# A U-Net Approach for InSAR Phase Unwrapping and Denoising

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## Abstract

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## 1. Introduction

- $\mathsf{\Omega}$ = wrapped phase, with range [−$\pi $, $\pi $];
- $\mu $ = unwrapped phase;
- k = integer.

#### 1.1. Phase Unwrapping in Single-Baseline Images

#### Classical Methods to Solve SBPU

_{$\psi $}are the absolute phase and wrap phase at a pixel location s, and p is the power of the norm. According to Ghiglia et al. [19], the optimization problem, when solved considering $p=0$, provides the best solution, but it is an NP-hard problem. Constantine et al. [20] provided a procedure to solve with $p=1$ which results in a minimum cost problem. The solution is smooth but does not maintain the integer count relationship between the wrap and unwrap phases. According to Ghiglia et al. [21], a solution considering $p=2$ results in an L

^{2}norm problem. This method is affected significantly by noise. A lower value of p provides better results but they are often difficult to solve. Adaptive methods include the combination of different Lp problems proposed by Yu et al. [22] for better results, but this increases the computational time and complexity. Yann et al. [23] proposed using additional constraints of the scalar properties of phase in the temporal domain when three differential interferograms are available. This improves the result but requires larger computational resources. Chartrand et al. [24] proposed the use of a sparse optimization algorithm to solve the phase unwrapping problem. In Equation (5), $p=0$ provides an optimal solution but is NP-hard. As the residues in the noisy wrap image tend to be sparse, the alternating directions, method of multipliers (ADMM) algorithm was utilized by the authors to perform faster and more reliable unwrapping. However, the authors provide only a few qualitative comparisons against SNAPHU [25].

## 2. Deep Learning for Phase Unwrapping

#### U-Net for Phase Unwrapping

## 3. Materials and Methods

- r is the refinement factor calculated from the second stage U-Net.

## 4. Results

- We generate the SLC images S1 and S2 with random Gaussian bubbles as the synthetic motion signals. We make sure that the generated SLCs satisfy Itoh’s condition [6] using a set of control parameters in the simulator.
- We add random additive white Gaussian noise at a random signal level to both SLC images to generate noisy SLC images.
- Using the SLCs, we generate clean and noisy interferometric phases and calculate the ground truth coherence.

- M = height of the interferogram;
- N = width of the interferogram;
- ${\varphi}_{xy}^{{}^{\prime}}$ = recovered absolute phase at pixel location [x, y];
- ${\varphi}_{xy}$ = reference absolute phase at pixel location [x, y].

- X = pixels of a window of the reference absolute phase image;
- Y = pixels of a window of the recovered absolute phase image;
- $\mu $ = mean of the pixel values of window;
- $\sigma $ = standard deviation of pixel values of window;
- ${\sigma}_{xy}$ = cross-correlation of pixels of two windows;
- c = stabilizing factor.

- M = height of the interferogram;
- N = width of the interferogram;
- ${\varphi}_{xy}^{{}^{\prime}}$ = recovered absolute phase at pixel location [x, y];
- ${\varphi}_{xy}$ = reference absolute phase at pixel location [x, y].

## 5. Discussion

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Boundary artifacts observed when trained with noisy input. Noisy wrap input (

**left column**); the output produced by Sica et al. [43] when trained with noisy wrap phase input and noiseless ground truth (

**middle column**); and ground truth (

**right column**). The x- and y-axes correspond to the azimuth and range spatial dimensions. The color bars indicate the scale of the wrap input and unwrap output.

**Figure 7.**Qualitative comparison of unwrapped outputs of various deep learning algorithms (starting from row 3), with our proposed method in row 3. Each column shows one sample example. Row 1 is the wrapped phase interferogram. Row 2 is the ground truth. Rows 4, 6, and 8 are the error maps for rows 3, 5, and 7 unwrap outputs. [a] and [b] refer to outputs of Sica et al. [43] and Perera et al. [42] respectively. (i), (ii) and (iii) indicate the color bar for the scale of wrap input, unwrap output, and error map respectively.

**Figure 8.**Qualitative comparison of unwrapped outputs of various traditional algorithms (starting from row 1). Each column shows one sample example and has the same wrap phase interferogram and ground truth (row 1 and row 2 respectively) as Figure 7. Rows 2, 4, 6, and 8 are the error maps for 1, 3, 5, and 7 unwrap outputs. [c], [d], [e], and [f] refer to outputs of BM3D [52] + SNAPHU [25], SNAPHU [25], Chartrand et al. [24] and, Herraez et al. [18] respectively. (i), (ii) and (iii) indicate the color bar for the scale of wrap input, unwrap output, and error map respectively.

