# SAR Images Statistical Modeling and Classification Based on the Mixture of Alpha-Stable Distributions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{0}[15], Alpha-stable and heavy-tailed Rayleigh [16,17], and recently proposed HG

^{0}[18], have been applied to the statistical modeling of SAR images. Gao [19] summarizes the development history and the current researching state of statistical modeling and discusses relevant issues in his work. However, as the resolution of the images increases, multimodal statistical properties become apparent due to the different types of backscatter from a single class. For instance, in an SAR image, building areas contain strong reflectors, textures, and shadows. In this case, the previously mentioned unimodal distributions fail to capture such multimodal statistical properties. Thus, a model with the ability to describe these attributes, as well as a heavy tail and sharp peak, is necessary. Alpha-stable distribution is a flexible model that can fit many other distributions into it. The appeal of Alpha-stable distribution also derives from theoretical reason, which states that stable processes arise as limiting processes of sums of independent identically distributed (i.i.d.) random variables via the generalized central limit theorem. Actually, the only possible nontrivial limit of normalized sums of i.i.d. terms is Alpha-stable. Moreover, Alpha-stable distribution has been used as a successful alternative for modeling non-Gaussian data and has also been applied to understand SAR images, e.g., for image restoration [17], object detection [20,21], image classification [22], and image fusion [23]. Applications of Alpha-stable distribution in other areas such as watermark detection, network traffic and stock returns have also been reported [24–26]. Driven by the multimodal and impulsive histogram of high resolution SAR image, as well as the theoretical appeal and successful applications of Alpha-stable family itself, we take the mixture of Alpha-stable distributions that can fit any situation for the statistical modeling of high resolution SAR images.

## 2. Mixture of Alpha-Stable Distributions

#### 2.1. Alpha-Stable Distribution

^{+}and μ ∈ ℝ, representing the characteristic exponent, skewness parameter, dispersion parameter, and location parameter, respectively. The parameter α sets the level of impulsiveness (smaller α gives a more picked PDF and a heavier tail); the parameter β controls the skewness of the PDF (symmetric if β = 0, negatively skewed if β = −1, and positively skewed if β = 1); the dispersion parameter and location parameter are similar to the variance and mean, respectively, in a Gaussian distribution. There are three special cases: the distribution reduces to a Gaussian distribution when α = 2, a Cauchy distribution when α = 1, β = 0, and a Levy distribution when α = 0.5, β = 1. Figure 1 illustrates the Alpha-stable distributions given by various parameter values.

#### 2.2. Mixture of Alpha-Stable (MAS) Distributions

_{k}is the weight of the k

^{th}component. In a Bayesian scheme, we can estimate the distribution parameters via prior information and observations using the Bayesian rule

_{1}, … , x

_{j}, … , x

_{M}), p(θ) is the prior probability, p(X|θ) is thte likelihood of X given θ, and p(X) is the prior probability of X.

#### 2.3. PSA Estimator for MAS Distributions

_{1}, … , x

_{j}, … , x

_{M}) has been randomly drawn from K subpopulations. For each variable x

_{j}, let z

_{j}be an allocation variable for the unobserved or missing variable that indicates to which component x

_{j}belongs. Thus, z

_{j}will be equal to 1 if x

_{j}belongs to the k

^{th}subpopulation, or 0 otherwise.

_{j}is considered to be drawn from the k

^{th}component described by θ

_{k}= (α

_{k}, β

_{k}, γ

_{k}, μ

_{k}) with probability

_{j}can be defined as follows

_{j}}. The candidate parameter ${\theta}_{k}^{\mathit{new}}=\left({\alpha}_{k}^{\mathit{new}},{\beta}_{k}^{\mathit{new}},{\gamma}_{k}^{\mathit{new}},{\mu}_{k}^{\mathit{new}}\right)$ is sampled from a proposal distribution q(·|·), and is accepted with probability ${A}_{{\theta}_{k}^{\mathit{new}}}$, where

_{0}and b

_{0}are parameters of the inverse gamma prior for γ, and κ and ξ are parameters of the normal prior for μ. Moreover, a normal proposal distribution q(·|·) = N(·|δ, σ) is utilized in our algorithm, and its parameters are adaptively updated using estimations obtained in previous iterations: δ is set to the estimation of the previous iteration, and σ is set to the standard deviation of the previous L estimations. Adaptively updating the parameters of the proposal distribution ensures that new candidates can properly explore the entire parameter space, which further ensures that estimations rapidly converge to the true values.

_{1}, x

_{2}, … , x

_{M}), the maximum iteration number Iter, the number of components K, starting temperature T(0), the value of L for updating the parameters of the proposal distribution, and the initial parameters ${\theta}_{k}^{0}$, ${\omega}_{k}^{0}$ for each component.

