Elsevier

Signal Processing

Volume 75, Issue 3, June 1999, Pages 277-301
Signal Processing

A novel fuzzy entropy approach to image enhancement and thresholding

https://doi.org/10.1016/S0165-1684(98)00239-4Get rights and content

Abstract

Image processing has to deal with many ambiguous situations. Fuzzy set theory is a useful mathematical tool for handling the ambiguity or uncertainty. In order to apply the fuzzy theory, selecting the fuzzy region of membership function is a fundamental and important task. Most researchers use a predetermined window approach which has inherent problems. There are several formulas for computing the entropy of a fuzzy set. In order to overcome the weakness of the existing entropy formulas, this paper defines a new approach to fuzzy entropy and uses it to automatically select the fuzzy region of membership function so that an image is able to be transformed into fuzzy domain with maximum fuzzy entropy. The procedure for finding the optimal combination of a, b and c is implemented by a genetic algorithm. The proposed method selects the fuzzy region according to the nature of the input image, determines the fuzzy region of membership function automatically, and the post-processes are based on the fuzzy region and membership function. We have employed the newly proposed approach to perform image enhancement and thresholding, and obtained satisfactory results.

Zusammenfassung

Die Bildverarbeitung muß sich mit vielen mehrdeutigen Sachverhalten auseinandersetzen. Die Theorie der unscharfen Mengen ist ein nützliches mathematisches Werkzeug im Umgang mit der Mehrdeutigkeit oder Unsicherheit. Für die Anwendung der Fuzzy-Theorie stellt die Auswahl des Fuzzy-Bereichs der Zugehörigkeitsfunktion eine grundlegende und wichtige Aufgabe dar. Die meisten Forscher verwenden einen Ansatz mit vorausbestimmtem Fenster, der inhärente Probleme aufweist. Es gibt mehrere Formeln zur Berechnung der Entropie einer unscharfen Menge. Um die Schwächen der herkömmlichen Entropieformeln zu überwinden, definiert dieser Aufsatz einen neuen Ansatz der Fuzzy-Entropie und benutzt ihn, um den Fuzzy-Bereich der Zugehörigkeitsfunktion automatisch auszuwählen, so daß ein Bild in ein Fuzzy-Gebiet mit maximaler Fuzzy-Entropie transformiert werden kann. Das Verfahren zum Auffinden der optimalen Kombination von a, b und c wird mit einem genetischen Algorithmus implementiert. Die vorgeschlagene Methode wählt den Fuzzy-Bereich gemäß der Natur des Eingangsbildes aus, bestimmt automatisch den Fuzzy-Bereich der Zugehörigkeitsfunktion, und die Nachverarbeitung baut auf den Fuzzy-Bereich und die Zugehörigkeitsfunktion auf. Wir haben den neu vorgeschlagenen Ansatz zur Bildverbesserung und -begrenzung angewendet und haben dabei zufriedenstellende Ergebnisse erzielt.

Résumé

Le traitement des images doit faire face à de nombreuses situations ambiguës. La théorie des ensembles flous est un outil mathématique utile pour manipuler l'ambiguı̈té et l'incertitude. Afin d'appliquer la théorie floue, la sélection de la région floue de la fonction d'appartenance est une tâche fondamentale et importante. La plupart des chercheurs utilisent une approche par fenêtres prédéterminées, qui présente des problèmes inhérents. Il existe plusieurs formules pour le calcul de l'entropie d'un ensemble flou. Afin de dépasser les faiblesses des formules d'entropie existantes, cet article définit une nouvelle approche de l'entropie floue et l'utilise pour sélectionner automatiquement la région floue de la fonction d'appartenance, de sorte qu'il est possible de transformer une image en un domaine flou avec une entropie floue maximale. La procédure de recherche de la combinaison optimale de a, b et c est mise en œuvre par algorithme génétique. La méthode proposée sélectionne de région floue selon la nature de l'image d'entrée, détermine automatiquement la région floue de la fonction d'appartenance, et les post-traitements reposent sur la région floue et la fonction d'appartenance. Nous avons utilisé cette nouvelle approche pour effectuer du rehaussement et du seuillage d'images, et nous avons obtenu des résultats satisfaisants.

Introduction

In the field of image processing, people have to deal with many ambiguous situations. Ambiguity caused by projecting a 3-D object into a 2-D image or digitizing analog pictures into digital images, and the uncertainty related to boundaries and nonhomogeneous regions are very common. Some of definitions, such as edges, contrast, enhancement, etc., are fuzzy as well. Fuzzy set theory is a useful mathematical tool for handling the ambiguity or uncertainty.

However, selecting the fuzzy region of membership function is a fundamental and important task. In order to apply the fuzzy theory, most researchers use a predetermined approach. For instance, multi-level segmentation is determined by many local maxima of fuzzy entropy with a predetermined window (fuzzy region) which is shifted along the histogram 8, 9, 11, 12. Basically, the predetermined window approach has some inherent problems. First, there is no theoretical evidence to explain why the fuzzy region should be set like that. Second, when the range of intensity is large, it cannot effectively select the fuzzy region. The selection of fuzzy region should depend on the nature of the images. That is, different images with different characteristics need to use fuzzy regions with different sizes.

The purpose of this work is to automatically determine the fuzzy region based on the maximum fuzzy entropy principle and apply it to find the membership function. The search of the best combination of a, b and c of the membership function is conducted using a genetic algorithm. Then, we process the fuzzified images for enhancement and thresholding.

Section snippets

Fuzzy theory and maximum entropy principle

In this section, we give brief descriptions about the fuzzy set theory and entropy which is used as a measurement of fuzziness of a fuzzy set. A new fuzzy entropy definition used in our experiments will be introduced as well.

Genetic algorithm

Genetic algorithm (GA) maintains a set of possible solutions which are encoded as chromosomes. Like physiological reproduction, the algorithm generates next generations by crossovers and mutations. The crossover strategy involves the elitist model which significantly improves the performance on the unimodal surface problems.

For a genetic algorithm, several parameters need to be defined 2, 10. They are described below.

  • 1.

    Coding method. Coding method represents the problem parameters as a finite

Determine the parameters for S-function

In this section, we will explain how to use genetic algorithm to find a combination of (a,b,c) such that H(A,a,b,c) has the maximum value.

Experimental results

In this section, we test the performance of the proposed approach among different images. The original images are shown in panels (a) of Fig. 1(a)Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7(a). The corresponding histograms are shown in panels (b) of Fig. 1(b)–Fig. 7(b). Each image has 256 gray levels from 0 (the darkest) to 255 (the brightest). The first subsection describes the experimental results of N=3 and N=5. The second subsection describes the effect on an image by different N values.

Discussions

There are many formulas for computing the entropy of a fuzzy set as described in Section 2.2such as , , . Here we want to explain why we need to propose a new method. Notice that the membership function used in our experiments is an asymmetric S-function with three parameters (a,b,c) and our goal is to keep the entropy as large as possible. Besides, the solution of (a,b,c) should be related to the histogram distribution.

In Zadeh's formula Eq. (5), P(xi) is from the intensity distribution of an

Conclusions

An automatically selecting fuzzy region approach with a new definition of fuzzy entropy is developed. The new entropy definition overcomes the drawbacks of the existing entropy definitions. The fuzzy region is found by a genetic algorithm based on the maximum fuzzy entropy principle. The image can keep as much information as possible when the image is transformed from the intensity domain to the fuzzy domain. In most of the existing methods, the cross-over point of S-function is set as the

References (16)

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