Image Processing Reference
In-Depth Information
(Proof) The former half is obvious from that only 1-voxels are changed, and
0-voxels are kept unchanged. The latter half is immediately known from the
fact that only a deletable voxel is deleted.
The following property is expected to hold and has been confirmed exper-
imentally, although it has not been proved theoretically.
Property 5.2. (1) A shrunk skeleton of a simply connected figure (a figure
that has neither a hole nor a cavity) is an isolated voxel (a single 1-voxel).
(2) A shrunk skeleton of a figure having one hole and no cavity (torus) con-
sists of only such voxels as ( R, H, Y )=( 2 , 0 , 0 ) (= connecting voxel).
Intuitively it is a loop-like line figure in 3D space.
(3) A shrunk skeleton of a figure having only one cavity and no hole (sphere
shell) consists of only such voxels as ( R, H, Y )=( 1 , 1 , 0 ) (inner voxel on
a 2D surface). Intuitively it is a sphere shell with one voxel thickness.
Concerning (1) above, it is expected that a simply connected component
consisting of three or more voxels contains at least two deletable voxels. By
deleting a deletable voxel one by one a simply connected figure will be reduced
to an isolated voxel.
5.4 Surface thinning and axis thinning
5.4.1 Definition
In 2D image processing a process that extracts a center line of a figure with
the finite width is called thinning (or skeletonization ). This is a very impor-
tant procedure widely used in practical applications of 2D pattern recognition
such as character recognition and document analysis. It is also employed in
processing of a gray-tone image in extraction of borders and edges.
The meaning of thinning in 2D image processing seems to be clear. How-
ever, it is not so easy to define thinning exactly. The word thinning expresses
the concept of a centerline of a natural or a reasonable shape located at the
reasonable position. This depends on the subjective judgment of the observer,
however, and cannot be specified theoretically.
A new problem occurs in the extension of the thinning to 3D image pro-
cessing even in subjective description. For example, the concept of a center
line is naturally acceptable for an elongated long figure such as a wire. In such
a case thinning is easily understood. On the other hand, some problems still
remain for a figure like a plate which is of a finite thickness. Imagine that
we have a wide plate with the unit thickness after shaving an input figure
iteratively from both sides. Then should we shave (thin) it until a centerline
is reached or should we stop a thinning procedure because we have already
reached a center surface ?
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