pith. sign in

arxiv: 1906.10864 · v1 · pith:MMY7TQVLnew · submitted 2019-06-26 · 📡 eess.SP

Cross-correlated Contrast Source Inversion

Shilong Sun , Bert Jan Kooij , Tian Jin , Alexander G. Yarovoy This is my paper

Pith reviewed 2026-05-25 15:33 UTC · model grok-4.3

classification 📡 eess.SP
keywords contrast source inversioncross-correlated cost functionalinverse scatteringelectromagnetic imagingmicrowave tomographyregularizationstate errordata error
0
0 comments X

The pith

Adding a cross-correlated term to the cost functional interrelates state and data errors to improve contrast source inversion.

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

The paper proposes enhancing the contrast source inversion method by introducing a cross-correlated cost functional that connects the state error with the data error. This modification aims to achieve greater robustness and accuracy in reconstructing scatterers from electromagnetic measurements compared to standard CSI and its multiplicative regularized variant. The approach maintains similar computational demands while demonstrating advantages in two-dimensional test cases involving both transverse magnetic and transverse electric excitations. If effective, it could support more precise imaging in applications relying on inverse scattering solutions.

Core claim

By introducing a cross-correlated term into the cost functional of the contrast source inversion algorithm, the state error and data error become interrelated in the measurement domain. This modification allows the method to achieve better robustness and higher inversion accuracy than the classical CSI and the multiplicative regularized CSI. Additionally, the gradient of this modified cost functional can be derived without a substantial increase in computational cost. The benefits are illustrated through comparisons on two-dimensional benchmark problems using both transverse magnetic and transverse electric wave excitations.

What carries the argument

The cross-correlated cost functional, which interrelates the state error and data error in the measurement domain.

Load-bearing premise

That interrelating state and data errors via the cross-correlated term will improve reconstruction quality without introducing new instabilities or systematic biases outside the tested 2-D benchmarks.

What would settle it

A reconstruction on a three-dimensional scatterer or with noise levels beyond the 2-D benchmarks where cross-correlated CSI produces lower accuracy or robustness than classical CSI.

Figures

Figures reproduced from arXiv: 1906.10864 by Alexander G. Yarovoy, Bert Jan Kooij, Shilong Sun, Tian Jin.

Figure 1
Figure 1. Figure 1: The configuration of the inverse scattering problem. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The original “Austria” profile contained in a region of [ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The relative permittivity and conductivity of the contrast obtained by classical CSI, [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The reconstruction error curves of classical CSI, MR-CSI, and CC-CSI, in terms of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The relative permittivity and conductivity of the contrast obtained by classical CSI, [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The reconstruction error curves of classical CSI, MR-CSI, and CC-CSI, in terms [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The relative permittivity and conductivity of the contrast obtained by classical CSI, [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The reconstruction error curves of classical CSI, MR-CSI, and CC-CSI, in terms of [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The relative permittivity and conductivity of the contrast obtained by classical CSI, [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The reconstruction error curves of classical CSI, MR-CSI, and CC-CSI, in terms [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

In this paper, we improved the performance of the contrast source inversion (CSI) method by incorporating a so-called cross-correlated cost functional, which interrelates the state error and the data error in the measurement domain. The proposed method is referred to as the cross-correlated CSI. It enables better robustness and higher inversion accuracy than both the classical CSI and multiplicative regularized CSI (MR-CSI). In addition, we show how the gradient of the modified cost functional can be calculated without significantly increasing the computational burden. The advantages of the proposed algorithms are demonstrated using a 2-D benchmark problem excited by a transverse magnetic wave as well as a transverse electric wave, respectively, in comparison to classical CSI and MR-CSI.

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 the cross-correlated contrast source inversion (CC-CSI) method, which augments the classical CSI cost functional with a cross-correlation term between state and data errors in the measurement domain. It claims this yields improved robustness and higher inversion accuracy relative to both classical CSI and multiplicative regularized CSI (MR-CSI), provides an efficient gradient derivation for the modified functional, and demonstrates the advantages on 2-D TM and TE benchmark problems.

