# Multibaseline Interferometric Phase Denoising Based on Kurtosis in the NSST Domain

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Signal Model

#### 2.2. Denoising Based on NSST

#### 2.2.1. The Nonsubsampled Shearlet Transform

#### Step 1: Multiscale Decomposition

#### Step 2: Direction Localization

#### 2.2.2. Pre-Thresholded Wiener Filter

#### 2.3. Noise Level Estimation Based on Kurtosis

#### 2.3.1. Kurtosis

#### 2.3.2. Noise Level Estimation

- the vector $k\in {\mathbb{R}}^{N}$$$k={\left(\right)}^{\sqrt{\kappa \left({x}^{1}\right)}}T$$
- A is a diagonal matrix of $N\times N$ and the diagonal element is$${A}_{ii}=\sum _{i=0}^{M}{\left(\right)}^{\frac{{\left(\right)}_{{\widehat{\sigma}}_{n}^{2}}}{t}}-1$$
- B is a symmetric matrix$${B}_{ij}=\left(\right)open="\{"\; close>\begin{array}{cc}N-1\hfill & i=j;\hfill \\ -1\hfill & otherwise.\hfill \end{array}$$
- the vector $C\in {\mathbb{R}}^{N}$$${c}_{i}=\sum _{i=0}^{M}2\sqrt{\kappa \left({y}_{i}^{j}\right)}\left(\right)open="("\; close=")">\frac{{\left(\right)}_{{\widehat{\sigma}}_{n}^{2}}}{t}{\sigma}_{{y}_{i}}^{2}.$$

Algorithm 1 Estimating the local noise variance ${\left(\right)}_{{\left({\widehat{\sigma}}_{n}^{2}\right)}_{l}}^{}$ |

Input:$N\times M$ NSST coefficients ${\left(\right)}_{{Y}_{j}^{i}}^{}$ of the observed interferograms with N different baselines, the size of patch $m\times n$ and the maximum iteration number ${N}_{iter}$. |

Initialization:${\left(\right)}_{{({\widehat{\sigma}}_{n}^{2})}_{l}}^{}$ = 0. |

1: Divide all coefficients into L patches whose size is $m\times n$ and calculate the kurtosis ${\left(\right)}_{{\kappa}_{l}}^{\left({y}_{i}^{j}\right)}$ and variance ${\left(\right)}_{{({\sigma}_{{y}_{i}^{j}}^{2})}_{l}}^{}$ of eath patch. |

2: Repeat. |

3: Let ${\left(\right)}_{{\left({\sigma}_{n}^{2}\right)}_{l}}^{}$ equals the solution of the last optimization, update ${\left(\right)}_{{\widehat{\kappa}}_{l}}^{\left(x\right)}$ by optimization function 1. |

4: Let ${\left(\right)}_{{\kappa}_{l}}^{\left(x\right)}$ equals the solution of the step 3, update ${\left(\right)}_{{({\widehat{\sigma}}_{n}^{2})}_{l}}^{}$ by optimization function 2. |

5: Until ${\left(\right)}_{{({\widehat{\sigma}}_{n}^{2})}_{l}}^{}$ and ${\left(\right)}_{{\widehat{\kappa}}_{l}}^{(x)}$ converges or ${N}_{iter}$ is reached. |

6: Return ${\left(\right)}_{{({\widehat{\sigma}}_{n}^{2})}_{l}}^{}$. |

## 3. Results

- Goldstein: the filtering window size is $32\times 32$, $\alpha $ equals 0.9;
- OADF: the filtering window size is $7\times 7$;
- LFE: the local frequency estimation window and filtering window are set to $9\times 9$;
- NL-InSAR: the iterative number is 10;
- InSAR-BM3D: the parameters are consistent with [15];
- NSST: the decomposition scale equals 5. Each scale contains 16 different directions.

#### 3.1. Noise Estimation Experiments

#### 3.2. Simulated Experiments

- MSE: NSST>InSAR-BM3D>NL-InSAR>OADF>LFE>Goldstein
- GMSM: NSST>InSAR-BM3D>NL-InSAR≥LFE>OADF>Goldstein
- Computation efficiency: Goldstein>NSST>InSAR-BM3D>OADF>LFE>NL-InSAR

#### 3.3. Experiments on Real Interferograms

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The partition of frequency domain; (

**b**) Frequency structure of the shearlet ${\widehat{\psi}}_{j,l,k}({w}_{1},{w}_{2})$, for ${w}_{1}>0,{w}_{2}>0$.

**Figure 4.**Five different topography (top) and their typical interferogram (bottom): (

**a**)–(

**e**) represent cone, building, plain, basin and plateau respectively.

**Figure 6.**Boxplot of 100 noise level estimation experiments corresponding to each coherence (the black dotted line is the mean of true value).

**Figure 8.**The filter results of the interferogram generated by a cone with coherence of 0.5 (top) and the statistical result of pixels at the center row (bottom, the black solid line is the true value; the blue dotted line denotes the mean of 100 experiments; the pale blue shadow is the range of three times standard deviation near the mean.).

