COMPLEX EXPONENT MOMENTS

FFT ALGORITHM AND ITS APPLICATION

ZiLiang Ping

and YongJing Jiang

Centenary College, Beijing University of Post and Communication, Beijing, China

Inner Mongolia Normal University, Huhhot, China

Keywords: Multi-distorted invariance, FFT algorithm, Compex Exponent Moments (CEMs), Human face recognition.

Abstract: A fast and accurate algorithm for computation of multi-distorted invariant Complex Exponent Moments

(CEMs) is presented in the paper. An image function in polar coordinate system,

(, )

, was divided

into 2-D discrete image matrix in which the radial variables on lines and angle variables on columns. 2-D

Fast Fourier Transform (FFT) was excuted for the matrix and the Complex Exponent Moments (CEMs) can

be obtained. The multi-distorted invariance and the excellent performance of Complex Exponent Moments

(CEMs) were demonstrated. The Complex Exponent Moments (CEMs) were applied in human face

recognition

1 PREFERENCE

Orthogonal multi-distorted invariant moments have

been successfully used in the fields of image

analysis, pattern recognition, object classification

and image watermarking ete. (Papakostas and

Boutalis, 2007; Kan and Srinath, 2002; Kim and Lee,

2003). Many research works devote to orthogonal

invariant moments, such as Zernike Moments

(Teague, 1980), orthogonal Fourier-Mellin Moments

(Sheng and Shen, 1994), Chebyshev-Fourier

Moments (Ping et al., 1748-1754),

Radial-Harmonic-Fourier Moments (Ren and Ping,

2003) and Jacobi-Fourier Moments (Ping and Ren,

2007). All of those moments were calculated by the

integral operation in former algorithm literature.

Because of the transform between Cartesian

coordinate system and polar coordinate system and

calculating complex degree, the calculation of the

moments is of time waste and those moments are

lower precision. The fast and accurate algorithm for

Complex Exponent Moments (CEMs), 2D FFT

algorithm, was proposed in this paper.

2 THE DEFINITION OF

COMPLEX EXPONENT

MOMENTS (CEMS)

Radial-Harmonic-Fourier Moments (RHFMs) (Ren

and Ping, 2003) is definited in polar coordinate

system as:

( , ) ( )exp( )

nm n

rTr jmrdrd

θθθ

=−

∫∫

(1)

Here

() sin( 1)

co s

if n

T r n r if n is odd

n r if n is even

⎧

⎪

⎨

⎪

⎩

(2)

According to Euler’s fomular the radial harmonic

function can be transform to complex exponential

function:

( ) exp( 2 )

rjkr

(3)

And the relationship between

()

and

A()r

is as following:

465

Ping Z. and Jiang Y..

COMPLEX EXPONENT MOMENTS FFT ALGORITHM AND ITS APPLICATION.

DOI: 10.5220/0003700904650468

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 465-468

ISBN: 978-989-8425-95-9

 2012 SCITEPRESS (Science and Technology Publications, Lda.)



() () 0

() ( () ()) 2 1 1,2,3

() ( () ()) 2 1,2,3

nkk

Tr Ar n k

Tr Ar A r n k k

−

⎧

= = =

⎪

= − = − =

⎨

⎪

= + = =

⎪

⎩

(4)

The Radial-Harmonic-Fourier Moments (RHFMs)

can be rewrited as:

,,,

( , ) exp( 2 )exp( )

km c xy xy

fxy jkr jm dxdy

πθ

+≤

=−−

∫∫

(5)

Here，

,arctan

xy xy

rxy

=+ =

, and the

integral can be calculated in Cartesian coordinate

system. The

,km

is definited as Complex Exponent

Moments (CEMs) and can be calculated with FFT

algorithm.

3 ALGORITHM OF THE

COMPLEX EXPONENT

MOMENTS(CEMS) AND

IMAGE RECONSTRUCTION

WITH CEMS

According to formular (5) the Complex Exponent

Moments (CEMs) can be numerically calculated as

following:

N1 1

122

(, ) exp( )exp( )

km p u v

EFr jkujmv

MN M

ππ

−−

=−−

∑∑

(6)

The CEMs can also be calculated with 2D fast

Fourier transform (FFT) algorithm. Uniformly

sampling an image function in polar coordinate

system to make image function being a discrete

MN×

numetrical image matrix first. Here N and

M is the sample point number on the radial direction

and angular direction respectively. Then taking 2D

Fast Fourier Transform for this discrete

numetrical image matrix, the CEMs of the image can

be obtained. FFT algorithm of the CEMs possesses

the lower calculation complexity (

log )ON N( )

than integral algorithm (

()ON ), so CEMs are

more efficient algorithm and time-saving.

