Mathematical Morphology and Image Processing
by Henk Heijmans
In the processing and analysis of images it is important to be
able to extract features, describe shapes and recognize patterns.
Such tasks refer to geometrical concepts such as size, shape,
and orientation. Mathematical morphology uses concepts from set
theory, geometry and topology to analyze geometrical structures
in an image. A substantial part of CWI’s research theme Signals
and Images is connected with multiresolution methods, based on
the application of fractals, wavelets and morphology. One line
of research explores the relationships between wavelets and morphological
methods, aiming at a unified approach.
The word ‘morphology’? stems from the Greek words morfh and logos,
meaning ‘the study of forms’. The term is encountered in a number
of scientific disciplines including biology and geography. In
the context of image processing it is the name of a specific methodology
designed for the analysis of the geometrical structure in an image.
Mathematical morphology was invented in the early 1960s by Georges
Matheron and Jean Serra who worked on the automatic analysis of
images occurring in mineralogy and petrography. Meanwhile the
method has found applications also in several other fields, including
medical diagnostics, histology, industrial inspection, computer
vision, and character recognition.
Mathematical morphology examines the geometrical structure of
an image by probing it with small patterns, called ‘structuring
elements’, of varying size and shape, just the way a blind man
explores the world with his fingers or a stick. This procedure
results in nonlinear image operators which are well-suited to
exploring geometrical and topological structures. A succession
of such operators is applied to an image in order to make certain
features apparent, distinguishing meaningful information from
irrelevant distortions, by reducing it to a sort of caricature
(skeletonization). For example, in optical character recognition
one may transform the digital image of a symbol by reducing each
connected component to a one-pixel-thick skeleton retaining the
symbol’s shape. Such a skeleton suffices for recognition and can
be handled much more economically than the full symbol.
Experience in image processing and computer vision has shown that
for a comprehensive understanding of a scene analysis at a broad
range of resolution levels is necessary. The resulting multiresolution
techniques (quadtrees, pyramids, fractal imaging, scale-spaces,
etc.) all have their merits and limitations. For example, fractals
have been exploited with great success in image compression but
to a much lesser extent for segmentation problems.
In the earliest multiresolution approaches to signal and image
processing, the most popular way was to obtain a coarse level
signal by subsampling a fine resolution signal, after linear smoothing,
in order to remove high frequencies. A ‘detail pyramid’ can then
be derived by subtracting from each level an interpolated version
of the next coarser level. From a frequency point of view, the
resulting difference signals (known as detail signals) form a
signal decomposition in terms of bandpass-filtered copies of the
original signal. There is neurophysiological evidence that the
human visual system indeed uses a similar kind of decomposition.
This tool has been one of the most popular multiresolution schemes
used in image processing and computer vision. CWI is currently
investigating a general axiomatic pyramid scheme encompassing
most existing linear as well as nonlinear (morphological) pyramids.
The emergence of wavelet techniques has considerably boosted the
multiresolution approach. Unfortunately, application of wavelets
to problems in image processing and computer vision is sometimes
hindered by its linearity. Coarsening an image by means of linear
operators may not be compatible with a natural coarsening of some
image attribute of interest (shape of object, for example), and
hence use of linear procedures may be inconsistent in such applications.
Mathematical morphology (nonlinear) is complementary to wavelets
(linear) in that it considers images as geometrical objects rather
than as elements of a linear (Hilbert) space. Many of the existing
morphological techniques, such as granulometries, skeletons, and
alternating sequential filters, are essentially multiresolution
techniques. Bearing this in mind, one may wonder what are the
relationships between the existing linear (wavelets) and nonlinear
(morphological) multiresolution approaches. Is it possible to
unify both approaches into one mathematical framework? Does there
exist such a thing as a ‘morphological wavelet’? The somewhat
ambitious goal of CWI’s research effort is to answer such questions.
First promising steps have been made by the author in collaboration
with J. Goutsias (Johns Hopkins University).
Please contact:
Henk Heijmans - CWI
Tel: +31 20 592 4057
E-mail: Henk.Heijmans@cwi.nl
