MODELS OF IMAGES AND IMAGE OPERATIONS - Fundamentals of Three-Dimensional Digital Image Processing - page 44

g ijk = φ ijk ( g i 1 j 1 k 1 ,g i 2 j 2 k 2 ,...,g i m j m k m ,
f i m +1 j m +1 k m +1 ,...,f i n− 1 j n− 1 k n− 1 ,f i n j n k n ) ,
( i n ,j n ,k n ) ∈N ijk (( i, j, k )) , ∀ n ( N ≥ n ≥ 1 ) (2.43)

where φ ijk is an arbitrary n -variable function, and m and n are arbitrary
integers such that n 1 . The right-hand side of the Eq. 2.43 must not

contain g ijk itself (Fig. 2.12 (d),(e)).
The parallel and the sequential type of operations are characterized as
follows:

≥ m ≥

[Parallel type]

(1) An output value g ijk is determined using the gray values of an input image
F

{ f ijk }

only.
(2) Values of g ijk for different ( i, j, k )s can be calculated independently (and
concurrently if suitable hardware is available).

=

[Sequential type]

(1) An output value g ijk at a voxel ( i, j, k ) is determined by using both an

input image F = { f ijk } and part of an output image G = { g ijk } at voxels

for which values already have been obtained.
(2) Output values must be computed one by one, sequentially, in a predeter-
mined order.

Many image operations can be performed by either type of algorithm. In
such cases, we distinguish the operation (the function) in general and the
method of execution by using the term algorithm to refer to the latter case.
In some classes of sequential operations the order of calculation among voxels
depends upon an input image. In the case of operations called tracing type
we start at a suitably selected initial voxel and proceed to the adjacent voxel
sequentially according to the set of predetermined rules based upon densities
of an input and/or an output image (Fig. 2.12 (f)).
A shift invariant local operation , sometimes called local parallel operation ,
or simply filtering operation , is most important for practical applications.
Specific types of them utilizing the 3 3 neighborhood have been called

mask operation or simply neighborhood logic in the literature.
The order of processing among all voxels, that is, the order in which voxels
are processed, is of critical importance in a sequential operation (or a sequen-
tial algorithm). A frequently used order in the sequential operation is called
raster scan , examples of which are shown in Fig. 2.13. One of them (mode I,
for example) is called forward raster and the other (mode II) backward raster .

× 3 ×

Remark 2.8 (Existence of sequential operation). Whether a sequential
operation exists or not which is equivalent to an arbitrarily given parallel op-
eration is a theoretically interesting problem which has not been solved com-
pletely. The idea of the sequential type operation was shown by Rosenfeld first