Ever since Zadeh established the basis of fuzzy logic in his famous article Fuzzy Sets (Zadeh, 1965), an increasing number of research areas have used his technique to solve and model problems and apply it, mainly, to control systems. This proliferation is largely due to the good results in classifying the ambiguous information that is typical of complex systems. Success in this field has been so overwhelming that it can be found in many industrial developments of the last decade: control of the Sendai train (Yasunobu & Miyamoto, 1985), control of air-conditioning systems, washing machines, auto-focus in cameras, industrial robots, etc. (Shaw, 1998) Fuzzy logic has also been applied to computerized image analysis (Bezdek & Keller & Krishnapuram & Pal, 1999) because of its particular virtues: high noise insensitivity and the ability to easily handle multidimensional information (Sutton & Bezdek & Cahoon, 1999), features that are present in most digital images analyses. In fuzzy logic, the techniques that have been most often applied to image analysis have been fuzzy clustering algorithms, ever since Bezdek proposed them in the seventies (Bezdek, 1973). This technique has evolved continuously towards correcting the problems of the initial algorithms and obtaining a better classification: techniques for a better initialization of these algorithms, and algorithms that would allow the evaluation of the solution by means of validity functions. Also, the classification mechanism was improved by modifying the membership function of the algorithm, allowing it to present an adaptative behaviour; recently, kernel functions were applied to the calculation of memberships. (Zhong & Wei & Jian, 2003) At the present moment, applications of fuzzy logic are found in nearly all Computer Sciences fields, it constitutes one of the most promising branches of Artificial Intelligence both from a theoretic and commercial point of view. A proof of this evolution is the development of intelligent systems based on fuzzy logic. This article presents several fuzzy clustering algorithms applied to medical images analysis. We also include the results of a study that uses biomedical images to illustrate the mentioned concepts and techniques.
Fuzzy logic is an extension of the traditional binary logic that allows us to achieve multi-evaluated logic by describing domains in a much more detailed manner and by classifying better through searches in a more extensive area. Fuzzy logic makes it possible to model the real world more efficiently: for example, whereas binary logic merely allows us to state that a coffee is hot or cold, fuzzy logic allows us to distinguish between all the possible temperature fluctuations: very hot, lukewarm, cold, very cold, etc.
Techniques based on fuzzy logic have proven to be very useful for dealing with the ambiguity and vagueness that are normally associated to digital images analysis. At what grey level do we fixate the thresholding? Where do we locate the edge in blurred objects? When is a grey level high, low, or average?
The fuzzy processing of digital images can be considered a totally different focus with respect to the traditional computerized vision techniques. It was not developed to solve a specific problem, but describes a new class of image processing techniques and a new methodology to develop them: fuzzy edge detectors, fuzzy geometric operators, fuzzy morphological operators, etc.
These features make fuzzy logic especially useful for the development of algorithms that improve medical images analysis, because it provides a framework for the representation of knowledge that can be used in any phase of the analysis. (Wu & Agam & Roy & Armato, 2004) (Vermandel & Betrouni & Taschner & Vasseu & Rosseau, 2007)
Key Terms in this Chapter
Medical Image: A medical specialty that uses x-rays, gamma rays, high-frequency sound waves, and magnetic fields to produce images of organs and other internal structures of the body. In diagnostic radiology the purpose is to detect and diagnose disease, whereas in interventional radiology, imaging procedures are combined with other techniques to treat certain diseases and abnormalities.
Membership Function: Gives the grade, or degree, of membership within the fuzzy set, of any element of the universe of discourse. The membership function maps the elements of the universe onto numerical values in the interval [0, 1].
Fuzzy Operator: Operations that enable us to combine fuzzy sets. A fuzzy operator combines two fuzzy sets to give a new fuzzy set. The most frequently used fuzzy operators are the following: equality, containment, complement, intersection and union.
Fuzzification: The process of decomposing a system input and/or output into one or more fuzzy sets. Many types of curves can be used, but triangular or trapezoidal shaped membership functions are the most common.
Fuzzy Inference Systems: A sequence of fuzzy conditional statements which may contain fuzzy assignment and conditional statements. The execution of such instructions is governed by the compositional rule of inference and the rule of preponderant alternative.
Fuzzy Algorithm: An ordered sequence of instructions which may contain fuzzy assignments, conditional statements, repetitive statements, and traditional operations.
Segmentation: A process that partitions a digital image into disjoint (non-overlapping) regions, using a set of features or characteristics. The output of the segmentation step is usually a set of classified elements, such as tissue regions or tissue edges.