Automatic multilevel thresholding for image segmentation using stratified sampling and Tabu Search

Abstract

Image segmentation techniques have been widely applied in many fields such as pattern recognition and feature extraction. For the primate visual attention model, the perceptual organization is an important process to automatically extract the desirable features. In this article, we propose a new method called an automatic multilevel thresholding algorithm using the stratified sampling and Tabu Search (AMTSSTS) by imitating the primate visual perceptual behaviors. In the AMTSSTS algorithm, a gray image is treated as a population with the gray values of pixels as the individuals. First, the image is evenly divided into several strata (blocks), and a sample is drawn from each stratum. Second, a Tabu Search-based optimization is applied to each sample to maximize the ratio between mean and variance for each sample. The threshold number and threshold values are preliminarily determined based on the optimized samples, and are further optimized by a deterministic method which includes a new local criterion function with property of local continuity of an image. Results of extensive simulations on Berkeley datasets indicate that AMTSSTS can obtain more effective, efficient and smooth segmentation, and can be applied to complex and real-time environments.

Appendix: Proofs for Sect. 4

Proof of Theorem 1

Proof Let $\beta =\vert t_{r-1}^*\times P_{t_{r-1}^*} -A\vert$.

According to Eqs. (6) and (7), we have,

$$\begin{aligned}&\min \vert v_r^*\times P_{v_r^*} -\frac{\sum \nolimits _{i=t_{r-1}^*}^{t_{r+1} } {i\times P_i} }{t_{r+1} -t_{r-1}^*+1}\vert =\min \\&\vert v_{r-1}^*\times P_{v_{r-1}^*} -\frac{\sum \nolimits _{i=t_{r-2}^*}^{t_r } {i\times P_i} }{t_{r}-t_{r-2}^*+1}\vert =\beta , \quad v_r^*\in [t_{r-1}^*,t_r ). \end{aligned}$$

When $v_r^*\in [t_r,t_{r+1}]$, $f(v_r^*)$ is a strictly monotone increasing function, hence $\vert v_r^*\times P_{v_r^*} -A\vert >\beta$. So, we can obtain the function

$$\begin{aligned} \vert v_r^*\times P_{v_r^*} -\frac{\sum \nolimits _{i=t_{r-1}^*}^{t_{r+1} } {i\times P_i} }{t_{r+1} -t_{r-1}^*+1}\vert ,v_r^*\in [t_{r-1}^*,t_{r+1} ], \end{aligned}$$

which can get the minimum value when $v_r^*=t_{r-1}^*$ in Eq. (7); i.e., $t_r^*=t_{r-1}^*$. The similar method for the case$A\ge t_{r-1}^*\times P_{t_{r-1}^*}$.