Fast image thresholding by finding the zero(s) of the first derivative of between-class variance

  • Authors:

  • Ku Chin Lin

  • Affiliations:

  • Department of Mechanical Engineering, Kun Shan University of Technology, 949 Dar Wan Rd., Yun Kung City, Tainan County, 71003 Taiwan, R.O.C.

  • Venue:
  • Machine Vision and Applications
  • Year:
  • 2003

Quantified Score

Hi-index 0.00

Visualization

Abstract

Among many thresholding methods, Otsu's method is an attractive one due to its simplicity in computation. In Otsu's paper, the between-class variance (BCV) is defined, and the gray level at the BCV maximum determines an optimal threshold. However, Otsu's method could fail to apply in cases of images with multiple BCV peaks, where a BCV peak rather than the BCV overall maximum can be a better choice as an optimal threshold. This paper presents new thresholding methods by solving a nonlinear equation that was derived based on searching for the zero derivative of image BCV. The study of finding the BCV maximum (or peaks) is treated as solving for the root(s) of the nonlinear equation, using a numerical root finder with good convergence property. From our analytical derivation, the relationship between Otsu's method and Ridler's algorithm (Trussell's equation) can be built. The proposed methods are applicable for thresholding images with single BCV peak as well as multiple BCV peaks. One of the proposed methods is equivalent to Ridler's algorithm in the total computational cost, but it is three times faster than Otsu's method. For images with a single BCV peak, the convergence and uniqueness in searching for the peak using the bisection method are always guaranteed as long as the BCV slope is continuous. But due to the round-off error, Ridler's algorithm could prematurely converge, and the uniqueness of convergence may not be guaranteed. The adequacy of the proposed methods has been proven through extensive tests, and some examples are included for illustration.