A Total Variation-Morphological Image Edge Detection Approach

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1 A Total Variation-Morphological Image Edge Detection Approach Peter Ndajah, Hisakazu Kikuchi, Shogo Muramatsu, Masahiro Yukawa and Francis Benyah Abstract: We present image edge detection using the total variation functional and morphological methods. First, we derive the total variation functional from first principles and vector gradient method. The total variation functional is then minimized using the Euler-Lagrange optimization method. The steady state equation which results from the minimization of the total variation functional is then used as an anisotrpic filter on images. While the total variation method has proven to be a better edge detector than the Marr-Hildreth method, it also segments the image into regions delineated by strong edges. To achieve results similar to the Marr-Hildreth method, we apply morphological considerations. We develop new operations based on erosion, dilation, opening and closing to achieve morphological edge detection of total variation filtered images. Key Words: total variation, edge detection, morphological edge detection, euler lagrange, image filter 1 Introduction There are two basic philosophical schools of image edge detection. The first is the search-based approach which attempts to find image edges by employing the properties of first order differential expressions. The most prominent of these methods is the Canny edge detection method. The second school approaches the problem by using second order partial differential equations rather than the first order partial differential equations. The most popular method of this approach is the Marr-Hildreth method which attempts to find the edges in an image by searching for zerocrossings in the second order PDE filtered image. Our approach is to use the total variation functional as a starting point and then minimize it using the Euler- Lagrange method for optimizing functionals. The resulting steady state equation is nonlinear as opposed to the Laplacian used by Marr and Hildreth (1980) in their work. Total variation image edge detection [11] has proven to be better than the Marr-Hildreth method but even though the image edges are much sharper, this method introduces new characteristics into the filtered image. While the Marr-Hildreth method gives single-pixel edges after the zero crossing algorithm is applied, total variation filtered images do not respond well to the zero-crossings algorithm. The algorithm Graduate School of Science and Technology, Niigata university, Japan. Department of Electrical and Electronics Engineering, Niigata University, Japan. Department of Electrical and Electronics Engineering, Niigata University, Japan. Department of Electrical and Electronics Engineering, Niigata University, Japan. Department of Mathematics, University of the Western Cape, Cape town, South Africa. actually weakens the edges thereby making it counterproductive to use. Secondly, the image shows segmentation characteristics. By this we mean that the total variation method segments the image into regions which are delineated by strong edges. In this work, we proceed to obtain new results similar to the Marr- Hildreth single-pixel edges. To do this, we employ morphological methods. 2 Derivation of The Total Variation Functional We derive the total variation functional from first principles in one dimension and then extend it to two dimensions using vector gradient method. The two dimensional total variation functional can be applied to images since images are two dimensional mathematical objects. We define the total variation (TV ) as TV i0 f(x i1 ) f i (1) Equation (1) can be multiplied and divided by x to obtain TV f(x i1 ) f i x x (2) Equation (2) can be transformed into the continuous form if we take limits as x 0 and n i.e. lim f(x i1 ) f i x 0 x x n i0 f(xh) f(x) h dx f (x) dx (3) ISBN:

2 To find the total variation of a two-dimensional mathematical object, we consider the directional derivative of a scalar function f( x) f(x 1,x 2,...,x n ) along a unit vector u (u 1,...,u n ). The directional derivative is defined to be the limit u f( x) lim h 0 f( xh u) f( x) h (4) Let the function f to be differentiable at x. This implies that the directional derivatives exist along any unit vector u, and one has u f( x) f( x) u (5) For images we represent the directional derivatives by i, j so that { x f( x) f( x) i (6) y f( x) f( x) j The components x f( x) and y f( x) are orthogonal. So f( x) i j (7) We deduce the inner product of f( x) f( x), f( x) f 2 L 2 () ( ( ) 2 (8) Therefore, equation (3) in two dimensions can be written as f dxdy (9) which is the total variation of f over the entire image surface. 3 Digital Image Total Variation Equation (9) is a norm and the vectorial components of f are and. So, the norm f dxdy In discrete form, ( f(x1,y) f(x,y) x f(x,y 1) f(x,y) y ( ) 2 dxdy (10) In digital image processing, x is the unit distance between adjacent pixels given by i 1 i 1. So x 1. The same argument applies to y. So x y 1 Therefore, the discrete total variation is written as: f x y [f(x1,y) f(x,y)] 2 [f(x,y 1) f(x,y)] 2 x y 4 Minimization of the Total Variation Functional We restate the total variation functional as J(u) u dxdy (12) u is our image function and it depends spatially on two independent variables x and y. The total variation functional has one dependent variable (the image function u) and two indepedent variables x and y. Let I(f) F(x 1,x 2,...,x n,f,f x1,f x2,...,f xn )dx (13) where f xi i. A minimization of (13) satisifies the PDE: F F 0 (14) i xi The total variation functional can be written as J(x,y,u, u ) u dxdy (15) (15) is independent of u on the right hand side but depends on x and y because ( u u ( ) u 2 f (16) and there is no term inuon the right hand side. So the Euler equation for the total variation is: 2 u 0 (17) i i but u 0 since the functional J does not depend on u. So we have (11) ISBN:

