Enhanced Edge Detection Method using

International Journal of Applied Engineering Research, ISSN 0973-4562 Vol. 9 No. 20, 2014 © Research India Publications http://www.ripublication.com/ijaer.htm

Enhanced Edge Detection Method using Unconstrained Non-Linear Optimization Technique D. Surya prabha

J. Satheesh kumar*

Department of Computer Applications, Bharathiar University, Coimbatore, India (Email: [email protected])

Department of Computer Applications, Bharathiar University, Coimbatore, India (Email: [email protected])

Abstract— Edge detection is a key method in image segmentation for retrieving exact object information from an image. This paper describes an enhanced edge detection method using newton-raphson which is a derivative based non-linear optimization method. Conventionally, scalar based gradient is used to calculate the rate of change of pixel values on the gray scale image. Conversely this study use vector based color gradient directly on the RGB color image. An objective function is used over the color gradient for iteration in the newtonraphson method to extract accurate edge information. Performance of the proposed method is compared with existing methods like sobel, robert, prewitt, log, canny and susan. In addition, other derivative based non-linear methods like steepest descent and conjugate gradient methods are also compared. Receiving Operating Characteristics (ROC) curve analysis is done for performance evaluation and comparison. ROC analysis result shows that newton-raphson method is more accurate than the existing methods including other non-linear derivative methods. Keywords— Edge detection; newton-raphson; optimization; ROC curve analysis; objective evaluation

I. INTRODUCTION Edge detection is an essential phase in computer vision and machine vision as object identification from the entire image scene is much important [1,2,3]. Edges form the borders between different objects in an image. Marking these edge points are based on the factors like color intensity, texture and orientation [4]. Edges are identified by the unexpected changes in any of these factors. Though identifying these edges is a low level image processing task, the degree of accuracy of edge detection has a direct impact on the overall performance of higher level of image processing tasks like image analysis, scene understanding and object recognition [5]. Extraction of accurate edge map from an image scene is critical as it is more frequently interrupted by noise. Abundance of work has been proposed in literature for past two decades in edge detection. In most of the literature work, principle concept used for identifying the edges are based on either first order derivative or second order derivative methods. Sobel, Robert and prewitt methods are some of the earlier and familiar first order derivative methods.

Performances of these methods are poor when it is prone to noise as it has less immunity towards them. Gaussian filter a well known linear technique based on second order derivative was proposed by Marr and Hildreth to reduce the noise using zero-crossing in edge detection [6]. Canny proposed a gradient based edge detection method with the concept of Gaussian filter [7]. Canny edge method is a most commonly used method as it introduced the concept of non maximal suppression and hysteresis in edge detection. Non-linear filtering technique was introduced for edge detection through Susan in his proposed edge detection method [8]. Fuzzy rule based edge detection, fuzzy logic reasoning strategy, anisotropic diffusion, sequential contour tracing, and genetic algorithm are few recent edge detection techniques that are available in the literature [9, 10]. In spite of the availability of numerous edge detection methods, yet more algorithms are developed for edge detection. The reason for this continued exploration is that outputs of most of the edge detection techniques face problems like double edges, false edges, edge displacement, discontinuity in edges, missing true edges, noises and edge over-detection [11]. This paper attempts to overcome the abovementioned deficiencies by using the concept of unconstrained non-linear derivative based optimization technique that view edge detection as a minimization problem of objective function. Steepest descent, conjugate gradient and newton’s method are the non-linear derivative based optimization methods that perform an efficient search through iterations until an optimal solution point is reached for the edge regions. In this paper, newton raphson method has been proposed for finding the exact edge points by defining an objective function. Objective or fitness function is assigned to the vector based color gradient in an image for localization of edges. Vector based color gradient uses 2-D vector space instead of scalar space in the gradient operator calculation [12,13]. The reason for applying color gradient instead of gray-scale gradient is the higher ability of color gradient to detect edges even in the low contrast images. Evaluation and comparison of the performance of proposed edge detection method with other existing method was done using ROC curve analysis. Area under roc curve was used to measure the performance accuracy of our proposed method over existing

