Fractal Geometry in Image Processing

characteristic feature of nonlinear (as opposed to linear)
process is the interaction (coupling) of different modes,
which may lead to non-random signal phase structure. Such
collective phase properties of the signal cannot be detected
by linear spectral methods. Fractal dimension can be an
useful measure for the characterization of
electrophysiological time series (Klonowski, 2000).
b. 2-D fractals and image analysis
A digitized image is a pattern stored as a
rectangular data matrix. It is distinguished between binary
images, grayscale images and color images. The ultimate
goal of image analysis is the identification of a scene and all
objects in the image. Image analysis can be described as a
set of techniques required to extract symbolic information
from the image data. Different techniques are available for
performing image analysis, of which fractal methods are
promising in certain applications. The images can be studied
by comparing the fractal dimensions of the original and
transformed images. Another approach which has been put
forth is the local fractal operator method. In this method, the
fractal dimension corresponding to each pixel of the image
is computed. This is followed by segmentation technique to
extract the region of interest from the image.
Marchette et al., (1997) have employed fractal
based techniques in digital mammography to detect
tumorous tissues. The tumor region was better identified
when segmentation boundaries were incorporated into the
calculation. Similar techniques can be employed for the
detection of an object from remotely sensed images or in the
recognition of biological structures from plant images.
1.6.3. Fractal techniques in classification
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the recognition of skin samples to determine the age of a
person. They are also used in fingerprint recognition and
fabrics processing.
1.7 Conclusion
The fractal theory is still descriptive rather than inferential.
There are a lot of application for fractals in different fields.
The basic concepts and applications of fractals are discussed
in this chapter.
References
1. Philip, N.S., Wadadekar, Y., Kembhavi, A,, Joseph,
K.B.,'A difference boosting neural network for
automated star-galaxy classification', Astronomy and
Astrophysics, 385 : 1119-1126,2002.
2. Polvere, M. and Nappi, M., 'Speed-Up In Image
Coding : Comparison of Methods', IEEE Trans, on
image processing 9, 2000.
3. Revathy, K., and Nayar, S.R.P. 'Fractal Analysis of
Ionospheric Plasma Motion', Fractals, 6 : 127- 130,
1998.
4. Revathy, K., Raju, G., Nayar, S.R.P., 'Image Zooming
by Wavelets', Fractals, 8 : 247-253, 2000.
5. Ruhl, M., Hartenstein, H. and Saupe, D., 'Adaptive
Partitioning For Fractal Image Compression', IEEE
International Conference on Image Processing
(ICIP'97), Santa Barbara, 1997.
Fractal geometry is widely used in the study of
image characteristics. For recognition of regions and objects
in natural scenes, there is always need for features that are
invariant and provide a good set of descriptive values for
the region. There are many fractal features that can be
generated from an image (Chaudhuri and Sarkar, 1995).
The most commonly used fractal feature is the fractal
dimension. In some applications, fractal dimension alone is
capable of discriminating one object from one another. But
in certain applications, fractal dimension alone may not be
sufficient in identifying the desired object. Here, a set of
fractal features obtained on simple image transformations
can be used as the feature set. Another technique is based
on fractal signature. Fractal signature is designed using
epsilon blanket method. The fractal signature assigns a
unique signature to each sample. Samples belonging to each
class have similar signature which makes them distinguish
from one another easily.
Fractal features are commonly used in textures.
Images that possess homogeneous regions can be easily
classified using these features. Fractal features are used in
6. Saar, E., 'Towards fractal description of Structure',
332. Morphological Cosmology Proceedings,, Cracon,
Poland, 205-218, 1988.
7. Saha, S. and Vemuri, R., 'An analysis on the effect of
image features on Lossy Coding Performance', IEEE
Signal Processing Letters, 7 : 104-107,2000.
8. Sarkar, N. and Chaudhuri, B., 'An efficient approach to
estimate fractal dimension of textural images', Pattern
Recognition, 25(9) : 1035-1041,1992.
9. Saupe, D., 'Breaking the time-complexity of fractal
image compression', Technical Report 53, Institut fur
Informatik, Universitat Freiburg, 1994.
10. Saupe, D. and Hamzaoui, R., 'A review of the fractal
image compression literature', ACM Computer
Graphics, 28 : 268 - 276,1994.
11. Sato, T., Matsuoka, M. and Takayasu, H., 'Fractal
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