Significance. If the reported gains prove robust, the approach supplies a low-overhead functional modification that could enhance accuracy in electromagnetic inverse scattering without requiring new regularization parameters. The explicit derivation of the gradient and the direct comparison to established CSI variants are constructive elements.

major comments (2)
  1. [Numerical experiments] Numerical experiments section: performance is shown exclusively on standard 2-D TM/TE benchmarks with fixed noise levels and contrasts. The central claim that the cross-correlated term 'enables better robustness' is load-bearing and requires evidence that the term remains stable or beneficial when the 2-D scalar/vector wave assumptions are relaxed (e.g., 3-D geometries or higher contrasts).
  2. [Method / gradient derivation] Cost-functional and gradient derivation: the manuscript presents the cross term as an explicit modification whose gradient adds negligible cost, yet contains no analysis of whether the modified functional introduces systematic biases or instabilities outside the tested 2-D cases; this directly affects the generality of the robustness claim.
minor comments (2)
  1. [Abstract] Abstract: the unqualified statement that the method 'enables better robustness' should be scoped to the 2-D benchmarks actually examined.
  2. [Throughout] Notation: ensure consistent use of symbols for the cross-correlation term across equations and text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive suggestions. We address the major comments point by point below and propose revisions where appropriate.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments section: performance is shown exclusively on standard 2-D TM/TE benchmarks with fixed noise levels and contrasts. The central claim that the cross-correlated term 'enables better robustness' is load-bearing and requires evidence that the term remains stable or beneficial when the 2-D scalar/vector wave assumptions are relaxed (e.g., 3-D geometries or higher contrasts).

    Authors: The numerical validation in the manuscript follows the standard practice in electromagnetic inverse scattering literature, where 2-D TM and TE cases are used as benchmarks to compare new methods against established ones like CSI and MR-CSI. The cross-correlated term is introduced in a general manner applicable to 3-D problems, as the formulation does not rely on 2-D assumptions. However, we agree that additional experiments in 3-D would further support the robustness claim. Due to computational constraints, we will add a discussion on the method's expected performance in higher dimensions and note this as a direction for future work. revision: partial

  2. Referee: [Method / gradient derivation] Cost-functional and gradient derivation: the manuscript presents the cross term as an explicit modification whose gradient adds negligible cost, yet contains no analysis of whether the modified functional introduces systematic biases or instabilities outside the tested 2-D cases; this directly affects the generality of the robustness claim.

    Authors: The gradient of the cross term is derived explicitly in the manuscript and shown to be computable with minimal additional cost by leveraging existing forward and adjoint operators. Regarding potential biases or instabilities, in all conducted experiments the method exhibited stable convergence and improved accuracy without introducing new artifacts. A full theoretical analysis of biases for arbitrary contrasts and dimensions is indeed complex and not provided; we will include a short paragraph acknowledging this limitation and stating that no such issues were observed in the 2-D benchmarks. revision: partial

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper introduces an explicit cross-correlated term into the CSI cost functional that interrelates state and data errors, then derives the corresponding gradient directly from the modified functional. This modification and its gradient are presented as algebraic extensions without any reduction to fitted parameters, self-definitions, or load-bearing self-citations. The claimed improvements are validated through explicit numerical comparisons on standard 2-D TM/TE benchmarks rather than by construction from the inputs. No uniqueness theorems, ansatzes, or renamings are invoked that collapse back to prior author work or data fits. The derivation chain remains self-contained and independent of the target performance claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated premise that the new functional term is well-defined and differentiable for the chosen wave problems.

pith-pipeline@v0.9.0 · 5648 in / 971 out tokens · 19865 ms · 2026-05-25T15:33:31.256345+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    Crosshole GPR full-waveform inversion and waveguide amplitude analysis: Recent developments and new challenges,

    A. Klotzsche, J. van der Kruk, A. Mozaffari, N. Gueting, and H. Vereecken, “Crosshole GPR full-waveform inversion and waveguide amplitude analysis: Recent developments and new challenges,” in 2015 8th International Workshop on Advanced Ground Penetrating Radar (IWAGPR), pp. 1–6, IEEE, 2015

    work page 2015

  2. [2]

    Simultaneous multifrequency inversion of full- waveform seismic data,

    W. Hu, A. Abubakar, and T. M. Habashy, “Simultaneous multifrequency inversion of full- waveform seismic data,” Geophysics, vol. 74, no. 2, pp. R1–R14, 2009

    work page 2009

  3. [3]

    Acoustic inversion in optoacoustic to- mography: A review,

    A. Rosenthal, V. Ntziachristos, and D. Razansky, “Acoustic inversion in optoacoustic to- mography: A review,” Current medical imaging reviews, vol. 9, no. 4, pp. 318–336, 2013

    work page 2013

  4. [4]

    Microwave biomedical data inversion using the finite-difference contrast source inversion method,

    C. Gilmore, A. Abubakar, W. Hu, T. M. Habashy, and P. M. Van Den Berg, “Microwave biomedical data inversion using the finite-difference contrast source inversion method,” IEEE Transactions on Antennas and Propagation , vol. 57, no. 5, pp. 1528–1538, 2009

    work page 2009

  5. [5]

