# The Offset-Compensated Nonlocal Filtering of Interferometric Phase

^{1}

^{2}

^{*}

## 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.

## References

- Seymour, M.; Cumming, I. Maximum likelihood estimation for SAR interferometry. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Ottawa, ON, Canada, 8–12 August 1994; pp. 2272–2275. [Google Scholar]
- Bamler, R.; Hartl, P. Synthetic aperture radar interferometry. Inverse Probl.
**1998**, 14, R1–R54. [Google Scholar] [CrossRef] - Zebker, H.; Villasenor, J. Decorrelation in interferometric radar echoes. IEEE Trans. Geosci. Remote Sens.
**1992**, 30, 950–959. [Google Scholar] [CrossRef][Green Version] - Massonnet, D.; Feigl, K. Radar interferometry and its application to changes in the Earth’s surface. Rev. Geophys.
**1998**, 36, 441–500. [Google Scholar] [CrossRef] - Costantini, M. A novel phase unwrapping method based on network programming. IEEE Trans. Geosci. Remote Sens.
**1998**, 36, 813–821. [Google Scholar] [CrossRef] - Lee, J.S.; Papathanassiou, K.; Ainsworth, T.; Grunes, M.; Reigber, A. A new technique for noise filtering of SAR interferometric phase images. IEEE Trans. Geosci. Remote Sens.
**1998**, 36, 1456–1465. [Google Scholar] - Goldstein, R.; Werner, C. Radar interferogram filtering for geophysical applications. Geophys. Res. Lett.
**1998**, 25, 4035–4038. [Google Scholar] [CrossRef][Green Version] - Wu, N.; Feng, D.Z.; Li, J. A locally adaptive filter of interferometric phase images. IEEE Geosci. Remote Sens. Lett.
**2006**, 3, 73–77. [Google Scholar] [CrossRef] - Fu, S.; Long, X.; Yang, X.; Yu, Q. Directionally adaptive filter for synthetic aperture radar interferometric phase images. IEEE Trans. Geosci. Remote Sens.
**2013**, 51, 552–559. [Google Scholar] [CrossRef] - Chao, C.F.; Chen, K.S.; Lee, J.S. Refined filtering of interferometric phase from InSAR data. IEEE Trans. Geosci. Remote Sens.
**2013**, 51, 5315–5323. [Google Scholar] [CrossRef] - Vasile, G.; Trouvé, E.; Lee, J.S.; Buzuloiu, V. Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation. IEEE Trans. Geosci. Remote Sens.
**2006**, 44, 1609–1621. [Google Scholar] [CrossRef][Green Version] - Baran, I.; Stewart, M.P.; Kampes, B.M.; Perski, Z.; Lilly, P. A modification to the Goldstein radar interferogram filter. IEEE Trans. Geosci. Remote Sens.
**2003**, 41, 2114–2118. [Google Scholar] [CrossRef] - Song, R.; Guo, H.; Liu, G.; Perski, Z.; Fan, J. Improved Goldstein SAR interferogram filter based on empirical mode decomposition. IEEE Geosci. Remote Sens. Lett.
**2014**, 11, 399–403. [Google Scholar] [CrossRef] - Jiang, M.; Dingi, X.; Li, Z.; Tian, X.; Zhu, W.; Wang, C.; Xu, B. The improvement for Baran phase filter derived from unbiased InSAR coherence. IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens.
**2014**, 7, 3002–3010. [Google Scholar] [CrossRef] - Lopez-Martinez, C.; Fabregas, X. Modeling and reduction of SAR interferometric phase noise in the wavelet domain. IEEE Trans. Geosci. Remote Sens.
**2002**, 40, 2553–2566. [Google Scholar] [CrossRef][Green Version] - Bian, Y.; Mercer, B. Interferometric SAR phase filtering in the wavelet domain using simultaneous detection and estimation. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 1396–1416. [Google Scholar] [CrossRef] - Zha, X.; Fu, R.; Dai, Z.; Liu, B. Noise Reduction in Interferograms Using the Wavelet Packet Transform and Wiener Filtering. IEEE Geosci. Remote Sens. Lett.
**2008**, 5, 404–408. [Google Scholar] - Bioucas-Dias, J.; Katkovnik, V.; Astola, J.; Egiazarian, K. Absolute phase estimation: adaptive local denoising and global unwrapping. Appl. Opt.
