In two recent papers [1,2], we have discussed the role of Tsallis entropy  in bi-level and multi-level thresholding, in the framework of a maximum entropy principle . The Tsallis entropy depends on a dimensionless parameter, usually indicated by q, which is known as the entropic index. In , we have shown that, in some cases, when q is spanning interval the (0,1) we can observe an “image transition”, that is, an abrupt change in the appearance of bi-level or multi-level output images. It means that we can adjust the entropic index to have a result more suitable for the desired image thresholding and segmentation.
Actually, several other formulations of entropy are available , besides the above-mentioned Tsallis one. It is therefore natural to investigate and compare the results obtained using these different entropic forms in image processing and segmentation. Here we discuss the Kaniadakis entropy, also known as κ-entropy, for a bi-level thresholding. We will find that its entropic index can produce an image transition too. Moreover, for the proposed example, we will see that Kaniadakis entropy gives an output image which is better than that obtained using Tsallis entropy.
2. The κ-entropy and images
As we have seen in [1,2], the Tsallis entropic index q is fundamental for the image segmentation, because it is driving the features of the output images. In the κ-entropy we have a dimensionless parameter too, the κ-index. For κ=0, the Kaniadakis entropy, which is a deformed Shannon entropy, reduces to the original Shannon form [5,6].
Let us define this entropy in the framework of image processing.
Let us consider an image having g gray levels. The image is a matrix of pixels, each having a gray tone, and tones have values 0,1,2,…g. Usually, g=255. To evaluate the entropy, we need a distribution of probabilities, p1,p2,…,pg for the gray levels. This distribution is given by the normalized histogram hi=ni/ntot, where ni is the number of pixels with gray value i and ntot is the number of all pixels in the image (in Figure 1, we show an example of histogram).
The κ-entropy of the image is:
The entropy is a function of entropic index κ.
3. Bi-level thresholding
Let us assume a bi-level threshold τ for the gray levels. According to this threshold [1,2], we have two classes, A and B, and their probability distributions:
The κ-entropies, one for each distribution, are given by:
If we use the κ-composition law , we can define the total entropy as:
When this entropy, which is a function of threshold τ, is maximized, the corresponding gray level τ is considered the optimum threshold value. The resulting bi-level image is a black and white image, created in the following manner: if pixels have a gray tone larger than the threshold, they become white. If pixels have a lower value, they become black.
4. An image transition
As we have observed in discussing the use of Tsallis entropy , the appearance of the bi-level output image is depending on the given threshold. And this threshold is depending on the entropic index. Here again, using the Kaniadakis entropy, we have that the output bi-level image depends on its κ-entropic index.
Let us apply the Kaniadakis entropy to the thresholding of the image in Figure 1. The bi-level thresholding is given in the Figure 2. In the same Figure we can also see the κ-entropy as a function of threshold τ, for two values of the entropic index. Where the entropy has a maximum, the corresponding value of τ is considered the best for thresholding. As it is evident from the images, the final result depends on κ.
In the lower part of the Figure 2, we have the behavior of the threshold as a function of the entropic index. This function has an jump at +0.85 (and also -0.85). To this jump, it corresponds an analogous abrupt change in the appearance of the corresponding bi-level image and in the information provided by it. For a positive entropic index above 0.85, we have the possibility to see the red cells of blood, which are not visible when the positive entropic index is below this value.
Due to this abrupt change of the output image, we can tell we are observing an “image transition”. This transition resembles the texture transitions observed in polarized light microscopy of liquid crystals [8,9]. In these transitions, the texture of the liquid crystal changes abruptly its appearance, because the material, driven by the temperature, had changed its free energy. In image transitions, entropy is playing the role of free energy and its entropic index the role of temperature.
In this paper we have shown an example of the use of Kaniadakis entropy in determining a bi-level thresholding. We see that, for the image of the given example, the value of threshold has two jumps, when the entropic index is spanning interval (-1,1). When the threshold jumps, we observe an abrupt transition in the appearance of the corresponding output image. We can define this behavior as an “image transition”. For the proposed example, the image transition is not observed if Tsallis entropy is used. Moreover, the result we obtain using Kaniadakis entropy is better than that obtained with Tsallis entropy.
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