Improving the Recognition Performance of Moment Features by Selection - Feature Selection for Data and Pattern Recognition - page 298

high information redundancy making them less efficient in difficult problems where
more discriminative information needs to be captured, due to monomials' lack of
orthogonality increasing their information redundancy.
This fact has motivated scientists to develop the orthogonal moments, which
use as kernel functions orthogonal polynomials that constitute orthogonal basis.
The property of orthogonality gives to the corresponding moments the feature of
minimum information redundancy, meaning that different moment orders describe
different image content.
Initially, the orthogonal moments defined in the continuous space were intro-
duced [ 26 ], such as Zernike, Pseudo-Zernike, Fourier-Mellin and Legendremoments.
Although these moments were widely applied in many disciplines for a long time,
their performance is degraded by several approximation errors [ 32 ] generated mainly
due to coordinates normalization and space granulation procedures.
Recently, enhanced orthogonal moments free from approximation errors and
directly defined inside the discrete coordinate system of the image, were proposed
to overcome the disadvantages of the continuous moments. The most representative
moment families of discrete form are the Tchebichef [ 19 ], Krawtchouk [ 34 ] and dual
Hahn [ 14 , 37 ] moments.
It is worth noting that the main research directions across which most scientists
work with, in the field of image moments are the following: (1) the development
of new algorithms that accelerate the overall moments' computation time, (2) the
improvement of the moments' accuracy by reducing the quantization and approxi-
mation errors and (3) the embodiment of invariance capabilities into the moments'
computation regarding the major linear image's transformations (translation, rotation
and scaling). The last direction is relative to the capabilities of the moment features
to achieve high recognition rates exploiting invariant behaviour under the aforemen-
tioned three basic transformations. Herein, only the description capabilities of the
moment features in terms of recognition accuracy will be studied, without paying
any attention to the invariant versions of them.
The most representative orthogonal moment families of both continuous and dis-
crete coordinate space are hereafter described and analyzed experimentally.

13.2.1 Continuous Orthogonal Moments

In the previous section it has already been mentioned that the first type of orthogonal
moments for images (2-D) was defined in the continuous coordinate space of a
continuous intensity function f . However, in order to use those moments

with digital images, which are defined in the discrete domain, an approximation
was applied the so-called zeroth order approximation (ZOA). These two different
definitions are as follows:

( x , y )