C. De Mol

The Stability of Inverse Problems

Many inverse problems arising in optics and other fields like geophysics, medical diagnostics and remote sensing, present numerical instability: the noise affecting the data may produce arbitrarily large errors in the solutions. In other words, these problems are ill-posed in the sense of Hadamard.

Linear inverse problems with discrete data. I: General formulation and singular system analysis

The authors discuss linear methods for the solution of linear inverse problems with discrete data. Such problems occur frequently in instrumental science, e.g. tomography, radar, sonar, optical imaging, particle sizing and so on. They give a general formulation of the problem by extending the approach of Backus and Gilbert (1968) and by defining a mapping from an infinite-dimensional function space into a finite-dimensional vector space. The singular system of this mapping is introduced and used to define natural bases both in the solution and in the data space. They analyse in this context normal solutions, least-squares solutions and generalised inverses. They illustrate the wide applicability of the singular system technique by discussing several examples in detail. Particular attention is devoted to showing the many connections between this method and techniques developed in other topics like the extrapolation of band-limited signals and the interpolation of functions specified on a finite set of points. For example, orthogonal polynomials for least-squares approximation, spline functions and discrete prolate spheroidal functions are particular cases of the singular functions introduced. The problem of numerical stability is briefly discussed, but the investigation of the method developed for overcoming this difficulty, like truncated expansions in the singular bases, regularised solutions, iterative methods, and so on, is deferred to a second part of this work.

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

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.

Accelerating gradient projection methods for l1-constrained signal recovery by steplength selection rules

Abstract We propose a new gradient projection algorithm that compares favorably with the fastest algorithms available to date for l 1 -constrained sparse recovery from noisy data, both in the compressed sensing and inverse problem frameworks. The method exploits a line-search along the feasible direction and an adaptive steplength selection based on recent strategies for the alternation of the well-known Barzilai–Borwein rules. The convergence of the proposed approach is discussed and a computational study on both well conditioned and ill-conditioned problems is carried out for performance evaluations in comparison with five other algorithms proposed in the literature.

Regularized iterative and non-iterative procedures for object restoration from experimental data

A regularized algorithm for the recovery of band-limited signals from noisy data is described. The regularization is characterized by a single parameter. Iterative and non-iterative implementations of the algorithm are shown to have useful properties, the former offering the advantage of flexibility and the latter a potential for rapid data processing. Comparative results, using experimental data obtained in laser anemometry studies with a photon correlator, are presented both with and without regularization.

A Sparsity-Enforcing Method for Learning Face Features

In this paper, we propose a new trainable system for selecting face features from over-complete dictionaries of image measurements. The starting point is an iterative thresholding algorithm which provides sparse solutions to linear systems of equations. Although the proposed methodology is quite general and could be applied to various image classification tasks, we focus here on the case study of face and eyes detection. For our initial representation, we adopt rectangular features in order to allow straightforward comparisons with existing techniques. For computational efficiency and memory saving requirements, instead of implementing the full optimization scheme on tenths of thousands of features, we propose a three-stage architecture which consists of finding first intermediate solutions to smaller size optimization problems, then merging the obtained results, and next applying further selection procedures. The devised system requires the solution of a number of independent problems, and, hence, the necessary computations could be implemented in parallel. Experimental results obtained on both benchmark and newly acquired face and eyes images indicate that our method is a serious competitor to other feature selection schemes recently popularized in computer vision for dealing with problems of real-time object detection. A major advantage of the proposed system is that it performs well even with relatively small training sets.

Regularized iterative and noniterative procedures for object restoration in the presence of noise: an error analysis

An analysis is carried out, using the prolate spheroidal wave functions, of certain regularized iterative and noniterative methods previously proposed for the achievement of object restoration (or, equivalently, spectral extrapolation) from noisy image data. The ill-posedness inherent in the problem is treated by means of a regularization parameter, and the analysis shows explicitly how the deleterious effects of the noise are then contained. The error in the object estimate is also assessed, and it is shown that the optimal choice for the regularization parameter depends on the signal-to-noise ratio. Numerical examples are used to demonstrate the performance of both unregularized and regularized procedures and also to show how, in the unregularized case, artefacts can be generated from pure noise. Finally, the relative error in the estimate is calculated as a function of the degree of superresolution demanded for reconstruction problems characterized by low space–bandwidth products.

On the problems of object restoration and image extrapolation in optics

In this paper we consider the problems of object restoration and image extrapolation, according to the regularization theory of improperly posed problems. In order to take into account the stochastic nature of the noise and to introduce the main concepts of information theory, great attention is devoted to the probabilistic methods of regularization. The kind of the restored continuity is investigated in detail; in particular we prove that, while the image extrapolation presents a Holder type stability, the object restoration has only a logarithmic continuity.

Particle size distributions from spectral turbidity: a singular-system analysis

The problem of the inversion of spectral turbidity measurements to obtain distributions of droplet size in polydisperse aerosols is considered by introducing a recently developed generalised theory of information based on singular-system analysis. In this work the authors limit themselves to the anomalous scattering approximation but give for this case an extensive treatment, with examples, of the reconstruction of the distribution as a continuous function in both weighted and unweighted L2 spaces from finite sampled data sets. The methods are entirely parallel to those already used successfully in the related problems of particle size distribution by photon correlation spectroscopy and by Fraunhofer diffraction.

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.