1: | Input: |

X = {x_{j}},
${\theta}_{k}^{0}=\left({\alpha}_{k}^{0},{\beta}_{k}^{0},{\gamma}_{k}^{0},{\mu}_{k}^{0}\right)$,
${\omega}_{k}^{0}$, k = 1, 2, … , K, L, T(0), Iter | |

2: | for each t < Iter do |

3: | Decrease temperature $T\left(t\right)=\frac{T\left(0\right)}{\text{lg}\left(t+1\right)}$ |

4: | Assign initial parameters ${\theta}_{k}^{\mathit{old}}={\theta}_{k}^{t-1}$, ${\omega}_{k}^{\mathit{old}}={\omega}_{k}^{t-1}$, k = 1, 2, … , K |

5: | For each data sample x_{j}, obtain allocation variable z_{j} using Equation (6) |

6: | Update parameters of the proposal distribution q(·|·) = N (·|δ, σ): set δ to value of the previous iteration and set σ to the standard deviation of the previous L estimations if t > L, otherwise set σ to its initialization value |

7: | Sample new candidates ${\theta}_{k}^{\mathit{new}}=\left({\alpha}_{k}^{\mathit{new}},{\beta}_{k}^{\mathit{new}},{\gamma}_{k}^{\mathit{new}},{\mu}_{k}^{\mathit{new}}\right)$ from proposal distribution q(·|·) = N (·|δ, σ) for each component |

8: | Accept ${\theta}_{k}^{\mathit{new}}$ according to Equation (8) and set ${\theta}_{k}^{t}={\theta}_{k}^{\mathit{new}}$, otherwise set ${\theta}_{k}^{t}={\theta}_{k}^{t-1}$ |

9: | Obtain weight ω = (ω_{1}, … , ω_{k}, …, ω_{K}) of each component as in [27] by drawing samples from distribution ω ∼ D, where D(ζ + n_{1}, …, ζ + n_{k}, … ,ζ + n_{K}) is the Dirichlet distribution with ζ > 0, and n_{k} is the number of samples assigned to the k^{th} component |

10: | end for |

11: | Output: |

θ_{k} = (α_{k}, β_{k}, γ_{k}, μ_{k}), ω_{k}, k = 1, 2, …, K |

#### 2.4. Simulation Result for PSA Estimator on MAS Distributions

_{1∼3}= [1.7, 1.7, 1.7], β

_{1∼3}= [0.7, 0.7, 0.7], γ

_{1∼3}= [1.0, 1.0, 1.0], μ

_{1∼3}= [−4.0, 1.0, 4.0]. The parameters of inverse gamma prior for γ are a

_{0}= 2 and b

_{0}= 3, and those for the normal prior for μ are ξ = 0 and κ = 6. T(0) and L are set to 15 and 200, respectively. Experimental results for the PSA estimation with Iter = 1000 iterations are displayed in Figure 2(a–e). These show that the PSA estimator has a good convergence procedure. Figure 2(f) displays the discrete histogram of the originally simulated data, the PDF curve of the estimated MAS distributions with three components and the PDF curve of the true MAS distributions. This illustrates that the estimated MAS distributions fit the simulated data well.

_{1}, ω

_{2}, ω

_{3}and γ

_{3}are similar and the estimations of PSA for two other parameters μ

_{1}and μ

_{3}are slightly worse than those of Salas-Gonzalez’s method. In addition, the standard deviation obtained by the proposed estimator is smaller than those of Salas-Gonzalez’s method. These results demonstrate the effectiveness of the proposed PSA estimator. Therefore, it can be concluded that the proposed PSA estimator is accurate for MAS distributions.

## 3. MAS-Based Statistical Modeling of SAR Images

_{n}) is the CDF and F̂(x

_{n}) is the EDF. In addition, we also considered the acceptance probability, which is defined as the ratio of the number of samples that the KS test accepts for a distribution (at the 5% significance level) to the total number of samples in a class. The evaluation results are shown in Tables 2 and 3.

^{0}distribution can model marsh and building, but not river or farmland. This indicates that G

^{0}distribution is suitable for modeling heterogeneous but not for homogeneous images. The Alpha-stable distribution is robust for various SAR image classes, as its average KSDs are not larger than 0.046, and it provides the best model for marsh areas. In addition, the maximum KSDs of the Alpha-stable distribution are the smallest of the different distributions tested. However, the average KSDs of the Alpha-stable distribution are still large. Similar results are also displayed in Table 3, which shows that Gamma distribution and Weibull distribution have a high probability of acceptance for modeling river and farmland, whereas they are always rejected for modeling the other two classes. The K distribution is always rejected when modeling marsh areas. The Alpha-stable distribution has a high probability of acceptance, but is not always accepted for modeling all four classes. These results indicate that unimodal distributions are not sufficient for modeling high-resolution SAR images due to the multimodal statistical properties inherent in such images.