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methods. It was noted that the output of the derivative methods were better than existing methods, especially the output of newton raphson method performed better than other non-linear derivative methods. II. UNCONSTRAINED NON-LINEAR OPTIMIZATION METHODS Unconstrained non-linear programming is a model that works on continuous decision variable similar to that of linear programming but have non-linear constraints and non-linear objective functions. Objective function can be of either single variable or multi variable function. Unconstrained non-linear methods are found to be advantageous than the linear methods as it has more probability to have finite optimal solution. Another favorable point is that most of the non-linear methods are constraint free. If few simple constraints are found in the problem, it can also be neglected to attain a feasible solution. Standard non-linear optimization methods are golden search method, quadratic approximation method, nelder mead method, steepest descent method, conjugate gradient methods and newton’s method. Golden search and Quadratic approximation method is a one-dimensional search method for minimization of a function over a single variable. Golden search method uses golden section search, a unimodal objective function that rapidly constricts the interval value assured to have an optimum [14]. Quadratic approximation is a three point bracketing pattern for unimodal objective function containing three decision variables with objective value at its minimum or maximum to reach the optimum. Nelder mead method is a non-derivative based search method to minimize objective function over multi variable value [15]. Steepest descent method, conjugate gradient methods and newton raphson method are derivative based search method that accelerates the search progress substantially to search for local minima [16]. Steepest descent method is used to search with the step-size to find the minimization of an objective function for ‘N’ variables. Step-size is the number of iteration determined based on the distance to be moved in the gradient vector. It searches in the direction of greatest decrease (negative gradient vector) to reach the local minimum point. Drawback of this method is the zig-zag approach used to reach the minimum point. Newton raphson method finds the minimization of ‘N’ variables with the objective function using the gradient vector. This method uses both the first degree and second degree polynomials of Taylor’s approximation for higher efficiency [17]. Conjugate gradient method is similar to steepest descent and newton raphson method as it also uses the gradient vector to find the minimum value of an objective function. Drawback of this method is the requirement of more computing effort to reach the minimum point. In this paper, Newton raphson method was used to find the better edge points in an image as convergence rate is fast in this method when compared with other derivative based non-linear methods. Implementation of this method for edge detection techniques has solved the problem of finding true edges. This method is also capable to distinguish between local variations in noise and image structures.

III. MATERIALS AND METHODS In the proposed method, edges are viewed as closed boundary without any discontinuity and misinterpretation of edge information. Long, continuous and thin edges with the capability to segregate dissimilar regions in an image are major advantage of this proposed method. The steps involved in the proposed edge detection method are discussed in detail in this section and it’s summarized in the algorithm given below; Step1: Select an input image. Step2: Compute vector based color gradient directly on the RGB image. Step3: Calculate standard deviation value from the gradient image to fix a threshold value for edge localization. Step4: Initialize starting pixel point in the gradient image. Step5: Selection of pixels based on threshold condition. If threshold value satisfied proceed further else consider next pixel point. Step6: Derive a hypothetical objective function. Step7: Iterate the objective function with the pixel value selected in step 5 to reach local optimum point. Step8: Repeat step 5 to 7 for all pixel values. Step9: Suppress non-edge pixels using kernel operation. Step10: An improved edge detection output. A. Vector Based Color gradient calculation Color gradient method takes an advantage over the scalar gradient method as it works efficiently in noise affected image and low contrast image. Vector based color gradient is different from the scalar space gradient method as it applies the concept of vector space instead of scalar space [18]. The vector space based gradient is implemented through various ways. In the scalar function, gradient is maximum rate of change of the function at the pixel co ordinate points in the image [19]. This concept is applied over unit vectors of individual color components of RGB color space. An image I(m,n) is considered as an vector function comprising of three color component axis A(m,n), B(m,n) and C(m,n) and m,n are considered as the pixel points for the three components [20]. The horizontal and vertical direction gradients(x, y) are defined using the formula, (1)

(2) The dot product for these vectors are calculated from equation 1 & 2 as mentioned below,

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(3)