**Figure 9.**Clean interferograms and noisy interferograms generated by a cone with coherence ranging from 0.1 to 0.9.

**Figure 10.**The filter results of the interferogram generated by a cone with variable coherence (

**top**), the residuals graph (

**middle**) and the Gradient Magnitude Similarity (GMS) map (

**bottom**).

**Figure 11.**Clean interferograms and noisy interferograms generated by a mountain with coherence ranging from 0.1 to 0.9.

**Figure 12.**The filter results of the interferogram generated by a complex topography with variable coherence (

**top**), the residuals graph (

**middle**) and the GMS map (

**bottom**).

**Figure 13.**The real interferograms with different baseline (the length of baseline increase form left to right).

**Figure 15.**The filtered results of the low-coherence reagion (the upper right corner of real interferogram with the longest baseline (row: 1:1000, column: 4910:5910)).

**Figure 16.**The real interferogram with the longest baseline(the order of white lines increases from left to right, from top to bottom).

**Figure 17.**The phase profile of white lines in Figure 16 (the red, green and blue solid line represent the result of NL-InSAR, InSAR-BM3D and NSST, respectively.).

**Table 1.**Parameters of multibaseline interferometric synthetic aperture radar (InSAR) simulation system.

Parameters | Value |
---|---|

Height | 642 km |

Cental Frequency | 3∼9.6 GHz |

Bandwidth | 100 MHz |

Baseline | 50∼500 m |

Look Angle | $34.5$° |

Baseline Orientation Angle | 5° |

**Table 2.**The standard deviation of kurtosis corresponding to various topography (including Cone, Building, Plain, Basin, Plateau).

Topograpgy | Cone | Building | Plain | Basin | Plateau |

Standard Deviation | 0.006 | 0.0298 | 0.1056 | 0.0442 | 0.0064 |

Coherence | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 |
---|---|---|---|---|---|

Actual Value | 0.7044 | 0.6830 | 0.6351 | 0.5469 | 0.3632 |

Maximum Error Rate(%) | 3.35 | 2.37 | 1.62 | 3.31 | 8.76 |

MSE | Residues | Times (s) | |
---|---|---|---|

Noisy Image | 1.7897 | 34492 | – |

Goldstein | 1.853 | 21041 | 0.32 |

LFE | 0.7699 | 1454 | 113.39 |

OADF | 0.8951 | 389 | 59.45 |

NL-InSAR | 0.6577 | 290 | 459.33 |

InSAR-BM3D | 0.6014 | 0 | 38.02 |

NSST | 0.4954 | 0 | 12.68 |

MSE | Residues | GMSM | Times (s) | |
---|---|---|---|---|

Noisy Image | 2.1571 | 11679 | 0.8297 | – |

Goldstein | 1.9478 | 8490 | 0.8562 | 0.15 |

LFE | 1.5504 | 4660 | 0.8975 | 39.00 |

OADF | 1.4209 | 317 | 0.8684 | 20.95 |

NL-InSAR | 1.3339 | 1211 | 0.8994 | 96.66 |

InSAR-BM3D | 1.0631 | 14 | 0.9103 | 10.59 |

NSST | 0.9841 | 9 | 0.9343 | 5.07 |

MSE | Residues | GMSM | Times (s) | |
---|---|---|---|---|

Noisy Image | 2.1870 | 518970 | 0.6876 | – |

Goldstein | 1.9655 | 371334 | 0.7534 | 4.30 |

LFE | 1.5598 | 199348 | 0.8284 | 2784.72 |

OADF | 1.4402 | 9138 | 0.7828 | 901.24 |

NL-InSAR | 1.2885 | 18963 | 0.8648 | 5491.40 |

InSAR-BM3D | 1.0564 | 126 | 0.9026 | 543.13 |

NSST | 1.0386 | 148 | 0.9213 | 272.79 |

Residues | Residues Reduction Rate | Times(s) | |
---|---|---|---|

Noisy Image | 174198 | – | – |

Goldstein | 124397 | 28.59 | 0.9 |

LFE | 60714 | 65.15 | 720.00 |

OADF | 10899 | 93.74 | 433.47 |

NL-InSAR | 15866 | 90.89 | 2542.23 |

InSAR-BM3D | 16672 | 90.42 | 171.69 |

NSST | 1374 | 99.21 | 91.57 |

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**MDPI and ACS Style**

Liu, Y.; Li, S.; Zhang, H.
Multibaseline Interferometric Phase Denoising Based on Kurtosis in the NSST Domain. *Sensors* **2020**, *20*, 551.
https://doi.org/10.3390/s20020551

**AMA Style**

Liu Y, Li S, Zhang H.
Multibaseline Interferometric Phase Denoising Based on Kurtosis in the NSST Domain. *Sensors*. 2020; 20(2):551.
https://doi.org/10.3390/s20020551

**Chicago/Turabian Style**

Liu, Yanfang, Shiqiang Li, and Heng Zhang.
2020. "Multibaseline Interferometric Phase Denoising Based on Kurtosis in the NSST Domain" *Sensors* 20, no. 2: 551.
https://doi.org/10.3390/s20020551