**2008**, 47, 5358–5369. [Google Scholar] [CrossRef] [PubMed] - Hongxing, H.; Bioucas-Dias, J.M.; Katkovnik, V. Interferometric phase image estimation via sparse coding in the complex domain. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 2589–2602. [Google Scholar] [CrossRef] - Ferraiuolo, G.; Poggi, G. A Bayesian filtering technique for SAR interferometric phase fields. IEEE Trans. Image Process.
**2004**, 13, 1368–1378. [Google Scholar] [CrossRef] [PubMed] - Denis, L.; Tupin, F.; Darbon, J.; Sigelle, M. Joint regularization of phase and amplitude of InSAR data: Application to 3-D reconstruction. IEEE Trans. Geosci. Remote Sens.
**2009**, 47, 3774–3785. [Google Scholar] [CrossRef] - Buades, A.; Coll, B.; Morel, J.M. A review of image denoising algorithms, with a new one. Multiscale Model. Simul.
**2005**, 4, 490–530. [Google Scholar] [CrossRef] - Deledalle, C.A.; Denis, L.; Tupin, F. Iterative weighted maximum likelihood denoising with probabilistic patch-based weights. IEEE Trans. Image Process.
**2009**, 18, 2661–2672. [Google Scholar] [CrossRef] [PubMed] - Parrilli, S.; Poderico, M.; Angelino, C.; Verdoliva, L. A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage. IEEE Trans. Geosci. Remote Sens.
**2012**, 50, 606–616. [Google Scholar] [CrossRef] - Cozzolino, D.; Parrilli, S.; Scarpa, G.; Poggi, G.; Verdoliva, L. Fast adaptive nonlocal SAR despeckling. IEEE Geosci. Remote Sens. Lett.
**2014**, 11, 524–528. [Google Scholar] [CrossRef] - Deledalle, C.A.; Denis, L.; Tupin, F. NL-InSAR: Nonlocal interferogram estimation. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 1441–1452. [Google Scholar] [CrossRef] - Chen, R.; Yu, W.; Wang, R.; Liu, G.; Shao, Y. Interferometric Phase Denoising by Pyramid Nonlocal Means Filter. IEEE Geosci. Remote Sens. Lett.
**2013**, 10, 826–830. [Google Scholar] [CrossRef] - Lin, X.; Li, F.; Meng, D.; Hu, D.; Ding, C. Nonlocal SAR interferometric phase filtering through higher order singular value decomposition. IEEE Geosci. Remote Sens. Lett.
**2015**, 12, 806–810. [Google Scholar] [CrossRef] - Deledalle, C.A.; Denis, L.; Tupin, F.; Reigber, A.; Jäger, M. NL-SAR: A unified nonlocal framework for resolution-preserving (Pol)(In) SAR denoising. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 2021–2038. [Google Scholar] [CrossRef] - Sica, F.; Cozzolino, D.; Zhu, X.X.; Verdoliva, L.; Poggi, G. InSAR-BM3D: A Nonlocal Filter for SAR Interferometric Phase Restoration. IEEE Trans. Geosci. Remote Sens.
**2018**, 56, 3456–3467. [Google Scholar] [CrossRef] - Chen, J.; Chen, Y.; An, W.; Cui, Y.; Yang, J. Nonlocal filtering for polarimetric SAR data: a pretest approach. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 1744–1754. [Google Scholar] [CrossRef] - Hu, J.; Guo, R.; Zhu, X.; Baier, G.; Wang, Y. Non-local means filter for polarimetric SAR speckle reduction-experiments using TerraSAR-X data. In Proceedings of the ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Taipei, Taiwan, 31 August–4 September 2015. [Google Scholar]
- Su, X.; Deledalle, C.; Tupin, F.; Sun, H. Two-step multitemporal nonlocal means for synthetic aperture radar images. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 6181–6196. [Google Scholar] - Sica, F.; Reale, D.; Poggi, G.; Verdoliva, L.; Fornaro, G. Nonlocal adaptive multilooking in SAR multipass differential interferometry. IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens.
