Image Processing Reference
In-Depth Information
Where
A
is a CA array (facade),
H
and
W
are the height and width of the array,
respectively;
r
and
k
are the width and the number of vertical stripes in which the
entire array is subdivided, respectively. The summation over all CA patterns pro-
duced at each IC from the given SIC:
F
f
=
1
CFA
[
A
f
]
1
F
GDE
=
(11.4)
Where
F
is the number of facades' states which equals to the length of SIC,
A
f
is an
f
th
CA array corresponding to the
f
th
IC in SIC. If each of the
r
-wide sub-arrays of
a given CA pattern have the same average densities, the values of GDE and GFE are
equal. For further details see [69]. CA for shading must be both controllable, that is
having sufficiently low GDE, and visually appealing. Although the latter criterion is
rather arbitrary, in most of the cases, the necessary condition is that the CA pattern
is complex, that is of Wolfram class IV behavior [66].
11.3
One-Dimensional Cellular Automata Applied on Surfaces
Although BEs are two-dimensional (2D), the application of one-dimensional (1D)
CA for CASS seems the most practical. This is analogous to the common convention
for presenting a 1D CA by showing the history of its generation changes, where each
row corresponds to a time step in the history. Every row becomes IC for the next
row and so forth. This process continues in a cascade and propagates over the entire
array. Use of a 2D CA may seem more intuitive because this domain is greater than
for one dimensional automata, increasing the chances of finding the best one for
CASS. Moreover, the inter-cell wiring seems to be easier to fabricate. Nevertheless,
as mentioned in [66], there are major concerns regarding the application of 2D CA.
Although their behavior is often intriguing, it is very difficult to control the states
of cells. 2D CA continuously updates all the cells until an equilibrium is reached,
which almost always leads to an uninteresting, uniform state of the array. All the
cells become black or white, often with artifacts (small islands of the opposite state)
and usually with locally strobing cells (switching their state at every step forever). In
general the exact final state of the 2D CA array is difficult or impossible to predict
due to the computational irreducibility. Such 2D array are usually not useful for
shading due to strobing. Controlling the state of 2D arrays is difficult, or perhaps
impossible. This problem could be solved by freezing the array at a certain step and
not allowing it to evolve further, but at present it seems to be a difficult technical
problem. With the adopted common convention of displaying 1D CA, this problem
does not occur because every row displays the state at a certain step, and once set
the state is maintained. The second major problem with 2D CA is the setting of
ICs. How is the initial input given to the cells of a 2D array? A possible solution
where only cells on the edges of the array are used, as done in the 1D case, has been
demonstrated in [63].
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