Image edge detection method based on anisotropic diffusion and total variation models

The Journal of EngineeringVolume 2019, Issue 2 p. 455-460 Research ArticleOpen Access Image edge detection method based on anisotropic diffusion and total variation models Ali Abdullah Yahya, Corresponding Author Ali Abdullah Yahya aselwey1@hotmail.com School of Computer and Information, Anqing Normal University, Anqing Anhui, 246011 People's Republic of ChinaSearch for more papers by this authorJieqing Tan, Jieqing Tan School of Computer and Information, Hefei University of Technology, Tunxi Road No.193 Hefei, Anhui, China, People's Republic of ChinaSearch for more papers by this authorBenyu Su, Benyu Su School of Computer and Information, Anqing Normal University, Anqing Anhui, 246011 People's Republic of ChinaSearch for more papers by this authorKui Liu, Kui Liu School of Computer and Information, Anqing Normal University, Anqing Anhui, 246011 People's Republic of ChinaSearch for more papers by this authorAli Naser Hadi, Ali Naser Hadi School of Computer and Information, Hefei University of Technology, Tunxi Road No.193 Hefei, Anhui, China, People's Republic of ChinaSearch for more papers by this author Ali Abdullah Yahya, Corresponding Author Ali Abdullah Yahya aselwey1@hotmail.com School of Computer and Information, Anqing Normal University, Anqing Anhui, 246011 People's Republic of ChinaSearch for more papers by this authorJieqing Tan, Jieqing Tan School of Computer and Information, Hefei University of Technology, Tunxi Road No.193 Hefei, Anhui, China, People's Republic of ChinaSearch for more papers by this authorBenyu Su, Benyu Su School of Computer and Information, Anqing Normal University, Anqing Anhui, 246011 People's Republic of ChinaSearch for more papers by this authorKui Liu, Kui Liu School of Computer and Information, Anqing Normal University, Anqing Anhui, 246011 People's Republic of ChinaSearch for more papers by this authorAli Naser Hadi, Ali Naser Hadi School of Computer and Information, Hefei University of Technology, Tunxi Road No.193 Hefei, Anhui, China, People's Republic of ChinaSearch for more papers by this author First published: 24 January 2019 https://doi.org/10.1049/joe.2018.5345Citations: 14AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, a novel image edge detection technique based on the combination of total variation (TV) and anisotropic diffusion (PM) models is presented. In the proposed technique, the authors first use the gradient magnitude to eliminate the noise, then utilise the adaptive weight function to detect the edges of the image. The adaptive weight function has a high ability to adapt and change according to the areas information (edges or flats areas). More specifically, TV filter is applied on the areas which suffer from double and false edges, whereas, anisotropic diffusion filter is applied on the areas which suffer from weak and discontinuous edges. Applying TV filter on the double edges areas will allow one to remove most of the false edges, and thus to obtain much sharper edges. While, applying anisotropic diffusion filter on the discontinuous edges areas will lead to obtaining robust and continuous edges. Consequently, less false edges besides high localisation accuracy were obtained. Experimental results demonstrate the superiority of the new approach in terms of removing the false edges and improving the localisation accuracy of the edges. As objective quantitative performance measures, the peak signal-to-noise ratio (PSNR) and Pratt's figure of merit (FOM) were used. 1 Introduction Edges can be defined as local intensity changes in an image. Edges usually occur on the border those separate between two different areas. With regard to patterns recognition, edge detection approaches have proved their ability to solve various pattern recognition issues [1]. The tasks of edges detection can be classified into two basic categories: one is generating the line drawing and the other is extracting significant features. Edge detection techniques are still facing many problems, the most common problems are: discontinuity in the detected edges, sensitivity to noise, and the false edges [2]. As we know, edge carries the most significant information of image. For this reason, numerous edge detection techniques have been proposed in the literature [1, 3-8]. The importance of edges detection lies in several applications, for instance; feature detection, face recognition, computer vision, medical diagnosis, remote sensing, machine vision and artificial intelligence [9]. In recent years, image edge detection techniques such as neural network, sparse coding, watershed and learning have become very common. Meftah et al. [10] applied a spiking neural-network-clustering-based approach to detect the edges of the image. To achieve optimum edge detection, network architecture parameters and learning parameters have been set up for every specified image problems. In [11] a hybrid between a multi-scale approach and a convolutional neural network is used to detect the horizon line in maritime scenarios. To minimise the amount of the noisy edges and maintain the prominent edges simultaneously, multi-scale edge detection is carried off at each scale and integrates these results to get the single edge map. Based on sparse subspace clustering theory, Tian and Li [12] proposed a new community detection algorithm. According to sparse subspace representation theory, in a certain similarity measure space, each community