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DIGITAL IMAGE PROCESSING AND EDGE DETECTORS

John Schmeelk

Department of Mathematical Sciences Virginia Commonwealth University Doha, Qatar

Abstract

This paper provides an introduction to three dimensional image edge detection and its relationship to partial derivatives, convolutions and wavelets. We are especially addressing the notion of edge detection because it has far reaching applications in all areas of research including medical research. A patient can be diagnosed as having an aneurysm by studying an angiogram. An angiogram is the visual view of the blood vessels whereby the edges are highlighted through the implementation of edge detectors. This process is completed through convolution, wavelets and matrix techniques. Some illustrations included will be vertical, horizontal, Sobel and wavelet edge detectors.

I. Introduction

To help motivate this paper, we provide an introduction to some interesting problems in image processing implementing matrix techniques, partial derivatives and convolutions. Section (2) provides an introduction to matrix and partial derivatives and how they are applied to the pixels to obtain the gray level value. Section (3) introduces a few specific examples such as the vertical, horizontal and Sobel edge detectors. Section (4) provides the reader with a series of illustrations that demonstrate edging techniques in three- dimensional image processing.

II. Some Notions and Notations

A monitor displaying an image may contain approximately 1024 rows and 512 columns of pixels. Of course the number continues to grow everyday as technology progresses. Then each pixel location designated by the coordinates, (x1, y1), contains a gray level value indicating the shade of gray within the image at that point. The values are usually on a scale of 0 to 255 whereby 0 corresponds to pure white and 255 correspond to black. The value of the gray level at this lattice point, (x1, y1), will be designated by f(x1, y1). However before we continue with the edge detection analysis, we first review a few matrix and calculus techniques. We first recall the familiar dot product for two vectors, 2 x, y, to be x•y= ∑ xi y i . From this dot or inner product we define the norm to be • i =1 2 = ∑ xi yi . Then we obtain the familiar and very important result to many 2 x i =1 applications that the cosine of the angle between the two vectors, x and y, satisfy the

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Schmeelk, J. (2006, June), Digital Image Processing And Edge Detectors Paper presented at 2006 Annual Conference & Exposition, Chicago, Illinois. 10.18260/1-2--1463