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arxiv: 2510.24512 · v2 · submitted 2025-10-28 · 📡 eess.SP · physics.geo-ph· stat.AP

Heuristic Quality Coefficients for Interferometric Phase Linking

Magnus Heimpel , Irena Hajnsek , Othmar Frey This is my paper

Pith reviewed 2026-05-18 02:57 UTC · model grok-4.3

classification 📡 eess.SP physics.geo-phstat.AP
keywords phase linkingInSARdistributed scatterersquality coefficientscoherence matrixgoodness-of-fitclosure phasereliability indicators
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The pith

Three heuristic quality coefficients assess the reliability of phase linking estimates for distributed scatterers in multitemporal InSAR.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops practical reliability indicators for phase linking methods that lack the uncertainty measures available for maximum-likelihood estimation. It introduces a unified framework containing three normalized coefficients: one that measures how well the achieved objective fits relative to an upper bound and a modeled noise floor, one computed directly from the coherence matrix for advance screening, and one that checks the estimate against alternatives in the solution space. Simulations under different decorrelation models demonstrate that the goodness-of-fit version tracks actual phase error most reliably while the closure-phase version works well for pre-filtering. Application to real TerraSAR-X data produces spatial patterns consistent with urban versus vegetated terrain and shows the ambiguity coefficient supplies extra information where scattering changes over time. These tools address the need to decide which distributed-scatterer estimates can safely enter deformation analysis.

Core claim

We propose three heuristic quality coefficients within a unified mathematical framework that covers common PL methods: (1) a method-specific goodness-of-fit coefficient that normalizes the achieved PL objective between a method-consistent upper bound and an empirically modeled noise floor level; (2) a closure phase coefficient computed from the sample coherence matrix in advance; and (3) an ambiguity coefficient that compares the obtained PL estimate with the best alternative in its orthogonal complement in the solution space. All coefficients are normalized to the interval [0,1], where 1 indicates maximum reliability and 0 matches the behavior expected under pure noise.

What carries the argument

Three normalized heuristic quality coefficients (goodness-of-fit, closure phase, and ambiguity) inside a single mathematical framework that evaluates phase linking results from the sample coherence matrix without requiring Fisher-information matrices.

Load-bearing premise

The empirically modeled noise floor level used to normalize the goodness-of-fit coefficient accurately represents the behavior expected under pure noise.

What would settle it

Running the same simulations under exponential and seasonal decorrelation models, computing the normalized absolute phase error independently, and observing whether the goodness-of-fit coefficient loses its consistent tracking relationship with that error.

Figures

Figures reproduced from arXiv: 2510.24512 by Irena Hajnsek, Magnus Heimpel, Othmar Frey.

Figure 1
Figure 1. Figure 1: Temporal and orbital baselines of the TerraSAR-X acquisitions. [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Region of interest within the TerraSAR-X dataset over Visp, Switzerland. Top left: SLC of the acquisition [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: pixel-wise closure phase coefficient of the SLC stack. Right: histogram of all closure phase coefficients [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Phase-linked differential interferometric phase between the acquisitions on 31 Aug 2017 and 11 Sep [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Equal-weighted eigendecomposition. Top left: goodness-of-fit coefficient with [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Equal-weighted phase triangulation. Top left: goodness-of-fit coefficient. Top right: ambiguity coefficient. [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Coherence-weighted phase triangulation. Top left: goodness-of-fit coefficient. Top right: ambiguity [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Coherence-weighted eigendecomposition. Top left: goodness-of-fit coefficient. Top right: ambiguity [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Maximum-likelihood phase triangulation with coefficients based on Table [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Maximum-likelihood phase triangulation with coefficients based on Eq. [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Maximum-likelihood eigendecomposition with coefficients based on Table [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Maximum-likelihood eigendecomposition with coefficients based on Eq. [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
read the original abstract

