Stability and resolution in electromagnetic inverse problems

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On the regularization of linear inverse problems in Fourier optics

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Ill-posed Inverse Problems and Logarithmic Continuity in Electromagnetics

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The Stability of Inverse Problems

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Applied Inverse Problems in Optics

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Linear inverse problems with discrete data: II. Stability and regularization

For pt.I. see ibid. vol.1, p.301 (1985). In the first part of this work a general definition of an inverse problem with discrete data has been given and an analysis in terms of singular systems has been performed. The problem of the numerical stability of the solution, which in that paper was only briefly discussed, is the main topic of this second part. When the condition number of the problem is too large, a small error on the data can produce an extremely large error on the generalised solution, which therefore has no physical meaning. The authors review most of the methods which have been developed for overcoming this difficulty, including numerical filtering, Tikhonov regularisation, iterative methods, the Backus-Gilbert method and so on. Regularisation methods for the stable approximation of generalised solutions obtained through minimisation of suitable seminorms (C-generalised solutions), such as the method of Phillips (1962), are also considered.

SVD for linear inverse problems

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Superresolution in linear inverse problems

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Super-resolution in confocal scanning microscopy: generalized inversion formulae

It was shown in previous papers that the resolution of a confocal scanning microscope can be significantly improved by measuring, for each scanning position, the full diffraction image and by inverting these data to recover the value of the object at the confocal point. In the present work, the authors generalize the data inversion procedure by allowing, for reconstructing the object at a given point, to make use of the data samples recorded at other scanning positions. This leads them to a family of generalized inversion formulae, either exact or approximate. Some previously known formulae are re-derived here as special cases in a particularly simple way.

Inversion from redundant data

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Iterative inversion of experimental data in weighted spaces

We present iterative algorithms for solving linear inverse problems with discrete data and compare their performances with the method of singular function expansion, in view of applications in optical imaging and particle sizing.

Inversion of light scattering data by singular system analysis

We consider the problem of inverting experimental data obtained in light scattering experiments described by linear theories. We discuss applications to particle sizing and we describe fast and easy-to-implement algorithms which permit the extraction, from noisy measurements, of reliable information about the particle size distribution. © 1987, SPIE.

Super-resolution in confocal scanning microscopy: II :The incoherent case

For pt.I see ibid. vol.3, p.195 (1987). The authors have shown that the resolution of a confocal scanning microscope can be improved by recording the full image at each scanning point and then inverting the data. These analyses were restricted to the case of coherent illumination. They investigate, along similar lines, the incoherent case, which applies to fluorescence microscopy. They investigate the one-dimensional and two-dimensional square-pupil problems and they prove, by means of numerical computations of the singular value spectrum and of the impulse response function, that for a signal-to-noise ratio of, say 10%, it is possible to obtain an improvement of approximately 60% in resolution with respect to the conventional incoherent light confocal microscope. This represents a working bandwidth of 3.5 times the Rayleigh limit.

Stability Problems in Inverse Diffraction

Inverse diffraction consists in determining the field distribution on a boundary surface from the knowledge of the distribution on a surface situated within the domain where the wave propagates. This problem is a good example for illustrating the use of least-squares methods (also called regularization methods) for solving linear ill-posed inverse problem. We focus on obtaining error bounds For regularized solutions and show that the stability of the restored field far from the boundary surface is quite satisfactory: the error is proportional to ∊(ðŗ‚ ≃ 1) ,ðŗœ being the error in the data (Hölder continuity). However, the error in the restored field on the boundary surface is only proportional to an inverse power of │In∊│ (logarithmic continuity). Such a poor continuity implies some limitations on the resolution which is achievable in practice. In this case, the resolution limit is seen to be about half of the wavelength. Copyright © 1981 by The Institute of Electrical and Electronics Engineers, Inc.

Positive regularised solutions in electromagnetic inverse scattering

Fredholm integral equations of the first kind are the mathematical model common to several electromagnetic, optical and acoustical inverse scattering problems. In most of these problems the solution must be positive in order to satisfy physical plausibility. We consider ill-posed deconvolution problems and investigate several linear regularization algorithms which provide positive approximate solutions at least in the absence of errors on the data.