LOCAL PROCESSING OF 3D IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 58

difference filter ). Basic characteristics of difference filters are almost the same
as those of 2D images. We neglect the details here and will discuss the most
important features.
(i) The simplest form of the difference filter is one in which the difference
between densities of two voxels (denoted by P and Q for explanation) is
calculated. In this case, the output for an edge running in the direction
near the line connecting two voxels P and Q is small. On the other hand,
for an edge perpendicular to the direction connecting P and Q, the output
will be a larger difference value. Therefore, the response of a difference
filter strongly depends on both the direction of the difference and that
of an edge. This type of difference filter is called a directional-type filter.
A filter giving a significant output to any direction of edges is called
an omnidirectional-type filter.The directional type is effective if we have
precise knowledge of the edge direction or if we want to detect only edges of
restricted directions. However, the omnidirectional-type filter is preferred
if we do not have enough knowledge about the edge direction or if detecting
the edges of all directions is necessary.
(ii) We can derive an omnidirectional filter from directional filters. This is
called omnidirectionalization . Details of this procedure are explained in
Subsection 3.3.5.
(iii) The difference in density between two voxels separated in a definite dis-
tance from each other may be used instead of the difference between adja-
cent voxels. By adjusting the distance k between two voxels for which the
density difference is calculated, we can design a filter that may respond
to edges and borders ( line detection type ) and to a massive figure ( mass
detection type ). This feature is basically common to 2D image processing
[Preston79].
(iv) A disadvantage of the difference filter is that it is too sensitive to ran-
dom noise. This sensitivity is prevented by applying a smoothing filter
before applying a difference filter or by developing a filter that has both
smoothing and difference calculation functions. This type of filter we call
a smoothed difference filter . Formally this is regarded as the serial compo-
sition of a smoothing operator and a difference operator. If both operators
are linear, their serial composition is also linear.
3.3.5 Omnidirectionalization
Let us first apply a few relatively simple difference filters ( element filters )
to obtain the final output by calculating a suitable function of those out-
puts. We call this process the integration of element filters. An omnidirec-
tional filter may be created by integrating directional filters. This is called
omnidirectionalization .Denoting n operations with those n element filters by
O 1 , O 2 ,..., O n , the final output is the parallel composition of element oper-
ators O 1 , O 2 ,..., O n .