[Une unicité globale holographique en imagerie passive]
We consider a radiation solution $\psi $ for the Helmholtz equation in an exterior region in $\mathbb{R}^3$. We show that the restriction of $\psi $ to any ray $L$ in the exterior region is uniquely determined by its imaginary part $\Im \psi $ on an interval of this ray. As a corollary, the restriction of $\psi $ to any plane $X$ in the exterior region is uniquely determined by $\Im \psi $ on an open domain in this plane. These results have holographic prototypes in the recent work Novikov (2024, Proc. Steklov Inst. Math. 325, 218-223). In particular, these and known results imply a holographic type global uniqueness in passive imaging and for the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in the whole space) in the monochromatic case. Some other surfaces for measurements instead of the planes $X$ are also considered.
Nous considérons une solution de rayonnement $\psi $ pour l’équation de Helmholtz dans une région extérieure de $\mathbb{R}^3$. Nous montrons que la restriction de $\psi $ à tout rayon $L$ de la région extérieure est déterminée de manière unique par sa partie imaginaire $\Im \psi $ sur un intervalle de ce rayon. En corollaire, la restriction de $\psi $ à tout plan $X$ de la région extérieure est déterminée de manière unique par $\mathop {\mathrm{Im}}\psi $ sur un domaine ouvert de ce plan. Ces résultats ont des prototypes holographiques dans l’article récent de Novikov (2024, Proc. Steklov Inst. Math. 325, 218-223). En particulier, ces résultats et des résultats connus impliquent une unicité globale de type holographique en imagerie passive et pour le problème inverse de Gelfand-Krein-Levitan (à partir des valeurs au bord de la mesure spectrale dans l’espace entier) dans le cas monochromatique. D’autres surfaces de mesure que les plans $X$ sont également considérées.
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Keywords: Helmholtz equation, Schrödinger equation, radiation solutions, Gelfand-Krein-Levitan problem, passive imaging, holographic global uniqueness
Mots-clés : Équation de Helmholtz, équation de Schrödinger, solutions de rayonnement, problème de Gelfand-Krein-Levitan, imagerie passive, unicité globale holographique
Roman G. Novikov 1, 2

@article{JEP_2025__12__1069_0, author = {Roman G. Novikov}, title = {A holographic global uniqueness in passive~imaging}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1069--1081}, publisher = {\'Ecole polytechnique}, volume = {12}, year = {2025}, doi = {10.5802/jep.306}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.306/} }
TY - JOUR AU - Roman G. Novikov TI - A holographic global uniqueness in passive imaging JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 1069 EP - 1081 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.306/ DO - 10.5802/jep.306 LA - en ID - JEP_2025__12__1069_0 ER -
%0 Journal Article %A Roman G. Novikov %T A holographic global uniqueness in passive imaging %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 1069-1081 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.306/ %R 10.5802/jep.306 %G en %F JEP_2025__12__1069_0
Roman G. Novikov. A holographic global uniqueness in passive imaging. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1069-1081. doi : 10.5802/jep.306. https://jep.centre-mersenne.org/articles/10.5802/jep.306/
[1] - “Monochromatic identities for the Green function and uniqueness results for passive imaging”, SIAM J. Appl. Math. 78 (2018) no. 5, p. 2865-2890 | DOI | MR
[2] - “Global uniqueness in a passive inverse problem of helioseismology”, Inverse Problems 36 (2020) no. 5, article ID 055004, 21 pages | DOI | MR
[3] - “On Sommerfeld’s “radiation condition.””, Philos. Math. (7) 40 (1949), p. 645-651
