Parallel to the notion of non-parametric use of the shape of the histogram, a new notion of the entropy of the gray level histogram has been introduced by Pun [43] to the research domain of histogram based thresholding. Thresholds have been determined by an apriori maximization of entropy determined aposteriori. Successively, Pun [44] also suggested an automatic threshold selection related to the asymmetry of the gray level histogram that facilitated the derivation of entropy based thresholding. Pun [44] in his work has also advocated the use of this method for multi-thresholding applications. The fundamental notion of Pun [43, 44] has been analysed by
Kapur et al. [45] and a new algorithm based on entropy has been proposed for real and artifi-cially generated histograms. The entropy based notion has been exploited by many researchers and in this regard Pal et al. [46] has presented a new definition of entropy which could be viewed as modification of Shannon’s entropy but suitable for thresholding. This proved to be effective in many cases. The concept of entropy was further extended by Li et al. [47], Brink et al. [48], and Pal [49], where the optimal threshold has been selected while minimizing the cross entropy between the image and its segmented versions. The first cross entropy based thresholding has been introduced by Li et al. [47] and the proposed method provides an unbiased estimate of binarized version of the image in an information theoretic sense. Li et al.’s [47] cross entropy method has been analysed by Pal [49] and towards this end a new cross entropy based method is presented overcoming the limitation of Li et al.’s [47] method. Besides entropy and cross entropy, relative entropy based thresholding has been proposed by Chang et al. [50], where the entropy of the co-occurrence matrix of one image has been used. Extension of this work is carried out by Althouse [51], where local entropy and local relative entropy thresholding meth-ods have been described and compared with Otsu’s [5] and Kittler’s [35] method. An iterative method for cross entropy based thresholding has been proposed by Li et al. [52] and could successfully be tested for many real images further extension of this entropy based thresholding has been carried out by Sahoo et al. [53] and this thresholding is based on Renyi’s entropy.
By and large, the definition of image entropy has been associated with the probability distribution of the gray levels. This entropy measured has been modified by Brink [54], where the spatial information of image has been incorporated into the entropy measure to devise the criterion function that improved the result substantially. Jinsong et al. [55] have implemented the methods proposed by Kapur et al. [45] and Sahoo et al. [53] using Genetic algorithm.
Both single and multi-thresholding methods have been dealt using Genetic algorithm. All the entropy based methods described above are more or less based on Shannon’s entropy. Pavesic et al. [56] have devised thresholding criterion based on the sum of Havrda and Charvat entropy and have shown that this entropy based scheme results in better segmentation than that of using Shannon’s entropy. The computational burden of maximum entropy based thresholding has been reduced using Q-learning algorithm in the Reinforced Learning (RL) paradigm proposed by Yin [57]. In Yin’s method [57], it has also been shown that the algorithm is suitable for multilevel thresholding applications.
Besides, a thresholding algorithm using Tsallis entropy has also been proposed by
Albu-querque et al. [58] and local entropy based method for extraction of transition region has been proposed by Yan et al. [59]. In the sequel, local entropy based algorithm for blood vessel de-tection has been devised by Chanwimaluang et al. [60] and the method produced promising results in case of many examples. Relative entropy based thresholding algorithm has also been proposed by Zhu et al. [61], where two dimensional histogram instead of single dimensional histogram of the image has been used to obtain optimal threshold. Yang et al. [62] have pro-posed a fast threshold selecting algorithm based on one-dimensional entropy. Recently, texture Renyi entropy has been proposed by Shareha et al. [63] for determining accurate threshold and minimum cross entropy based thresholding [64] has been proposed for thresholding SAR images. A non-extensive relative entropy also known as Tsallis entropy has been employed to develop optimal threshold detection strategy [65] and the Tsallis entropy has been applied as a generalized entropy formalism for information theory. Entropy based thresholding algorithms have also been validated for biomedical images specifically ultrasound images [66].
It has been found out that the spatial correlation among the pixels do influence the un-derlying notion of separation of object from the background. In order to take into account the spatial correlation of the pixels together with the gray level distribution of images, two dimen-sional entropy based thresholding method was first introduced by Abutaleb [67]. The proposed 2-D entropy based approach produced appreciable result even when signal to noise ratio (SNR) is decreased. Subsequently, Chen et al. [68] have suggested a fast two-dimensional entropy based thresholding algorithm to reduce the computational burden. It has been shown that the processing time reduced drastically. Besides, a wavelet transform based fast 2D entropic thresh-olding algorithm also been proposed by Wang et al. [69]. Specifically for ultrasound images, a two-dimensional minimum cross entropy based algorithm has been developed by Zimmer et al. [70] and the algorithm could successfully be tested for ovarian cysts. The two-dimensional entropy based algorithm has also been extended [71] further to incorporate Tsallis-Havrda-Charvat entropy while devising optimal threshold algorithm. Recently, thresholding strategy has further been reinforced using 2D Tsallis entropy [72] and the resulting algorithm produced better segmentation result than the previously proposed two dimensional thresholding methods.
Recently, Tian et al. [73] has proposed a Tsallis-entropy image thresholding method using two-dimension histogram obque segmentation. The superiority of this method has also been shown to other methods.