Improving the Recognition Performance of Moment Features by Selection - Feature Selection for Data and Pattern Recognition - page 299

NF 1
A

M nm = Kernel nm ( x , y ) f ( x , y ) dxdy (13.1)

where A is the computation coordinate space, Kernel nm ( · )

corresponds to the
moment's kernel (product of two polynomials) consisting of specific orthogonal
polynomials of order n and m , which constitute the orthogonal basis and NF 1 is a
normalization factor. The type of Kernel's polynomials gives the name to the moment
family and thus a wide range of different moment types can be derived.
The zeroth order approximation of Eq. ( 13.1 )fora N × N image having intensity

function f ( x , y ) has the form:

N − 1 N − 1

( ZOA ) : M nm = NF 2 Kernel nm ( x , y ) f ( x , y ) (13.2)

x = 0 y = 0

where NF 2 is a normalization factor and the double integral of Eq. ( 13.1 ) is sub-
stituted by a double summation, by incorporating some approximation error. The
minimization of this error has been the subject of many works [ 31 ], which try to
propose a discrete computation form that converges to the theoretical values. Three
representative moment families of this category will be described in details in the
next sections.

13.2.1.1 Zernike Moments

Zernikemoments are themost widely used family of orthogonal moments due to their
inherent property of being invariant to an arbitrary rotation of the image they describe.
The main characteristic of this moment family is the usage of a set of complex
polynomials as basis, which forms a complete orthogonal set over the interior of the
unit circle x 2 y 2

+ = 1. These polynomials have the form:

e jm ʸ V nm ( r ,ʸ) = R nm ( r ) (13.3)

where n is a non-negative i nteger a nd m an integer subject to the constraints n −| m |

x 2

y 2

even and | m |≤ n , r ( r = + ) is the length of the vector from the origin to the

ʸ ʸ = ) is the angle between the vector r and x -axis in counter-

tan − 1 pixel and ( y / x

clockwise direction. R nm ( r ) , are the Zernike radial polynomials [ 35 ], in ( r ,ʸ) polar

coordinates defined as:

n −| m |

2

( − ) !

n s

r n − 2 s

s

R nm ( r ) =

0 ( −

1 ) n +| m |

2

s

n −| m |
2

s

(13.4)

s ! − ! − !

s =

Note that R n ( − m ) ( r ) = R nm ( r ) .