Extraction of polydispersity information from photon correlation data

Photon correlation spectroscopy (PCS) is a light-scattering technique for particle size diagnosis. It has been used mainly in the investigation of hydrosol particles since it is based on the measurement of the correlation function of the light scattered from the Brownian motion of suspended particles. Recently this technique also proved useful for studying soot particles in flames and similar aerosol systems. In the case of a polydispersed system the problem of recovering the particle size distribution can be reduced to the problem of inverting the Laplace transform. In this paper we review several methods introduced by the authors for the solution of this problem. We present some numerical results and we discuss the resolution limits characterizing the reconstruction of the size distributions. © 1989.

Stability and Resolution in Electromagnetic Inverse Scattering - Part I

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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.

Applied Inverse Problems in Optics

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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.

The Use of Singular Systems and other Inversion Methods in Weighted L2- spaces

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Extraction of polydispersity information in photon correlation spectroscopy

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Particle size distributions from Fraunhofer diffraction

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Particle sizing by inversion of extinction data

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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.

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.

A new inversion procedure for confocal scanning microscopy and other bandlimited signal processing problems

We find a simple analytic expression for the inverse of an infinite matrix related to the problem of data reduction in confocal scanning microscopy and other band-limited signal processing problems. Potential applications of this result to practical problems are outlined. The matrix arises from a sampling expansion approach to the integral equation.

Superresolution in linear inverse problems

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