ALGORITHM OF BINARY IMAGE PROCESSING - Fundamentals of Three-Dimensional Digital Image Processing - page 167

N ( k ) ]:

M [ F = { f ijk }→ G = { g ijk }

∈N ( k ) (( i, j, k )

g ijk =max { 0 , max { f pqr ;( p, q, r ) }− 1 }

∈N ( k ) (( i, j, k ))

=max { 0 ,f pqr − 1 ;( p, q, r ) } . (5.22)

F ( k ) an output image obtained after performing [STEP 1] in Representing by

Algorithm 5.13,

F ( α ) = M [ N ( k ) ]( F ( α− 1) ) ,α =1 , 2 ,...,k

(5.23)

F (0) ≡ F

(input image) .

Hence an output image G is given as follows using a suitable finite value α :

N ( k ) ] α ( G = M [ F ) . (5.24)

Here the constant α is determined, depending on an input image. A neighbor-
hood

N ( k ) is any one of the 6-, 18-, and 26-neighborhoods. Neglecting N ( k )

for the sake of simplicity of an expression,

= M α ( G F ) . (5.25)

Representing a skeleton image (binary image) of the distance value α by F α ,

that is, a binary image in which voxels of the skeleton with the distance value α
are given the value 1 and other voxels are given 0 , and an ordinary maximum
filter by M , the DT image F is expressed as

F 2 )+ M 2 ( + M M− 1 ( F = F 1 + M ( F 3 )+ ··· F M ) . (5.26)

5.5.11 Extraction of skeleton

A skeleton should be defined clearly before constructing an algorithm to ex-
tract it. A skeleton of a 2D image using the 4- and the 8-neighborhood has
been studied in detail [Toriwaki81]. By extending it directly to a 3D image,
a skeleton is defined clearly for the fixed neighborhood DT using the 6-, 18-,
and 26-neighborhood. It has been known in this case that a resulting skeleton
coincides exactly for all four ways of definition in Def. 5.5 (1)

∼

(4) except
for some ambiguity pointed out there. Therefore, we can obtain a skeleton by
simply extracting local maxima of the DT. That is, we extract a voxel ( i, j, k )
such that

∈N ( k ) (( i, j, k )) . (5.27)
Note here that the type of the neighborhood should be the same as was used
in the calculation of the DT.
These are almost the same, in principle, in the variable neighborhood DT.
A neighborhood that was employed at the moment the DT value was fixed
should be selected in Eq. (5.27). Thus, the neighborhood depends on a DT
value of a voxel ( i, j, k ) as was shown in 5.5.5. The above discussion cannot
be applied to the Euclidean DT.

f ijk ≥ f pqr , ∀ ( p, q, r )