insights collection – summaries of the excellent ......in inverse problems to the wider research...

Insights collection – summaries of the excellent interdisciplinary work published in Inverse Problems

iopscience.org/ip

Inverse Problems

http://iopscience.iop.org/0266-5611

Inverse Problems

Journal scopeInverse Problems is an interdisciplinary journal combining mathematical and experimental papers on inverse problems with theoretical, numerical and practical approaches to their solution.

In addition to applied mathematicians, physical scientists and engineers the readership includes those working in geophysics, radar, optics, biology, acoustics, communication theory, signal processing and imaging, among others.

The emphasis is on publishing original contributions to methods of solving mathematical, physical and applied problems. To be published in Inverse Problems, papers must meet the highest standards of scientificquality, contain significant and original new science and should present substantial advancement in thefield. Due to the broad scope of the journal, we require that authors provide sufficient introductory material to appeal to the wide readership and that articles that are not explicitly applied include a discussion of possible applications.

Article typesResearch papersReports of original research work that discuss the analysis or application of inverse problems.

Special issues or sectionsCollections of novel research articles on a specific topic commissioned by the Editorial Board and Guest Editors. Submissions may be by invitation or by an open call for papers. Papers submitted to these special issues undergo the same rigorous peer review process as all other manuscripts and are published within, or as a replacement for, regular journal issues.

Topical reviewsWritten by leading researchers in their respective fields, these articles present a review of the current state of the art of a particular field. Topical reviews are normally commissioned by the Editorial Board.

iopscience.org/ip

ISSN 0266-5611

InverseProblems

An international journal on the theory and practice of inverse problems, inverse methods and the computerized inversion of data

Volume 29 Number 9 September 2013

iopscience.org/ip


Inverse Problems

iopscience.org/ip 3

Insights articles are short news stories that highlight some of the excellent work published in Inverse Problems to the wider research community. The first Insights article was published on the journal homepage in May 2011 and we have now published a total of 29 articles.

With the help of the editorial team I choose suitable papers, taking care to select a range of different topics that represent the multidisciplinary nature of the journal. The authors are invited to prepare a short Insights article on their work, detailing the key advances in their paper in the context of the wider field. The Insights articles and details of the authors are then published free to read on the journal homepage. Each article includes links to the full text paper and to the authors’ more recent Inverse Problems publications. These short articles allow readers to quickly assess the main aims and advances of the work, and further details can be found by following a link to the full text paper.

I would like to thank all of the Insights authors for submitting their excellent work to the journal and for supporting this programme. I would also like to thank our editorial team for their support.

Finally, I would like to thank all of the journal’s authors, referees and Board Members for their valued contribution to the ongoing success of Inverse Problems.

I hope you enjoy reading the Inverse Problems Insights collection.

http://iopscience.iop.org/0266-5611/labtalk/1

Alfred K LouisEditor-in-Chief

Welcome

Society publisherInverse Problems is published by IOP Publishing, the publisher central to the Institute of Physics (IOP) and a world leader in professional scientific communications. IOP is a not-for-profit society and registered charity that works to advance physics research, application and education, and engages with policymakers and the public to develop awareness and understanding of physics. Any surplus from IOP Publishing goes to supporting science through the activities of the Institute.


http://iopscience.iop.org/0266-5611/labtalk/1

Inverse ProblemsInverse Problems

4 iopscience.org/ip

ContentsSeismic exploration without low frequencies 6Frank Natterer

Structured matrices in the controllability of the heat equation 6Faker Ben Belgacem and Sidi Mahmoud Kaber

Local regularization of linear ill-posed problems 7Cara D Brooks and Patricia K Lamm

The inexactness of ‘exact’ reconstructions from a few projections 8E Garduño, G T Herman and R Davidi

Feature reconstruction in inverse problems 9Alfred K Louis

Extrapolation in variable RKHSs with application to the blood glucose reading 10 V Naumova, S V Pereverzyev and S Sivananthan

Quantitative photoacoustic tomography with multiple illuminations 10Guillaume Bal and Kui Ren

A new approach to nonlinear constrained Tikhonov regularization 11Kazufumi Ito and Bangti Jin

An adaptive method for travel-time tomography 12Eric Chung, Jianliang Qian, Gunther Uhlmann and Hongkai Zhao

Unique recovery of inhomogeneity from transmission eigenvalues 13Tuncay Aktosun, Drossos Gintides and Vassilis G Papanicolaou

Reconstruction of missing data in social networks based on temporal patterns of interactions 14Alexey Stomakhin, Martin B Short and Andrea L Bertozzi

Best basis choice in geophysical tomography 14D Fischer and V Michel

Recovering hidden sparse signals using Bayes’ formula 15V Kolehmainen, M Lassas, K Niinimäki and S Siltanen

