## 2 Methodology of Mkcis
### Nonlocal Means Filter in PCA  
The redundancy information in images is adequately exploited by nonlocal means filter, which means that every small patch in an image have many similar patches in the same image. As a result, the speckle randomly distributed in the image can be suppressed by these similar local patches. If \(I\) is an SAR image with noises, \(x(i)\) is the observed image and \(NL(i)\) is the noise removed image by nonlocal means filter, the denoising operator can be defined as the weighted average of the pixels in the original image.  
\[NL\big{(}i\big{)}=\sum\
olimits_{j\in I}w\big{(}i,j\big{)}x\big{(}j\big{)} \tag{1}\]  
\[w(i,j)=\frac{1}{\sqrt[]{x}\big{(}i\big{)}}e^{\frac{\|x\big{(} NB_{i}\big{)}\cdot\big{(}NB_{i}\big{)}\|_{x}}{x^{i}}},\forall\ (i)=\sum\
olimits_{j}e^{\frac{\|x\big{(} NB_{i}\big{)}\cdot\big{(}NB_{j}\big{)}\|_{x}}{x^{i}}} \tag{2}\]  
The weights in Equation (1) measure the similarity of pixel \(i\) and their neighborhoods \(NB_{i}\), \(v(NB_{i})\) is the vector of neighboring pixel around \(i\), \(a\) is the standard deviation of the Gaussian kernel, and \(h\) controls the decay of the exponential function.  
Recently, the introduction of lower dimensional subspace of neighboring vectors combing with nonlocal mean (NLM) filter can save computational cost and performs more robust to noise than distances calculated in the full-dimensional space. The covariance matrix of neighboring pixel can be listed as follows.  
\[C_{NB}=\frac{1}{i}\sum\limits_{i\in i}\Big{(}v\big{(}NB_{i}\big{)}-\overline{ v\big{(}NB_{i}\big{)}}\Big{)}\Big{(}v\big{(}NB_{i}\big{)}-\overline{v\big{(}NB_{i} \big{)}}\Big{)}^{T} \tag{3}\]  
\(i\) is the set of neighboring pixel around \(i\). Compute the eigenvectors \(E=\{e_{1}\), \(e_{2}\), \(\ldots\), \(e_{N}\}\) and eigenvalues \(\Phi=\{\)\(\lambda_{1}\), \(\lambda_{2}\), \(...\), \(\lambda_{N}\}\) by de-compositing above matrix. If the \(d\) larger eigenvectors are  
Figure 1: The procedure of the synthetic aperture radar (SAR) image segmentation algorithm by using the nonlocal means filter in principal component analysis (PCA) and multi-objective kernel clustering in artificial immune system (AIS).  
selected to span _d_-dimensional space, the original neighboring vector in (2) can be denoted in the low dimensional space.  
\[v_{d}\left(NB_{i}\right)=\sum_{i=1}^{d}\left\langle v\left(NB_{i}\right),e_{i} \right\rangle e_{i} \tag{4}\]  
Hence, if we use Equation (4) to replace the item \(v(NB_{i})\) in Equation (2). \(<\)\(v(NB_{i})\), \(e\)\(>\) denotes the inner product of the two vectors. The meaning of Equation (1) can be updated by non-local mean estimator in low-dimensional space.  
How to select the dimensionality of the subspace is the essential issue of the NLM in PCA. Here, knee point detection using maximum gradient tracing to select the changing point of the curve of eigenvalues is used here [11,18]. Here, local image patches are selected and their covariance matrixes are decomposed. By choosing different number of eigenvalues to reconstruct the local patches, the differences or map of the reconstructed patches and the patches rebuilt by only one largest eigenvector are plot and maximum gradient of the knee point can be found. The main idea is shown in Figure 2.