    An accurate equivalent behavioral model of antenna radiation using a mode-matching technique based on spherical near field measurements,

    M. Serhir, P. Besnier, and M. Drissi, “An accurate equivalent behavioral model of antenna radiation using a mode-matching technique based on spherical near field measurements,” IEEE Transactions on Antennas and Propagation , vol. 56, pp. 48–57, Jan 2008

    work page 2008

  6. [6]

    Aperture antenna modeling by a finite number of elemental dipoles from spherical field measurements,

    M. Serhir, J.-M. Geffrin, A. Litman, and P. Besnier, “Aperture antenna modeling by a finite number of elemental dipoles from spherical field measurements,” IEEE Transactions on Antennas and Propagation , vol. 58, pp. 1260–1268, April 2010

    work page 2010

  7. [7]

    Quantitative imaging with incident field modeling from multistatic measurements on line segments,

    S. Nounouh, C. Eyraud, A. Litman, and H. Tortel, “Quantitative imaging with incident field modeling from multistatic measurements on line segments,” Antennas and Wireless Propagation Letters, IEEE, vol. 14, pp. 253–256, 2015

    work page 2015

  8. [8]

    A simple method using morozov’s discrepancy principle for solving inverse scattering problems,

    D. Colton, M. Piana, and R. Potthast, “A simple method using morozov’s discrepancy principle for solving inverse scattering problems,”Inverse Problems, vol. 13, no. 6, pp. 1477– 1493, 1997

    work page 1997

  9. [9]

    A simple method for solving inverse scattering problems in the resonance region,

    D. Colton and A. Kirsch, “A simple method for solving inverse scattering problems in the resonance region,” Inverse problems, vol. 12, no. 4, pp. 383–393, 1996

    work page 1996

  10. [10]

    A contrast source inversion method,

    P. M. Van Den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse problems, vol. 13, no. 6, pp. 1607–1620, 1997

    work page 1997

  11. [11]

    Nonlinear inversion of a buried object in transverse electric scattering,

    B. Kooij, M. Lambert, and D. Lesselier, “Nonlinear inversion of a buried object in transverse electric scattering,” Radio Science, vol. 34, no. 6, pp. 1361–1371, 1999

    work page 1999

  12. [12]

    Extended contrast source inversion,

    P. M. van den Berg, A. Van Broekhoven, and A. Abubakar, “Extended contrast source inversion,” Inverse Problems, vol. 15, no. 5, pp. 1325–1344, 1999

    work page 1999

  13. [13]

    New tools and series for forward and inverse scat- tering problems in lossy media,

    T. Isernia, L. Crocco, and M. D’Urso, “New tools and series for forward and inverse scat- tering problems in lossy media,” Geoscience and Remote Sensing Letters, IEEE , vol. 1, no. 4, pp. 327–331, 2004

    work page 2004

  14. [14]

    The contrast source-extended Born model for 2D subsurface scattering problems,

    L. Crocco, M. D’Urso, and T. Isernia, “The contrast source-extended Born model for 2D subsurface scattering problems,” Progress In Electromagnetics Research B, vol. 17, no. 1, pp. 343–359, 2009

    work page 2009

  15. [15]

    A finite-difference contrast source inversion method,

    A. Abubakar, W. Hu, P. Van Den Berg, and T. Habashy, “A finite-difference contrast source inversion method,” Inverse Problems, vol. 24, no. 6, p. 065004, 2008. 17

    work page 2008

  16. [16]

    Three- dimensional seismic full-waveform inversion using the finite-difference contrast source in- version method,

    A. Abubakar, G. Pan, M. Li, L. Zhang, T. Habashy, and P. van den Berg, “Three- dimensional seismic full-waveform inversion using the finite-difference contrast source in- version method,” Geophysical Prospecting, vol. 59, no. 5, pp. 874–888, 2011

    work page 2011

  17. [17]

    Finite-element contrast source inversion method for microwave imaging,

    A. Zakaria, C. Gilmore, and J. LoVetri, “Finite-element contrast source inversion method for microwave imaging,” Inverse Problems, vol. 26, no. 11, p. 115010, 2010

    work page 2010

  18. [18]

    The finite-element method contrast source inversion algorithm for 2D transverse electric vectorial problems,

    A. Zakaria and J. LoVetri, “The finite-element method contrast source inversion algorithm for 2D transverse electric vectorial problems,” IEEE Transactions on Antennas and Prop- agation, vol. 60, no. 10, pp. 4757–4765, 2012

    work page 2012

  19. [19]

    Contrast source extended Born inversion in non- canonical scenarios via FEM modeling,

