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• W1986899128 abstract "We study X-ray and Divergent beam transforms of Trkalian fields and their relation with Radon transform. We make use of four basic mathematical methods of tomography due to Grangeat, Smith, Tuy, and Gelfand-Goncharov for an integral geometric view on them. We also make use of direct approaches which provide a faster but restricted view of the geometry of these transforms. These reduce to well known geometric integral transforms on a sphere of the Radon or the spherical Curl transform in Moses eigenbasis, which are the members of an analytic family of integral operators. We also discuss their inversion. The X-ray (also Divergent beam) transform of a Trkalian field is Trkalian. Also the Trkalian subclass of X-ray transforms yields Trkalian fields in the physical space. The Riesz potential of a Trkalian field is proportional to the field. Hence, the spherical mean of the X-ray (also Divergent beam) transform of a Trkalian field over all lines passing through a point yields the field at this point. The pivotal point is the simplification of an intricate quantity: Hilbert transform of the derivative of Radon transform for a Trkalian field in the Moses basis. We also define the X-ray transform of the Riesz potential (of order 2) and Biot-Savart integrals. Then, we discuss a mini-twistor representation, presenting a mini-twistor solution for the Trkalian fields equation. This is based on a time-harmonic reduction of wave equation to Helmholtz equation. A Trkalian field is given in terms of a null vector in $mathbb {C}^{3}$C3 with an arbitrary function and an exponential factor resulting from this reduction." @default.
• W1986899128 created "2016-06-24" @default.
• W1986899128 creator A5039446708 @default.
• W1986899128 date "2013-10-01" @default.
• W1986899128 modified "2023-09-28" @default.
• W1986899128 title "Trkalian fields: ray transforms and mini-twistors" @default.
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• W1986899128 doi "https://doi.org/10.1063/1.4826106" @default.
• W1986899128 hasPublicationYear "2013" @default.
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