In multitemporal InSAR, phase linking (PL) refers to the estimation of a single-reference interferometric phase history for distributed scatterers (DS) from the information contained in the sample coherence matrix. Because the phase information in this matrix is typically inconsistent, DS processing needs practical reliability indicators to decide whether a pixel's PL estimate is sufficiently supported by the data for subsequent deformation analysis. For maximum-likelihood estimation, uncertainty can be quantified via Fisher-information-based covariance estimates, but no analogous, generally applicable uncertainty quantification is available for the broad range of non-ML methods. We propose three heuristic quality coefficients within a unified mathematical framework that covers common PL methods: (1) a method-specific goodness-of-fit coefficient that normalizes the achieved PL objective between a method-consistent upper bound and an empirically modeled noise floor level; (2) a closure phase coefficient computed from the sample coherence matrix in advance; and (3) an ambiguity coefficient that compares the obtained PL estimate with the best alternative in its orthogonal complement in the solution space. All coefficients are normalized to the interval $[0,1]$, where 1 indicates maximum reliability and 0 matches the behavior expected under pure noise. Simulations under exponential and seasonal decorrelation models show that the goodness-of-fit coefficient tracks the normalized absolute phase error most consistently, whereas the closure phase coefficient provides an a priori indicator for pre-screening. Experiments on a TerraSAR-X stack over Visp, Switzerland, reveal plausible spatial patterns across urban and vegetated areas and show that the ambiguity coefficient provides complementary information, especially in regions with temporally varying scattering mechanisms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes three heuristic quality coefficients for evaluating the reliability of interferometric phase linking (PL) estimates in multitemporal InSAR for distributed scatterers. Within a unified framework applicable to common PL methods, it introduces (1) a method-specific goodness-of-fit coefficient that normalizes the achieved PL objective between a method-consistent upper bound and an empirically modeled noise floor, (2) a closure phase coefficient derived from the sample coherence matrix, and (3) an ambiguity coefficient comparing the PL estimate to alternatives in its orthogonal complement. All are scaled to [0,1] with 1 indicating high reliability and 0 corresponding to pure noise behavior. Simulations under exponential and seasonal decorrelation models indicate that the goodness-of-fit coefficient most consistently tracks the normalized absolute phase error, while the closure phase coefficient serves as an a priori pre-screening tool. Real-data experiments on a TerraSAR-X stack over Visp, Switzerland, demonstrate plausible spatial patterns in urban and vegetated areas, with the ambiguity coefficient providing complementary information in regions with varying scattering mechanisms.

Significance. If validated, the coefficients address a practical gap by supplying reliability indicators for non-ML PL methods lacking Fisher-information covariance estimates, potentially aiding DS selection in deformation analysis. The unified framework covering multiple PL methods and the simulation results showing consistent tracking of phase error by the goodness-of-fit coefficient are positive contributions. The real-data spatial patterns add empirical support, though the heuristic and empirically normalized nature limits immediate theoretical guarantees.

major comments (2)
  1. [Abstract and Section 3] Abstract and Section 3 (goodness-of-fit definition): The central claim that the goodness-of-fit coefficient 'tracks the normalized absolute phase error most consistently' depends on the [0,1] scaling produced by subtracting an empirically modeled noise floor from the achieved PL objective. The manuscript models this floor empirically but provides no derivation, closed-form expectation under pure noise, or verification against coherence-matrix statistics for the specific PL estimators; without this, the reported consistency may be sensitive to the chosen floor rather than a general property of the data.
  2. [Simulation results] Simulation results (exponential and seasonal decorrelation models): The reported superiority of the goodness-of-fit coefficient over the other two in tracking normalized absolute phase error is load-bearing for the practical utility claim, yet the results are presented without sensitivity tests to perturbations in the empirical noise floor parameter; this leaves open whether the tracking holds under alternative floor choices consistent with the same coherence statistics.
minor comments (2)
  1. [Abstract] The abstract refers to a 'method-consistent upper bound' without specifying its exact functional form for each PL method; adding a brief equation or reference in the main text would improve clarity for readers.
  2. Notation for the three coefficients could be introduced with consistent symbols early in the manuscript to aid cross-referencing between the unified framework and the individual definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Abstract and Section 3] Abstract and Section 3 (goodness-of-fit definition): The central claim that the goodness-of-fit coefficient 'tracks the normalized absolute phase error most consistently' depends on the [0,1] scaling produced by subtracting an empirically modeled noise floor from the achieved PL objective. The manuscript models this floor empirically but provides no derivation, closed-form expectation under pure noise, or verification against coherence-matrix statistics for the specific PL estimators; without this, the reported consistency may be sensitive to the chosen floor rather than a general property of the data.

