Image Processing Reference
In-Depth Information
11.3.1
Elementary Cellular Automata
There are 256 1D 2C, radius 1 (r1), so called elementary cellular automata (ECA).
However, due to symmetries, the number of fundamentally inequivalent ECAs is
only 88 [29]. For an interactive demonstration of CASS based on ECAs see [62].
11.3.2
The Original CA for Shading
Since none of ECAs meet both “shading” criteria, for the original CASS 2C1Dr2
CA code {3818817080,2,2} has been introduced in [66]. The first, second and third
values in the latter code represent the decimal enumeration of the TR outputs, the
number of possible states (colors) and the neighborhood size, respectively. Figure
11.3 shows the best possible, that is the ideal SIC for a 13
×
13 array at fixed bound-
ary conditions with this CA.
Fig. 11.3
13
×
13 array {3818817080,2,2} CA at SIC*: {2 5 2 175415411}rendersthe
opacity transition with the lowest GDE = 0.58 and GFE = 0.42. From the top: 1): Black dots
indicate the value of grayness (the average density) of each CA pattern. Dashed line indicates
the referential proportionality. Dark gray filling indicates GFE at particular IC: excessive and
insufficient densities are shown over and under the reference line, respectively. Similarly, light
gray filling indicates GDE at particular IC. 2): The sequence of patterns. 3): The sequence of
gray levels equivalent to the graynesses of the CA patterns shown above.
11.3.3
Half-Distance Automata
So called half-distance rules can be created by shifting the successive rows, so
the number of input cells becomes even. If an underlying cell is placed between
the two cells above, it is called radius
2
(r-
2
)CA.Forradii
2
(r-
2
)and
2
(r-
2
), the
underlying cell is placed between the corresponding 4 and 6 cells, respectively. For
an interactive demonstration see [40]. There are only 16 r-
2
1D2C such automata,
none of them particularly interesting visually. However, the number of correspond-
ing r-
2
automata is 65,536. Half of them, that is 32,768, being EN which is rather
manageable for simple analysis. As Fig. 11.4 indicates many of them have low GFE.
As Fig. 11.4 indicates, there are many, namely 720, r-
2
CA with very low GFE, that
is below 0.05. The six automata whose GF plots are shown on the right have been
selected according to the following criteria:
1
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