# The Offset-Compensated Nonlocal Filtering of Interferometric Phase

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Nonlocal Filtering

#### 2.2. Signal Model

#### 2.3. InSAR Nonlocal Filtering

## 3. Nonlocal Filtering with an Offset-Compensated Similarity Measure

- for each phase patch in the neighborhood of the target, estimate the phase offset that minimizes the distance between the target patch and the offset-compensated predictor;
- perform nonlocal filtering with offset-compensated patches in place of the original patches.

#### 3.1. Cosine Dissimilarity

#### 3.2. Preliminary Experiments with Offset-Compensated NLM

Algorithm 1 OC-Switch. | |

Require:$\mathbf{\Psi},{F}_{\mathrm{min}},{R}_{\mathrm{max}}$ | ▹ input patch, decision thresholds |

Ensure: OC-switch | ▹ output on-off switch for offset compensation |

1: set OC-switch to OFF | |

2: $G=PowerSpectrum(\mathbf{\Psi})$ | ▹ compute the power spectrum of $\mathbf{\Psi}$ |

3: $({i}_{M},{j}_{M})=arg{max}_{i,j}G(i,j)$ | ▹ find the peak of G |

4: $F=\sqrt{{\left({i}_{M}\right)}^{2}+{\left({j}_{M}\right)}^{2}}$ | ▹ distance of peak from origin, proxy for frequency |

5: $(I,J)=\{(i,j):G({i}_{M},{j}_{M})-G(i,j)<10$ | ▹ large-G region, within 10 dB of maximum |

6: $R={max}_{(i,j)\in (I,J)}\sqrt{{(i-{i}_{M})}^{2}+{(j-{j}_{M})}^{2}}$ | ▹ radius of large-G region |

7: if $F>{F}_{\mathrm{min}}$ AND $R<{R}_{\mathrm{max}}$ then | |

8: set OC-switch to ON | ▹ activate offset compensation |

9: end if |

## 4. Experimental Results

#### 4.1. Experiments on Simulated Data

#### 4.2. Experiments on Real-World Data

#### 4.3. Computational Performance

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. ENL of Nonlocal Means with Constant Slope Signals

**Figure A1.**The 1d geometry used for ENL computation. Left: small slope, no phase wrapping in the search window. Right: large slope, and phase wrapping. Distances are normalized to the window search size. The target patch, at the center, is highlighted in red, a predictor patch in highlighted in black.

**Figure A2.**Distribution of the normalized nonlocal means weights on the search window (only right half) as a function of phase slope and decay parameter. The four subfigures, from left to right, refer to the normalized slopes $\beta =0$, $\beta =1/2$, $\beta =1$, $\beta =2$. The three plots are for $\lambda =1$ (green), $\lambda =2$ (blue), $\lambda =4$ (red).

**Figure A3.**Normalized ENL as a function of the normalized slope for three values of the decay parameter. The maximum is obtained for $\beta =0$. For $\beta >1$ the ENL oscillates around the value obtained at integers, with decreasing amplitude of oscillations.

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**Figure 1.**Bias-variance trade-off and nonlocal filtering. From left to right: original noisy image, output of 3 × 3 boxcar filter, output of 9 × 9 boxcar filter and output of nonlocal means.

**Figure 3.**Two synthetic examples to motivate the need for offset-compensated similarity measures. Top: simple noiseless signals, characterized by small fluctuations (

**left**), or a constant gradient (

**right**). Middle: squared error between the local patch and the target (red) patch. Bottom: patches (black) similar to the target are rare in the presence of a significant gradient. Many more would be available with offset compensation.

**Figure 4.**Examples of test phase images (size 105 × 105 pixels), used in the preliminary experiments. From top to bottom: large ($\nabla \psi =\pi /4$), and medium phase gradient ($\nabla \psi =2\pi /32$). From left to right: clean image, noisy images at low ($\rho =0.3$), medium ($\rho =0.6$), and high ($\rho =0.9$) coherence.

**Figure 5.**Sample filtering results with various versions of NLM. From top to bottom: large gradient-low coherence, large gradient-medium coherence, large gradient-high coherence, medium gradient-low coherence. From left to right: noisy original, filtered with NLM, OC-NLM, prefiltered adaptive OC-NLM.

**Figure 6.**Cone image (

**top**). Filtered images (

**left**) and error images (

**right**) for all methods under comparison.

**Figure 7.**Ramp image (

**top**). Filtered images (

**left**) and error images (

**right**) for all methods under comparison.

**Figure 8.**Peaks image (

**top**). Filtered images (

**left**) and error images (

**right**) for all methods under comparison.

**Figure 9.**Cone image. Sliding-window RMSE (

**left**) and residues (

**right**) for all methods under comparison.

**Figure 10.**Ramp image. Sliding-window RMSE (

**left**) and residues (

**right**) for all methods under comparison.

**Figure 11.**Peaks image. Sliding-window RMSE (

**left**) and residues (

**right**) for all methods under comparison.

**Figure 12.**Kamchatka Clip #1. Filtered (

**right**) and method noise images (

**left**) for all methods under comparison.

**Figure 13.**Kamchatka Clip #2. Filtered (

**right**) and method noise images (

**left**) for all methods under comparison.

RMSE | Residues | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Cone | Ramp | Peaks | Cone | Ramp | Peaks | |||||||

boxcar | 0.414 | (0.009) | 0.536 | (0.011) | 0.440 | (0.008) | 166.3 | (18.8) | 486.9 | (23.2) | 223.4 | (22.2) |

SpInPHASE | 0.495 | (0.009) | 0.381 | (0.012) | 0.426 | (0.008) | 654.1 | (51.9) | 372.9 | (46.2) | 442.5 | (27.1) |

NL-SAR | 0.268 | (0.045) | 0.608 | (0.065) | 0.377 | (0.063) | 65.8 | (52.9) | 1206.7 | (417.3) | 355.1 | (231.6) |

NL-InSAR | 0.195 | (0.007) | 0.981 | (0.013) | 0.325 | (0.018) | 0.0 | (–) | 445.2 | (53.6) | 37.0 | (10.8) |

InSAR-BM3D | 0.141 | (0.004) | 0.254 | (0.012) | 0.152 | (0.004) | 0.0 | (–) | 11.9 | (5.2) | 0.2 | (0.6) |

OC-InSAR-BM3D | 0.119 | (0.006) | 0.126 | (0.005) | 0.120 | (0.004) | 0.0 | (–) | 0.0 | (–) | 0.0 | (–) |

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## Share and Cite

**MDPI and ACS Style**

Sica, F.; Cozzolino, D.; Verdoliva, L.; Poggi, G.
The Offset-Compensated Nonlocal Filtering of Interferometric Phase. *Remote Sens.* **2018**, *10*, 1359.
https://doi.org/10.3390/rs10091359

**AMA Style**

Sica F, Cozzolino D, Verdoliva L, Poggi G.
The Offset-Compensated Nonlocal Filtering of Interferometric Phase. *Remote Sensing*. 2018; 10(9):1359.
https://doi.org/10.3390/rs10091359

**Chicago/Turabian Style**

Sica, Francescopaolo, Davide Cozzolino, Luisa Verdoliva, and Giovanni Poggi.
2018. "The Offset-Compensated Nonlocal Filtering of Interferometric Phase" *Remote Sensing* 10, no. 9: 1359.
https://doi.org/10.3390/rs10091359