    Authors: We acknowledge that the noise floor is obtained empirically via Monte Carlo simulation under noise-only conditions rather than from a closed-form expectation. Because the proposed coefficients are heuristic and the PL estimators are nonlinear, a general closed-form derivation for arbitrary estimators is not available. We will revise Section 3 to expand the description of the empirical estimation procedure, report the specific simulation settings used to obtain the floor, and include additional verification steps that compare the modeled floor against the observed distribution of the PL objective and the statistical properties of the sample coherence matrix. revision: partial

  2. Referee: [Simulation results] Simulation results (exponential and seasonal decorrelation models): The reported superiority of the goodness-of-fit coefficient over the other two in tracking normalized absolute phase error is load-bearing for the practical utility claim, yet the results are presented without sensitivity tests to perturbations in the empirical noise floor parameter; this leaves open whether the tracking holds under alternative floor choices consistent with the same coherence statistics.

    Authors: We agree that explicit sensitivity tests would strengthen the simulation results. In the revised manuscript we will add a dedicated sensitivity analysis (new figure or table) that perturbs the noise-floor value within a range consistent with the coherence statistics of the simulated data and demonstrates that the relative performance of the goodness-of-fit coefficient in tracking normalized absolute phase error remains stable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; coefficients defined from coherence matrix and objectives with separate simulation validation

full rationale

The paper defines the three heuristic quality coefficients directly from the sample coherence matrix, PL objective values, and solution-space comparisons, with the goodness-of-fit normalization using a method-consistent upper bound and an empirically modeled noise floor. The claim that the goodness-of-fit coefficient tracks normalized absolute phase error is supported by separate simulations under exponential and seasonal decorrelation models rather than by algebraic construction or by fitting the coefficient to the phase-error values themselves. No self-citation load-bearing steps, uniqueness theorems imported from prior author work, or ansatz smuggling appear in the described framework. The derivation remains self-contained as a proposal of heuristics whose performance is assessed externally to their definition.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central proposal rests on an empirical noise floor model and standard InSAR assumptions about coherence matrix inconsistency; no new physical entities are introduced.

free parameters (1)
  • empirically modeled noise floor level
    Used to set the lower bound for normalizing the goodness-of-fit coefficient to the [0,1] interval.
axioms (1)
  • domain assumption Phase information in the sample coherence matrix is typically inconsistent for distributed scatterers.
    Invoked to justify the need for reliability indicators in phase linking.

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Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    Permanent scatterers in SAR interferometry

    A. Ferretti, C. Prati, and F. Rocca. “Permanent scatterers in SAR interferometry”. In:IEEE Transactions on Geoscience and Remote Sensing39.1 (2001), pp. 8–20.issn: 0196-2892. doi: 10.1109/36.898661

    work page doi:10.1109/36.898661 2001

  2. [2]

    A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers

    Andrew Hooper et al. “A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers”. In:Geophysical Research Letters31.23 (Dec. 2004).issn: 1944-8007. doi: 10.1029/2004gl021737

    work page doi:10.1029/2004gl021737 2004

  3. [3]

    Interferometric point target analysis for deformation mapping

    C. Werner et al. “Interferometric point target analysis for deformation mapping”. In:2003 IEEE International Geoscience and Remote Sensing Symposium. Vol. 7. IGARSS-03. IEEE, 2003, pp. 4362–4364.doi: 10.1109/ igarss.2003.1295516

    work page arXiv 2003

  4. [4]

    Persistent Scatterer Interferometry: A review

    Michele Crosetto et al. “Persistent Scatterer Interferometry: A review”. In:ISPRS Journal of Photogrammetry and Remote Sensing115 (May 2016), pp. 78–89.issn: 0924-2716. doi: 10.1016/j.isprsjprs.2015.10.011

    work page doi:10.1016/j.isprsjprs.2015.10.011 2016

  5. [5]

    Differential tomography: a new framework for SAR interferometry

    F. Lombardini. “Differential tomography: a new framework for SAR interferometry”. In:IEEE Transactions on Geoscience and Remote Sensing43.1 (Jan. 2005), pp. 37–44.issn: 0196-2892.doi: 10.1109/tgrs.2004.838371

    work page doi:10.1109/tgrs.2004.838371 2005

  6. [6]

    Four-Dimensional SAR Imaging for Height Estimation and Monitoring of Single and Double Scatterers

