A Fast Algorithm for Multilevel Thresholding

PING-SUNG LIAO, TSE-SHENG CHEN* AND PAU-CHOO CHUNG+

Department of Electrical Engineering ChengShiu Institute of Technology Kaohsiung, 833 Taiwan * Department of Engineering Science National Cheng Kung University Tainan, 701 Taiwan + Department of Electrical Engineering National Cheng Kung University Tainan, 701 Taiwan

Otsu reference proposed a criterion for maximizing the between-class variance of pixel intensity to perform picture thresholding. However, Otsu’s method for image segmentation is very time-consuming because of the inefficient formulation of the between-class variance. In this paper, a faster version of Otsu’s method is proposed for improving the efficiency of computation for the optimal thresholds of an image. First, a criterion for maximizing a modified between-class variance that is equivalent to the criterion of maximizing the usual between-class variance is proposed for image segmentation. Next, in accordance with the new criterion, a recursive algorithm is designed to efficiently find the optimal threshold. This procedure yields the same set of thresholds as the original method. In addition, the modified between-class variance can be pre-computed and stored in a look-up table. Our analysis of the new criterion clearly shows that it takes less computation to compute both the cumulative probability (zeroth order moment) and the mean (first order moment) of a class, and that determining the modified between-class variance by accessing a look-up table is quicker than that by performing mathematical arithmetic operations. For example, the experimental results of a five-level threshold selection show that our proposed method can reduce down the processing time from more than one hour by the conventional Otsu’s method to less than 107 seconds. Keywords: Otsu’s thresholding, image segmentation, picture thresholding, multilevel thresholding, recursive algorithm

1. INTRODUCTION

Thresholding is an important technique for image segmentation that tries to identify and extract a target from its background on the basis of the distribution of gray levels or texture in image objects. Most thresholding techniques are based on the statistics of the one-dimensional (1D) histogram of gray levels and on the two-dimensional (2D) co-occurrence matrix of an image. Many 1D thresholding methods have been investigated [1-9]. Locating the thresholds can be proceed in parametric or nonparametric approaches [1, 4, 13]. In parametric approaches, the gray level distribution of an object class leads to finding the thresholds. For instance, in Wang and Haralick’s study [5], Received May 5, 1999; revised August 24, 1999; accepted December 30, 1999. Communicated by Wen-Hsiang Tsai.

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PING-SUNG LIAO, TSE-SHENG CHEN AND PAU-CHOO CHUNG

the pixels of an image are first classified as either edge non-edge pixels. According to their local neighborhoods, edge pixels are then classified as being relatively dark or relatively bright. Next, one histogram is obtained for those edge pixels which are dark and another for those edge pixels which are bright. The highest peaks of these two histograms are chosen as the thresholds. Moment preserving thresholding is a parametric method which segments the image based on the condition that the thresholded image has the same moments as the original image [3]. In nonparametric approaches, the thresholds are obtained in an optimal manner according to some criteria. For instance, Otsu’s method chooses the optimal thresholds by maximizing the between-class variance with an exhaustive search [2]. In Pun’s method [7], as modified by Kapur et al. [8], the picture threshold is found by maximizing the entropy of the histogram of gray levels of the resulting classes. Other, some 1D thresholding techniques extend from bi-level threshold selection to multilevel threshold selection...