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Fuzzy Image Segmentation
The different theoretical components of fuzzy image processing
provide us with diverse possibilities for development of new segmentation
techniques. The following table gives a brief overview of different
fuzzy approaches to image segmentation:
Tabel 1. The most important fuzzy approaches to image segmentation
(See Tizhoosh, Fuzzy Image Processing, 1997).
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Fuzzy approach
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brief description
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Fuzzy Clustering Algorithms
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Fuzzy clustering is the oldest fuzzy approach to image segmentation.
Algorithms such as fuzzy c-means (FCM, Bezdek) and possibilistic
c-means (PCM, Krishnapuram & Keller) can be used to build
clusters (segments). The class membership of pixels can be interpreted
as similarity or compatibility with an ideal object or a certain
property.
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Fuzzy Rule-Based Approach
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If we interpret the image features as linguistic
variables, then we can use fuzzy if-then rules to segment the
image into different regions. A simple fuzzy segmentation rule
may seem as follows:
IF the pixel is dark
AND its neighbourhood is also dark AND homogeneous
THEN it belongs to the background.
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Fuzzy Integrals
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Fuzzy integrals can be used in different forms:
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Segmentation by weightening the features (fuzzy measures
represent the importance of particular features)
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Fusion of the results of different segmentation algorithms (optimal
use of individual adavatages)
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Segmentation by fusion of different sensors (e.g. multispectral
images, fuzzy measures represent the relevance/importance
of each sensor)
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Measures of Fuzziness and image information
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Mesaures of fuzziness (e.g. fuzzy entropy) and image information
(e.g. fuzzy divergence) can be also used in segmentation and thresholding
tasks (see the example below).
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Fuzzy Geometry
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Fuzzy geometrical measures such as fuzzy compactness
(A. Rosenfeld) and index of area coverage (S.K. Pal and A. Ghosh)
can be used to measure the geometrical fuzziness of different
regions of an image. The optimization of these measure (e.g. minimization
of fuzzy compactness regarding to the cross-over point of membership
function) can be applied to make fuzzy and/or crisp pixel classifications.
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Example: Thresholding by minimizaion of fuzziness
There are many (classical) thresholding techniques. In recent years,
some authors have also used the idea of image fuzziness to develop new
thresholding techniques (see the papers of Pal/Murthy 1990, Rosenfeld/Pal
1988, Huang/Wang 1995). For example, a membership function (standard S
function) is moved pixel by pixel over the existing range of gray levels
(see Fig.1). In each position, a measure of fuzziness
is calcultaed. The position with a minimum amount of fuzziness can be
regarded as a suitable threshold. Fig. 2 shows a comparison between this
method and the Otsu thresholding algorithm.
Fig. 1. The detection of minimum fuzziness as a tool for threshold selection.
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test image
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thresholded by fuzzy method
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thresholded by Otsu algorithm
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Fig. 2. Example for fuzzy thresholding by minimization
of image fuzziness.
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