    G. Fornaro, D. Reale, and F. Serafino. “Four-Dimensional SAR Imaging for Height Estimation and Monitoring of Single and Double Scatterers”. In:IEEE Transactions on Geoscience and Remote Sensing47.1 (Jan. 2009), pp. 224–237.issn: 1558-0644. doi: 10.1109/tgrs.2008.2000837

    work page doi:10.1109/tgrs.2008.2000837 2009

  7. [7]

    Single-Look SAR Tomography as an Add-On to PSI for Improved Deformation Analysis in Urban Areas

    Muhammad Adnan Siddique et al. “Single-Look SAR Tomography as an Add-On to PSI for Improved Deformation Analysis in Urban Areas”. In:IEEE Transactions on Geoscience and Remote Sensing54.10 (Oct. 2016), pp. 6119–6137.issn: 1558-0644. doi: 10.1109/tgrs.2016.2581261

    work page doi:10.1109/tgrs.2016.2581261 2016

  8. [8]

    InSAR Deformation Analysis with Distributed Scatterers: A Review Complemented by New Advances

    Markus Even and Karsten Schulz. “InSAR Deformation Analysis with Distributed Scatterers: A Review Complemented by New Advances”. In:Remote Sensing 10.5 (May 2018), p. 744. issn: 2072-4292. doi: 10.3390/rs10050744

    work page doi:10.3390/rs10050744 2018

  9. [9]

    A New Algorithm for Surface Deformation Monitoring Based on Small Baseline Differential SAR Interferograms

    P. Berardino et al. “A New Algorithm for Surface Deformation Monitoring Based on Small Baseline Differential SAR Interferograms”. In:IEEE Transactions on Geoscience and Remote Sensing40.11 (Nov. 2002), pp. 2375–

    work page 2002

  10. [10]

    doi: 10.1109/tgrs.2002.803792

    issn: 0196-2892. doi: 10.1109/tgrs.2002.803792

    work page doi:10.1109/tgrs.2002.803792 2002

  11. [11]

    On the Exploitation of Target Statistics for SAR Interferometry Applications

    Andrea Monti Guarnieri and Stefano Tebaldini. “On the Exploitation of Target Statistics for SAR Interferometry Applications”. In:IEEE Transactions on Geoscience and Remote Sensing46.11 (Nov. 2008), pp. 3436–3443. issn: 0196-2892. doi: 10.1109/tgrs.2008.2001756. 34

    work page doi:10.1109/tgrs.2008.2001756 2008

  12. [12]

    A New Algorithm for Processing Interferometric Data-Stacks: SqueeSAR

    Alessandro Ferretti et al. “A New Algorithm for Processing Interferometric Data-Stacks: SqueeSAR”. In: IEEE Transactions on Geoscience and Remote Sensing49.9 (Sept. 2011), pp. 3460–3470.issn: 1558-0644. doi: 10.1109/tgrs.2011.2124465

    work page doi:10.1109/tgrs.2011.2124465 2011

  13. [13]

    Mathematical Framework for Phase-Triangulation Algorithms in Distributed-Scatterer Interferometry

    Ning Cao, Hyongki Lee, and Hahn Chul Jung. “Mathematical Framework for Phase-Triangulation Algorithms in Distributed-Scatterer Interferometry”. In:IEEE Geoscience and Remote Sensing Letters12.9 (Sept. 2015), pp. 1838–1842.issn: 1558-0571. doi: 10.1109/lgrs.2015.2430752

    work page doi:10.1109/lgrs.2015.2430752 2015

  14. [14]

    Efficient Phase Estimation for Interferogram Stacks

    Homa Ansari, Francesco De Zan, and Richard Bamler. “Efficient Phase Estimation for Interferogram Stacks”. In: IEEE Transactions on Geoscience and Remote Sensing56.7 (July 2018), pp. 4109–4125.issn: 1558-0644. doi: 10.1109/tgrs.2018.2826045

    work page doi:10.1109/tgrs.2018.2826045 2018

  15. [15]

    CAESAR: An Approach Based on Covariance Matrix Decomposition to Improve Multibaseline–Multitemporal Interferometric SAR Processing

