Ingeniería, investigación y tecnología. - Facultad de Ingeniería ...

3-D Carte sian Geometric Moment Compu ta tion using Morpho log ical Oper a tions and ...
The reaser can easily show that:
m1 00 = m 000 X c
(12a)
m 0 10 = m 000 Y c
(12b)
m 0 01 = m 000 Z c
(12c)
m
m
m
102
201
012
m 000 2
= X c( 3 Z c + t ( t + 1)) (12o)
3
m 000 Z X
2
=
c( 3
c
+ t ( t + 1)) (12p)
3
m 000 2
= Y c( 3 Z c + t ( t + 1)) (12q)
3
m
2 00
m 00 0 2
= ( 3X
+ t( t + 1)) (12d)
c
3
m
021
m 000 2
= Z
c( 3 Y
c
+ t ( t + 1)) (12r)
3
m
0 20
m 00 0 2
= ( 3Yc
+ t( t + 1)) (12e)
3
m = m X Y Z
(12s)
111 000
c c c
m
0 02
m
00 0 2
= ( 3Zc
+ t( t + 1)) (12f)
3
m = m Y = m X Y (12g)
1 10 100 c 00 0 c c
m = m Z = m X Z (12h)
1 01 100 c 00 0 c c
m = m Y = m Y Z (12i)
0 11 001 c 00 0 c c
m = m X ( X + t( t +1 )) (12j)
3 00 000
c
2
c
m = m Y ( Y + t( t +1)) (12k)
0 30 000
c
2
c
m = m Z ( Z + t( t +1)) (12l)
m
0 03 000
1 20
c
2
c
m 000 X Y
2
= c( 3 c + t ( t + 1 )) (12m)
3
Method based on iter ated erosions
to get the parti tion
The following method to compute the
geometric moments of a 3-D object R ⊂ Z
3 ,
using morphological erosions is a direct
extension to the one described in Sossa et al.
(2001). It is composed of the following steps:
1. Initialize 20 accu mu la tors Ci=0,
for i=1,2,...,20, one for each geometric
moment.
2. Make A= R and, B=
{( ± a , ± b , ± c) a, b , c∈ { −1 , 0, 1 }}, B is a 3 × 3×
3
pixel neigh bor hood in Z 3 .
3. Assign A ← A θ B iteratively until
the next erosion results in ∅ (the null
set). The number of iter a tions of the
erosion oper a tion before set ∅ appears,
is the radius t of the maximal cube
completely contained in the original
region R. The center of this cube is found
in set A just before set ∅ appears.
m
2 10
m
00 0 Y c X
2
= ( 3 c + t ( t + 1 )) (12n)
3
4. Select one of the points of A and
given that the radius t of the maximal
cube is known, we use the formulae
116 INGENIERIA Investigación y Tecnología FI-UNAM