## 4. MAS-Based MRF Classification Algorithm

#### 4.1. MRF-Based Segmentation

_{1}, … , x

_{j}, … , x

_{M}) be a random observation and Y = (y

_{1}, … , y

_{j}, … , y

_{M}) be the expected labeling result, with y

_{j}∈ {1, 2, … , K}. In the Bayesian estimation framework, the SAR image classification could be described as a MAP problem, Ŷ = arg max

_{Y}P(Y|X). Based on the Bayesian theory, P(Y|X) ∝ P(X|Y)P(Y), it is equivalent with Ŷ = arg max

_{Y}P(X|Y)P(Y). The restoration method proposed in [38] assumes that the conditional probability of labeling Y = y, given an observation X = x, can be modeled by a Markov random field. Using the Hammersley–Clifford theorem, this probability can be written under Gibbs field formalism as Equation (11):

_{y}is the set of cliques of the selected neighborhood, and V

_{c}is the potential of label configuration. In this paper, we focus solely on the data term, which is evaluated by mixture of Alpha-stable distributions. The second term is simply modeled by the Potts model [39], which can be written as Equation (13):

#### 4.2. MAS-Based MRF Classification Algorithm

## 5. Experiment and Result Analysis

#### 5.1. Experimental Data

#### 5.1.1. Wuhan Data

#### 5.1.2. Foshan Dataset

#### 5.2. Classification Result

#### 5.3. Result Analysis

## 6. Conclusions

## Acknowledgments

**Conflict of Interest**The authors declare no conflict of interest.

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**Figure 1.**PDF of Alpha-stable distributions (μ = 0). (

**a**) Symmetric Alpha-stable case: α = [0.4,0.7,1.0,1.3,2.0], β = 0.0, γ = 1.0; (

**b**) Enlargement of the tails in (a); (

**c**) Skewness parameter: α = 1.2, β = [0.0,0.2,0.5,0.8,1.0], γ = 1.0; (

**d**) Dispersion parameter: α = 1.6, β = 0.0, γ = [0.3,0.6,1.0,2.0,3.0].

**Figure 2.**Simulation results for the PSA estimator. (

**a**)–(

**e**) Evolution procedure of the three components for parameters α, β, γ, μ and ω, respectively (red: component 1, green: component 2, blue: component 3); (

**f**) Fitting results of the estimated MAS distributions to simulated data histogram and the PDF curve obtained by the true parameters.

