Simplified expression and recursive algorithm of multi-threshold Tsallis entropy.

• Propose and proved simplified representation of multi-threshold Tsallis entropy and Renyi entropy. • Proved that Tsallis entropy and Renyi entropy are equivalent for multi-threshold segmentation with the same parameters. • Proposed recursive algorithm of simplified Tsallis entropy and Renyi entropy under multiple thresholds. • Compared with the traditional Tsallis entropy, the time complexity of the simplified representation is lower, and the time complexity of the recursive simplified representation is the lowest. Image segmentation is an important step in obtaining image information, and it has always been an extremely critical link in the fields of computer vision and pattern recognition. Because of its simplicity and effectiveness, threshold-based image segmentation technology has received significant attention and research by researchers. Tsallis entropy and Renyi entropy are two important global threshold selection methods in image thresholding. They not only have a special correspondence in mathematical expression, but also have a certain relationship in the field of image segmentation. Considering the number of thresholds increases, the complexity of the expression of Tsallis entropy criterion function also increases sharply, which is difficult to apply in multiple thresholds. Given this, this paper proposes a simplified expression of Tsallis entropy with multiple thresholds, which is proved by mathematical induction. Simultaneously, it is concluded that the multiple-threshold segmentation of Tsallis entropy and Renyi entropy with the same parameters is equivalent. To overcome the inconvenience of high calculation cost under multi-threshold and to improve calculation efficiency, recursive algorithms based on simplified Tsallis entropy and Renyi entropy under multi-threshold are given. The calculation costs of the traditional Tsallis entropy and Renyi entropy threshold, simplified Tsallis entropy and Renyi entropy threshold, and the recursive simplified Tsallis entropy and Renyi entropy are compared through experiments. The results have revealed that the calculation time of simplified Tsallis entropy and Renyi entropy is shorter than the traditional methods, and the calculation time of the recursive simplified Tsallis entropy and Renyi entropy is much lower than that of the simplified expression. Finally, we use the three optimization algorithms including particle swarm, differential evolution and Harris hawks algorithm to solve the optimal solutions of the six expressions in the case of multilevel thresholds. Experiments also show that recursive simplified Tsallis entropy and Renyi entropy have the lowest calculation time cost under the same parameters. [ABSTRACT FROM AUTHOR]