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Fuzzy Image Processing: Potentials and State of the Art
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Fuzzy Image Processing:
Potentials and State of the Art
Hamid R. Tizhoosh
University of Magdeburg, Faculty of Electrical Engineering (IPE)
P.O. Box 4120, D-39016 Magdeburg, Germany
In recent years, many researchers have applied the fuzzy
logic to develop new image processing algorithms.
Meanwhile, the fuzzy image processing is one of the
important application areas of fuzzy logic. This paper
gives a brief overview of potentials of fuzzy techniques.
The state of the art is described by referring to some
practical examples.
well-known methodologies is that fuzzy techniques
operate on membership values. Therefore, image fuzzification (generation of suitable membership values) is
always the first processing step. Topological relationships
such as connectedness, surroundedness and adjacency
were defined for binary images. The extension of digital
topology to fuzzy sets [40] builds the necessary framework for many fuzzy local operations.
1 Introduction
3 Structure of fuzzy image processing
There are many reasons why fuzzy logic should be investigated regarding to its applicability in image processing.
The most important reason is that fuzzy logic provides us
with a mathematical framework for representation and
processing of expert knowledge. Here, the concept of
linguistic variables and the fuzzy if-then rules play a key
role [54]. The second reason is that the uncertainties
within image processing tasks are not always due to
randomness but often to vagueness and ambiguity. Fuzzy
techniques enable us to manage these problems efficiently
[21,48].
Fuzzy image processing consists generally of three steps
(Fig. 1): fuzzification (image coding), operations in the
membership plane, and finally, defuzzification (decoding
of results).
2 Fuzzy image understanding
Fig. 1. Structure of fuzzy image processing systems [48]
Abstract
To apply the idea of fuzzy sets [52] to image processing
problems, one should develop a new image understanding.
We need a new image definition, some ways to fuzzify the
images and their features, and finally, an extension of
digital topology which plays a pivotal role in image
representation and local operations, respectively.
An image X of size M×N with L gray levels g = 0, 1, 2, ...,
L-1 can be defined as an array of fuzzy singletons (a fuzzy
set with only one supporting point) indicating the membership values µmn of each image point xmn regarding to a
predefined image property (e.g. brightness, homogeneity,
noisiness, edginess etc.) [33,48]:
M
X
N
Pmn
x
m 1n 1
(1)
Fuzzification does mean that we assign the image (its gray
levels, features, segments, ...) with one or more membership values regarding to the interesting properties (e.g.
brightness, edginess, homogeneity, ...). Generally, there
are three ways for image fuzzification [48]: histogrambased gray-level fuzzification, local fuzzification, and
feature fuzzification. After transformation of image into
the membership plane (fuzzification), a suitable fuzzy
approach aggregates and/or modifies the membership
values. To achieve new results (modified gray-levels,
segmented image regions, classified objects etc.), the
output of membership plane should be decoded (defuzzification). It means that the membership values are retransformed into the gray-level plane.
mn
The definition of the membership values µmn depends on
the specific requirements of actual application and the
corresponding expert knowledge.
Fuzzy image processing is a kind of nonlinear and knowledge-based image processing. The difference to other
4 Theory of fuzzy image processing
Fuzzy image processing is knowledge-based and nonlinear. It is based on fuzzy logic and uses its logical, settheoretical, relational and epistemic aspects. The most
important theoretical frameworks that can be used to
Published in: 5th International Conference on Soft Computing, Iizuka, Japan, October 16-20, 1998, vol. 1, pp. 321-324
construct the foundations of fuzzy image processing are:
fuzzy geometry, measures of fuzziness/image information,
rule-based approach, fuzzy clustering algorithms, fuzzy
mathematical morphology, fuzzy measure theory, and
fuzzy grammars. Any of these topics can be used either to
develop new techniques, or to extend the existing algorithms. Following, a brief description of each field is
given. Here, the soft computing techniques (e.g. neural
fuzzy, fuzzy genetic etc.) are not mentioned because of
space limitations.
4.1 Fuzzy geometry
The geometrical relationships between the image components play a key role in the intermediate image processing.
Many geometrical categories such as area, perimeter,
diameter etc. are already extended to fuzzy sets
[34,35,41]. The geometrical fuzziness arising during
segmentation tasks can be handled efficiently if we
consider the image or its segments as fuzzy sets. The main
application areas of fuzzy geometry are feature extraction
(e.g. in image enhancement) [35,3], image segmentation
[42,48] and image representation [31].
4.2 Measures of fuzziness/image information
Measures such as index of fuzziness, fuzzy entropy, fuzzy
divergence, fuzzy correlation etc. can be used to quantify
the image information when the image is defined as a
fuzzy set [29,32,38,48]. The main application area of
measures of fuzziness is feature extraction (e.g. in image
enhancement [48,49,50,51] and image thresholding
[2,6,16,36,37]).
