A holographic global uniqueness in passive imaging
[Une unicité globale holographique en imagerie passive]
Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1069-1081.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.306
Classification : 35J05, 35J08, 35P25, 35R30
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

1 CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris, Palaiseau, France
2 & IEPT RAS, 117997 Moscow, Russia
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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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/

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