Cellular Automaton Shading for Building Envelopes - Cellular Automata in Image Processing and Geometry - page 204

Fig. 11.2 An example of a sequence of 13 initial conditions (SIC) with the corresponding
encoded SIC*

any sub-matrix, row of cells or group of cells. A pattern with all white (transparent)
cells has grayness 0, with all black (opaque) cells the grayness is 1.

Grayness Function (GF) links the grayness of IC with the grayness of the entire
CA array. In other words, GF is a series of graynesses for consecutive ICs. For
shading, GF should be monotonic, ideally the grayness of the entire array should
be proportional to the grayness of IC. It should also render values from the entire
range, that is from 0 to 1. Such GF ensures that the average opacity of the array can
be fully controlled. For an interactive demonstration see [41].

Grayness Function Error(GFE) is the difference between the ideal (proportional)
GF and the GF of a particular CA at given SIC. For a single CA array A at the given
IC it is expressed as:

W
w = 1

H
h = 1 a h , w
WH

W
w

]= ∑

∑

− ∑

1 a 1 , w
W

=

[

GFE A (11.1)

Where H and W are the height and width of an array (facade) respectively. GFE for
SIC is the result of the summation for all ICs:

F
i = 1 | GFE [ A i ] |

1
F

GFE = (11.2)

Where F is the number of facades' states which equal to the length of SIC, A i is the
i th CA array (for the i th IC of SIC). GFE is usually used for preliminary search for
automata suitable for CASS.
Grayness Monotonicity and Pattern Distribution Error(GDE) . As discussed in
[67], finding the best SIC is a computationally challenging problem. GDE is used
for fine-tuning the SIC as it evaluates each CA pattern considering both the grayness
function monotonicity and the uniformity of the pattern distribution:

1 a 1 , w

( k + 1 ) r

1
Hr

H
h

1
W

W
w

W r −

1

1 a h , w −

∑ ∑ ∑

x = kr = =

k = 1

GDE [ A , r ]= (11.3)

W
r