4.3 Rule-based systems
The fuzzy if-then rules have become increasingly important regarding to their applicability in image processing.
They can be applied to different problems in image
processing such as image enhancement, segmentation and
edge detection [17,27,43,48]. Fuzzy inference systems are
also a powerful tool for the extension and improvement of
the performance of classical algorithms because they offer
an excellent framework for representation and processing
of expert knowledge.
4.4 Fuzzy/possibilistic clustering
Fuzzy clustering is the oldest fuzzy approach to pattern
recognition and image segmentation, respectively. Fuzzy c
means and its extensions, noise clustering, possibilistic
clustering, and mixed clustering are the most important
algorithms for data classification and image segmentation
[1,6,7,8,20,22,24,30,48]. Actually, fuzzy and possibilistic
clustering algorithms are the most frequently applied
fuzzy approach in image processing.
4.5 Fuzzy mathematical morphology
Mathematical morphology is one of the important areas in
image processing. The basic image transformations
dilation and erosion and their different combinations can
be applied to developed new algorithms for low level/
intermediate level image processing. The main application
areas of mathematical morphology are: image description/representation (thinning, thickening and skeletonization etc.). The extension of mathematical morphology to
fuzzy sets is investigated by some researchers
[4,9,10,12,28,44,48]. Also some practical results are
available [5,25].
4.6 Fuzzy measure theory
Fuzzy measures can be considered as a generalization of
classical (probability) measures [45]. Fuzzy integrals are
nonlinear aggregation operators for combination of
different sources of uncertain information [18,47]. Fuzzy
integrals can be applied to pattern classification and image
segmentation [6,13,14,19,48,55]. Generally, a fusion of
images, features, algorithms is carried out by fuzzy
integrals. The corresponding fuzzy measure denotes the
importance of each information source.
4.7 Fuzzy grammars
In some applications, the patterns can be described using
their structural information rather than numerical values
(e.g. in OCR systems). In such situations, it can be more
appropriate to apply the formal language theory [11].
However, the real patterns are often ill-defined. This
problem is usually due to the inherent pattern fuzziness.
The fuzzy languages and fuzzy grammars can be used to
overcome these difficulties [26,33,39,48,53].
4.8 Extension of classical algorithms
The aforesaid topics (4.1-4.7) can also be used to extend
the existing image processing algorithms and improve
their performance. Some examples are: fuzzy Hough
transform [15], fuzzy mean filtering [23] and fuzzy
median filtering [46].
5 State of the art
Among all publications on fuzzy approaches to image
processing, fuzzy clustering algorithms and rule-based
approaches have the greatest share. Measures of fuzziness
and fuzzy geometrical measures are usually used as
features within the selected algorithms. Fuzzy measures
and fuzzy integrals seem to become more and more an
interesting subject of research. The theoretical research on
fuzzy mathematical morphology seems still to be more
important than practical reports; only a few number of
applications of fuzzy morphology can be found in the
literature. Fuzzy grammars, finally, seem to be still as
Published in: 5th International Conference on Soft Computing, Iizuka, Japan, October 16-20, 1998, vol. 1, pp. 321-324
unpopular as its classical counterpart. Table 1 gives an
overview of theoretical/practical ripeness of different
fuzzy approaches (here, the ripeness may also be interpreted as degree of popularity measured with number of
corresponding publications) .
Table 1. Theoretical and practical ripeness of fuzzy
techniques in image processing [48]
Approach
Fuzzy clustering
Rule-based approach
Fuzzy geometry
Measures of fuzziness
Fuzzy measure theory
Fuzzy morphology
Fuzzy grammars
Theoretical/practical ripeness
in great detail investigated
investigations still necessary
Besides numerous publications on new fuzzy techniques,
the literature on introduction to fuzzy image processing
can be divided into overview papers [21,32,55], collections of related papers [1], and textbooks [6,24,33,48].
6 Future view
Fuzzy clustering algorithms and rule-based approaches
will certainly play an important role in developing new
image processing algorithms. Here, the potentials of fuzzy
if-then rule techniques seem to be greater than already
estimated. The disadvantage of rule-based approach,
however, is its expensive computing in local operations.
Hardware developments will be presumably a subject of
investigations. Fuzzy integrals will find more and more
applications in image data fusion. The theoretical research
on fuzzy morphology will be completed regarding to its
fundamental questions, and more practical reports will be
published in this area. Fuzzy geometry will be further
investigated and play an indispensable part of fuzzy image
processing.
It is not possible (and also not meaningful) to do everything in image processing with fuzzy techniques. Fuzzy
image processing will mainly play a supplementary role in
computer vision. Its part will be possibly small in many
applications, its role, nevertheless, will be a pivotal and
decisive one.
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