(4) (10) The second derivative or Hessian matrix is defined as (5) The maximum rate of change of the function I(m,n) is given by, (6) The rate of change of I(m,n) in the direction of θ(m,n) is calculated using, (7)

(11) Finally from the above equations, the newton’s method is defined as

As a final step, the magnitude of I(m,n) is calculated using, (12) (8) The final value ‘V’ is the vector based color gradient image for the given input RGB image. B. Finding edge points using Newton - Raphson method Newton method is an extension of the quadratic approximation method to process functions with multiple independent variables. If an appropriate objective function is chosen, then convergence of the objective function to an optimal solution is definite. Unlike steepest descent method, this method uses the second-order Taylor approximation along with the first-order Taylor approximation [21, 22]. First order approximation of Taylor polynomial is linear in approach over the directional components of gradient whereas second order approximation is non-linear in approach. First derivatives or gradients give detailed information about the changes of objective function near a current solution. It explains the slope or rate of change in the function with small variations in objective function. Second derivatives or Hessian matrix provide information about the function’s nature in the neighborhood of a current solution. Second derivatives are used for multi variable objective optimization to find the optimum of the gradient function. The Taylor polynomial of degree two to minimize the function,f(a) is defined as (9) where ‘i’ is the iteration value, f(a) is an objective function with respect to the independent variable ‘a’ i.e., the intensity value at each pixel co-ordinate points in the image and ‘ gi’ is the gradient vector and ‘hi’ is the Hessian matrix. The first derivative or gradient is defined as,

Intensity value of the pixels ‘(a i)’ of the vector based color gradient was considered for input to the objective function ‘f(a)’ to determine the optimal edge region [23, 24]. The objective or fitness function for pixel value (a) manipulation is defined in this proposed work as follows: (13) This equation is iterated at each pixel position until the final pixel position was reached. Thresholding technique was used to select the pixels for optimization. Standard deviation value of the gradient image was calculated to fix the threshold value. Pixel values that satisfy the threshold value was iterated through the fitness function until convergence is reached and other pixels are not considered for iteration. In such a manner, pixels related to edge region are taken one at a time and processed through newton-raphson iteration. Resultant image after applying this equation form the entire edge map. As the last step, kernel was used to manipulate over neighborhood pixel region to suppress the unwanted edge regions that are marked. The edge output achieved is long and continuous and its accuracy is better when compared with other derivative based optimization methods and existing methods for edge detection. IV. RESULTS AND DISCUSSION The main purpose of this proposed method is to fit better segmentation. This method is developed not only for specific application or for specific image group but it is developed for all types of images such as synthetic images and real time images. To prove the efficiency and effectiveness of the proposed method over the existing methods, the proposed algorithm is applied over different set of images and images obtained from Berkley Segmentation Dataset (BSD) [25]. There is not much standard performance index to evaluate the performance of the edge detection methods. In most of the existing works, subjective evaluation by human subjects is done to compare the performance of different edge methods.

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Subjective evaluation is not preferred as a standard index for performance as its result varies from person to person and is biased. In this paper, performance evaluation and comparison of the proposed method is done using objective method of evaluation where output of edge detection methods are compared with golden standard or ground truth image. Objective method of evaluation is chosen for image set which consist of ground truth image. Receiver Operating Characteristics (ROC) curve, a classical method and Area under ROC (AUC) curve, a similarity method of supervised evaluation method is used for evaluating the performance of edge methods and other derivative based non-linear methods.

Fig. 2. a) Input image, Output Image of (b) Sobel, (c) Robert, (c) Prewitt, (d) LoG, (e) Canny, (f) Susan, (g) proposed method using newton raphson, (h) conjugate, (i) steepest descent method.

Fig. 1.(a) Input image, Output Image of (b) Sobel, (c) Robert, (c) Prewitt, (d) LoG, (e) Canny, (f) Susan, (g) Proposed method using newton raphson, (h) conjugate, (i) Steepest descent method.