    Gianfranco Fornaro et al. “CAESAR: An Approach Based on Covariance Matrix Decomposition to Improve Multibaseline–Multitemporal Interferometric SAR Processing”. In:IEEE Transactions on Geoscience and Remote Sensing53.4 (Apr. 2015), pp. 2050–2065.issn: 1558-0644. doi: 10.1109/tgrs.2014.2352853

    work page doi:10.1109/tgrs.2014.2352853 2015

  16. [16]

    Sequential Estimator: Toward Efficient InSAR Time Series Analysis

    Homa Ansari, Francesco De Zan, and Richard Bamler. “Sequential Estimator: Toward Efficient InSAR Time Series Analysis”. In:IEEE Transactions on Geoscience and Remote Sensing55.10 (Oct. 2017), pp. 5637–5652. issn: 1558-0644. doi: 10.1109/tgrs.2017.2711037

    work page doi:10.1109/tgrs.2017.2711037 2017

  17. [17]

    Non-linear phase linking using joined distributed and persistent scatterers

    Sara Mirzaee, Falk Amelung, and Heresh Fattahi. “Non-linear phase linking using joined distributed and persistent scatterers”. In:Computers & Geosciences171 (Feb. 2023), p. 105291.issn: 0098-3004. doi: 10. 1016/j.cageo.2022.105291

    work page arXiv 2023

  18. [18]

    Phase Estimation for Distributed Scatterers in InSAR Stacks Using Integer Least Squares Estimation

    Sami Samiei-Esfahany et al. “Phase Estimation for Distributed Scatterers in InSAR Stacks Using Integer Least Squares Estimation”. In:IEEE Transactions on Geoscience and Remote Sensing54.10 (Oct. 2016), pp. 5671–5687.issn: 1558-0644. doi: 10.1109/tgrs.2016.2566604

    work page doi:10.1109/tgrs.2016.2566604 2016

  19. [19]

    Covariance Fitting Interferometric Phase Linking: Modular Framework and Optimization Algorithms

    Phan Viet Hoa Vu et al. “Covariance Fitting Interferometric Phase Linking: Modular Framework and Optimization Algorithms”. In:IEEE Transactions on Geoscience and Remote Sensing63 (2025), pp. 1–18. issn: 1558-0644. doi: 10.1109/tgrs.2025.3550978

    work page doi:10.1109/tgrs.2025.3550978 2025

  20. [20]

    A Unified Approach of Multitemporal SAR Data Filtering Through Adaptive Estimation of Complex Covariance Matrix

    Jie Dong et al. “A Unified Approach of Multitemporal SAR Data Filtering Through Adaptive Estimation of Complex Covariance Matrix”. In:IEEE Transactions on Geoscience and Remote Sensing56.9 (Sept. 2018), pp. 5320–5333.issn: 1558-0644. doi: 10.1109/tgrs.2018.2813758

    work page doi:10.1109/tgrs.2018.2813758 2018

  21. [21]

    Accurate Estimation of Correlation in InSAR Observations

    H.A. Zebker and K. Chen. “Accurate Estimation of Correlation in InSAR Observations”. In:IEEE Geoscience and Remote Sensing Letters2.2 (Apr. 2005), pp. 124–127.issn: 1545-598X. doi: 10.1109/lgrs.2004.842375

    work page doi:10.1109/lgrs.2004.842375 2005

  22. [22]

    Coherenceestimationinsyntheticapertureradardatabasedonspeckle noise modeling

    CarlosLópez-MartínezandEricPottier.“Coherenceestimationinsyntheticapertureradardatabasedonspeckle noise modeling”. In:Applied Optics46.4 (Feb. 2007), p. 544.issn: 1539-4522. doi: 10.1364/ao.46.000544

    work page doi:10.1364/ao.46.000544 2007

  23. [23]

    Robust Estimators for Multipass SAR Interferometry

    Yuanyuan Wang and Xiao Xiang Zhu. “Robust Estimators for Multipass SAR Interferometry”. In:IEEE Transactions on Geoscience and Remote Sensing54.2 (Feb. 2016), pp. 968–980.issn: 1558-0644. doi: 10.1109/ tgrs.2015.2471303

    work page arXiv 2016

  24. [24]

    Robust Phase Linking in InSAR

    Phan Viet Hoa Vu et al. “Robust Phase Linking in InSAR”. In:IEEE Transactions on Geoscience and Remote Sensing 61 (2023), pp. 1–11.issn: 1558-0644. doi: 10.1109/tgrs.2023.3289338

    work page doi:10.1109/tgrs.2023.3289338 2023

  25. [25]