**Figure 4.**Selected building area and fitting results. (

**a**) Building area in SAR image; (

**b**) Fitting results by different models; (

**c**) Three components of the estimated MAS distributions.

**Figure 6.**TerraSAR-X data of Wuhan city. (

**a**) SAR image of Wuhan, Hubei, China; (

**b**) Optical image from Google Earth (© Google 2013); (

**c**) Visually interpreted map; (

**d**) Classification result of MAS + MRF classifier.

**Figure 7.**TerraSAR-X data of Foshan city. (

**a**) SAR image of Foshan, Guangdong, China; (

**b**) Optical image from Google Earth (© Google 2013); (

**c**) Visually interpreted map; (

**d**) Classification result of MAS + MRF classifier.

**Figure 10.**Classifications of localized area (

**top**, left–right: SAR image, result from GAM + MRF, result from AS + MRF;

**bottom**, left–right: Visually interpreted map, result from MGAM + MRF, result from MAS + MRF).

Parameter | PSA | Salas-Gonzalez | ||||
---|---|---|---|---|---|---|

Name | True | Starting | Estimated | std | Estimated | std |

α_{1} | 1.20 | 1.70 | 1.20 | 0.02 | 1.27 | 0.09 |

β_{1} | 0.50 | 0.70 | 0.53 | 0.01 | 0.65 | 0.08 |

γ_{1} | 1.00 | 1.00 | 1.01 | 0.02 | 0.98 | 0.06 |

μ_{1} | −4.25 | −4.00 | −4.11 | 0.19 | −4.30 | 0.60 |

ω_{1} | 0.40 | 0.33 | 0.40 | 0.01 | 0.40 | 0.02 |

α_{2} | 1.20 | 1.70 | 1.28 | 0.07 | 1.30 | 0.17 |

β_{2} | 0.00 | 0.70 | −0.03 | 0.07 | 0.04 | 0.30 |

γ_{2} | 0.50 | 1.00 | 0.47 | 0.02 | 0.45 | 0.05 |

μ_{2} | 0.30 | 1.00 | 0.25 | 0.07 | 0.40 | 0.30 |

ω_{2} | 0.20 | 0.33 | 0.20 | 0.01 | 0.20 | 0.02 |

α_{3} | 1.50 | 1.70 | 1.45 | 0.04 | 1.37 | 0.12 |

β_{3} | 0.50 | 0.70 | 0.51 | 0.08 | 0.34 | 0.20 |

γ_{3} | 0.30 | 1.00 | 0.30 | 0.01 | 0.30 | 0.02 |

μ_{3} | 3.25 | 4.00 | 3.28 | 0.02 | 3.24 | 0.06 |

ω_{3} | 0.40 | 0.33 | 0.40 | 0.01 | 0.40 | 0.02 |

Average KSD (Maximum KSD) | ||||
---|---|---|---|---|

Model | River | Marsh | Farmland | Building |

Gamma | 0.042 (0.213) | 0.205 (0.427) | 0.044 (0.172) | 0.280 (0.590) |

Weibull | 0.058 (0.207) | 0.109 (0.171) | 0.047 (0.164) | 0.085 (0.152) |

G^{0} | 0.118 (0.915) | 0.051 (0.100) | 0.145 (0.806) | 0.035 (0.287) |

K | 0.024 (0.133) | 0.081 (0.164) | 0.014 (0.069) | 0.064 (0.287) |

Alpha-stable | 0.046 (0.079) | 0.027 (0.071) | 0.042 (0.072) | 0.039 (0.100) |

Acceptance Probability at 5% Significance Level (Percentage) | ||||
---|---|---|---|---|

Model | River | Marsh | Farmland | Building |

Gamma | 81.17 | 0.17 | 71.67 | 0.00 |

Weibull | 55.33 | 0.00 | 70.33 | 0.67 |

G^{0} | 83.83 | 45.44 | 81.69 | 83.33 |

K | 97.00 | 4.17 | 99.83 | 43.83 |

Alpha-stable | 68.83 | 96.50 | 70.83 | 67.50 |

Average KSD (Maximum KSD) | ||||
---|---|---|---|---|

Components | River | Marsh | Farmland | Building |

1 | 0.046 (0.079) | 0.027 (0.071) | 0.042 (0.072) | 0.039 (0.100) |

3 | 0.025 (0.050) | 0.024 (0.088) | 0.012 (0.049) | 0.026 (0.099) |

5 | 0.021 (0.041) | 0.023 (0.113) | 0.011 (0.025) | 0.024 (0.115) |

7 | 0.021 (0.074) | 0.022 (0.110) | 0.010 (0.023) | 0.023 (0.115) |

Acceptance Probability at 5% Significance Level (Percentage) | ||||
---|---|---|---|---|

Components | River | Marsh | Farmland | Building |

1 | 68.83 | 96.50 | 70.83 | 67.50 |

3 | 99.83 | 96.83 | 100.00 | 87.17 |

5 | 100.00 | 97.17 | 100.00 | 88.67 |

7 | 100.00 | 98.00 | 100.00 | 89.33 |

Dataset | Wuhan data (Average accuracy: 84.35) | |||
---|---|---|---|---|

Class | River | Marsh | Farmland | Building |

River | 95.7 | 0.7 | 1.2 | 2.4 |

Marsh | 22.2 | 63.7 | 2.5 | 11.6 |

Farmland | 0.6 | 1.3 | 82.6 | 15.5 |

Building | 3.6 | 0.7 | 9.6 | 86.1 |

Dataset | Foshan data (Average accuracy: 83.22) | |||
---|---|---|---|---|

Class | River | Marsh | Farmland | Building |

River | 80.3 | 11.2 | 2.0 | 6.5 |

Marsh | 4.0 | 87.3 | 3.4 | 5.3 |

Farmland | 0.3 | 0.9 | 74.3 | 24.5 |

Building | 0.7 | 10.0 | 2.5 | 86.8 |

© 2013 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Peng, Y.; Chen, J.; Xu, X.; Pu, F.
SAR Images Statistical Modeling and Classification Based on the Mixture of Alpha-Stable Distributions. *Remote Sens.* **2013**, *5*, 2145-2163.
https://doi.org/10.3390/rs5052145

**AMA Style**

Peng Y, Chen J, Xu X, Pu F.
SAR Images Statistical Modeling and Classification Based on the Mixture of Alpha-Stable Distributions. *Remote Sensing*. 2013; 5(5):2145-2163.
https://doi.org/10.3390/rs5052145

**Chicago/Turabian Style**

Peng, Yijin, Jiayu Chen, Xin Xu, and Fangling Pu.
2013. "SAR Images Statistical Modeling and Classification Based on the Mixture of Alpha-Stable Distributions" *Remote Sensing* 5, no. 5: 2145-2163.
https://doi.org/10.3390/rs5052145