GEOMETRICAL PROPERTIES OF 3D DIGITIZED IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 91

)=
h =1 , 3

Nc (6) (

x x h, 0 ( 1 − x h,k · x 2 ,k )

k∈S 1

+
k∈S 1

x 2 ,k { 1 − x 2 ,k +1 · x 2 ,k +2

( 1 − x h, 0 · x h,k · x h,k +1 · x h,k +2 ) } . (4.29)

h =1 , 3

x 3 , 0 +
k∈S 1

Nc (18 ) ( x x 1 , 0 + x 2 ,k ( 1 − x 2 ,k +1 · x 2 ,k +2 )

)=

− [ x h, 0 · x h,k · x 2 ,k ( 1 − x h,k +2 · x 2 ,k +1 · x 2 ,k +2 )

k∈S 1 h =1 , 3

+ x h,k +1 { x h, 0 · x 2 ,k +1 · ( x h,k · x 2 ,k +2 · x 2 ,k · x h,k +2

+ x 2 ,k · x h,k +2 · x h,k · x 2 ,k +2 )

+ x h, 0 · x 2 ,k +1 · x h,k · x 2 ,k · x 2 ,k +2 · x h,k +2 } ] . (4.30)

Nc (18) ( x )= 2 − x 1 , 0 − x 3 , 0 − x 2 ,k ( 1 − x 2 ,k +1 · x 2 ,k +2 )

k∈S 1

+
k∈S 1

[ x h, 0 · x h,k · x 2 ,k ( 1 − x h,k +2 · x 2 ,k +1 · x 2 ,k +2 )

h =1 , 3

+ x h,k +1 { x h, 0 · x 2 ,k +1 · ( x h,k · x 2 ,k +2 · x 2 ,k · x h,k +2

+ x 2 ,k · x h,k +2 · x h,k · x 2 ,k +2 )

+ x h, 0 · x 2 ,k +1 · x h,k · x 2 ,k · x 2 ,k +2 · x h,k +2 } ] . (4.31)

Nc (26) ( x )= 2 − x h, 0 ( 1 − x h,k · x 2 ,k )

h =1 , 3 k∈S 1

+
k∈S 1

x 2 ,k { 1 − x 2 ,k +1 · x 2 ,k +2

− x h, 0 · x h,k · x h,k +1 · x h,k +2 ) } ,

( 1 (4.32)

h =1 , 3

where S 1 = { 1 , 3 , 5 , 7 } , x a, 9 ≡ x a, 1 ( a = 1 , 2 , 3 , 4 )and x a,b ≡ 1 − x a,b .

(Proof) Eqs. 4.29 4.32 are derived from Eq. 4.11 in Def. 4.10 and values of

the Euler number. Calculation of the Euler number will be discussed in detail
in the next section.

∼

4.7 Calculation of the Euler number

The Euler number is a kind of global feature characterizing the whole of a
figure and a binary image. It is necessary to consider the whole of an input
image or input figure for calculating the Euler number.