ALGORITHM OF BINARY IMAGE PROCESSING - Fundamentals of Three-Dimensional Digital Image Processing - page 118

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Fig. 5.2. Concept of surface/axis thinning.

For the moment we will take into consideration both cases above, and
we call the former case surface thinning and the latter case axis thinning
(Fig. 5.2). Thus, we will define or specify surface/axis thinning as follows.

Definition 5.1 (Surface thinning). Surface thinning is a process that
transforms a 3D figure with a finite thickness to the center surface (a figure of
the unit thickness located at the center of an input figure) while keeping the
topology unchanged. In this sense, a surface figure means a figure such that
all voxels

x

in it satisfy at least one of (i) and (ii) below.
(i) No 3D simplex (= arrangements of voxels shown in Fig. 5.3) is contained
in the 3

× 3 × 3 neighborhood of x .

(ii) x is not deletable.

Definition 5.2 (Axis thinning). Axis thinning is a process that transforms
a figure with a finite thickness and/or a finite width into a line figure with the
unit thickness being located at the center of the figure without changing the
topology. By a line figure we mean a figure that does not contain a deletable
voxel except for at the end points.

These are conceptually straightforward extensions of those in 2D image
processing to a 3D image. As was stated above, there are two cases of thinning:
transformation into a center surface or into a centerline. This is a problem
specific to 3D image processing.
However, an output of the axis thinning still may not become a line fig-
ure due to topology preservation requirement. An example is an input figure
which has a cavity. Therefore, the difference between the axis thinning and
the surface thinning may not always be clear for some kinds of input figures.
For a long plate-like figure of the unit thickness, there are clearly two possi-
bilities: whether it is further unchanged (surface thinning or medial surface
extraction) or is transformed until it becomes a line figure (axis thinning or
medial axis extraction) (Fig. 5.2). Either of them may be desired according
to each application. We will present both types of algorithms here.