    Phase Inconsistencies and Multiple Scattering in SAR Interferometry

    Francesco De Zan, Mariantonietta Zonno, and Paco Lopez-Dekker. “Phase Inconsistencies and Multiple Scattering in SAR Interferometry”. In:IEEE Transactions on Geoscience and Remote Sensing53.12 (Dec. 2015), pp. 6608–6616.issn: 1558-0644. doi: 10.1109/tgrs.2015.2444431

    work page doi:10.1109/tgrs.2015.2444431 2015

  26. [26]

    On Closure Phase and Systematic Bias in Multilooked SAR Interferometry

    Yujie Zheng et al. “On Closure Phase and Systematic Bias in Multilooked SAR Interferometry”. In:IEEE Transactions on Geoscience and Remote Sensing(2022). doi: 10.1109/tgrs.2022.3167648

    work page doi:10.1109/tgrs.2022.3167648 2022

  27. [27]

    Vegetation and soil moisture inversion from SAR closure phases: First experiments and results

    Francesco De Zan and Giorgio Gomba. “Vegetation and soil moisture inversion from SAR closure phases: First experiments and results”. In:Remote Sensing of Environment217 (Nov. 2018), pp. 562–572.issn: 0034-4257. doi: 10.1016/j.rse.2018.08.034

    work page doi:10.1016/j.rse.2018.08.034 2018

  28. [28]

    LaMIE: Large-Dimensional Multipass InSAR Phase Estimation for Distributed Scatterers

    Yusong Bai et al. “LaMIE: Large-Dimensional Multipass InSAR Phase Estimation for Distributed Scatterers”. In: IEEE Transactions on Geoscience and Remote Sensing61 (2023), pp. 1–15.issn: 1558-0644. doi: 10.1109/ tgrs.2023.3330971

    work page arXiv 2023

  29. [29]

    Compressed SAR Interferometry in the Big Data Era

    Dinh Ho Tong Minh and Yen-Nhi Ngo. “Compressed SAR Interferometry in the Big Data Era”. In:Remote Sensing 14.2 (Jan. 2022), p. 390.issn: 2072-4292. doi: 10.3390/rs14020390

    work page doi:10.3390/rs14020390 2022

  30. [30]

    Golub and C.F

    G.H. Golub and C.F. Van Loan.Matrix Computations. 4th ed. Baltimore: The Johns Hopkins University Press,

    work page

  31. [31]

    isbn: 9781421407944. 35

    work page

  32. [32]

    Trefethen and D

    L.N. Trefethen and D. Bau.Numerical Linear Algebra. Philadelphia: Society for Industrial and Applied Mathematics, 1997. isbn: 9780898719574

    work page 1997

  33. [33]

    Statistics: textbooks and monographs 185

    Leandro Pardo.Statistical inference based on divergence measures. Statistics: textbooks and monographs 185. Boca Raton: Chapman & Hall/CRC, 2006. 492 pp.isbn: 9781584886006

    work page 2006

  34. [34]

    Phase Unwrapping in InSAR: A Review

    Hanwen Yu et al. “Phase Unwrapping in InSAR: A Review”. In:IEEE Geoscience and Remote Sensing Magazine 7.1 (Mar. 2019), pp. 40–58.issn: 2373-7468. doi: 10.1109/mgrs.2018.2873644

    work page doi:10.1109/mgrs.2018.2873644 2019

  35. [35]

    Angular synchronization by eigenvectors and semidefinite programming

    A. Singer. “Angular synchronization by eigenvectors and semidefinite programming”. In:Applied and Compu- tational Harmonic Analysis30.1 (Jan. 2011), pp. 20–36.issn: 1063-5203. doi: 10.1016/j.acha.2010.02.001

    work page doi:10.1016/j.acha.2010.02.001 2011

  36. [36]

    Nonconvex Phase Synchronization

    Nicolas Boumal. “Nonconvex Phase Synchronization”. In:SIAM Journal on Optimization26.4 (Jan. 2016), pp. 2355–2377.issn: 1095-7189. doi: 10.1137/16m105808x

    work page doi:10.1137/16m105808x 2016

  37. [37]

