# Analysis of Stochastic Distances and Wishart Mixture Models Applied on PolSAR Images

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## Abstract

**:**

## 1. Introduction

- (a)
- Data set type (numerical real, numerical complex, categorical);
- (b)
- Data set normalization need;
- (c)
- Outliers, and how to deal with them;
- (d)
- Number of clusters;
- (e)
- Cluster shape;
- (f)
- Similarity measure;
- (g)
- The initial centroid location choice.

## 2. PolSAR Image Representation

## 3. Stochastic Distances

- Bhattacharyya$${d}_{{W}_{B}}({\mathsf{\Sigma}}_{\mathit{X}},{\mathsf{\Sigma}}_{\mathit{Y}})=L\left(\right)open="["\; close="]">\frac{log|{\mathsf{\Sigma}}_{\mathit{X}}|+log|{\mathsf{\Sigma}}_{\mathit{Y}}|}{2}-log\left|{\left(\frac{{\mathsf{\Sigma}}_{\mathit{X}}^{-1}+{\mathsf{\Sigma}}_{\mathit{Y}}^{-1}}{2}\right)}^{-1}\right|$$
- Kullback-Leibler$${d}_{{W}_{KL}}({\mathsf{\Sigma}}_{\mathit{X}},{\mathsf{\Sigma}}_{\mathit{Y}})=L\left(\right)open="["\; close="]">\frac{Tr({\mathsf{\Sigma}}_{\mathit{X}}^{-1}{\mathsf{\Sigma}}_{\mathit{Y}}+{\mathsf{\Sigma}}_{\mathit{Y}}^{-1}{\mathsf{\Sigma}}_{\mathit{X}})}{2}-q$$
- Hellinger$${d}_{{W}_{H}}({\mathsf{\Sigma}}_{\mathit{X}},{\mathsf{\Sigma}}_{\mathit{Y}})=1-{\left[\frac{|{2}^{-1}{({\mathsf{\Sigma}}_{\mathit{X}}^{-1}+{\mathsf{\Sigma}}_{\mathit{Y}}^{-1})}^{-1}|}{\sqrt{|{\mathsf{\Sigma}}_{\mathit{X}}\left|\right|{\mathsf{\Sigma}}_{\mathit{Y}}|}}\right]}^{L}$$
- Rényi of order $\beta $$$\begin{array}{cc}\hfill {d}_{{W}_{R}^{\beta}}({\mathsf{\Sigma}}_{\mathit{X}},{\mathsf{\Sigma}}_{\mathit{Y}})& =\frac{log2}{1-\beta}+\frac{1}{\beta -1}log\left\{\right[|{\mathsf{\Sigma}}_{\mathit{X}}{|}^{-\beta}|{\mathsf{\Sigma}}_{\mathit{Y}}{|}^{\beta -1}|{(\beta {\mathsf{\Sigma}}_{\mathit{X}}^{-1}+(1-\beta ){\mathsf{\Sigma}}_{\mathit{Y}}^{-1})}^{-1}{\left|\right]}^{L}+\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left[\right|{\mathsf{\Sigma}}_{\mathit{Y}}{|}^{-\beta}|{\mathsf{\Sigma}}_{\mathit{X}}{|}^{\beta -1}|{(\beta {\mathsf{\Sigma}}_{\mathit{Y}}^{-1}+(1-\beta ){\mathsf{\Sigma}}_{\mathit{X}}^{-1})}^{-1}{\left|\right]}^{L}\}\hfill \end{array}$$
- Chi-square$$\begin{array}{cc}\hfill {d}_{{W}_{{\chi}^{2}}}({\mathsf{\Sigma}}_{\mathit{X}},{\mathsf{\Sigma}}_{\mathit{Y}})& =\frac{1}{4}\{{\left(\right)}^{\frac{|{\mathsf{\Sigma}}_{\mathit{X}}|}{|{\mathsf{\Sigma}}_{\mathit{Y}}{|}^{2}}}L+\hfill \end{array}$$

