Optimal Differential Filter on Hexagonal Lattice - Computational Intelligence Paradigms in Advanced Pattern Classification - page 75

f ex 1 is composed mostly of low frequencies. It has smooth changes in luminance
given as:

R 1 − (

x 2 y 2

f ex 1 ( x , y ) ≡ + ) . (81)

The gradient intensity and orientation of this image are

x 2

y 2

+

f int

( , ) ≡− R 1 − (

x y (82)

ex 1

x 2 y 2

+ )

f ori
ex 1

( x , y ) ≡ arctan ( x , y ) , (83)

where R 1 is a constant.
f ex 2 is an image composed mostly of high frequencies. It has periodical changes
in luminance given as:

( ω 2 x 2 cos 2

y 2

f ex 2 ( x , y ) ≡ A 2 · + ) . (84)

Its gradient intensity and orientation are

ω 2 x 2 f int

ex 2

( x , y ) ≡| A 2 · ω 2 · sin ( 2 + y 2 ) | (85)

f ori
ex 2

( x , y ) ≡ arctan ( x , y ) , (86)

where A 2 and ω 2 are constants.

f ex 3 is composed of low to high frequencies given as

cos 2 x 2 y 2

f ex 3 ( x , y ) ≡ A 3 · ( ω 3 ( + )) . (87)

Its gradient intensity and orientation are

x 2

f int
ex 3

x 2 y 2 y 2

( x , y ) ≡ 2 | A 3 · ω 3 · sin ( 2 ω 3 ( + )) | + (88)

f ori
ex 3

( x , y ) ≡ arctan ( x , y ) , (89)

where A 3 and ω 3 are constants.

For a given input image f , Int ( f ) is the gradient intensity computed from the

filtered image and Ori is the orientation computed from the filtered image. To

evaluate the accuracy of the filters, the errors are then calculated as follows:

( f )

f int

| − ( ) |

Int f (90)

and

f ori | − Ori ( f ) |. (91)