# Bias Removal for Goldstein Filtering Power Using a Second Kind Statistical Coherence Estimator

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Adaptive Goldstein Filtering

#### 2.2. Coherence Estimation Based on the Second Kind Statistic

#### 2.2.1. Definition of the Coherence Estimator

#### 2.2.2. Weight Determination Weight

#### 2.2.3. Bias Removal Using Second Kind Statistic Estimator

#### 2.3. Modeling Filtering Power Using Unbiased Coherence

#### 2.4. Filter Development

- (1)
- For each image pixel P, a coherence estimate window of size n is defined. The similarity between P and its neighbors is compared individually on an average intensity SAR image with patch size m (Figure 1a), and the weight of the coherence estimate window is confirmed using Equation (6). The sample coherence for pixel P is then estimated using Equation (3).
- (2)
- The procedure is repeated until the sample coherence of the last pixel in the whole image is estimated and coherence map is generated.
- (3)
- Goldstein filtering is performed on the interferogram in which the coherence sample with size k in each filtering patch is first averaged to obtain $\overline{\widehat{\gamma}}$ using Equation (11), and is further corrected to unbiased coherence $\tilde{\gamma}$ according to Equation (12).
- (4)
- The filtering power is adjusted according to Equation (13) and Goldstein filtering is performed on such patches.
- (5)
- Steps (3) and (4) are repeated until the whole interferogram has been filtered.

## 3. Results and Discussion

#### 3.1. Synthetic Data

#### 3.2. Real Data

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Similarity measurement using the patch-based Anderson-Darling test: (

**a**) The procedure of AD estimation; (

**b**) some similar patches (green) with respect to the reference patch (red); (

**c**) the corresponding AD values of the whole window.

**Figure 3.**The best filtering power $\alpha $ under each coherence level. The dashed line denotes the empirical model presented in [9].

**Figure 5.**Synthetic data with 400 $\times $ 400 pixels. (

**a**) Noise-free SAR intensity; (

**b**) original coherence; (

**c**) noise-free interferogram; and (

**d**) noise-added interferogram after the conjugation operator ${z}_{1}{z}_{2}^{\ast}$.

**Figure 6.**The results estimated from different methods. Each column corresponds to a filtering procedure: first (Baran filter), second (AGFC), third (AGFP), and last (new filter). Each row corresponds to a parameter: first (coherence), second (filtering power), and third (filtered phase).

**Figure 9.**Interferogram filtering over incoherent area. (

**a**) Baran filter; (

**b**) AGFC; (

**c**) AGFP; (

**d**) New.

**Figure 11.**Interferogram filtering over coherent area. (

**a**) Baran filter; (

**b**) AGFC; (

**c**) AGFP; (

**d**) New.

**Figure 12.**The ROC curves for evaluating the filtering performances on edge preservation over the coherent area.

Sensor | Track | Acquisition Date | Perp. Baseline | Imaging Area | Interferometric Feature | Mean Coherence |
---|---|---|---|---|---|---|

Sentinel-1A | 128 | 30 July 2017 | 35 m | Jiuzhaigou, China | Incoherent | 0.25 |

11 August 2017 | ||||||

TanDEM-X | - | 1 January 2013 | 286 m | Hong Kong, China | Coherent | 0.76 |

1 January 2013 |

**Table 2.**Statistics of SPD and residue reduction in interferograms estimated from different filters using Sentinel-1A data over the Jiuzhaigou area.

Method | $\mathit{SPD}\text{}(\times {10}^{7})$ | Residue Number | Residue Reduction (%) |
---|---|---|---|

Unfiltered | 2.88 | 393,230 | - |

Baran | 1.80 | 136,828 | 65.20 |

AGFC | 1.72 | 125,197 | 68.16 |

AGFP | 1.05 | 17,647 | 95.51 |

New | 1.35 | 94,460 | 75.98 |

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**MDPI and ACS Style**

Tian, X.; Jiang, M.; Xiao, R.; Malhotra, R.
Bias Removal for Goldstein Filtering Power Using a Second Kind Statistical Coherence Estimator. *Remote Sens.* **2018**, *10*, 1559.
https://doi.org/10.3390/rs10101559

**AMA Style**

Tian X, Jiang M, Xiao R, Malhotra R.
Bias Removal for Goldstein Filtering Power Using a Second Kind Statistical Coherence Estimator. *Remote Sensing*. 2018; 10(10):1559.
https://doi.org/10.3390/rs10101559

**Chicago/Turabian Style**

Tian, Xin, Mi Jiang, Ruya Xiao, and Rakesh Malhotra.
2018. "Bias Removal for Goldstein Filtering Power Using a Second Kind Statistical Coherence Estimator" *Remote Sensing* 10, no. 10: 1559.
https://doi.org/10.3390/rs10101559