[4] - “The uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation”, Trudy Moskov. Mat. Obshch. 7 (1958), p. 1-62, English translation: Transl., Ser. 2, Am. Math. Soc. 35, 167–235 (1964) | MR
[5] - Inverse wave problems of acoustic tomography. Part 2, Inverse problems of acoustic scattering, LENAND, Moscow, 2019, in Russian
[6] - “The use of low-frequency noise in passive tomography of the ocean”, Acoustical Physics 48 (2008) no. 1, p. 42-51 | DOI
[7] - Inverse acoustic and electromagnetic scattering theory, Applied Math. Sciences, vol. 93, Springer, Cham, 2019 | DOI | MR
[8] - Lectures on linear partial differential equations, Graduate Studies in Math., vol. 123, American Mathematical Society, Providence, RI, 2011 | DOI | MR
[9] - Quantum scattering theory for several particle systems, Mathematical Physics and Applied Math., vol. 11, Kluwer Academic Publishers Group, Dordrecht, 1993, Translated from the 1985 Russian original | DOI | MR
[10] - “Passive sensor imaging using cross correlations of noisy signals in a scattering medium”, SIAM J. Imaging Sci. 2 (2009) no. 2, p. 396-437 | DOI | MR
[11] - “Computational helioseismology in the frequency domain: acoustic waves in axisymmetric solar models with flows”, Astronomy and Astrophysics 600 (2017), article ID A35 | DOI
[12] - “New stability estimates for the inverse acoustic inhomogeneous medium problem and applications”, SIAM J. Math. Anal. 33 (2001) no. 3, p. 670-685 | DOI | MR
[13] - The analysis of linear partial differential operators. II, Classics in Math., Springer-Verlag, Berlin, 2005 | DOI | MR
[14] - “A convergent ’farfield’ expansion for two-dimensional radiation functions”, Comm. Pure Appl. Math. 14 (1961), p. 427-434 | DOI
[15] - Iterated helioseismic holography, Ph. D. Thesis, University of Göttinger, 2023, https://ediss.uni-goettingen.de/handle/11858/15455
[16] - “Quantitative passive imaging by iterative holography: the example of helioseismic holography”, Inverse Problems 40 (2024) no. 4, article ID 045016, 32 pages | DOI | MR
[17] - “On reconstruction from imaginary part for radiation solutions in two dimensions”, 2024 | arXiv
[18] - “A holographic uniqueness theorem for the two-dimensional Helmholtz equation”, J. Geom. Anal. 35 (2025) no. 4, article ID 123, 14 pages | DOI | MR
[19] - “A multidimensional inverse spectral problem for the equation ”, Funktsional. Anal. i Prilozhen. 22 (1988) no. 4, p. 11-22 | DOI
[20] - “The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential”, Comm. Math. Phys. 161 (1994) no. 3, p. 569-595 | DOI | MR
[21] - “Inverse scattering without phase information”, in Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2014–2015, Editions de l’École Polytechnique, Palaiseau, 2016, p. XVI.1-XVI.13
[22] - “Multipoint formulas for phase recovering from phaseless scattering data”, J. Geom. Anal. 31 (2021) no. 2, p. 1965-1991 | DOI | MR
[23] - “A holographic uniqueness theorem”, Proc. Steklov Inst. Math. 325 (2024), p. 218-223 | DOI
[24] - “Phase recovery from phaseless scattering data for discrete Schrödinger operators”, Inverse Problems 39 (2023) no. 12, article ID 125006, 12 pages | DOI | MR
[25] - “Fixed-distance multipoint formulas for the scattering amplitude from phaseless measurements”, Inverse Problems 38 (2022) no. 2, article ID 025012, 22 pages | DOI | MR
[26] - “Multipoint formulas in inverse problems and their numerical implementation”, Inverse Problems 39 (2023) no. 12, article ID 125016, 25 pages | DOI | MR
[27] - “Extracting the Green’s function of attenuating heterogeneous acoustic media from uncorrelated waves”, J. Acoust. Soc. Am. 121 (2007), p. 2637-2643 | DOI
[28] - “Ultrasonics without a Source: Thermal Fluctuation Correlations at MHz Frequencies”, Phys. Rev. Lett. 87 (2001), article ID 134301 | DOI
[29] - “A generalization of theorems of Rellich and Atkinson”, Proc. Amer. Math. Soc. 7 (1956), p. 271-276 | DOI | MR
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