# Identification of Statistically Homogeneous Pixels Based on One-Sample Test

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Signal Suppression

#### 2.2. Outlier Removal

#### 2.3. One-Sample Test

## 3. Experiments and Discussion

#### 3.1. Monte Carlo Simulation

- (i)
**Temporal change: No; Outlier: No**The two samples possess neither temporal changes nor outliers. Three types of distributions are analyzed. The distributional parameters are shown in Table 1.- (ii)
**Temporal change: No; Outlier: Yes**The two samples possess outliers without temporal changes. The distributional parameters are the same as Table 1; however, both samples include 5% outliers. The magnitudes of outliers are set to be $\mu +5\sigma $, where μ and σ represent the mean and standard deviation of the sample.- (iii)
**Temporal change: Yes; Outlier: No**A temporal change is designed in the first sample. The time of change is always at $N/2$ for different sample sizes, meaning that observations before and after $N/2$ follow different distributional parameters. The corresponding parameters are shown in Table 2.- (iv)
**Temporal change: Yes; Outlier: Yes**A temporal change is designed in the first sample. The distributional parameters are the same as Table 2; however, both samples include 5% outliers. The magnitudes of outliers are set to be $\mu +5\sigma $.

#### 3.2. Experiments with SAR Data Stack

## 4. Conclusions

- Removing signals and outliers before conducting any hypothesis tests is advantageous. By reducing the impacts of these measurements, the hypothesis testing can deal only with the stochastic processes. This not only makes the parametric tests applicable, but also augments the power of the test operation.
- Considering temporal variabilities is pragmatically necessary, especially when dealing with data stacks crossing through a large temporal spacing. The proposed approach helps to identify SHP family even with temporal variations.
- Since having large sample sizes can lower the probability of conducting type I and II errors, the proposed approach becomes useful in mitigating the impact of temporal variabilities while keeping the large temporal spacing.
- The difference of two time series (i.e., $\psi \left({P}_{ij}\right)$) keeps the temporal sequence that would be lost when building sample CDF in tests like KS or AD. This also helps to improve the effectiveness of hypothesis tests.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Research flowchart. The main procedures include three steps: (1) signal suppression; (2) outlier removal; and (3) one-sample test.

**Figure 6.**The area of interest (white box) in Hong Kong retrieved from World Imagery (Esri, Redlands, CA, USA). Two areas were selected for detailed comparisons.

**Figure 7.**Number of statistically homogeneous pixels (SHP) identified at each pixel based on different tests and stack sizes. The significance level is 5%. (

**a**–

**c**) represent the SHP number of the Anderson-Darling (AD) test using 10, 30, 75 images, respectively; (

**d**–

**f**) correspond to the SHP number of Robust-T (TR) test using 10, 30, 75 images, respectively.

**Figure 8.**Histograms of the SHP number. Horizontal axis is the number of SHP based on a 15 × 15 window. Vertical axis is the pixel counts at the corresponding SHP number; (

**a**–

**f**) correspond to Figure 7a–f, respectively.

**Figure 9.**Amplitude denoising of Area 1 based on different approaches and stack sizes. (

**a**) an optical image retrieved from Google Earth; (

**b**) an unfiltered amplitude map acquired on the 25 October 2008; (

**c**) a reflectivity map generated by the original data stack (75 images); (

**d**–

**g**) SHP number correspond to ${\mathrm{AD}}^{10imgs}$, ${\mathrm{AD}}^{75imgs}$, ${\mathrm{TR}}^{10imgs}$, and ${\mathrm{TR}}^{75imgs}$, respectively; (

**h**–

**k**) denoised amplitude maps correspond to ${\mathrm{AD}}^{10imgs}$, ${\mathrm{AD}}^{75imgs}$, ${\mathrm{TR}}^{10imgs}$, and ${\mathrm{TR}}^{75imgs}$, respectively; and (

**l**–

**o**) denoised reflectivity maps correspond to ${\mathrm{AD}}^{10imgs}$, ${\mathrm{AD}}^{75imgs}$, ${\mathrm{TR}}^{10imgs}$, and ${\mathrm{TR}}^{75imgs}$, respectively.

