# Parallel Compact Differential Evolution for Optimization Applied to Image Segmentation
## 3 Parallel Compact Differential Evolution
Table 3 shows the test results, where '+' represents that the performance of the



\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Function** & **cDE** & **pCDE (Optimal Elite)** & **pCDE (Mean Elite)** \\ \hline \hline  & & \multicolumn{2}{c}{n = 10} \\ \hline



mathbf{0.000\times

\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Function** & **cDE** & **pCDE (Optimal Elite)** & **pCDE (Mean Elite)** \\ \hline \hline  & \multicolumn{3}{c}{n = 10} \\ \hline



mathbf{0.000\times 10^{0}\pm 0.00\times 10^{0}}\) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & \multicolumn{2}{c}{**Scheme 1: Rand/1/Bin**} & \multicolumn{2}{c}{**Scheme 2: Best/1/Bin**} & \multicolumn{2}{c}{**Scheme 3: Rand-to-Best/1/Bin**} \\ \hline  & **pcDE (Optimal Elite)** & **pcDE (Mean Elite)** & **pcDE (Optimal Elite)** & **pcDE (Mean Elite)** & **pcDE (Optimal Elite)** & **pcDE (Mean Elite)** \\  & **and pcDE** & **and pcDE** & **and pcDE** & **and pcDE** & **and pcDE** & **and pcDE** \\ \hline  & & & & & \\ \hline \multicolumn{5}{c}{n = 30} \\ \hline \(1\) & \(+\) & \(+\) & \(+\) & \(-\) & \(+\) & \(+\) \\ \hline \(2\) & \(+\) & \(-\) & \(+\) & \(-\) & \(+\) & \(-\) \\ \hline \(3\) & \(+\) & \(+\) & \(-\) & \(-\) & \(-\) & \(-\) \\ \hline \(4\) & \(+\) & \(-\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \(5\) & \(+\) & \(-\) & \(+\) & \(-\) & \(+\) & \(+\) \\ \hline \(6\) & \(+\) & \(-\) & \(+\) & \(-\) & \(+\) & \(+\) \\ \hline \(7\) & \(+\) & \(-\) & \(+\) & \(-\) & \(+\) & \(-\) \\ \hline \(8\) & \(+\) & \(-\) & \(+\) & \(-\) & \(+\) & \(-\) \\ \hline \(9\) & \(+\) & \(+\) & \(-\) & \(+\) & \(+\) & \(+\) \\ \hline \(10\) & \(+\) & \(+\) & \(+\) & \(-\) & \(-\) & \(-\) \\ \hline \(11\) & \(-\) & \(-\) & \(+\) & \(-\) & \(+\) & \(+\) \\ \hline \(12\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \(13\) & \(-\) & \(-\) & \(+\) & \(-\) & \(+\) & \(+\) \\ \hline \(14\) & \(+\) & \(-\) & \(+\) & \(-\) & \(+\) & \(+\) \\ \hline \(15\) & \(+\) & \(-\) & \(+\) & \(-\) & \(-\) & \(-\) \\ \hline \(16\) & \(+\) & \(+\) & \(+\) & \(-\) & \(+\) & \(+\) \\ \hline \(17\) & \(+\) & \(-\) & \(+\) & \(-\) & \(+\) & \(-\) \\ \hline \multicolumn{5}{c}{n various} \\ \hline \(18\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \(19\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \(20\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \(21\) & \(-\) & \(-\) & \(+\) & \(+\) & \(+\) \\ \hline \(22\) & \(+\) & \(-\) & \(+\) & \(+\) & \(-\) & \(-\) \\ \hline \(23\) & \(-\) & \(-\) & \(-\) & \(+\) & \(+\) \\ \hline \(24\) & \(+\) & \(+\) & \(+\) & \(+\) & \(-\) & \(+\) \\ \hline \(25\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \end{tabular}
\end{table} 
Table 3: Two communication strategies under three different schemes. Figure 4: _Cont._

## 5 Case of Study: Parallel Compact Differential Evolution for Image Segmentation

In this section, we employ the pcDE algorithm to implement image threshold segmentation. The differences between pcDE and traditional methods are compared. For the image Img(\(x\), \(y\)), we suppose that its size is \(width*height\) and the gray level is L, the neighborhood average gray value g of \((2*n+1)^{2}\) at the \((x,y)\) point is:

\[g(x,y)=\frac{1}{(2*n+1)^{2}}\sum_{i=-n}^{n}\sum_{j=-n}^{n}Img(x+i,y+j) \tag{10}\]

We use the point gray value and \((2*n+1)*(2*n+1)\) neighborhood average gray value to establish a two-dimensional gray histogram F(\(x\), \(y\)), then segment F(\(x\), \(y\)) with a two-dimensional threshold (s, t). Let \(N_{i,j}\) be the number of pixels whose gray value is i and neighborhood average gray value is j, \(P_{i,j}\) be the probability, then

\[P_{i,j}=\frac{N_{i,j}}{width*height} \tag{11}\]

\(\{P_{i,j},i,j=1,2,3...L\}\) is the two-dimensional gray histogram of the image, which is shown in Figure 5. Figure 5 shows that the peak of the probability of the point-neighbor average gray is mainly distributed near the diagonal of the plane, and the overall appearance is bimodal.