Table of contents

Imaging is a rapidly growing area in applied sciences. It has an interdisciplinary character and a wide range of applications such as medicine, nondestructive evaluation, microscopy and astronomy, as well as many industrial processes. The increasing demand on imaging is due to the change of role of vision. Today vision is not through eyes only but complemented, for instance, by ultrasound, x-ray computerized tomography (CT), electrical impedance tomography (EIT), to name but a few. Moreover, traditional imaging systems such as microscopes and telescopes are now equipped with detection instruments (CCD cameras) and the resulting digital images are currently processed and enhanced. Finally, the relevance of imaging for industry is best documented by a recent feature by Robert West (West R 2003 In industry, seeing is believing Physics World June 2003). Today the scope of imaging has broadened and plays a central role in many different areas ranging, for instance, from remote sensing to seismology.

In most cases the new imaging techniques are based on indirect measurements of physical parameters; therefore they quite naturally lead to the demand of solving (linear or nonlinear) inverse problems. This indicates the central role that inverse problems have in imaging science.

This special section highlights several topics of recent advances in imaging. The first five papers concern problems originating from medical imaging which can have important applications in other domains. The paper by Ji et al covers a new and promising diagnostic tool in medicine: the identification of abnormal tissues by elastic shear wave properties. The two subsequent papers by Louis and by Defrise et al concern 3D cone beam tomography which is the most recent and advanced technique in x-ray CT. Both the case of circular and helical scanning are considered. The paper by Natterer et al is also about tomography but is intended to exploit the mathematical analogies between x-ray CT and synthetic aperture radar, achieving a unified approach to the important problem of estimating resolution in these two completely different imaging techniques. Electrical impedance tomography is an imaging technique originally proposed for medical applications which can be usefully applied also to problems of nondestructive evaluation. Recent progress in the mathematical treatment of this problem is presented in the paper by Hanke and Brühl.

The next three papers are about scattering problems, a fundamental topic in imaging techniques based for instance on ultrasound and microwave sounding. The papers by Kress and by Colton et al are concerned with inverse obstacle problems presenting two different concepts: Kress gives a survey of Newton methods while Colton et al discuss linear sampling methods. Borcea et al cover the problem of detecting and imaging small or extended objects embedded in inhomogeneous media.

Finally Strong and Chan discuss the application of the total variation regularization method to denoising problems and present new results which enlighten the edge-preserving and scale-dependent properties of this method.

We review and present new results on the transient elastography problem, where the goal is to reconstruct shear stiffness properties using interior time and space dependent displacement measurements. We present the unique identifiability of two parameters for this inverse problem, establish that a Lipschitz continuous arrival time satisfies the eikonal equation, and present two numerical algorithms, simulation results, and a reconstruction example using a phantom experiment accomplished by Mathias Fink's group (the Laboratoire Ondes et Acoustique, ESPCI, Université Paris VII). One numerical algorithm uses a geometrical optics expansion and the other utilizes the arrival time surface.

The aim of this paper is to present a general approach to derive inversion formulae for 3D cone beam tomography. Starting from the approximate inverse we derive reconstruction kernels using the symmetries of the scanning geometry to derive fast algorithms. Reconstructions from real data show the ability of the method to produce reliable images where the data noise is reduced as much as possible.

This paper deals with three-dimensional (3D) image reconstruction for helical computerized tomography with multi-row detectors. We describe a method to improve the accuracy of the rebinning algorithms, which separate 3D reconstruction into independent 2D reconstructions for a set of oblique slices. Each oblique slice is reconstructed from an estimate of its 2D Radon transform obtained from the cone-beam projections acquired while the x-ray source moves along a segment of the helix. Because this helix segment is not contained within the oblique slice, the estimated 2D Radon transform is approximate. In theory, exact rebinning could be achieved by solving John's partial differential equation to virtually move the x-ray source within the oblique slice. In contrast with previous work by Patch (2002 IEEE Trans. Med. Imaging 21 801–13), we do not attempt to solve John's equation exactly. Instead, we use John's equation to compute a first order correction to the rebinning algorithm. Tests with simulated data demonstrate a significant improvement of image quality, obtained with a negligible increase of the computation time and of the sensitivity to noise.

