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The reliability of a recently introduced nonlinear estimation method for tomographic imaging is discussed in full detail. It is shown how a proper choice of the functional spaces to which unknown quantities belong and the exploitation of the expected properties of the object under test and of all the available a priori information positively affect robustness against false solutions of the inversion procedure. The developed arguments allow the authors to understand causes of possible false solutions, suggesting possible countermeasures. In particular, it is shown how the proposed approach allows the authors to achieve accurate and reliable reconstructions in a set of cases larger than the range of applicability of other "false solutions free" approaches. Numerical analyses confirm the validity of the approach and the effectiveness of developed inversion procedures through practical examples.

Signal reconstruction from phaseless data occurs in electromagnetics both in the diagnostics of large reflector antennas and in the near-field testing of microwave and millimeter wave antennas. The main difficulty in solving such problems arises from its inherent nonlinearity. Because of the latter, in fact, the solution procedure can converge to a function which has nothing to deal with the actual solution. In this paper we review and summarize the properties of a recently proposed approach to phase retrieval problems which allows to understand and possibly avoid the false solution problem. To this end, ill-posedness of the problem is briefly recalled, and a convenient generalized solution is introduced as the global minimum of a proper functional whose local minima are indeed the false solutions of the problem. Then the geometrical meaning of such a functional is considered, and properties of the solution are discussed according to the formulation of the problem as a quadratic inverse one. In particular, the relevance of using nonredundant representations for the unknown of the problem and as many independent data as possible are discussed in full detail from both a theoretical and a numerical point of view. Finally, it is shown how the proposed solution procedure, descending from the adopted formulation based on a weaker nonlinearity with respect to the other ones, has indeed a better behavior as far as local minima problem is concerned.

The retrieving of the phase of a field radiated by a planar source over a plane in near zone is investigated by formulating it as the inversion of a quadratic operator. To solve the problem, a representation of the near field is adopted which takes into account both the dimension and the location of the source. This makes it possible to derive a new uniqueness result which allows one of the main Limitations of previous methods to be removed. The conditions to be met in order to obtain a completely reliable solution procedure are then briefly illustrated by exploiting the quadraticity of the operator to be inverted. Finally, the iterative solution algorithm is described and examples are shown.

In this series of two papers, we consider in its maximum generality the problem of determining radiated electromagnetic fields starting from phaseless distributions on one or more surfaces surrounding the source. In this first part, theoretical aspects of the problem and the basic point of an appropriate formulation are examined. The problem is conveniently tackled as the inversion of the nonlinear, in particular 'quadratic', operator mapping the set of radiated fields into square amplitude distributions over prescribed surfaces. Next, the compactness property of the set of the unknown field is exploited, leading to an introduction of the finite dimensional representations for it and the concept of its "essential dimension". Furthermore, for the square amplitude data distributions finite dimensional representations are introduced and their "essential dimension" considered, also for more than one observation domain. Finally, due to the ill-posed nature of the problem, a generalized solution is defined as t...

In this series of two papers we consider in its maximum generality the problem of determining electromagnetic fields starting from phaseless distributions on one or more surfaces. In this second part, novel uniqueness conditions for the solution of the problem, requiring in principle a single scanning surface, are introduced. Then, all questions related to the attainment of the generalized solution are discussed. The geometrical properties of the functional set corresponding to the range of the quadratic operator relating the unknowns to the data are examined. It is shown how to avoid both contingent local minima problems and ill-conditioning questions. The crucial role of the dimension of the data representation space is emphasized. The main benefits of the proposed approach are in the numerical search for the solution, that is the exact, closed form line search step in the minimization procedure and the introduction of a proper metric change in the functional, are stressed. A critical review of the resu...

In this paper, we investigate whether and how one can synthesize given far field patterns by acting on the dielectric profile of a lens antenna. In particular, an approach to design a radiating system consisting of a variable dielectric profile and a single or a low number of feeds is proposed and discussed. To properly take into account physical and practical feasibility issues, the proposed method splits the problem into two steps. The first one is aimed to fix design goals (i.e., the nominal field), in such a way to avoid super-directive or physically unfeasible antennas and properly exploits known properties of electromagnetic fields radiated by finite extent sources. The second step, dealing with the dielectric profile synthesis, is based on suitable modifications of inverse scattering techniques and can take into account constraints on the spatial behavior as well as of dielectric characteristics of the profile to be synthesized. To show the interest of the subject, as well as effectiveness of the proposed approach, the synthesis of new dielectric antennas, such as a high directivity source and a lens radiating a steerable degradation-free Chebyshev pattern, is considered in the two dimensional scalar case.