ROC curve is used to determine the best edge detection methods over other existing methods. It is formed by analyzing edge and non-edge regions between the output of edge method and the ground truth image [26]. Edges that are correctly identified between two images are referred as true positives and edges that are incorrectly identified as edges are referred as false positives. Non edges that are correctly identified between two images are referred as true negatives and non-edges that are incorrectly identified are referred as false negatives. ROC curve is calculated by using the measures, sensitivity and fall out or 1-specificity. Sensitivity is true positive rate and is defined as

(14) and 1-Specificity is false positive rate and is defined as

(15)

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had a higher peak towards left-top region followed by steepest descent and conjugate gradient methods as in Fig. 3. The output of derivative based non-linear methods also had a better performance than the existing methods. Highest AUC value was noted in the newton’s method for better edge detection as in Fig. 4. The second best performance was noted in steepest descent method and third best was attained by conjugate gradient method.

Fig. 3. ROC curve analysis for evaluating proposed method with existing edge methods and other derivative based non-linear optimization methods.

Area under ROC curve (AUC) is a popular analysis to measure the performance of an edge method. Its value ranges between 0 to 1 [27]. Higher measurement value of AUC indicates good performance of the edge method and lower measurement value of AUC value indicates poorer performance. It is defined as

V. CONCLUSION This paper proposes a novel edge detection method using a derivative based non-linear newton raphson method. Vector based color gradient was used instead of gradient from grayscale image to get accurate information directly from RGB images. These gradient value influenced by newton-raphson method was capable to produce an appropriate edge map. Other derivative based non-linear methods such as steepest descent and conjugate gradient method were also experimented for analysis in this paper. Edges are continuous, clear and definite in the derivative based non-linear optimization methods. Performance of the proposed method was evaluated and compared using ROC analysis. It was noted that the output of derivative based methods show a higher performance when compared with other existing methods. Newton-raphson method have higher accuracy rate than all other derivative based methods.

References (16) [1]

where ‘a’ and ‘b’ are the minimum and maximum axis points in the curve with ‘f(x)’ a function partly above and below the curve. In simple words, AUC is the difference between the area above the curve and area below the curve.

Fig. 4. Performance analysis of proposed edge method with other existing methods and derivative based non-linear optimization methods.

The result of proposed method has a better performance when compared with other existing methods and other derivative based non-linear methods as in Fig 1 & Fig. 2. Existing methods like sobel, Robert, prewitt, log, canny and susan was used for comparison. ROC curve of the newton’s

D. Surya Prabha, and J. Satheesh Kumar, “Assessment of banana fruit maturity by image processing technique”. J.Food Sci. Tech., DOI: 10.1007/s13197-013-1188-3, 2013. [2] D. Surya Prabha, and J. Satheesh Kumar, “Three dimensional object detection and classification methods: a study”. Int. J. Engg. Res. Sci. & Tech., Vol. 2(1), pp. 33-42, 2013. [3] D. Surya Prabha, and J. Satheesh Kumar, “Study on Banana Leaf Disease Identification Using Image Processing Methods”, Intern. J. Res. Comp. Sci. & Inform. Technol., vol 21, pp. 89-94, 2014. [4] D. Surya Prabha, and J. Satheesh Kumar, “Image processing methods and its role in agricultural sector – A study”. Int. J. Business Intelligents., Vol. 3(1), pp. 366-373, 2014. [5] D.R. Martin, C.C. Fowlkes, and J.Malik, “Learning to detect natural image boundaries using local brightness, color, and texture cues,” IEEE Trans. Patt. Anal. Machine Intel., vol.26 (1), pp. 1-20, November 1986. [6] E. C. Marr, Hildreth, Theory of edge detection. Proc. Roy. Soc. London b207, 187–217, 1980. [7] J. Canny, “A computational approach to edge detection,” IEEE Trans. Patt. Anal. Machine Intel., vol.6, pp. 679-698, November 1986. [8] S.M. Smith, and J.M. Brady, SUSAN – A new approach to low level image processing. Internat. J. Comput. Vision 23 (1), 45–78, 1997. [9] M.N. Ahmed, S.M. Yamany, N. Mohamed, A.A. Farag, and T. Moriarty, “A modified fuzzy c-means algorithm for bias field estimation and segmentation of MRI data”, IEEE Trans. Medical Imaging, vol. 21, pp. 193-199, 2002 [10] Y.W. Lim, and S.U. Lee, “On the color image segmentation algorithm based on the thresholding and the fuzzy c-means techniques”, Pattern Recog., vol. 23, pp. 935-952, 1990.