## 4. Stochastic Clustering Algorithm

## 5. Expectation Maximization of Wishart Mixture Model

- The Expectation or E-step. In E-step the log-likelihood of the observed data ${\mathit{Z}}_{\mathit{n}}$, given the estimated parameter ${\Psi}^{t}$, is calculated as:$$Q(\Psi ,{\Psi}^{t})=\sum _{n=1}^{N}\sum _{k=1}^{K}{u}_{n,k}\left(\right)open="["\; close="]">log\left({\pi}_{k}^{t}\right)-Llog|{\mathsf{\Sigma}}_{\mathit{k}}^{\mathbf{t}}|+(L-q)log\left|{\mathit{Z}}_{\mathit{n}}\right|-tr\left({\mathsf{\Sigma}}_{\mathit{k}}^{\mathbf{t}}{\mathit{Z}}_{\mathit{n}}\right)$$
- The Maximization or M-step. The M-step finds the new ${\Psi}^{t+1}$ estimation by maximizing $Q(\Psi ,{\Psi}^{t})$:$${\Psi}^{t+1}:\frac{\partial Q(\Psi ,{\Psi}^{t})}{\partial \Psi}=0$$Since the parameter $\Psi $ is composed of ${\pi}_{k}$ and ${\Sigma}_{k}$, the parameter optimization is done by setting the respective partial derivative to zero. The optimization with respect to ${\pi}_{k}^{t+1}$ can be summarized as:$${\pi}_{k}^{t+1}=\frac{1}{N}\sum _{n=1}^{N}{u}_{n,k}$$$${\mathsf{\Sigma}}_{\mathit{k}}^{\mathbf{t}+\mathbf{1}}=\frac{{\sum}_{n=1}^{N}{u}_{n,k}{\mathit{Z}}_{\mathit{n}}}{{\sum}_{n=1}^{N}{u}_{n,k}}$$

## 6. Applications

- Expectation-Maximization for Wishart mixture model distribution (EM-W);
- Stochastic Clustering using Bhattacharyya distance (SC-B);
- Stochastic Clustering using Kullback-Leibler distance (SC-KL);
- Stochastic Clustering using Hellinger distance (SC-H);
- Stochastic Clustering using Rényi of order $\beta $ distance (SC-R). The selected value of the Rényi’s order ($\beta $) was 0.9;
- Stochastic Clustering using Chi-square distance (SC-C).
- K-means using Euclidean distance (KM-E);

#### 6.1. Experiment I

#### 6.1.1. Image Simulation

#### 6.1.2. Monte Carlo Simulation Results

- S01: All six initial centroids were selected from the one class;
- S02: The six initial centroids are distributed over three class;
- S03: The six initial centroids were picked from the borders of two classes;
- S04: Three initial centroids were selected in three different class, and the other three comes from the borders of two classes;
- S05: All initial centroids comes from overlays;
- S06: One initial centroid were picked per class.

#### 6.2. Experiment II

#### 6.2.1. ALOS PALSAR Image Description

#### 6.2.2. Results

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Resolution Cell. (

**b**) Constructive sum of scatters returns. (

**c**) Destructive sum of scatters returns.

**Figure 4.**(

**a**) PolSAR image in 1-look, obtained by R99B sensor, used to simulate PolSAR data. (

**b**) The phantom image (

**c**) Simulated Image example, containing 36 segments and 6 classes.

**Figure 7.**Confusion matrix of the simulated PolSAR image classification results presented in Figure 6.

**Figure 10.**Study area of Tapajós National Forest, located in the state of Pará. The PolSAR color composition ($R={S}_{hh}$, $G={S}_{hV}$ and $B={S}_{vv}$) image and the Truth Map image with the spatial distribution of classes.

**Figure 12.**Confusion matrix of the PalSAR image classification results presented in Figure 11.

**Table 1.**Average complex covariance matrix samples calculated from five samples of each class taken from R99B Sensor Image.

