A review and examination of results on uniqueness in inverse problems

Abstract

Inverse source and inverse scattering problems represent distinct situations with regard to the uniqueness of their solutions, although they are often treated the same. This paper examines the reasons why the distinction between these two problems is critical in determining uniqueness. What is known about sufficient (and, in some cases, necessary) information to insure uniqueness for real‐world inverse problems is reviewed. Finally, although Hoenders' result is clear for the inverse scattering problem, there is some controversy over what set of constraints are necessary and sufficient to insure uniqueness in the inverse source problem. This is examined, and the origins of questions which arise from the usual derivations of the fields associated with nonradiating sources are explained.

Light emission by accelerated electric, toroidal, and anapole dipolar sources

“…In practice, this means that if, given a field distribution, a suitable source charge-current configuration can be found, it will not be unique, since one can always add a nonradiating configuration to it without changing the emitted field. This feature of Maxwell's equations, also known as the inverse source problem [2][3][4], impacts many branches of science where light is used to interrogate distant or otherwise inaccessible objects, from medical imaging, to astronomy and radar science [4]. Here we will show that neutral nonradiating configurations can, in fact, be made to radiate by using acceleration.…”

“…We therefore will refer to the excitations defined in Eqs. (1)(2)(3) as 'anapole dipole' throughout this paper.…”

“…1, an anapole corresponds to a superposition of electric and toroidal dipoles. The charge (ρ) and current (J ) densities of an anapole particle stationary in the lab-frame are [8]: (3) (1)…”

“…Here, c is the speed of light, t and r are time and the position of the observer, whilst p = p (t) and T = T (t) are the electric and toroidal dipole moments (respectively). The three-dimensional delta function is denoted with δ (3) = δ (3) (r − r), where r is the position of the point-particle. The time-derivative is denoted by 'overdot', i.e.…”

A Comparision Among Some Local Approximation in One-Dimensional Profile Reconstruction

“…In addition, inverse problems are well known to be nonunique. The nonuniqueness of electromagnetic inverse problems have been well documented in the literature [1][2][3][4][5][6][7][8][9][10] and will not be considered here. Rather, we shall investigate the effectiveness of the localized approximations as originally proposed by Habashy et al [11], and extensions thereto.…”