[Page No. 4701]

International Journal of Applied Engineering Research, ISSN 0973-4562 Vol. 9 No. 20, 2014 © Research India Publications http://www.ripublication.com/ijaer.htm [11] H.D. Cheng, X.H. Jiang, and J. Wang, “Color image segmentation based on homogram thresholding and region merging”, Pattern Recog., vol. 35, pp. 373-393, 2002. [12] G. Dong, and M. Xie, “Color clustering and learning for image segmentation based on neural networks”, IEEE Trans. Neural Networks, vol 16, pp. 925-936, 2005. [13] N. Pal, R. Sankar, and K. Pal, “A review on image segmentation techniques”, Pattern Recog., vol. 26, pp.1277-1294, 1993. [14] J. Shi, and J. Malik, “Normalized cuts and image segmentation”, IEEE Trans. Pattern Anal. Machine Intel., vol 22, pp. 888-905, 2000. [15] D. Surya Prabha, and J. Satheesh Kumar, “Improved edge detection method using non-linear constrained optimization,” in Data Science and Engineering, S. Arumugaperumal Eds. Elsevier 2014, pp. 195-204. [16] Q. Ji, and R.M. Haralick, “Efficient facet edge detection and quantitative performance evaluation,” Pattern Recogn., vol. 35, pp. 689 -700, 2002. [17] R.L. Rardin, Optimization in Operations Research, 3rd ed, Pearson Education(Singapore) Pvt. Ltd: India, 2005. [18] J.H. Mathews, and K.D. Fink, Numerical methods using MATLAB, 4 th ed, Prentice-Hall Pvt. Ltd: India, 2005. [19] D. Surya Prabha, and J. Satheesh Kumar, “Hybrid Segmentation of Peel Abnormalities in Banana Fruit”. In: IJCA Proceedings of the International Conference on Research Trends in Computer Technologies, Coimbatore, Tamil Nadu, India, 30-31, January 2013, pp. 38-42.

[20] S. Zhu, K.N. Plataniotis, and A. N. Venetsanopoulos, “Comprehensive analysis of edge detection in color image processing,” Opt. Eng., vol.38 (4), pp. 612-625, April 1999. [21] R.C. Gonzalez, R.E. Woods, and S.L. Eddins, Digital Image Processing Using MATLAB, 2 nd ed, Tata McGraw Hill Education Pvt. Ltd: India, 2010. [22] W.Y.Yang, W. Cao, T.Chung and J. Morris, Applied numerical methods using MATLAB, Wiley India (P) Ltd: India, 2007. [23] C. Kang, and W. Wang, “A novel edge detection method based on the maximizing objective function,” Pattern Recognition, vol. 40, pp. 609 – 618, 2007. [24] P.N. Happ, R.Q. Feitosa, and A. Street, “Assessment of optimization methods for automatic tuning of segmentation parameters,” In: Proceedings of the 4th GEOBIA, Rio de Janeiro, Brazil, 7-9 May 2012, pp. 490-495. [25] P Arbelaez, M. Maire, C. Fowlkes, and J. Malik, “Contour Detection and Hierarchical Image Segmentation”, IEEE Trans. Patt. Anal. Machine Intel., Vol. 33, pp. 898-916, May 2011. [26] M. Heath, S. Sarkar, T. Sanocki, and K. Bowyery, “Comparison of edge detectors - a methodology and initial study,” Com. Vision. Image. Unders., Vol. 69, pp. 38-54, January 1998. [27] Y. Yitzhaky,,and E. Peli, “A method for objective edge detection evaluation and detector parameter selection.” IEEE Trans. Patt. Anal. Machine Intel., vol.25, pp. 1-7, October 2003.

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