can be extended across a subspace. From original dataset space that is based on sparse subspace clustering, authors tried to reformulate community detection's problem as a detecting low-dimensional subspace. Inspired by the idea of blending multi-level information obtained from different intermediate layers, Hu et al. [13] proposed a new convolutional neural-network-based pipeline for edge detection. The edge in the proposed detector is detected in the way of image-to-image without the need for any post-processing. To assess their proposed algorithm, the authors utilised BSDS500, NYUD, Multicue, and Pascal VOC'12 datasets. Salman [14] proposed to combine k -means clustering, watershed, and difference in strength map to get a new edge detector. Based on the mean intensity value, the author proposed to utilise watershed with new merging procedures to segment the image regions, while that, edge strength technique is used to get precise edge maps without broken lines. Over the last few decades, applying partial differential equations (PDEs) in image edge detection has grown rapidly [15]. Barcelos and Pires [16] have presented a new intelligent computational mechanism for edge detection based on non-linear diffusion equations. The authors incorporated a non-linear diffusion equation to the Canny edge detector. Ndajah and Kikuchi [17] proposed a new edge detection algorithm based on total variation (TV) anisotropic. In this algorithm, the authors began from the TV functional for a two-dimensional image, which the Euler–Lagrange method has been used to minimise the functional. Bazan and Blomgren [18] proposed image smoothing and edge detection technique based on a combination of non-linear diffusion and bilateral filtering. In this technique, the diffusion stopping criterion is based on the moving average of the second derivative of the correlation between the noisy image and the filtered image. Fontaine and Basu [19] addressed the problem of edges detection by solving an anisotropic diffusion equation with a wavelet basis. The reason to adopt the wavelet basis is that; in partial differential equations where the solutions tend to form discontinuities, the wavelet basis will be more efficient. In this paper, we first minimise the functionals of TV and anisotropic diffusion by Euler–Lagrange method, then combine the minimiser of these two functionals to get our new model. In our proposed method, we use a suitable weight function to combine the TV filter and anisotropic diffusion (PM) filter. In the proposed algorithm, first, we adopt a gradient magnitude strategy to reduce the noise. This strategy can be implemented as follows: In case of the gradient is large, the diffusion will be low which will result in less noise smoothing on the edges of the image. In case of the gradient is small, the diffusion will be high which will result in more noise smoothing on the flat areas. Second, we use the zero-crossing detector to distinguish the edges areas from the flat areas, and apply an adaptive weight function to detect the edges of the image. The proposed adapting weight function has ability to change and adapt according to the areas information (edges or flat areas). The rest of this paper is organised as follows: In Sections 2 and 3, we briefly describe the TV and PM filters. The proposed method is described in Section 4. In Section 5, we present our experimental results that confirm the efficiency of proposed model. Some concluding remarks are presented in Section 6. 2 Total variation model Total variation filter was first proposed by Rudin et al. [20] in 1992 and so is today known as the Rudin-Osher-Fatemi (ROF) filter. In addition to image enhancing, TV filter is focused on edge detection and segmentation problems. Rudin et al. proposed the TV functional as follows: (1) and the energy functional: (2) where is the gradient and is the domain of the image. By letting (3) the partial derivatives of the integrand (4) will be as follows: , , and . The Euler–Lagrange equation [21] corresponding to (2) is (5) The Euler–Lagrange equation of the TV model can be expressed as follows: (6) Equation (1) is a norm and the vectorial components of are and [17, 22, 23]. Therefore, the norm (7) The main advantage of TV model is that during the filtration process single edges are formed. However, these edges are weak and discontinuous. 3 Anisotropic diffusion model Anisotropic diffusion filter was first proposed by Perona and Malik (PM) [24]. Perona and Malik formulated the problem of edge detection as one of solving the non-linear anisotropic diffusion equation, which is also known as a Fick's law [19]: . The energy functional of the anisotropic diffusion filter can be expressed as follows: (8) where the diffusivity function is given by (9) Then energy functional can be rewritten as (10) Now let us suppose that (11) Then the partial derivatives of the integrand (12) will be as follows: and So the Euler–Lagrange equation corresponding to (8) is (13) The major advantage of anisotropic diffusion filter is to smooth the homogeneous areas of the image and at the same time enhance the edges [25]. However, anisotropic diffusion model usually yield double and false edges, so the image that is filtered by this model looks very messy. 