Because of the orthogonal performance of kernel

function the CEMs is orthogonal image moments

and an image can be reconstructed with its CEMs

according to following equation:

( , ) ( )exp( )

km k

rEArjm

+∞ +∞

=−∞ =−∞

∑∑

(7)

The similar degree between the reconstructed image

and the original image will increase with the

increasing of moment’s numbers. Former research

(Ren and Ping, 2003) works verified that RHFMs

possess the best image reconstruction performance

in all of orthogonal moments. Produced from

RHFMs, CEMs possess the best reconstruction

performance too. The experiments verified that

CEMs calculated by FFT algorithm possess better

reconstruction performance than calculated by

integral algorithm. Fig.1 shows comparation of

integral algorithm with FFT algorithm. From Fig.1 it

can be seen that the quality of reconstructed images

via FFT algorithm is much better than reconstructed

images via integral algorithm.

4 THE ROTATION AND SCALING

INVARIANT PERFORMANCE

OF CEMS

The complex Fourier factor of the kernel function in

angular direction will maintain the rotated-invariant

performance of the modular value of the moments.

Through unifying process the performance of CEMs

will be scaled-distorted invariant and

intensity-distorted invariant. So, Complex Exponent

Moments (CEMs) are multi-distorded invariant

Figure 1: The image reconstruction with CEMs: (a)-(d) reconstructed images with integral algorithm, (e)-(h) reconstructed

images with FFT algorithm.

ICAART 2012 - International Conference on Agents and Artificial Intelligence

466

(Ping et al., 1748-1754). Fig.2 shows the

rotation-distorted invariant performance of Complex

Exponent Moments (CEMs): The line-up is original

image of Lena and its modular value distribution, the

line-down is rotated image of Lena and its modular

value distribution. From Fig.2 it can be seen that the

modular values of CEMs for rotated image are

invariant.



Figure 2: The rotated invariant performance of CEMs. The

line up is standart image and its CEMs modular value

distribution. The line down is rotated image and its CEMs

value distribution.

缩放前的图像放大1.5倍后的图像

Figure 3: The scaling invariant performance of CEMs. The

left column is normal image and its CEMs modular value

distribution. The right column is scaled image and its

CEMs modular value distribution.

Fig.3 shows the scaling invariant performance of

modular value of CEMs

The left column is Lena normal image and

modular value distribution of its CEMs, the right

column is distorted image scaled 1.5 times and its

modular value distribution of its CEMs. From

Fig.3 it can be seen that the modular value of CEMs

of scaled image possesses the same modular value

distribution with the normal image.

5 APPLICATION OF CEMS FOR

PATTERN RECOGNITION

Using the CEMs of image to be feature vectors for

Suppot Vector Machine (SVM) algorithm pattern

recognition was performed for human face of 20

persons, for whom each one has 10 various visions and

different facial expression. Fig.4 shows the experiment

images of two persons of the recognized human faces.

Each face image was rotated for 10°、20°、30° and has

40 pieces of image for one person, and there are 800

pieces of variable image for 20 persons.

Take mass center of image to be original point of

the image coodinate system and calculate the CEMs

of the image via FFT algorithm. Using CEMs of the

image to be feature vectors of Support Vector

Machine (SVM), the human face recognition

experiment was performed. The 24 pieces of face

image for each person and 480 pieces of face image

for 20 persons were taken as trining sample image

set, and the other 320 pieces of face images were

taken as testing sample image set, and the one to

more SVM algorithm was applied in the experiment.

Table-1 shows the experiment result. From the data

of Table-1 it can be seen that taking 36 CEMs as

image feature vectors the recognition rate is the

highest, achieving 92.5%, as number of CEMs

increasing the recognition rate of image is

decreasing down, this is because of the over-learning

problem for Suport Vector Machine(SVM)



Figure 4: The various face expression for two persons in experiment, upper: face of male person; lower: face of female

person.

COMPLEX EXPONENT MOMENTS FFT ALGORITHM AND ITS APPLICATION

467

Table 1: The recognition rate human face via Complex Exponent Moments (CEMs).

Momen

Number

Recognition

rate （%）

Momen

Number

Recognition

rate （%）

Momen

Number

Recognition

rate （%）

88.75

90.75

92.5

90.63

87.97

81.25

78.59

76.25

6 CONCLUSIONS

In the paper a novel orthogonal multi-distorted

invariant moments (CEMs) was presented and a fast

and accurate algorithm, 2D Fast Fourier Transform

(FFT) algorithm, was performed for this moments.

The theoretical analysis and experiment results have

verified that 2D FFT algorithm of CEMs possesses

multi-distorted invariance, lower image

reconstruction error, higher quality of reconstructed

image and lower calculating complicity degree

compared to the integral method. The CEMs were

applied in human face recognition and experiment

result has verified efficience of CEMs and the

2D-FFT algorithm for it.

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(6) (1994) 1748-1757

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(2002) 1748-1754

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