**Figure 10.**Amplitude denoising of Area 2 based on different approaches and stack sizes. (

**a**) an optical image retrieved from Google Earth; (

**b**) an unfiltered amplitude map acquired on 25 October 2008; (

**c**) a reflectivity map generated by the original data stack (75 images); (

**d**–

**g**) SHP number corresponds to ${\mathrm{AD}}^{10imgs}$, ${\mathrm{AD}}^{75imgs}$, ${\mathrm{TR}}^{10imgs}$, and ${\mathrm{TR}}^{75imgs}$, respectively; (

**h**–

**k**) denoised amplitude maps correspond to ${\mathrm{AD}}^{10imgs}$, ${\mathrm{AD}}^{75imgs}$, ${\mathrm{TR}}^{10imgs}$, and ${\mathrm{TR}}^{75imgs}$, respectively; and (

**l**–

**o**) denoised reflectivity maps correspond to ${\mathrm{AD}}^{10imgs}$, ${\mathrm{AD}}^{75imgs}$, ${\mathrm{TR}}^{10imgs}$, and ${\mathrm{TR}}^{75imgs}$, respectively.

**Table 1.**Distributional parameters for case (i) and (ii). λ, k, θ, Ω, m, u, and σ represent the scale, shape, scale, spread, shape parameters, log mean, and log standard deviation, respectively.

Rayleigh | Gamma | Nakagami | Lognormal | Inverse Gaussian | Exponential | |
---|---|---|---|---|---|---|

Sample 1 | $\lambda =0.20$ | $k=1.00$ $\theta =0.20$ | $\mathsf{\Omega}=1.00$ $\mu =0.20$ | $\sigma =1.00$ $\mu =0.20$ | $m=1.00$ $\mu =0.20$ | $\mu =1.00$ |

Sample 2 | $\lambda =0.24$ | $k=1.00$ $\theta =0.26$ | $\mathsf{\Omega}=1.00$ $\mu =0.25$ | $\sigma =1.00$ $\mu =0.50$ | $m=1.00$ $\mu =0.23$ | $\mu =1.50$ |

**Table 2.**Distributional parameters for case (iii) and (iv). λ, k, θ, Ω, m, u, and σ represent the scale, shape, scale, spread, shape parameters, log mean, and log standard deviation, respectively. Subscripts b and a indicate the parameters before and after the temporal change, respectively.

Rayleigh | Gamma | Nakagami | Lognormal | Inverse Gaussian | Exponential | |
---|---|---|---|---|---|---|

Sample 1 | ${\lambda}_{b}=0.20$ ${\lambda}_{a}=0.24$ | $k=1.00$ ${\theta}_{b}=0.20$ ${\theta}_{a}=0.26$ | $\mathsf{\Omega}=1.00$ ${\mu}_{b}=0.20$ ${\mu}_{a}=0.25$ | $\sigma =1.00$ ${\mu}_{b}=0.20$ ${\mu}_{a}=0.50$ | $m=1.00$ ${\mu}_{b}=0.20$ ${\mu}_{a}=0.23$ | ${\mu}_{b}=1.00$ ${\mu}_{a}=1.50$ |

Sample 2 | $\lambda =0.24$ | $k=1.00$ $\theta =0.26$ | $\mathsf{\Omega}=1.00$ $\mu =0.25$ | $\sigma =1.00$ $\mu =0.50$ | $m=1.00$ $\mu =0.23$ | $\mu =1.50$ |

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Lin, K.-F.; Perissin, D. Identification of Statistically Homogeneous Pixels Based on One-Sample Test. *Remote Sens.* **2017**, *9*, 37.
https://doi.org/10.3390/rs9010037

**AMA Style**

Lin K-F, Perissin D. Identification of Statistically Homogeneous Pixels Based on One-Sample Test. *Remote Sensing*. 2017; 9(1):37.
https://doi.org/10.3390/rs9010037

**Chicago/Turabian Style**

Lin, Keng-Fan, and Daniele Perissin. 2017. "Identification of Statistically Homogeneous Pixels Based on One-Sample Test" *Remote Sensing* 9, no. 1: 37.
https://doi.org/10.3390/rs9010037