M. Bertero

Introduction to inverse problems in imaging

Part 1 Image deconvolution: some mathematical tools examples of image blurring the ill-posedness of image deconvolution regularization methods iterative regularization methods statistical methods. Part 2 Linear inverse problems: examples of linear inverse problems singular value decomposition (SVD) inversion methods revisited Fourier based methods for specific problems comments and concluding remarks. Part 3 Mathematical appendices. References.

Ill-posed problems in early vision

Mathematical results on ill-posed and ill-conditioned problems are reviewed and the formal aspects of regularization theory in the linear case are introduced. Specific topics in early vision and their regularization are then analyzed rigorously, characterizing existence, uniqueness, and stability of solutions. A fundamental difficulty that arises in almost every vision problem is scale, that is, the resolution at which to operate. Methods that have been proposed to deal with the problem include scale-space techniques that consider the behavior of the result across a continuum of scales. From the point of view of regulation theory, the concept of scale is related quite directly to the regularization parameter lambda . It suggested that methods used to obtained the optimal value of lambda may provide, either directly or after suitable modification, the optimal scale associated with the specific instance of certain problems. >

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.

Image deblurring with Poisson data: From cells to galaxies.

A semiconductor device including an N-type semiconductor substrate which includes arsenic as an impurity, a first electrode formed on a main surface of the N-type semiconductor substrate, a ground surface formed on another surface of the N-type semiconductor substrate, a second electrode formed on the ground surface and ohmically-contacted with the N-type semiconductor substrate, a semiconductor element formed in the N-type semiconductor substrate and flowing current between the first electrode and the second electrode during ON-state thereof. The device has a reduced ON-resistance thereof.

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.

Super-resolution in computational imaging.

Super-resolution is a word used in different contexts but mainly in the case of methods aimed at improving the resolution of an optical instrument beyond the diffraction limit. Such a result may be achieved by means of specific instrumental techniques (such as, for instance, interferometry) or by means of a suitable processing of a digital image; in the latter case we will use the expression computational super-resolution (CSR). In this paper we describe the basic concepts underlying CSR without using the mathematics required for establishing its theoretical validity. The aim is to introduce a wide audience to this topic, to specify the situations where CSR is feasible and to emphasize the point that unlimited CSR is not possible.

Efficient gradient projection methods for edge-preserving removal of Poisson noise

Several methods based on different image models have been proposed and developed for image denoising. Some of them, such as total variation (TV) and wavelet thresholding, are based on the assumption of additive Gaussian noise. Recently the TV approach has been extended to the case of Poisson noise, a model describing the effect of photon counting in applications such as emission tomography, microscopy and astronomy. For the removal of this kind of noise we consider an approach based on a constrained optimization problem, with an objective function describing TV and other edge-preserving regularizations of the Kullback–Leibler divergence. We introduce a new discrepancy principle for the choice of the regularization parameter, which is justified by the statistical properties of the Poisson noise. For solving the optimization problem we propose a particular form of a general scaled gradient projection (SGP) method, recently introduced for image deblurring. We derive the form of the scaling from a decomposition of the gradient of the regularization functional into a positive and a negative part. The beneficial effect of the scaling is proved by means of numerical simulations, showing that the performance of the proposed form of SGP is superior to that of the most efficient gradient projection methods. An extended numerical analysis of the dependence of the solution on the regularization parameter is also performed to test the effectiveness of the proposed discrepancy principle.

Projected Landweber method and preconditioning

The projected Landweber method is an iterative method for solving constrained least-squares problems when the constraints are expressed in terms of a convex and closed set . The convergence properties of the method have been recently investigated. Moreover, it has important applications to many problems of signal processing and image restoration. The practical difficulty is that the convergence is too slow. In this paper we apply to this method the so-called preconditioning which is frequently used for increasing the efficiency of the conjugate gradient method. We discuss the significance of preconditioning in this case and we show that it implies a modification of the original constrained least-squares problem. However, when the original problem is ill-posed, the approximate solutions provided by the preconditioned method are similar to those provided by the standard method if the preconditioning is suitably chosen. Moreover, the number of iterations can be reduced by a factor of 10 and even more. A few applications to problems of image restoration are also discussed.

On the Recovery and Resolution of Exponential Relaxation Rates from Experimental Data: A Singular-Value Analysis of the Laplace Transform Inversion in the Presence of Noise

The problem of numerical inversion of the Laplace transform is considered when the inverse function is of bounded, strictly positive support. The recent eigenvalue analysis of McWhirter and Pike for infinite support has been generalized by numerical calculations of singular values. A priori knowledge of the support is shown to lead to increased resolution in the inversion, and the number of exponentials that can be recovered in given levels of noise is calculated.