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Sparavigna, A. (2015). An example of image thresholding obtained by means of  Kaniadakis entropy. PHILICA.COM Article number 461.

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An example of image thresholding obtained by means of Kaniadakis entropy

Amelia Carolina Sparavignaunconfirmed user (Department of Applied Science and Technology, Politecnico di Torino)

Published in compu.philica.com

In this paper we are proposing an example of image thresholding, obtained by means of Kaniadakis entropy, in the framework of a maximum entropy principle. We discuss the role of its entropic index in determining the threshold and in driving an image transition, that is, an abrupt transitions in the appearance of the corresponding bi-level image. In the proposed example, Kaniadakis entropy works better than Tsallis entropy.

Article body

1. Introduction

In two recent papers [1,2], we have discussed the role of Tsallis entropy [3] in bi-level and multi-level thresholding, in the framework of a maximum entropy principle [4]. The Tsallis entropy depends on a dimensionless parameter, usually indicated by q, which is known as the entropic index. In [2], 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 [5], 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  [7], 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 [2], 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.

6. Conclusion

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.



1.      Sparavigna, A.C.  Tsallis entropy in bi-level and multi-level image thresholding, International Journal of Sciences, 2015, 4, 40-49.

2.      Sparavigna, A.C. Gray-level image transitions driven by Tsallis entropic index, submitted for publication, 2015.

3.      Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 1988, 52, 479-487.

4.      Gull, S.F.; Skilling, J. Maximum entropy method in image processing. Communications, Radar and Signal Processing, IEE Proceedings F, 1984, 131, 646-659

5.      Beck, C. Generalised information and entropy measures in physics, Contemporary Physics, 2009, 50, 495-510.

6.      Kaniadakis, G. Statistical mechanics in the context of special relativity, Phys. Rev. E, 2002, 66, 056125.

7.      Kaniadakis, G. Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions, Entropy, 2013, 15, 3983-4010.

8.      Sparavigna, A; Mello, A; Montrucchio, B.  Texture transitions in the liquid crystalline alkyloxybenzoic acid 6OBAC, Phase Transitions,  2006, 79,  293-303.

9.      Montrucchio, B.; Sparavigna, A.; Strigazzi, A. A new image processing method for enhancing the detection sensitivity of smooth transitions in liquid crystals. Liquid crystals, 1998, 24, 841-852.




Figure 1 -  The image shows some blood cells (Courtesy Wikimedia Commons)  in gray tones. On the right, we can see the histogram, with tones ranging from 0 to 255. The histogram allows obtaining the probability of a gray tone and, from it, the κ-entropy of the image.



 Figure 2 - Effects of a bi-level thresholding obtained  by maximizing κ-entropy, on the image of blood  cells. Pixels having a gray tone larger than the threshold become white; pixels having a lower value become black. The κ-entropy as a function of threshold τ is also given for two values of the entropic index κ.  In the lower part of the Figure, we can see  the optimal threshold as a function of  κ in the interval (-1,1). Note the symmetry of the result. Moreover, we can see that an “image transition” exists,  driven by the entropic index, at values -0.85 and +0.85. The output bi-level image has an abrupt change, with red blood cells appearing and disappearing in it.  



 Figure 3 – Here the effect of  thresholding obtained by maximizing Tsallis entropy, on the same image of Figure 2.. The value of the optimized threshold is quite constant when q is spanning interval (0,1), and no “image transition” is observed. Let us note that, in this case, we are not able to see the red blood cells, therefore the κ-entropy is giving a better result.

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This Article was published on 15th February, 2015 at 19:55:45 and has been viewed 2667 times.

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The full citation for this Article is:
Sparavigna, A. (2015). An example of image thresholding obtained by means of Kaniadakis entropy. PHILICA.COM Article number 461.

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1 Author comment added 15th February, 2015 at 20:27:17

Unfortunately, I made a typo in the kappa-composition formula.
All the entropies that you see in the abovementioned formula have as subscript kappa. These are Kaniadakis entropies and have entropic index kappa.

2 Author comment added 20th February, 2015 at 09:48:54

Since it is impossible to emend the text, I would like to invite the reader to see a new version of the paper at arxiv.
Bi-Level Image Thresholding obtained by means of Kaniadakis Entropy

3 Author comment added 24th February, 2015 at 09:03:48

A more extended version (revised formula and images, according canonical approach) on arxiv at “Shannon, Tsallis and Kaniadakis entropies in bi-level image thresholding”, link http://arxiv.org/abs/1502.06556

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