EDGE DETECTION USING MINMAX MEASURES

1. EDGE DETECTION USING MINMAX MEASURES SOUNDARARAJAN EZEKIEL Matthew Lang Department of Computer Science Indiana University of Pennsylvania Indiana, PA USA 15705 SEZEKIEL@IUP.EDU

2. INTRODUCTION • In this paper, we present a minmax measure-based algorithm for edge detection for images. • Edge detection is a tool that has been widely used in image processing and computer vision for a variety of reasons. • In an image, edges are areas where there is a strong change in contrast. • Typically, edges represent the boundaries of objects within an image, and therefore, determining the locations of these boundaries is important in further machine analysis of image content.

3. In most cases, edge detection requires smoothing and differentiation of the image. Smoothing results in the loss of image information, and differentiation is an ill-conditioned problem. • In this paper, we calculate the local fractal dimension to estimate the roughness of that region. • To calculate the local fractal dimension, we use the minmax measure. • We then form an image, called the slope image, from the fractal dimension. • We then apply simple threshold techniques on the slope image to extract edge information. • The results suggest that this method is more effective than traditional methods and that is has the capability to be applied to a broad range of image categories.

4. MINMAX MEASURE • we describe the basics of the minmax approach to image edge detection. • The essential difference between this approach and classical methods (Canny edge detector, mathematical morphology etc.,) lies in the way they handle irregularity. • Our study of edge detection analysis starts with the following definitions due to Vehel, Canus, and Vojak.

9. Methodology • We know that the Hausdorff dimension cannot be computed.

11. Minmax Mask: • We define a special square mask of size 9 by 9 for the purpose of calculating local fractal dimension. This mask can be expanded to various sizes. Figure 1 shows the mask who’s elements are the distance values from the center (* denotes a power of ½). This mask has an advantage during computation due to the fact that distances are fixed and do not need to be computed for every pixel.

12. Algorithm:- Minmax • The minmax algorithm for calculating local fractal dimension is as follows: • Step 1: Start with the image that has to be analyzed with the edge detection algorithm. If the image is a color image, convert it to grayscale or analyze the red, green, and blue images individually.  • Step 2: Choose the mask size and resize the original image by padding the boundaries so that the mask can be applied to the boundary pixels.

13. Step 3: Center the mask on a pixel and extract those pixels that lie at a distance d, for every d in the mask. Compute the maximum value and minimum value for those pixels extracted. Compute the slope by plotting log (max) vs. log (d). Repeat the same process for minimum values. The local fractal dimension can be calculated from the following expression: fractal dimension (fd) = topological dimension (T) + 1 – slope. • Step 4: Form the slope image(s) by computing the slope values for every pixel in the image [11]. Figure 2 shows the slope images for max and min of Lena (512 by 512) and Figure 3 shows the original image. • Step 5: Extract the edge features from the slope image(s) by applying a threshold value.

14. Slope image for Max Slope image for Min

15. EXPERIMENTAL RESULTS • We illustrate our proposed maxmin method for edge detection by applying it to a set of images. • We apply our maxmin method and create a slope image, and then we extract the edge information by applying a threshold value. • The resulting mages are the edge features of the original image. Lena Original Rice Original

16. Max Edges Min Edges

17. Max slope Min Slope Max Edges Min Edges

18. Max Edge

19. Max Edge