![](bg4.png)
where p≥0, q=0,± 1, ± 2,...
By using an infinite number of moments Φpq,
-M
≤
q
≤
M, 0
≤
p
≤
N, where M, N are positive integers,
the original image can be reconstructed through the
following formula
()
∑∑
=−=
∧
Φ=
N
p
M
Mq
iq
ppq
eQrf
0
,
θ
θ
(18)
As in the case of Zernike and Pseudo-Zernike
moments, the magnitudes of the OFMMs are also
rotation invariant. The majority of OFMMs in
contrast to the other orthogonal moments is focused
on the fact, that they can describe the high spacial
frequency components of an image more accurately
(Kan, 2002). This capability comes from the number
of zeros of their radial polynomials, which is greater
than the other moments.
The number of linearly independent OFMMs is
(p+1)
2
, so the degree p of Q
p
in the OFMMs required
to represent an image can be much lower than a
representation using ZMs and PZMs.
Because the Zernike, Pseudo-Zernike and
Fourier-Mellin moments are only rotationally
invariant, additional properties of translation and
scale invariance must be given to these moments in
some way. We can ensure these
invariances by
converting the absolute pixel coordinates
(Khotanzad, 1990).
3 MOMENT COMPRESSION
In this section a predefined algorithm that consists of
two complementary paths, involving moment
computation and a compression method, is
presented.
In Fig.1 this algorithm is depicted in a generic
form, in order to maintain a systematic procedure
that performs a feature extraction method, while the
inverse process is also provided.
The concerned algorithm, which is presented in
details in (Papakostas, 2002, 2004), can be
summarized in the following steps:
Direct path
Step 1: The original image is being pre-
processed, (filtering, binarization).
Step 2: Computation of the orthogonal
moments to be compressed, with the
additional ensuring of translation,
scaling invariance, and finally the
computation of the so called “moment
signal”. This 1-D signal consists of the
resulted moments, in the order they
have been produced.
Step 3: Application of the Wavelet transform,
or an alternative one (Fourier), to the
“moment signal”.
Step 4: Compression by thresholding of the
resulted wavelet (Fourier) coefficients.
Original
Image
Image
Pre-Processing
Transformation
Inverse
Transformation
Image
Post-Processing
Computation of
Orthognal
Moments
Image
Reconstruction
Final
Reconstructed
Image
Normalized
Reconstruction
Error
Compression
Figure 1: Generic compression of moment features.
Inverse path
Step 1: Application of the inverse transform,
upon the compressed coefficients, in
order to construct the compressed
“moment signal”.
Step 2: Image reconstruction using the
compressed moments, by applying the
inverse formula of the corresponding
moment family.
Step 3: Image post-processing, including
mapping into the range [0-255],
binarization or histogram equalization.
The direct path of the above algorithm is applied,
in order to generate feature vectors with an as small
as possible dimension, but with an increasing
amount of image information. The resulted feature
vectors are consisted of wavelet coefficients that
describe the compressed moment signal.
The inverse path being used to verify the
effectiveness of the moment based feature vectors,
by means of the normalized reconstruction error.
ON THE RECONSTRUCTION PERFORMANCE OF COMPRESSED ORTHOGONAL MOMENTS
471