ALGORITHM OF BINARY IMAGE PROCESSING - Fundamentals of Three-Dimensional Digital Image Processing - page 123

else f ijk ← 10 ( FPV )

endif
else no operation is performed
endif
enddo

[STEP 3] (Examine the terminating rule)
if no point changed through [STEP 2]
then stop
else go to [STEP 1] (repetition of the main cycle)
endif

Remark 5.4. In this type of algorithm, which includes deletion of voxels
performed sequentially, a border voxel that was not deletable at one time of
processing may become deletable afterwords due to the change in the local
arrangement of voxels caused by the processing in the subsequent cycles. This
fact is worth being taken into consideration when designing an algorithm. In
Fig. 5.4 (a), for example, the voxel B satisfies the condition C 2 (not deletable
because a hole is generated by the deletion). After the voxel A was deleted,
however, B is deletable because its state is the same as A before its deletion. As
this situation is repeated, an unnatural result as shown in Fig. 5.4 (b) might
occur. This is a typical example of the phenomenon called degeneration .In
Algorithm 5.3, we prevent this type of excessive degeneration by employing
the deletability test (=the preservation test) at a specified point in the exe-
cution and by preserving a voxel satisfying a given condition without testing
deletability in sequent procedures. Such voxels are called a finally preserved
voxel ( FPV ). In [STEP 2] of Algorithm 5.3, FPV is first detected and given
the mark in [STEP 2.1]. After that conditions C 1and C 2 are tested only at a
voxel having no FPV mark in [STEP 2.2] and a voxel is deleted, if deletable.
In this step, a voxel that was deemed deletable was immediately replaced by
a 0-voxel (deleted).

Let us show a property of Algorithm 5.3 below:

Property 5.3. (1) A line figure on a surface figure that is contained in a plane
perpendicular to one of the coordinate axes is never deleted because any
voxel in such a figure is regarded as either an inside pixel or a connecting
pixel or an edge pixel of a 2D figure [Toriwaki85a, Toriwaki85b]. An inside
voxel and a connecting voxel become a finally preserved voxel according to
the condition C 1and C 2. An edge voxel also becomes a finally preserved
voxel according to the condition C 1 (Fig. 5.5).
(2) When an input figure is a parallelepiped of the size (the number of voxels)
I K ( I>J>K ), and all edges are parallel to one of the coordinate

axes, an output figure is a plane of the size I ×

× J ×

J × K (voxels) where

I ≤

( I − K +1) ≤ ( I − K +2)