Two new approaches to the optimal synthesis of difference patterns are proposed that can deal in an effective fashion with arbitrary sidelobe bounds. The first approach, which amounts to solving a convex programming problem, can be applied to completely arbitrary (fixed geometry) arrays and it is capable of taking into account additional constraints. The second approach, which amounts to solving a simpler linear programming problem, can be applied to uniformly spaced linear or planar arrays, and allows results about the uniqueness of the solution to be inferred. Numerical examples show the flexibility and effectiveness of the proposed procedures.

A new approach providing new theoretical results to the optimal far-field focusing of uniformly spaced arrays, subject to a completely arbitrary mask for sidelobe bounds, is presented and discussed. In both cases of linear or planar arrays (with rectangular boundaries), it is first shown that the problem can be formulated, without any loss to performances on the maximum, as a linear programming one, guaranteeing a globally optimal solution. Second, a sufficient uniqueness criterion for the solution of the overall problem is also developed, which shows how the solution may not be unique (as is actually the case) when planar arrays are considered. In addition, further globally effective optimization procedures are proposed for the latter case in order to optimize directivity, smoothness of excitations, or other performance parameters in the set of equivalent solutions. Last, an extension to planar arrays with a nonrectangular boundary is also given. A thorough numerical analysis confirms the effectiveness of the approach proposed and of the solution codes developed.

The aim of this paper is twofold. First, why and how exploitation of multifrequency information is of great usefulness in inverse scattering problems is discussed. Second, three different solution strategies, all based on a recently introduced approach, are presented, discussed, and compared in the actual case of noise affected data in the two-dimensional (2D) scalar case. The first one is a (nonlinear) frequency-hopping technique, which favorably compares with approaches of the same kind that use linear inversion steps. As a second approach, the contemporary use of the different frequencies data is considered. Contrary to common assumptions, it is shown that such an approach may perform better than the previous one, the different performances between the two being dictated by the spatial frequency content of the unknown contrast. Finally, a hybrid and, to the best of the authors' knowledge, novel strategy, which exploits the advantages of the the previous ones, is introduced and discussed.

With reference to inverse scattering from an unknown object of limited extension embedded in a homogeneous background at a fixed frequency, we show that only a finite-dimensional representation of the unknown contrast can be hopefully retrieved. Exploiting the quasi-band-limitedness property of scattered fields, an accurate upper bound to the dimension of such a space is evaluated in both the single incidence and multiview cases. Moreover, effective schemes are given to collect all the information available from the scattering experiments in a nonredundant manner. As a by-product, an optimal (minimally redundant) sampling strategy for the monostatic radar cross section is also provided. Finally, we briefly discuss how the requirement for a globally effective and reliable solution scheme can lead to a reduction of the actually retrievable information.

Radiation pattern determination from phaseless near field data distributions collected over limited domains on planar surfaces is addressed as a nonlinear inverse problem. Numerical results show the good performances of the considered approach suggesting new possibilities for simplifying measurement setups in near field techniques.

A new method for the estimation of the far field from two amplitude distributions in near zone is presented. The crucial point of the proposed approach relies on the choice of an appropriate function of the data, which is minimised in order to find the solution. It turns out that, although the problem is nonlinear, the global minimum can be found efficiently by a Newton-like minimisation procedure.

After briefly reviewing the properties of the squared amplitudes of radiated fields, a simple and effective necessary condition to test if a source of given size and structure can radiate or not a power pattern lying in a given mask is furnished. The criterion is shown to be also sufficient for linear arrays, thus allowing to reduce the overall problem to the synthesis of a nominal pattern.Then, new synthesis procedures, strongly relying on the properties of squared amplitude distributions and on the quadraticity of the operator relating the source to the power pattern, are introduced. They are implemented and tested in the cases of linear and planar uniform arrays, showing that thanks to a full exploitation of the properties of quadratic operators and of those of squared amplitude distributions, they allow one to achieve very efficient solutions to mask constrained synthesis problems.