Radar imaging and x-ray computed tomography (CT) are both based on inverting the Radon transform. Yet radar imaging can make images from as little as two degrees of aperture while x-ray CT typically requires an aperture of at least 120°. Our discussion addresses this phenomenon.

We consider the inverse problem of finding cavities within some body from electrostatic measurements on the boundary. By a cavity we understand any object with a different electrical conductivity from the background material of the body. We survey two algorithms for solving this inverse problem, namely the factorization method and a MUSIC-type algorithm. In particular, we present a number of numerical results to highlight the potential and the limitations of these two methods.

The inverse scattering problem of how to image the shape of a scatterer D from the far-field pattern for the scattering of time-harmonic waves can be interpreted as a nonlinear ill-posed operator equation with the operator F mapping the boundary onto the far field. We will review recent results on regularized Newton iteration methods as applied to the above equation and present first ideas of an alternative approach that resembles a least-squares method for the solution of inverse obstacle scattering problems due to Kirsch and Kress and does not require a forward solver.

We survey the linear sampling method for solving the inverse scattering problem for time-harmonic electromagnetic waves at fixed frequency. We consider scattering by an obstacle as well as scattering by an inhomogeneous medium both in and . Included in our discussion is the use of regularization methods for ill-posed problems and numerical examples in both two and three dimensions.

In time reversal, an array of transducers receives the signal emitted by a localized source, time reverses it and re-emits it into the medium. The emitted waves back-propagate to the source and tend to focus near it. In a homogeneous medium, the cross-range resolution of the refocused field at the source location is λ₀L/a, where λ₀ is the carrier wavelength, L is the range and a is the array aperture. The refocusing spot size in a homogeneous medium is independent of the bandwidth of the pulse, but broad-band can help in reducing spurious Fresnel zones. In a noisy (random) medium, the cross-range resolution is improved beyond the homogeneous diffraction limit because the array can capture waves that move away from it at the source, but get scattered onto it by the inhomogeneities. We refer to this phenomenon as super-resolution of the time reversal process in random media. Super-resolution implies in particular that, because of multipathing, the array appears to have an effective aperture a_e that is greater than a. Since a_e depends on the scattering medium, it is not known. In this paper we present a brief review of time reversal theory in a remote sensing regime and a robust procedure for estimating a_e from the signals received at the array. Knowing a_e permits assessing quantitatively super-resolution in time reversal for applications in spatially localized communications with reduced interference. We also review interferometric imaging and its relation to time reversal and to matched field imaging. We show that a_e quantifies in an explicit way the loss of resolution in interferometric array imaging.

We give and prove two new and fundamental properties of total-variation-minimizing function regularization (TV regularization): edge locations of function features tend to be preserved, and under certain conditions are preserved exactly; intensity change experienced by individual features is inversely proportional to the scale of each feature. We give and prove exact analytic solutions to the TV regularization problem for simple but important cases. These can also be used to better understand the effects of TV regularization for more general cases. Our results explain why and how TV-minimizing image restoration can remove noise while leaving relatively intact larger-scaled image features, and thus why TV image restoration is especially effective in restoring images with larger-scaled features. Although TV regularization is a global problem, our results show that the effects of TV regularization on individual image features are often quite local. Our results give us a better understanding of what types of images and what types of image degradation are most effectively improved by TV-minimizing image restoration schemes, and they potentially lead to more intelligently designed TV-minimizing restoration schemes.

This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series.

There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task, and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties.

A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four task-specific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks, i.e., no terms corresponding to coupled tasks are ever computed. This inversion in stages provides a tremendous practical advantage. The achievement of a task is a form of useful information exploited in the redefined and restarted problem; and the latter represents a critically important step in the logic and overall strategy. The individual subseries are analysed and their strengths, limitations and prerequisites exemplified with analytic, numerical and field data examples.