    Statistical Modeling of Polarimetric SAR Data: A Survey and Challenges

    Xinping Deng et al. “Statistical Modeling of Polarimetric SAR Data: A Survey and Challenges”. In:Remote Sensing 9.4 (Apr. 2017), p. 348.issn: 2072-4292. doi: 10.3390/rs9040348

    work page doi:10.3390/rs9040348 2017

  38. [38]

    DEM-Based SAR Pixel-Area Estimation for Enhanced Geocoding Refinement and Radiometric Normalization

    Othmar Frey et al. “DEM-Based SAR Pixel-Area Estimation for Enhanced Geocoding Refinement and Radiometric Normalization”. In:IEEE Geoscience and Remote Sensing Letters10.1 (Jan. 2013), pp. 48–52. issn: 1558-0571. doi: 10.1109/lgrs.2012.2192093

    work page doi:10.1109/lgrs.2012.2192093 2013

  39. [39]

    Automated terrain corrected SAR geocoding

    U. Wegmuller. “Automated terrain corrected SAR geocoding”. In:IEEE 1999 International Geoscience and Remote Sensing Symposium. IGARSS’99 (Cat. No.99CH36293). Vol. 3. IGARSS-99. IEEE, 1999, pp. 1712–1714. doi: 10.1109/igarss.1999.772070

    work page doi:10.1109/igarss.1999.772070 1999

  40. [40]

    Phase-Based Similarly Decorrelated Pixel Selection and Phase-Linking in InSAR Using Circular Statistics

    Shuyi Yao and Timo Balz. “Phase-Based Similarly Decorrelated Pixel Selection and Phase-Linking in InSAR Using Circular Statistics”. In:IEEE Transactions on Geoscience and Remote Sensing62 (2024), pp. 1–12. issn: 1558-0644. doi: 10.1109/tgrs.2024.3422171

    work page doi:10.1109/tgrs.2024.3422171 2024

  41. [41]

    W. H. Press et al.Numerical Recipes. The Art of Scientific Computing. 3rd ed. Cambridge University Press,

    work page

  42. [42]

    Cheap, Valid Regularizers for Improved Interferometric Phase Linking

    S. Zwieback. “Cheap, Valid Regularizers for Improved Interferometric Phase Linking”. In:IEEE Geoscience and Remote Sensing Letters19 (2022), pp. 1–4.issn: 1558-0571. doi: 10.1109/lgrs.2022.3197423

    work page doi:10.1109/lgrs.2022.3197423 2022

  43. [43]

    Horn and C.R

    R.A. Horn and C.R. Johnson.Matrix Analysis. 2nd ed. New York: Cambridge University Press, 2013.isbn: 9780521548236

    work page 2013

  44. [44]

    Johnson, S

    N.L. Johnson, S. Kotz, and N. Balakrishnan.Continuous Univariate Distributions, Volume 2. 2nd ed. Wiley Series in Probability and Statistics. New York: Wiley, 1995.isbn: 9780471584940

    work page 1995

  45. [45]

    Wright.Numerical optimization

    Jorge Nocedal and Stephen J. Wright.Numerical optimization. 2nd ed. Springer Series in Operations Research and Financial Engineering. New York, NY: Springer, 2006.isbn: 9780387400655

    work page 2006

  46. [46]

    Youcef Saad.Iterative Methods for Sparse Linear Systems. 2nd ed. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2007.isbn: 9780898715347

    work page 2007

  47. [47]

    On the shortest spanning subtree of a graph and the traveling salesman problem

    Joseph B. Kruskal. “On the shortest spanning subtree of a graph and the traveling salesman problem”. In: Proceedings of the American Mathematical Society7.1 (Feb. 1956), pp. 48–50.issn: 1088-6826. doi: 10.1090/s0002-9939-1956-0078686-7

    work page doi:10.1090/s0002-9939-1956-0078686-7 1956

  48. [48]

    Shortest Connection Networks And Some Generalizations

    R. C. Prim. “Shortest Connection Networks And Some Generalizations”. In:Bell System Technical Journal 36.6 (Nov. 1957), pp. 1389–1401.issn: 0005-8580. doi: 10.1002/j.1538-7305.1957.tb01515.x. 36

    work page doi:10.1002/j.1538-7305.1957.tb01515.x 1957