Edge Detection in Presence of Impulse Noise - Advances in Image and Graphics Technologies - page 8

Edge Detection in Presence of Impulse Noise

Yuying Shi 1 ,FengGuo 2 , Xinhua Su 3 , and Jing Xu 4

1 Department of Mathematics and Physics, North China Electric Power University,
Beijing, 102206 China
2 Department of Mathematics and Physics, North China Electric Power University,
Beijing, 102206 China
3 School of Mathematical Sciences, University of Electronic Science and Technology
of China, Chengdu, Sichuan, China
4 School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou,
China

Abstract. Edge detection in image processing is a dicult but meaning-
ful problem. In this paper, we propose a variational model with L 1 -norm
as the fidelity term based on the well-known Mumford-Shah functional.
To solve it, we devise fast numerical algorithms through applying the bi-
nary label-set method. Numerical experiments on gray-scale images are
given. By comparing with the famous Ambrosio-Tortorelli model with
L 1 -norm as the fidelity term, we demonstrate that our model and algo-
rithms show advantages in eciency and accuracy for impulse noise.

Keywords: Mumford-Shah model, binary level set method, edge detec-
tion, split Bregman method.

1 Introduction

Edge detection shows how important it is in image processing, computer vision,
and in many other fields, such as material science and physics [24,3]. Accord-
ing to the contexts of image processing, edge detection means extracting the
boundaries of some objects from a given image. A lot of methods have been
proposed for this purpose, such as gradient operator (Roberts operator, Sobel
operator, Prewitt operator) (see, e.g., [16]), second-order derivatives (LOG op-
erator, Canny operator) (see, e.g., [1]) and some new methods (using wavelets,
fuzzy algorithms etc.) (see, e.g., [7]). The capability of using gradient operator
is limited, as the accuracy of edge identification is usually deteriorated in pres-
ence of noise. Though the second-order derivative operators have advantages in
denoising and smoothing the edge, they can blur the images which do not have
noise. We also notice the recent researches have favor in using a variety of filter
banks to improve the accuracy of edge detection, and the interested readers are
referred to [4,13,12,18,22] and the references therein.
The seminal Mumford-Shah model [15] aims to simultaneously solve the prob-
lems of denoising and edge detection. The Mumford-Shah (MS) model is:

E ( u,ʓ )= μ ,

2 d x + ʽ
2

I ) 2 d x +

min
u,ʓ

ʩ\ʓ |∇

u | ( u − | ʓ | (1)

ʩ