3 2. (18) i i An image is two dimensional, therefore x 1 x and x 2 y. Which means (18) becomes 2 i i. (19) This is equivalent to differentiating u with respect toxandysince we made u f for simplicity. So differentiating with respect to x, we get Also, [ ( u [ ( u [ ( u 1 ( u 2 [ ( u 1 ( u 2 ( ) u 2 2 u 1 u u (20) ( ) u 2 2 u 1 u u (21) Using the forward difference discretization approach, the Laplacian can be discretized as follows: 2 u 2 u(x1,y)u(x 1,y) 2u(x,y) ( x (25) and 2 u u(x,y 1)u(x,y 1) 2u(x,y) 2 ( y (26) but x y 1 for digital images. Therefore adding the two equations, we get the discrete Laplacian for digital images 2 f(x,y) f(x1,y)f(x 1,y)f(x,y 1) f(x,y 1) 4f(x,y) (27) Table 1: Mask 1 which filters the image. Equation (27) can also be implemented by convolving the filter mask in Table 1 with the image. Figure 2 shows the effect of this mask on the image in Figure 1. The mask is isotropic i.e. it filters uniformly in all directions. From the foregoing, the Euler equation of the total variation functional is minj(u) J min (u) u u (22) 5 Laplacian Image Filters Marr and Hildreth[1] used the Laplacian to filter images. They tested the filtered images for zero crossings. When the Laplacian operator L (23) operates on the image u, it produces a Laplacian filtered image of u. This is achieved on digital images by first discretizing the Laplacian of u: 2 u 2 u 2 2 u 2. (24) Figure 1: Original Image with Sharp Edges 6 Total Variation Image Filter We obtained a minimization of the total variation functional by means of the Euler-Lagrange minimization method. We use the resulting nonlinear steady state partial differential equation given by ISBN:

4 laplace1shapes 7 Laplacian and TV Edge Detection We follow a similar process to the method employed by Marr and Hildreth to achieve edge detection using total variation. It consists of first convolving the image u(x,y) with a Gaussian function Figure 2: Laplacian Filtered Image Showing Edges J min u 2 u u (28) to filter images in place of the Laplacian. While the Laplacian is isotropic, the total variation filter is anisotropic because of the denominator u. This anisotropic property makes the total variation filter also texture sensitive. This can be seen in figure 3. The changes in the image texture are well captured in the filtered image. Also, the Laplacian forms double edges during image filteration process. The total variation filter overcomes this disadvantage by producing only single a edge. While the Laplacian filtered image is scalable, total variation filtered images are not. i.e. G(x,y) e x2 y 2 2σ 2 (29) D(x,y) G(x,y) u(x,y). (30) σ which represents the standard deviation acts as a scaling factor. All noise and image features below the scale σ are smoothed out in the process. For edge detection using the Laplace operator, we apply the Laplacian operator 2 to the smoothed imaged(x,y) i.e. 2 D(x,y). The resulting image is called the Laplacian of of Gaussian (LoG) image. We carry out the same operation for the total variation nonlinear operator 2 D(x,y) u. TV filter Figure 4: Edge Detection in Total variation Filtered Image with Thresholding at 0 and σ 2 8 Morphological Edge Detection Figure 3: TV Filtered Image Showing Edges Applying the method of total variation to image edge detection produces images with strong edges but it also has some segmentation characteristics. This is because the algorithm separates regions of an image by strong edges. However, further processing is possible. Our aim here is to obtain edges that are at least one pixel thick. The choice of the edge thickness will depend on the structuring element used in the morphological operation. For the case of binary edge detection we convert the grayscale image to a binary image and then apply the operations of erosion, dilation, ISBN:

