#### The Discrete Walsh Analysis

We briefly outline the discrete Walsh analysis here (

Figure 1). For details of the mathematical procedures, see previous studies [

3,

4,

6]. The Walsh function

$wal\left(r,x\right)$ of order

$r$ and argument

$x$ can be represented over the interval

$0\le x<1$ as follows:

where

${r}_{i}=0\text{or}1$,

$r={\sum}_{i=0}^{m-1}{r}_{i}{2}^{i}$, and

$m$ is the smallest positive integer such that

${2}^{m}>r$. Dyadic addition of non-negative integers

r and

s is defined as

$r\oplus s={\sum}_{i=0}^{f}\left|{r}_{i}-{s}_{i}\right|{2}^{i}$, where

$r={\sum}_{i=0}^{f}{r}_{i}{2}^{i}$ and

$s={\sum}_{i=0}^{f}{s}_{i}{2}^{i}$. Consequently, the product of the two Walsh functions is given by

$wal\left(r,x\right)wal\left(s,x\right)=wal(r\oplus s,x)$. From this calculation, the Walsh function can be shown to form an orthonormal set. In this case, we can define a Walsh transform, similar to a Fourier transform.

**Figure 1.**
(**a**) Examples of the two-dimensional discrete Walsh functions for $m=n=8$. Black represents +1, and white represents –1; and (**b**) four types of symmetry in terms of the discrete Walsh function.

**Figure 1.**
(**a**) Examples of the two-dimensional discrete Walsh functions for $m=n=8$. Black represents +1, and white represents –1; and (**b**) four types of symmetry in terms of the discrete Walsh function.

The two-dimensional discrete Walsh function can be represented in matrix form as $\left[{W}_{mn}\left(i,j\right)\right]$, where ${W}_{mn}\left(i,j\right)$ is the amount of the (m,n)th -order Walsh function in the ith row cell in the jth column.

This pattern can be written as

$\left[{x}_{ij}\right]$, where

${x}_{ij}$ is the amount of the gray level in the

${i}^{th}$ row cell in the

jth column and

$i,j=0,1,\dots ,N-1.$ If just two gray levels exist, for example, “black” and “white”, then

${x}_{ij}$ is typically represented by 1 and 0, respectively (

Figure 1a). The two-dimensional discrete Walsh transform of the pattern

$\left[{x}_{ij}\right]$ is given by:

where

$m,n=0,1,\dots ,N-1.$ The functions

${a}_{mn}$ and

${\left({a}_{mn}\right)}^{2}$ are the two-dimensional Walsh spectrum and power spectrum, respectively.

The Walsh power spectrum can be normalized as follows:

where

$K={\sum}_{m=0}^{N-1}{\sum}_{n=0}^{N-1}{\left({a}_{mn}\right)}^{2}-{\left({a}_{00}\right)}^{2}$. In this case, we obtain:

where the sum is taken over all ordered pairs

$\left(m,n\right)$ for

$0\le m,n\le N-1$, except for

$\left(m,n\right)=\left(0,0\right)$.

The spatial pattern is considered as an information source, consisting of dot patterns. The dot patterns emitted from the source are assumed to occur with the corresponding probabilities given by Equation (4).

Next, we consider the information entropy resulting from the symmetry of the pattern. Because the two-dimensional Walsh functions can be easily divided into four types of symmetry (

Figure 1b), Equation (4) can be rewritten as:

where vertical symmetric component:

horizontal symmetric component:

centro symmetric component:

and double symmetric component:

When the spatial pattern is regarded as an information source consisting of four types of symmetry, the corresponding probabilities are given by Equations (6)–(9). Applying the entropy function from information theory to these four symmetric components, we obtain:

As shown in Equations (5) and (10),

$S$ ranges from 0 to 2 bits. Since this entropy relates to the symmetry, it is called the “symmetropy” [

6]. The symmetropy can be considered as a quantitative and objective measure of symmetry. If the amount of a certain component is larger than the amounts of the other three components, then the pattern is rich in symmetry related to the particular component. In this case, Equation (10) shows that

$S$ decreases. Conversely, if the amounts of the four components are almost equal to one another, then the pattern is poor in symmetry, and

$\text{}S\text{}$ increases. Previous studies have concentrated on symmetry [

4]. Following this idea, in this study, we measured the symmetry of the initial condition and the final pattern to examine the effects on final patterns of the initial condition of symmetry in CA patterns.