We present an inverse scattering approach for computing n-peakon solutions of the Degasperis–Procesi equation (a modification of the Camassa–Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be amenable to analysis using the isospectral deformations induced from the n-peakon solution, and the inverse problem is solved by a method generalizing the continued fraction solution of the peakon sector of the CH equation.

This paper addresses the scattering of acoustic waves from an inhomogeneous compactly supported scatterer. It is proved that knowledge of one incident wave and the corresponding scattered field at a plane outside the scatterer for all times is sufficient for the unique determination of the field and wave velocity. A layer stripping method is used. In each strip the wave velocity is uniquely determined by Neumann-to-Dirichlet mappings defined on the boundary of the strip. The field in a strip is uniquely determined by integral equations. The stability of the ill-posed field problems is discussed. A numerical example is investigated.

The authors study the regularization of projection methods for solving linear ill-posed problems with compact and injective linear operators in Hilbert spaces. The smoothness of the unknown solution is given in terms of general source conditions, such that the framework of variable Hilbert scales is suitable.

The structure of the error is analysed in terms of the noise level, the regularization parameter and as a function of other parameters, driving the discretization. As a result, a strategy is proposed which automatically adapts to the unknown source condition, uniformly for certain classes, and provides the optimal order of accuracy.

We present in this study some three-dimensional numerical results that validate the use of the linear sampling method as an inverse solver in electromagnetic scattering problems. We recall that this method allows the reconstruction of the shape of an obstacle from the knowledge of multi-static radar data at a fixed frequency. It does not require any a priori knowledge of the physical properties of the scatterer nor any nonlinear optimization scheme. This study also contains some analytical results in the simplified case of a spherical scatterer that somehow make the link between known abstract theoretical results and the numerical scheme. Special attention has been given to pointing out the influence of the frequency on the inversion accuracy.

We present a mathematical treatment of time reversal. Two mathematical models describing approximately the propagation of the time-reversed field are proposed and discussed. Zero initial conditions are exploited in the first model, whereas the method of quasi-reversibility is adopted when constructing the second model. Since computer simulation of time reversal requires knowledge of material properties of a propagating medium, such as the sound speed or electrical conductivity, the general problem of time reversal is nonlinear and ill posed. The ill-posedness is due to the nonuniqueness and instability. To treat this problem, a two-stage procedure is proposed and justified. In the first stage, the unknown material properties of a propagating inhomogeneous medium are approximately determined. Since weak scattering is not assumed, the convexification approach is adopted to estimate such properties. In the second stage, the time-reversed field is approximately determined from the solution of the Cauchy problem for a hyperbolic equation with the lateral data by the method of quasi-reversibility.

Inverse problems in option pricing are frequently regarded as simple and resolved if a formula of Black–Scholes type defines the forward operator. However, precisely because the structure of such problems is straightforward, they may serve as benchmark problems for studying the nature of ill-posedness and the impact of data smoothness and no arbitrage on solution properties. In this paper, we analyse the inverse problem (IP) of calibrating a purely time-dependent volatility function from a term-structure of option prices by solving an ill-posed nonlinear operator equation in spaces of continuous and power-integrable functions over a finite interval. The forward operator of the IP under consideration is decomposed into an inner linear convolution operator and an outer nonlinear Nemytskii operator given by a Black–Scholes function. The inversion of the outer operator leads to an ill-posedness effect localized at small times, whereas the inner differentiation problem is ill posed in a global manner. Several aspects of regularization and their properties are discussed. In particular, a detailed analysis of local ill-posedness and Tikhonov regularization of the complete IP including convergence rates is given in a Hilbert space setting. A brief numerical case study on synthetic data illustrates and completes the paper.

We consider a deconvolution problem with a random noise. The noise is a product of a Gaussian stationary random process by a weight function with constant >0. We study the asymptotically minimax and Bayes settings . In the minimax model a priori information is given that the solution belongs to a ball in Sobolev space W₂^β(R¹). For such a priori information we find an asymptotically minimax estimator of the solution. In the Bayes setting the noise is the same. The solution is a realization of a random process defined as a product of a Gaussian stationary random process by a weight function . We show that the standard Wiener filters remain asymptotically Bayes estimators for this modification of Wiener filtration. The introduction of weight functions is the main difference from the standard settings. This allows us not to make the traditional assumptions that the powers of noise and solutions are infinite or tend to infinity.

We consider a two-dimensional inverse scattering problem of determining a sound-soft obstacle from the far field pattern and, under a non-trapping condition, we establish uniqueness within a class of polygonal domains by means of a single incoming plane wave. Moreover, we will show similar uniqueness in the sound-hard case by means of two incoming plane waves. The key is the analyticity of the solution of the scattering problem and the reflection of solutions.

First we consider the two-dimensional version of the following. Consider a rectangular parallelepiped which is a model of a three-dimensional multilayered material. Assume that the conductivity of the material depends only on the height from the bottom face and is given by a piecewise constant function; the material contains an unknown crack whose connected components may exist in the planes parallel to the bottom face where the conductivity may have a jump. Extract information about the location of the unknown crack by an observation datum that is a single set of the electric current density and the voltage potential on the boundary of the material.

In this paper, the material is given by a rectangle, while the conductivity depends on the height from the bottom line and is given by a piecewise constant function. The observation datum is given by a single set of Cauchy data on the boundary of the rectangle of a solution of a governing equation in two dimensions. The unknown crack is given by the disjoint union of closed segments; all the segments are contained in the lines parallel to the bottom line where the conductivity may have a jump.

We show that the enclosure method introduced by the author himself is applicable to the problem provided the support of the electric current density is localized in the upper line (or bottom line). We give a formula that extracts information about the convex hull of the unknown crack from the observation datum.

Second, we apply the enclosure method to an inverse problem for the Helmholtz equation.

We consider the identification of a parameter in an elliptic equation which—in its weak formulation—can be described by a strictly monotone and Lipschitz continuous operator from knowledge of the physical state. Taking advantage of the special structure, we develop a derivative-free Landweber iteration for solving this nonlinear inverse problem in a stable way. Thereby, the Fréchet differentiability of the parameter-to-output map as well as conditions restricting its nonlinearity are no longer required. As a result, the convergence analysis can be performed using more natural assumptions associated with the solvability of the direct problem, which can also be nonlinear. Numerical experiments are presented.

Ground-based astronomical imaging at thermal-infrared wavelengths requires a differential technique, known as chopping and nodding, to extract the weak astronomical signal from the huge background due to the atmosphere and telescope emission. The resulting image is the second difference of the intensity distribution of the astronomical target, and leads to an image restoration problem that can be formulated as the inversion of a second-difference operator. In general, the problem is affected by a huge non-uniqueness, but the degeneracy is reduced when convenient boundary conditions can be used. In particular, if the target field is surrounded by empty sky, it is natural to require that the solution is zero at the boundary of the image. In this paper we investigate the problem of inverting a second-difference operator with the addition of Dirichlet boundary conditions. We show that the related discrete problem can be reduced to the inversion of a non-singular positive definite matrix whose eigenvalues and eigenvectors can be explicitly given. We also give an inversion formula and we investigate the numerical stability of the solution. Since in most practical situations the inversion problem is ill-conditioned, we give a reformulation as a least-squares problem. The advantage is that it is possible to introduce additional constraints such as the non-negativity of the solution. Moreover, we introduce an iterative algorithm converging to the unique non-negative least-squares solution. Since the latter can be still affected by numerical instability, we show that early stopping of the iterations has a regularization effect. We conclude with a discussion of the observational implications of our analysis.

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