**Figure 9.**Comparison of the unwrapped output of algorithms: Perera et al. [42] (column (a)) and the proposed method. Row 1 and row 2 show two samples. The rectangular boxes highlight the errors in boundaries in Perera et al. [42] not observed in the proposed method. The color bar indicates the scale of the unwrap outputs.

Reference | Methodology | Key Highlights | Limitation |
---|---|---|---|

Huntley [9] | Minimum length cut algorithm to avoid residues | Works well in high-quality and low-noise phase maps | May not generate output at all if phase maps are too noisy |

Goldstein [10] | Integration paths encircle an equal number of positive and negative residues | Branch-cut is used to find a solution | NP-Hard |

Flynn [12], Zhong [13], and Zhao [14] | Quality-maps-guided unwrapping | Coherence was used as measure of quality | Requires many SLCs to generate reliable quality maps |

Ching et al. [15] | Segmentation-guided unwrapping | Each segment is unwrapped independently | Boundary errors |

An et al. [17] | Segmentation-guided unwrapping | Faster minimum spanning tree algorithm | Boundary errors |

Herraez et al. [18] | Reliability-guided unwrapping | Unwraps from most reliable to least reliable pixel | Poor noise immunity |

Constantine et al. [20] | Lp norm optimization | P = 1, minimum cost problem | Unwrap solution is smooth but not best |

Ghiglia et al. [21] | Lp norm optimization | P = 2, results in smooth unwrap solution | Affected by noise |

Chartrand et al. [24] | Sparse optimization | Faster computation | Poor noise immunity |

SNAPHU [25] | MAP estimation | Captures dependencies with factors such as amplitude | The accuracy depends on the estimates of probability density functions |

Reference | Network Architecture | Loss Function | Limitation |
---|---|---|---|

Zhou et al. [34] | GAN | Adversarial loss | Dataset has no noise |

Xu et al. [37] | MNet | MAE and SSIM | Only works in low concentration of speckle noise |

Qin et al. [38] | ResUNet | MAE | Works only in low noise |

Spoorthly et al. [40] | Dense-UNet | Cross-entropy + Mean absolute error (MAE) | Incorrect unwrapping boundaries |

Zhang et al. [41] | EESANET | Cross-entropy + L1 loss + residue loss | Poor noise immunity |

Sica et al. [43] | U-Net | Cross-entropy + Jaccard loss + Mean absolute error (MAE) | Poor noise immunity |

Perera et al. [42] | Spatial Quad-Directional Long Short-Term Memory (LSTM) | Variance error | Incorrect unwrapping boundaries |

**Table 3.**Comparison of our method with other methods: average/worst RMSE, average/worst SSIM, and UFR. The best-performing method for each metric is highlighted in bold text.

Method | RMSE (Average/Worst) | SSIM (Average/Worst) | UFR (%) |
---|---|---|---|

Herraez et al. [18] | 1.425/9.36 | 0.89/0.33 | 3.52 |

Chartrand et al. [24] | 0.93/14.12 | 0.93/0.61 | 1.25 |

SNAPHU [25] | 0.96/12.59 | 0.91/0.63 | 1.2 |

BM3D [52] + SNAPHU [25] | 0.87/14.69 | 0.93/0.65 | 1.1 |

Sica et al. [43] | 0.54/2.91 | 0.91/0.64 | 0.01 |

Perera et al. [42] | 1.01/5.66 | 0.98/0.92 | 0.75 |

Our method | 0.11/2.18 | 0.99/0.95 | 0.006 |

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## Share and Cite

**MDPI and ACS Style**

Vijay Kumar, S.; Sun, X.; Wang, Z.; Goldsbury, R.; Cheng, I.
A U-Net Approach for InSAR Phase Unwrapping and Denoising. *Remote Sens.* **2023**, *15*, 5081.
https://doi.org/10.3390/rs15215081

**AMA Style**

Vijay Kumar S, Sun X, Wang Z, Goldsbury R, Cheng I.
A U-Net Approach for InSAR Phase Unwrapping and Denoising. *Remote Sensing*. 2023; 15(21):5081.
https://doi.org/10.3390/rs15215081

**Chicago/Turabian Style**

Vijay Kumar, Sachin, Xinyao Sun, Zheng Wang, Ryan Goldsbury, and Irene Cheng.
2023. "A U-Net Approach for InSAR Phase Unwrapping and Denoising" *Remote Sensing* 15, no. 21: 5081.
https://doi.org/10.3390/rs15215081