Improving the Recognition Performance of Moment Features by Selection - Feature Selection for Data and Pattern Recognition - page 302

13.2.2 Discrete Orthogonal Moments

The aforementioned drawback of the continuous orthogonal moments, has motivated
scientists in the field of image moments to develop more accurate moment families.
This goal has been achieved by the introduction of the discrete orthogonal moments
being defined directly on the discrete image coordinates space. Some of the most
representative discrete moment families are discussed herein.

13.2.2.1 Tchebichef Moments

This moment family is the first proposed in the literature by Mukundan et al. [ 19 ].
The ( n + m ) th order Tchebichef moment of a N × N image having intensity function

f ( x , y ) is defined as:

N − 1 N − 1

1

0 t n (

) t m (

T nm = x y ) f ( x , y ) (13.15)

ˁ( n , N ) ˁ( m , N )

x = 0 y =

where t n ( is the n th order normalized Tchebichef polynomial, introduced in order

to ensure numerical stability and moments' limited dynamical range, defined as
follows:

x )

t n (

)

x

t n (

) =

x (13.16)

ʲ( n , N )

where the ordinary Tchebichef polynomial of n order has the form:

t n ( x ) = ( 1 − N ) n 3 F 2 ( − n , − x , 1 + n ; 1 , 1 − N ; 1 )

k N

n x

k

.

n

− 1 − k + k

n −

=

0 ( −

1 ) (13.17)

n − k n

k =

In the above formulas, 3 F 2 , is the generalized hypergeometric function, n , x =

0 is a suitable constant independent

of x that serves as a scaling factor, such as N n . Moreover,

, 1 , 2 ,..., N − 1, N is the image size and ʲ( n , N )

ˁ( n , N ) is the normalized

norm of the Tchebichef polynomials defined as:

ˁ( n , N )

ˁ( n , N ) = (13.18)

2

ʲ( n , N )

with

N

,

+ n

ˁ( n , N ) = ( 2 n ) ! n = 0 , 1 ,..., N − 1 . (13.19)

2 n + 1