4 New model Inspired by the TV and anisotropic diffusion edge detection filters we proposed a novel edges detection algorithm. Amazing edges detection results will be obtained if we integrate the advantages of these two filters. In the proposed algorithm, we adopt a new appropriate adaptive weight function to integrate the TV and anisotropic diffusion models to create our new model. By (2) and (8) the energy functional of our model will be (14) where (15) and (16) (17) where P is the pixels at position . By using the gradient descent method, the new model can be expressed as follows: (18) We can rewrite the Euler–Lagrange equations of TV model and anisotropic diffusion model as follows: (19) (20) In discrete form, we replace the first-order derivatives by the first-order central divided differences and the second-order derivatives by forward divided differences as follows: (21) (22) (23) (24) and (25) From (16), (17) and (18) we can expect that: First, in terms of noise removal: In the areas that contains more image features (such as edges etc.), the new model will highlight the role of (16). In this case, will be large. This will result in decreasing the smoothing on the edges of the image, which in turn will lead to maintaining the significant information of the image (such as edges). In the flat areas of the image, which contain less image features, the new model will highlight the role of (17). In this case, will be small. Consequently, the diffusion will be high which will result in more smoothing on the flat areas, which in turn will lead to a more effective noise reduction. Second, in terms of edges detection: In the areas that have double edges, the new algorithm will play good role to dramatically reduce the false edges, namely the new algorithm will highlight the role of TV model (since is close to one in this case) which will lead to achieving a pleasant result with less false edges with increasing the localisation accuracy. In the areas that have weak (discontinuous) edges caused by noise, the new algorithm will play a good role to make the edges stronger (continuous), namely the new algorithm will highlight the role of PM model (since is close to zero in this case). This will contribute to improving the detection results, which in turn will lead to more robust detection results. In the proposed scheme, we exploit the advantages of the TV and the anisotropic diffusion filters and utilise an appropriate adaptive weight function to integrate the two filters. Figs. 1-5 show that the proposed scheme has achieved obvious improvements in the edge localisation. Fig. 1Open in figure viewerPowerPoint From top to bottom: Noisy image (noise variance = 0.01), image filtered with anisotropic diffusion model, image filtered with Ref. [17] model, image filtered with Ref. [26] model, image filtered with Ref. [27] model, image filtered with Ref. [28] model and image filtered with the new model Fig. 2Open in figure viewerPowerPoint From left to right and from top to bottom: Original image (Lena), image filtered with anisotropic diffusion model, image filtered with Ref. [17] model, image filtered with Ref. [26] model, image filtered with Ref. [27] model, image filtered with Ref. [28] model, and image filtered with the new model Fig. 3Open in figure viewerPowerPoint From left to right and from top to bottom: Gaussian smoothed image with standard deviation , result of anisotropic diffusion edge detection model at , result of Ref. [17] edge detection model at , result of Ref. [26] edge detection model at , result of Ref. [27] edge detection model at , result of Ref. [28] edge detection model at and result of the new edge detection model at Fig. 4Open in figure viewerPowerPoint From left to right and from top to bottom: Gaussian smoothed image with standard deviation , result of anisotropic diffusion edge detection model at , result of Ref. [17] edge detection model at , result of Ref. [26] edge detection model at , result of Ref. [27] edge detection model at , result of Ref. [28] edge detection model at and result of the new edge detection model at Fig. 5Open in figure viewerPowerPoint From left to right and from top to bottom: Original image, image filtered with anisotropic diffusion model, image filtered with Ref. [17] model, image filtered with Ref. [26] model, image filtered with Ref. [27] model, image filtered with Ref. [28] model and image filtered with the new model 5 Experimental results and analysis In our experiments, we experimented the proposed technique on different types of common images. It is worth noting that all test images in this paper have the size of . The experimental results are shown in Figs. 1-5. Figs. 1-5, respectively, illustrate, the original (or noisy) images, the result by anisotropic diffusion algorithm, the result by algorithm in [17], the result by algorithm in [26], the result by algorithm in [27], the result by algorithm in [28] and the result by the new algorithm. In this paper, peak signal-to-noise ratio (PSNR) and figure of merit (FOM) have been used as objective assessment measurements of the proposed model. In this work, PSNR is calculated as follows: (26) where N is the maximal variation in the input image data, G is the edge of the image and L is the original image. Experimental results show that the images filtered by the new model have single and robust edges. In Fig. 1 we added the salt and pepper noise with variance 0.01. While, in Figs. 2-5 the images are naturally noisy images. From Fig. 1, we can observe that applying the gradient magnitude suppressed most of the noise while well preserved the edges. While, applying the adaptive weight function successfully eliminated most false edges and achieved very high edge localisation performance. In