    E. A. Attardo, G. Vecchi, and L. Crocco, “Contrast source extended Born inversion in non- canonical scenarios via FEM modeling,” IEEE Transactions on Antennas and Propagation, vol. 62, no. 9, pp. 4674–4685, 2014

    work page 2014

  20. [20]

    The linear sampling method as a way to quantitative inverse scattering,

    L. Crocco, I. Catapano, L. D. Donato, and T. Isernia, “The linear sampling method as a way to quantitative inverse scattering,” IEEE Transactions on Antennas and Propagation, vol. 60, no. 4, pp. 1844–1853, 2012

    work page 2012

  21. [21]

    A two-stage method for inverse medium scattering,

    K. Ito, B. Jin, and J. Zou, “A two-stage method for inverse medium scattering,” Journal of Computational Physics , vol. 237, pp. 211–223, 2013

    work page 2013

  22. [22]

    A recursive approach to improve the image quality in well- logging environments,

    Y.-H. Kuo and J.-F. Kiang, “A recursive approach to improve the image quality in well- logging environments,” Progress In Electromagnetics Research B, vol. 60, pp. 287–300, July 2014

    work page 2014

  23. [23]

    Three-dimensional near-field microwave imaging using hybrid linear sampling and level set methods in a medium with compact support,

    M. Eskandari, R. Safian, and M. Dehmollaian, “Three-dimensional near-field microwave imaging using hybrid linear sampling and level set methods in a medium with compact support,” IEEE Transactions on Antennas and Propagation , vol. 62, pp. 5117–5125, Oct 2014

    work page 2014

  24. [24]

    Colton and R

    D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory , vol. 93. New York: Springer, 2013

    work page 2013

  25. [25]

    Simultaneous TE and TM polarization inversion based on FDFD and frequency hopping scheme in ground penetrating radar,

    S. Sun, B. J. Kooij, T. Jin, and A. Yarovoy, “Simultaneous TE and TM polarization inversion based on FDFD and frequency hopping scheme in ground penetrating radar,” in 8th International Workshop on Advanced Ground Penetrating Radar (IWAGPR), 2015 , pp. 1—5, IEEE, 2015

    work page 2015

  26. [26]

    Contrast source inversion method: state of art,

    P. Van Den Berg and A. Abubakar, “Contrast source inversion method: state of art,” Journal of Electromagnetic Waves and Applications , vol. 15, no. 11, pp. 1503–1505, 2001

    work page 2001

  27. [27]

    R. P. Brent, Algorithms for minimization without derivatives . New Jersey: Englewood Cliffs, 1973

    work page 1973

  28. [28]

    G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations. Prentice-Hall, 1976

    work page 1976

  29. [29]

    Simultaneous nonlinear reconstruction of two-dimensional permittivity and conductivity,

    T. M. Habashy, M. L. Oristaglio, and A. T. Hoop, “Simultaneous nonlinear reconstruction of two-dimensional permittivity and conductivity,” Radio Science, vol. 29, no. 4, pp. 1101– 1118, 1994

    work page 1994

  30. [30]

    Using multiple frequency information in the iterative solution of a two-dimensional nonlinear inverse problem,

    K. Belkebir and A. Tijhuis, “Using multiple frequency information in the iterative solution of a two-dimensional nonlinear inverse problem,” inProceedings Progress in Electromagnet- ics Research Symposium, PIERS 1996, 8 July 1996, Innsbruck, Germany, p. 353, University of Innsbruck, 1996. 18

    work page 1996

  31. [31]

    Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set,

    A. Litman, D. Lesselier, and F. Santosa, “Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set,” Inverse problems, vol. 14, no. 3, pp. 685– 706, 1998

    work page 1998

  32. [32]

    Multiplicative regularization for contrast profile inversion,

    P. M. van den Berg, A. Abubakar, and J. T. Fokkema, “Multiplicative regularization for contrast profile inversion,” Radio Science, vol. 38, no. 2, pp. 1–10, 2003

    work page 2003

  33. [33]

    MaxwellFDFD Webpage,

    W. Shin, “MaxwellFDFD Webpage,” 2015. https://github.com/wsshin/maxwellfdfd

    work page 2015

  34. [34]

    Shin, 3D finite-difference frequency-domain method for plasmonics and nanophotonics

    W. Shin, 3D finite-difference frequency-domain method for plasmonics and nanophotonics . PhD thesis, Stanford University, 2013

    work page 2013

  35. [35]

    Nonlinear total variation based noise removal algorithms,

    L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena , vol. 60, no. 1, pp. 259–268, 1992. 19

    work page 1992