Reconstruction of the interfaces using complex geometrical optics solutions 16Mourad Sini and Kazuki Yoshida

Time reversal-defocusing may not be a bad thing 16Jianlong Li, Wen Xu and Liling Jin

Complex transmission eigenvalues for real-valued refractive indices 17Yuk-J Leung and David Colton

Enhanced approximate cloaking by SH and FSH lining 18Jingzhi Li, Hongyu Liu and Hongpeng Sun

http://iopscience.iop.org/0266-5611/


iopscience.org/ip 5

Locating small conductivity inhomogeneities with two electrodes 19Nuutti Hyvönen and Otto Seiskari

An extended-DORT method and its application in a cavity configuration 20X Y Zhang, H Tortel, A Litman and J-M Geffrin

Wave-equation-based linearized seismic inversion 21Fons ten Kroode

A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators 22Thomas Peter and Gerlind Plonka

Convergence rates in ,1-regularization if the sparsity assumption fails 22Martin Burger, Jens Flemming and Bernd Hofmann

The factorization method for inverse acoustic scattering in a layered medium 23Oleksandr Bondarenko, Andreas Kirsch and Xiaodong Liu

A new algorithm for discrete tomography 24 E Gouillart, F Krzakala, M Mézard and L Zdeborová

Reconstruction of piecewise constant fluorophore concentrations in optical tomography 25A Laurain, M Hintermüller, M Freiberger and H Scharfetter

An exact inversion formula for cone beam vector tomography 26Alexander Katsevich and Thomas Schuster

Kalman-based methods for the solution of inverse problems 27Marco A Iglesias, Kody J H Law and Andrew M Stuart

Blind deconvolution for astronomical imaging 28M Prato, A La Camera, S Bonettini and M Bertero

Quantitative photoacoustic tomography using the radiative transfer equation 29T Saratoon, T Tarvainen, B T Cox and S R Arridge

Cover image: an artistic interpretation inspired by figures from the article “Detection of multiple inclusions from sweep data of electrical impedance tomography” N Hyvönen and O Seiskari 2012 Inverse Problems 28 095014.



6 iopscience.org/ip

ArticlesSeismic exploration without low frequencies

In seismic exploration one finds structures such as oil fields, salt domes and rocks by exposing the subsurface to sound waves and measuring the reflected signals. One of the biggest problems with this technique is that the seismic sound sources are unable to produce low frequency signals. Typically the lowest frequencies available are about 5 Hz. In the recent discipline of low frequency seismics one tries to overcome this difficulty by employing transducers that deliver smaller frequencies.

We tried a different approach that does not need low frequency data, based purely on mathematics. The idea is to apply a data completion procedure to the spectrally incomplete data. In principle, such a completion procedure is necessarily highly unstable. However, we are able to show that for media that have only a restricted dip angle the completion procedure is reasonably stable. Our analysis is based on linearization and numerical evidence. We are continuing our work in this area to extend the technique to the fully non-linear problem to allow this method to be used in practice.

2011 Inverse Problems 27 035011

Frank Natterer became Professor for Applied Mathematics in Saarbrücken in 1973. In 1981 he became Professor for Applied Mathematics in Münster and chair from 1984 to 1985 and from 2003 to 2005. Frank Natterer works on the mathematics and numerical analysis of image reconstruction.

Frank Natterer

Structured matrices in the controllability of the heat equation

Controllability or trajectory control by boundary controls aims to realize a desired state at a fixed time by operating on the boundary of the system. Problems that arise from the controllability of diffusive processes are often ill-posed. The numerical counterparts are highly ill-conditioned. Understanding the control operator may help users to choose appropriate computational tools to secure efficient results.

The control operator has singular values that are clustered around zero and infinity. Fourier calculations transform the ill-posedness into one related to linear algebra, connected with the spectrum of some infinite structured matrices. The Gershgorin–Hadamard theorem and Collatz–Wielandt formula provide lower and upper bounds for the largest singular value. After checking that they are also solutions of symmetric Lyapunov equations with a low displacement rank, we use a Penzl estimate to bound the ratio smallest/largest singular values. Computations on the truncated control operator confirm the bounds predicted. According to a Galerkin procedure, the control is hunted for in a vector space,

Faker Ben Belgacem

Sidi Mahmoud Kaber


http://iopscience.iop.org/0266-5611/labtalk-article/45865



iopscience.org/ip 7

with dimension N. The explicit formulas for the inverse provide a formal exponential series representation of the Dirichlet control. Afterwards, that series must be checked as to whether it is convergent to discover if the desired state is reachable.


Faker Ben Belgacem is a member of the Génie-Informatique Department at the Université de Technologie de Compiègne and an associated member of LAMSIN, ENIT, Tunisia. Sidi Mahmoud Kaber is a member of the Laboratoire Jacques-Louis Lions (Université Pierre & Marie Curie, Paris, France).

Local regularization of linear ill-posed problems

In contrast to most regularization methods, local regularization is based on two essential ideas: (1) only the data most relevant to the desired solution should be used; (2) there are advantages to imposing regularization over local regions of the solution’s domain.

In generalizing what has been known as local (sequential) regularization, defined previously in the context of specific inverse problems, a larger class of methods is now defined based on these ideas. Such methods have the potential for greater efficiencies and accuracy in reconstruction of the unknown solution. Furthermore, a functional regularization parameter can be employed to provide variable control of regularization throughout the domain.

The generalized theory applies to linear ill-posed problems of the form Au = f in Banach function spaces. In the generalized framework, regularization operators take the form

Rα = (a

αLα + A

α)–1 T

α;

The authors focus on zeroth order regularization only. As an example, they apply the generalized theory to Volterra convolution equations in Lp (0, 1), 1 ≤ p ≤ 1, establishing rates of convergence beyond those previously known and under more general source conditions. In subsequent papers, the authors address the issues of a posteriori selection of the regularization parameter and higher order generalized local regularization, and applications to non-Volterra problems.


Cara Brooks is a research associate at Michigan State University, but has moved to Florida Gulf Coast University (fall 2011). Patricia (Patti) Lamm is a member of the Department of Mathematics at Michigan State University, and is the organizer and Editor-in-Chief of the Inverse Problems Network (IPNet).

Cara Brooks

Patricia Lamm





8 iopscience.org/ip

The inexactness of ‘exact’ reconstructions from a few projections

Image reconstruction from projections aims to acquire knowledge of the interior of an object from physically obtained approximations of line integrals of some spatially varying physical parameter. In practice, measurements are taken for a number of lines. The aim is to reconstruct the distribution of the physical parameter from the measured data; such a distribution is referred to as an image.

An ideal projection comprises all line integrals of the image for lines in a fixed direction. The practical problem is to reconstruct the image from measured samples of some of its projections. In some applications it is desirable to take the measurements in such a way that the number of sampled projections is small. The possibility of obtaining good reconstructions from a few projections is based on the fact that images in any application area are not random samples of all possible images, but have some common attributes. If these attributes are reflected in the smallness of an objective function, then the aim of satisfying the projections can be complemented with the aim of having a small objective value.

In previous work we demonstrated that such claims of ‘exact reconstructions’ by TV minimization are invalid when physically realistic projection data are used. Unfortunately, our current investigation shows that using the ,1-norm of the Haar transform instead of TV does not improve the quality of the reconstructions from 60 realistic projections. On the other hand, reconstructing from 360 realistic projections by a standard method (ART with blob basis functions) produces diagnostically efficacious reconstructions.


Edgar Garduño, Gabor T Herman and Ran Davidi have been members of the Discrete Imaging and Graphics (DIG) Group at the City University of New York. Edgar Garduño is also working at the Universidad Nacional Autónoma de México. Gabor T Herman heads the DIG group and is a distinguished professor of computer science. Ran Davidi is a postdoctoral research fellow at Stanford University.

Edgar Garduño

Gabor T Herman

Ran Davidi

Did you know?On average, Insights articles receive more than 250 visits each

Did you know?Our most popular individual Insights article received more than 700 page views




iopscience.org/ip 9

Feature reconstruction in inverse problems

To enhance the information content of reconstructed images, feature extraction methods are applied to find edges or to determine optical flow in movies. We present a methodology to combine the reconstruction step and the feature extraction in one algorithm, which strongly improves the overall performance and allows for applications in many fields.

The algorithm is based on the concept of the approximate inverse, where starting from a prescribed mollifier (considered as a mathematical lens) a reconstruction kernel is computed by solving an auxiliary problem independent of the data. In that way, the reconstruction kernel can be precomputed, and the solution of the problem is reduced to the application of the reconstruction kernel to the data. If the same problem has to be solved repeatedly for different sets of data, the procedure is extremely efficient. This is the case if the data are produced in a measuring device using different specimens, as for example in tomography.

Here the auxiliary problem depends on the evaluation operator. The criteria for the selection of the mollifier depend both on the smoothing properties of the original problem and the evaluation, and on invariances allowing for fast algorithms. The overall costs for the calculation of the enhanced image are the same as for the original reconstruction step.

We derive the algorithm, provide a theoretical framework of order-optimality in Banach spaces to determine the needed properties of the mollifier, and present reconstructions from synthetic and real data from different tomographic modalities.


Alfred K Louis has been a full professor of applied mathematics at Saarland University in Saarbrücken since 1990. From 1997 to 1999 he was Vice-President of that university and in 2005 he received the French order ‘Chevalier de l’Ordre des Palmes Académiques’.