Fig. 2, we can see that the edges of the images that are filtered with [17, 26] models almost disappear, and the images which are filtered with [27] model, [28] model and anisotropic diffusion model are suffered from false edges (looks very messy). Unlike these five models, the new model has eliminated most of the false edges and produced edges with high localisation accuracy. This means that the new model is the most capable of sensing the image edges. In Figs. 3 and 4, images first smoothed by Gaussian filter with different standard deviations and then detected the edges by different methods. From these two figures, we can see that the greater the standard deviation the more blurry the edges of the images. Accordingly, the detection results will be worse. Nevertheless, it is quite obvious that the proposed model has achieved optimal results, while the images those filtered by the new model seem to have the sharpest edges compared with other images edges. In Figs. 3 and 4, we first smooth an image with Gaussian function by convolving the image and Gaussian function , i.e. , then apply the six models operators to convolve the smoothed image . Here is the standard deviation of Gaussian function. From Figs. 1-5, we can see that the images filtered by Ref. [17] (TV) and [26] models suffer from weak and discontinuous edges, which means that many of true edges have been missed, which is clearly shown in Fig. 4. The images filtered by anisotropic diffusion model have significant double and false edges. Therefore, the images look very messy when compared with the results of the TV model, [26] model and the results of the new model. However, our proposed model has achieved satisfactory results with less false edges and high localisation accuracy. As shown in Table 1, the objective assessment FOM proposed by Pratt [29] is used to evaluate the performance of our proposed algorithm (27) where NI and NA are the numbers of the actual edges and the detected edges, respectively. is the displacement of actual edge points from detected edge and is a scaling constant. From Table 1, we can obviously notice that the FOM results of the proposed algorithm outperforms those obtained by PM, [17, 26-28] algorithms. This demonstrates the superiority of the proposed method in terms of edge detection. Table 1. Pratt's figure of merit (FOM) of Fig. 1 Detector Pratt figure of merit (FOM) PM 0.7743 [17] 0.7421 [26] 0.7750 [27] 0.5042 [28] 0.4475 new 0.8113 Table 2 compares between our proposed algorithm and the other reference algorithms in terms of the estimated running time (in second). From this table, we can see that the proposed algorithm takes less time-consuming, which means that our proposed algorithm is more economical and faster. Table 2. Computational cost of the Refs. [26-28] and the proposed algorithms Figures [26] Model [27] Model [28] Model New model Fig. 1 2.35s 2.52s 2.49s 2.33s Fig. 2 3.11s 3.62s 0.3.33s 3.03s Fig. 3 2.51s 2.97s 2.85s 2.47s Fig. 4 2.53s 3.01s 2.94s 2.49s Fig. 5 2.42s 2.87s 2.82s 2.38s Table 3 shows the PSNR results of the six algorithms. Despite a higher PSNR commonly indicates that the reconstruction of the image compression is with higher quality. Nevertheless, in some situation such as edge-detection PSNR should be with low decibel to get an effective edge of the image detected [30]. In other words, the lower PSNR, the higher quality of the edge detected. From this perspective, we can say that our proposed algorithm has achieved the best edge localisation accuracy with few false edges. Table 3. PSNR (dB) of different edge detectors with different images Images PM [17] [26] [27] [28] New Fig. 1 8.1811 8.3665 8.3167 10.8703 11.2265 8.0819 Fig. 2 6.5958 6.6206 6.6125 11.8701 7.9836 6.5705 Fig. 3 8.4055 8.4067 8.4066 15.0722 15.7152 8.4043 Fig. 4 8.4061 8.4068 8.4067 14.9645 15.5983 8.4053 Fig. 5 6.7899 6.8036 6.8006 7.2183 7.2972 6.7735 The proposed algorithm can be carried out as follows: Step 1 : Input original (or noisy) image u Step 2 : Applying the gradient magnitude strategy to reduce the noise of u image. Step 3 : Utilising the zero crossing detector to distinguish the edges areas from the flat areas. Step 4 : Calculate M by (16). Step 5 : Calculate S by (17). Step 6 : Calculate the adaptive weight function by (15) Step 7 : For each calculate and by (19) and (20), respectively. Step 8 : Calculate (18). Step 9 : Output image. 6 Conclusion In this paper, a novel model of image edge detection based on anisotropic diffusion and TV filters has been presented. This model has a high potential to acclimate and change according to areas information (edges or flats areas). In the proposed model, first, the gradient magnitude was applied to reduce the noise and utilised the zero-crossing detector to distinguish the edges areas from the flat areas. Finally, the adaptive weight function was applied to detect the edges of the image. Consequently, our proposed scheme has succeeded in removing most of the false edges and achieved a high edge localisation performance. Experimental results showed that applying the adaptive weight function has made a great improvement in terms of increasing the edge localisation accuracy. As a consequence, the false edges caused by noise have been mitigated which in turn led to better edge detection results. Our experiments have been conducted on four common images to detect the edges of these images, and four different algorithms have been utilised. 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