On the inversion of the Radon transform: standard versusM²approach

Abstract

We compare the Radon transform in its standard and symplectic formulations and argue that the inversion of the latter can be performed more efficiently.

Tomography: mathematical aspects and applications

“…Now we want to obtain an inversion formula, namely we want to prove that one can recover a function f on R N from the knowledge of its Radon transform. In order to get this result we need a preliminary lemma, whose proof can be found in [20], which suggests an interesting physical interpretation.…”

“…In quantum mechanics the Radon transform of the Wigner function [15,16,17] was considered in the tomographic approach to the study of quantum states [18,19] and experimentally realized with different particles and in diverse situations. For a review on the modern mathematical aspects of classical and quantum tomography see [20]. Good reviews on recent tomographic applications can be found in [21] and in [22], where particular emphasis is given on maximum likelihood methods, that enable one to extract the maximum reliable information from the available data can be found.…”

Robustness of raw quantum tomography

“…The Radon transform of the Wigner function (3) has been generalized to the following symplectic, or M 2 , transform [13,26] W (X, µ, ν) = R 2 W (p, q) δ (X − qµ − pν) dp dq, (4) with µ, ν ∈ R. Its complete equivalence with (3) is expressed by the relation [27,28] W (X, r cos ϕ, r sin ϕ)…”

“…The linear dependence is obtained analogously by replacing T with R and Z N with Z. By Fourier inverting (27) one gets…”

Reconstruction of time-dependent coefficients: A check of approximation schemes for non-Markovian convolutionless dissipative generators

“…In particular we will use the symplectic transform, or M 2 -transform [20], that finds its natural implementation in experiments with massive particles, as for example those to detect the longitudinal motion of neutrons [21] and to reconstruct the transverse motional states of helium atoms [22]. We also note that the symplectic transform is equivalent to the Radon transform [23] which is experimentally implemented by homodyne detection in the context of quantum optics [24]. We now recall a procedure introduced in [3] allowing to connect, at any fixed time t, tomographic measurements and the first two cumulants of a Gaussian probe by means of a small number of detections.…”