5 opening and closing to obtain edge detection results with edge thickness of our choice. At this point it is necessary to point out that the Marr-Hildreth images do not respond well to basic operations of mathematical morphology. This is because the edges are only a pixel thick and an erosion of the image simply destroys practically all image features. Because of this, it is quite difficult to carry out secondary morphological operations such as opening and closing since they are derived from the operations of erosion and dilation. Figures 5 to 7 show the effect of eroding a Marr- Hildreth image. Figure 7: Eroded Binary Marr-Hildreth Image for A and B in Z 2 (the two dimensional set of integers). To obtain edges, therefore we apply the algorithm: A (A B) (32) to the binary image. (see Figure 8). Figure 5: Edge Detection Using The Marr-Hildreth Method Figure 8: Erosion of TV-Filtered Image at Scaleσ 2 The dilation of A byb is given as A B {z ( ˆB) z A } (33) Figure 6: Binary Marr-Hildreth Image The erosion of an image A by a structuring element B is defined as A B {z (B) z A} (31) where ˆB is the reflection of B. To obtain edges in the case of dilation, we carry out the operation: (A B) A (34) We also apply the operation of closing and opening to the total variation-morphological method of ISBN:

6 9 Conclusion We have shown that total variation filtered images can be subjected to morphological edge detection operations such as we have introduced in this work. This is not easily achievable with Laplacian filtered images introduced by Marr and Hildreth[1]. However, not all morphological approaches give good results. For example, the operation in (38) failed to give an appreciable result. Also, attempts to find edges using the operations in (34) and (35) required very careful choice of parameters. Figure 9: Binary Edge Detection in TV-Filtered Image at Scale σ 4 edge detection. The opening of an image set A by a structuring element B is denoted as A B (A B) B. (35) That is, we erode A by B and dilate the result by B. To obtain edges, we carry out the operation: A (A B) (36) Figure 10 represents the operation in (36). Figure 10: Binary Edge Detection in TV-Filtered Image Using Opening Operation at Scale σ 2 Similarly, the closing of A by B is denoted as A B (A B) B (37) From the closing operation, we can obtain edges by carrying out the following operation: (A B) A (38) References: [1] D. Marr and E. Hildreth, Theory of Edge Detection, Proc. R. Soc. Lond. B 207, ,, [2] Tony F. Chan and Jianchong Shen, Image Processing and Analysis: Variational, PDE, Wavelet and Stochastic Methods, SIAM [3] Peter Ndajah and Hisakazu Kikuchi, Total Variation Image Edge Detection, 10th International Conference on Signal Processing, Robotics and Automation, Feb , 2011, University of Cambridge, pp [4] Rafael Gonzalez and Richard E. Woods, Digital Image Processing, Pearson Prentice Hall, [5] Al Bovik, Handbook of Image and Video Processing, Academic Press, [6] Mark Nixon and Alberto Aguado, Feature Extraction and Image Processing, Second Edition, Academic Press, [7] J.R. Parker, Algorithms for Image Processing and Computer Vision, Wiley Computer Publishing, [8] Fritz John, Partial Differential Equations, Fourth Edition, Springer, [9] J.D. Logan, An Introduction to Nonlinear Partial Differential Equations, Second Edition, John Wiley and Sons, Inc., [10] Rosenfeld A. and Kak A.C. Digital Picture Processing, Academic Press, [11] Peter Ndajah and Hisakazu Kikuchi, Scaled Image Edge Detection Based on The Total Variation Functional, Inter.J. of Appl. Math. and Informatics, Issue 2, Vol. 5, 2011, pp ISBN: