captured in the first criterion since when there are two responses to the same edge, one of them must be considered false. However, the mathematical form of the first criterion did not capture the multiple response requirement and it had to be made explicit. Canny edge detector make two further approaches to enhance the detection result. First, it set up two thresholds, and define weak edge to be the points lying between these two thresholds [6]. Then the decision of whether to maintain these weak edges is made by checking if the weak edge belongs to some natural extension of the nearby strong edges. In practice, this can be done by calculating the edge direction of the strong edges and see if there are neighboring weak edge lies on the direction. If there is, then mark the relative weak edge to be true edge. And it also introduces non maximal edge suppression to ensure there is only one response for each real edge. As Canny derived in 1986 in his cornerstone paper [3], it is natural for the edge detecting filter to get multiple local noise maximals near the true edge location as the distance between the noise edges(zero crossing point) is ( x (zc) (f) = π f 2 1/2 (x)dx) f (6) 2 Where f(x) is the filtered image function and the operator width W is multiple times larger than the distance x zc. W = N n x zc (7) Where N n is the expected number of noise maxima appearing in region W. The Canny Edge fulfills the non maximal suppression function by eliminating the neighboring non significant edges lying in the perpendicular direction of the edge [3], which is described as minor edge in [10]. III. COMBINATION OF WAVELET TRANSFORM WITH CANNY EDGE DETECTOR In this section, we come up with three mechanism to combine wavelet transform with Canny edge detector with detailed algorithm for implementation. A. Method Emphasize on Discrete Wavelet Transform We perform Discrete Wavelet Transform on different scales of the image and perform multiplication to the result of different levels of transform. Then we perform the inverse wavelet transform and the addition of different scales to get the synthesized edge image. And we perform the hysteresis component of the canny detector to post process the edge detected. The time complexity is the supremum of DWT and Canny Detector, which is computational efficient. B. Balanced Approach between Wavelet Transform and Canny Edge detector We perform Discrete Wavelet Transform on different scales of the image and perform multiplication of different levels of transform. And we perform the inverse wavelet transform and the addition of different scales to get the synthesized edge image. Then we perform non maximal suppression component of Canny Edge detector combined with hysteresis thresholding to the results to get better performance. The time complexity is also the supremum of DWT and Canny Detector. C. Method Emphasize on Canny Edge Detector We perform Discrete Wavelet Transform on the image from which we get the LH (Horizontal Low Pass and Vertical High Pass) and HL (Horizontal High Pass and Vertical Low Pass) information of the image. We calculate the gradient direction n for each pixel of the image via the element wise division of matrix LH and matrix HL. Then we detect the edge location with n G n f = 0 (8)

3 Algorithm 1 Emphasize on DWT Require: The wavelet and scaling operator h(n) and g(n) Require: Gaussian filtered original image I 1: for (each iteration of decomposition) do 2: for (each row in Image I) do 3: convolute with h(n) and g(n) horizontally 4: store the resulting image to be H(n). 5: end for 6: for (each column in Image H(n)) do 7: convolute with h(n) and g(n) vertically 8: store the resulting image to be M(n). 9: end for 10: convolute M(n) with high pass filter H to be P(n) 11: perform inverse wavelet transform to P(n) 12: store the result to be Q(n) 13: end for 14: for (each adjacent iteration level n and n+1) do 15: resize Q(n+1) to be Q (n+1) the same size as Q(n) 16: perform element wise multiplication on Q(n) and Q (n+1) 17: store the result to be S(n) 18: end for 19: Define: Matrix T to store synthesized edge graph 20: for (each decomposition level) do 21: Resize S(n) to be S (n) as the same size with I 22: T=T+S (n) 23: end for 24: Define: row and col to be the number of rows and columns of original graph I 25: Define: Threshold t1 to identify strong edge and threshold t2 to identify weak edge 26: Define Matrix Mark to denote the edge type 27: for (Each T[i][j] in T) do 28: if (T[i][j])>t1 then 29: Mark[i][j]=Strong 30: end if 31: if (T[i][j])<t1&&T[i][j])>t2 then 32: Mark[i][j]=Weak 33: end if 34: if (T[i][j])<t2 then 35: Mark[i][j]=NULL 36: end if 37: end for 38: Define: Mark[i*][j*] the neighboring point of Mark[i][j] 39: while (weak edge change to strong edge) do 40: for (each Mark[i][j] in Mark) do 41: check Mark[i*][j*] in the edge direction 42: if (Mark[i*][j*]=Weak) then 43: Mark[i*][j*]=Strong 44: end if 45: end for 46: end while After which we continue to perform non maximal suppression and hysteresis thresholding, the same as described in Algorithm 2 Balance between DWT and Canny Detector Require: The wavelet and scaling operator h(n) and g(n) Require: Gaussian filtered original image I 1: the first part same as Algorithm I Line 1 to 28 2: for (each T[i][j] in T) do 3: search adjacent T[i*][j*] orthogonal to edge direction 4: if ( T[i*][j*] for T[i][j] < T[i*][j*]) then 5: T[i][j]=0 6: end if 7: end for 8: Define row, col, t1, t2 and Mark same as Algorithm 1 9: for (i from 1 to row) do 10: for (j from 1 to col) do 11: if (T[i][j])>t1 then 12: Mark[i][j]=Strong 13: end if 14: if (T[i][j])<t1&&T[i][j])>t2 then 15: Mark[i][j]=Weak 16: end if 17: if (T[i][j])<t2 then 18: Mark[i][j]=NULL 19: end if 20: end for 21: end for 22: while (weak edge change to strong edge) do 23: for (each Mark[i][j] in Mark) do 24: check Mark[i*][j*] in the edge direction 25: if (Mark[i*][j*]=Weak) then 26: Mark[i*][j*]=Strong 27: end if 28: end for 29: end while Algorithm 2. We can see this combination method is trivial compared with previous two methods as the edge detection result will be the same with the classic canny edge detector, so we omit the detailed implementation of this method. We will implement Algorithm 1 and Algorithm 2 in the next section and compare these two methods with the performance of classic Wavelet Transform. A. Experimental Setup IV. EXPERIMENT In order to fully illustrate the strength brought by the combined methods and based on the close relationship with the common convolution masks used in image processing, we use Haar wavelet as the base wavelet function in our experiment. We make comparisons of the results on original images, image added with gaussian noise and image added with Salt Pepper noise. The detailed experimental setting is shown in Table I For comparison, we first get the raw result directly from DWT and then perform a single value thresholding on the edge image to eliminate noise. The edge intensity is reversed to get better perception of the edges. The result is presented in

4 Figure 1: From top right to bottom left: (1) The original lena image. (2) The image added with Gaussian noise. (3) The image added with Salt Pepper noise. (4)(5)(6) The result directly after DWT on original image, Gaussian noise image and Salt Pepper noise image. (7)(8)(9) The result based on Algorithm 1 on original image, Gaussian noise image and Salt Pepper noise image. (10)(11)(12) The result based on Algorithm 2 on original image, Gaussian noise image and Salt Pepper noise image. Figure 2: From top right to bottom left: (1) The original carriage image. (2) The image added with Gaussian noise. (3) The image added with Salt Pepper noise. (4)(5)(6) The result directly after DWT on original image, Gaussian noise image and Salt Pepper noise image. (7)(8)(9) The result based on Algorithm 1 on original image, Gaussian noise image and Salt Pepper noise image. (10)(11)(12) The result based on Algorithm 2 on original image, Gaussian noise image and Salt Pepper noise image. Table I: Experimental Setup Experimental Setup Image Imposed Noise Var of Gaussian Noise Prob of Salt Pepper Noise Range of Edge Intensity High Threshold t1 Low Threshold t2 Single Threshold t Decompostion Level Neighbor Definition Lena, Carriage Gaussian, Salt Pepper 20 5% [0, 255] Neighbor the second row of Figure1 and Figure 2. We can see the edge of the original image is quite clear and can reflect the structure of the whole image, but the edge quality is still suffering from fake edges. For the gaussian noise image, the normal thresholding did a good job in eliminating noise but the noise is still significant for the impulse noise image. And the noise is affecting the recognition of benchmark characteristics in the image as we can see the number 12 on the back of the carriage in Figure?? is blurred at this time, and we move on to the combination of wavelet transform with canny edge detector. B. Implementation of Combined Algorithms 1) Algorithm 1: We implemented hysteresis thresholding in Canny Edge Detector after DWT to suppress the spurious edges. We performed the hysteresis thresholding on the results of original image, the Gaussian noise image and Salt Pepper noise image, the result is shown in the third row of Figure 1 and Figure 2. We can see the edge detection results on both image is improved and the benchmark character of Figure 2, the number 12 at the back of the carriage, is now clear and easy to perceive. 2) Algorithm 2: We further combined wavelet transformation with canny edge detector by introducing non maximal suppression, which is based on the third criteria of canny edge detector. The result is shown in the last row of Figure 1 and Figure 2. We can see under this implementation method, the true edges become more significant, sharp and accurate, the number 12 at the back of the carriage is clear. Such results is more readily to use for further computer vision tasks including object recognition and image understanding.

5 V. CONCLUSION In this article, we provided the unified framework for the combination of Discrete Wavelet Transform with Canny Edge Detector. The three combination methods under this framework have different bias on DWT and Canny Detector and can serve in different applications. We provide detailed computational efficient algorithms for implementation of these methods and the experiment result shows the combination methods is robust against noise and can get different resolution edge information, which integrate the strength of DWT and Canny Detector. REFERENCES [1] Amara Graps et al. An introduction to wavelets. IEEE Computer Society, [2] A.S.Yakovlev et al. Window fourrier and wavelet transforms. St. Petersburg State University. [3] John Canny et al. A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, (6): , [4] K.K.Shukla et al. Efficient Algorithms for Discrete Wavelet Transform with Applications to Denoising and Fussy Inference Systems. Springer, Springer London Heidelberg New York Dordrecht, [5] Sumitra. P et al. Development of jpeg2000 with gamma code based on discrete wavelet transform for still image compression standard. International Journal of Computer Applications, (71): , [6] William McIIhagga et al. The canny edge detector revisited. International Journal on Computer Vision, (91): , [7] He Ping Liu et al Jing Lu. Discrete meyer wavelet transform features for online hangul script recognition. Research Journal of Applied Sciences, Engineering and Technology, (4(20)): , [8] Sheenam Malhotra Khushpreet Kaur. Image compression using haar wavelet transform and discrete cosine transform. International Journal of Computer Applications, (125):28 31, [9] Paul Bao Lei Zhang. Edge detection by scale multiplication in wavelet domain. Pattern Recognition Letters, (23): , [10] Ardeshir Goshtasby Lijun Ding. On the canny edge detector. Pattern Recognition, (34): , [11] Patrick J. VanFleet. Discrete Wavelet Transformations. Wiley, 111 River Street, Hoboken, NJ, 2008.