Class Name | Covariance Matrix |
---|---|

Class 1 | $\left[\begin{array}{ccc}0.000761& -0.0000749-i0.000229& 0.000138+i0.000839\\ & 0.002485& -0.000590-i0.000045\\ & & 0.003227\end{array}\right]$ |

Class 2 | $\left[\begin{array}{ccc}0.012859& 0.001219-i0.00071& 0.003911+i0.001879\\ & 0.033695& -0.000849-i0.001182\\ & & 0.015434\end{array}\right]$ |

Class 3 | $\left[\begin{array}{ccc}0.002963& 0.000486+i0.000155& 0.000341+i0.000143\\ & 0.008689& -0.000203-i0.000824\\ & & 0.004335\end{array}\right]$ |

Class 4 | $\left[\begin{array}{ccc}0.001405& -0.0000257-i0.00014& 0.000436+i0.000941\\ & 0.006056& -0.000492-i0.000216\\ & & 0.004237\end{array}\right]$ |

Class 5 | $\left[\begin{array}{ccc}0.000489& -0.0000522-i0.0000627& 0.000138+i0.000529\\ & 0.001211& -0.000330-i0.0000858\\ & & 0.002567\end{array}\right]$ |

Class 6 | $\left[\begin{array}{ccc}0.001870& 0.0000812-i0.000172& 0.000126+i0.000608\\ & 0.0032809& -0.000301-i0.000167\\ & & 0.002586\end{array}\right]$ |

EM-W | SC-B | SC-KL | SC-H | SC-R | SC-C | KM-E | |
---|---|---|---|---|---|---|---|

Average Accuracy | 54.34 | 72.21 | 70.91 | 72.29 | 35.22 | 41.72 | 57.99 |

Average STD | 15.98 | 17.05 | 16.79 | 17.06 | 6.15 | 5.22 | 10.48 |

EM-W | SC-B | SC-KL | SC-H | SC-R | SC-C | KM-E | |
---|---|---|---|---|---|---|---|

S01 | 36.41 | 48.13 | 31.20 | 48.23 | 41.36 | 40.14 | 47.37 |

S02 | 36.06 | 48.02 | 34.45 | 48.12 | 45.46 | 30.74 | 45.72 |

S03 | 49.03 | 64.26 | 65.49 | 63.26 | 46.86 | 33.95 | 59.17 |

S04 | 65.30 | 58.68 | 60.62 | 58.78 | 30.35 | 35.77 | 56.67 |

S05 | 55.72 | 51.21 | 54.75 | 51.31 | 45.23 | 31.81 | 47.20 |

S06 | 95.29 | 94.67 | 93.91 | 94.77 | 47.79 | 44.15 | 62.97 |

EM-W | SC-B | SC-KL | SC-H | SC-R | SC-C | KM-E | |
---|---|---|---|---|---|---|---|

Time | 30.35 | 22.93 | 24.47 | 24.07 | 23.95 | 26.522 | 21.52 |

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**MDPI and ACS Style**

Carvalho, N.C.R.L.; Sant’Anna Bins, L.; Siqueira Sant’Anna, S.J.
Analysis of Stochastic Distances and Wishart Mixture Models Applied on PolSAR Images. *Remote Sens.* **2019**, *11*, 2994.
https://doi.org/10.3390/rs11242994

**AMA Style**

Carvalho NCRL, Sant’Anna Bins L, Siqueira Sant’Anna SJ.
Analysis of Stochastic Distances and Wishart Mixture Models Applied on PolSAR Images. *Remote Sensing*. 2019; 11(24):2994.
https://doi.org/10.3390/rs11242994

**Chicago/Turabian Style**

Carvalho, Naiallen Carolyne Rodrigues Lima, Leonardo Sant’Anna Bins, and Sidnei João Siqueira Sant’Anna.
2019. "Analysis of Stochastic Distances and Wishart Mixture Models Applied on PolSAR Images" *Remote Sensing* 11, no. 24: 2994.
https://doi.org/10.3390/rs11242994