**2015**, 8, 1727–1742. [Google Scholar] [CrossRef] - Ferretti, A.; Fumagalli, A.; Novali, F.; Prati, C.; Rocca, F.; Rucci, A. A new algorithm for processing interferometric datastacks: SqueeSAR. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 3460–3470. [Google Scholar] [CrossRef] - Fornaro, G.; Verde, S.; Reale, D.; Pauciullo, A. CAESAR: An approach based on covariance matrix decomposition to improve multibaseline-multitemporal interferometric SAR processing. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 2050–2065. [Google Scholar] [CrossRef] - Chierchia, G.; El Gheche, M.; Scarpa, G.; Verdoliva, L. Multitemporal SAR image despeckling based on block-matching and collaborative filtering. IEEE Trans. Geosci. Remote Sens.
**2017**, 55, 5467–5480. [Google Scholar] [CrossRef] - Deledalle, C.A.; Denis, L.; Poggi, G.; Tupin, F.; Verdoliva, L. Exploiting patch similarity for SAR image processing: the nonlocal paradigm. IEEE Signal Process. Mag.
**2014**, 31, 69–78. [Google Scholar] [CrossRef] - Sica, F. Non-Local Methods for InSAR Parameters Estimation. Ph.D. Thesis, University of Naples Federico II, Napoli, Italy, 2016. [Google Scholar]
- Dabov, K.; Foi, A.; Katkovnik, V.; Egiazarian, K. Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process.
**2007**, 16, 2080–2095. [Google Scholar] [CrossRef] [PubMed] - Salmon, J.; Strozecki, Y. Patch reprojections for non-local methods. Signal Process.
**2012**, 92, 477–489. [Google Scholar] [CrossRef] - Hagberg, J.O.; Ulander, L.M.; Askne, J. Repeat-pass SAR interferometry over forested terrain. IEEE Trans. Geosci. Remote Sens.
**1995**, 33, 331–340. [Google Scholar] [CrossRef] - Touzi, R.; Lopes, A.; Bruniquel, J.; Vachon, P.W. Coherence estimation for SAR imagery. IEEE Trans. Geosci. Remote Sens.
**1999**, 37, 135–149. [Google Scholar] [CrossRef] - Zebker, H.A.; Chen, K. Accurate estimation of correlation in InSAR observations. IEEE Trans. Geosci. Remote Sens.
**2005**, 2, 124–127. [Google Scholar] [CrossRef] - Goodman, N. Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). Ann. Math. Stat.
**1963**, 3, 152–177. [Google Scholar] [CrossRef] - Just, D.; Bamler, R. Phase statistics of interferograms with applications to synthetic aperture radar. Appl. Opt.
**1994**, 33, 4361–4368. [Google Scholar] [CrossRef] [PubMed] - Lee, J.S.; Hoppel, K.; Mango, S.; Miller, A. Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery. IEEE Trans. Geosci. Remote Sens.
**1994**, 32, 1017–1028. [Google Scholar] - Lopez-Martinez, C.; Pottier, E. On the extension of multidimensional speckle noise model from single-look to multilook SAR imagery. IEEE Trans. Geosci. Remote Sens.
**2007**, 45, 305–320. [Google Scholar] [CrossRef] - Mardia, K.V.; Jupp, P.E. Directional Statistics; John Wiley & Sons: Hoboken, NJ, USA, 2000. [Google Scholar]
- Pepe, A.; Yang, Y.; Manzo, M.; Lanari, R. Improved EMCF-SBAS processing chain based on advanced techniques for the noise-filtering and selection of small baseline multi-look DInSAR interferograms. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 4394–4417. [Google Scholar] [CrossRef] - Pepe, A.; Mastro, P. On the use of directional statistics for the adaptive spatial multi-looking of sequences of differential SAR interferograms. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium 2017, Fort Worth, TX, USA, 23–28 July 2017; Volume 2, pp. 3791–3794. [Google Scholar]

**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 | (–) |

© 2018 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## 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