Alfred K Louis




10 iopscience.org/ip

Extrapolation in variable RKHSs with application to the blood glucose reading

Recent progress in diabetes technology is related to the continuous glucose monitoring (CGM) systems estimating the blood glucose (BG) from the electric current measured in the interstitial fluid.

Mathematically the problem can be formulated as follows: we are given a data set, where each element is a value of an unknown function paired with a point at which this value is attained. The function values may be blurred by noise, and the problem is to approximate the unknown function for the whole range of relevant values of the argument. This problem can be seen as an extrapolation, since it is not guaranteed that given data points will span the required range.

It is well known that extrapolation is ill-posed and requires special regularization that trades off between data fitting and a complexity of a data fitter. The latter one is often measured by the norm in some reproducing kernel Hilbert space (RKHS), such as a Sobolev space, for example. We propose and theoretically justify a data-driven choice of RKHS for the extrapolation. Experiments with clinical data show that the proposed choice allows essential reduction of erroneous BG-estimations as compared to commercially available CGM-devices. Moreover, the proposed approach can be used for BG-prediction and is a part of a Patent Application.


Valeriya Naumova, Sergei V Pereverzyev and Sivananthan Sampath are all members of the Inverse Problems group at Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences.

Valeriya Naumova

Sergei V Pereverzyev

Sivananthan Sampath

Quantitative photoacoustic tomography with multiple illuminations

Photoacoustic tomography (PAT) combines the large contrast of optical imaging with the high resolution of ultrasound imaging. PAT is based on the photoacoustic effect; as optical radiation propagates, a fraction of its energy is absorbed and generates a local heating of the underlying medium. The resulting mechanical expansion is the source of acoustic signals that propagate through the domain of interest. Ultrasonic transducers located at the boundary of the domain then record the emitted pressure waves as a function of time. The first step in PAT thus consists of reconstructing the absorbed radiation map from the recorded pressure measurements.

The second step, called quantitative photoacoustic tomography (qPAT), aims to reconstruct the unknown thermal expansion, optical absorption and diffusion coefficients from knowledge of the absorbed radiation map recovered in the first step. It is shown that we can uniquely reconstruct only two of the three aforementioned coefficients, even when data are collected using an arbitrary number of illuminations. If one of the coefficients

Guillaume Bal





iopscience.org/ip 11

A new approach to nonlinear constrained Tikhonov regularization

Nonlinear inverse problems arise frequently, especially distributed parameter identifications for differential equations, e.g. corrosion detection, electrical impedance tomography and inverse scattering. Often Tikhonov regularization is adopted for its stable and accurate numerical solution. The standard approach for analyzing nonlinear Tikhonov formulations, developed by H W Engl, K Kunisch and A Neubauer in 1989, hinges essentially on two conditions: (1) the source condition: the true solution lies in the range of the adjoint of the linearized operator; (2) the nonlinearity condition: the operator is (Gateaux) differentiable with a Lipschitz continuous derivative and the Lipschitz constant is sufficiently small.

The approach has been successfully applied to some practical examples, but its application to concrete nonlinear inverse problems remains inconvenient. According to optimization theory, the source condition represents a necessary optimality condition for the true solution. Meanwhile, the theory suggests an alternative nonlinearity condition: the operator satisfies a sufficient optimality condition of the Tikhonov functional at the true solution.

This condition proves much weaker than the classical one, and yet it is sufficient for deriving a priori/a posteriori error estimates for popular parameter choice rules including discrepancy and balancing principles. Also, for a general class of parameter identification problems, the source condition can be significantly weakened by exploiting specific structures of the adjoint operator, and it is often more readily interpretable. Further, for some bilinear problems, surprisingly, one crucial term in validating the new nonlinearity condition actually admits a nice form that is independent of the (inaccessible) source representer. Thanks to these structural properties, a number of nonlinear inverse problems can be easily analysed in the framework. Their potential impacts likely go beyond Tikhonov theory, e.g. to iterative regularization methods, which however, are still to be explored.


Kazufumi Ito is Professor of Mathematics and a member of the Center for Research in Scientific Computing at North Carolina State University. Bangti Jin is a KAUST fellow at the Institute for Applied Mathematics and Computational Science, and a visiting assistant professor at the Department of Mathematics at Texas A&M University.

Bangti Jin

is known, however, the other two coefficients can be reconstructed stably from data collected from only two well-chosen illuminations. The mathematical investigation leads to an explicit numerical reconstruction procedure that is non-iterative even though the inverse problem is nonlinear. The accuracy of the reconstruction algorithm is demonstrated with synthetic data.


Guillaume Bal is a professor of applied mathematics at Columbia University. Kui Ren is an assistant professor of applied mathematics at the University of Texas, Austin.






An adaptive method for travel-time tomography

Travel-time tomography aims to determine the internal properties of a medium by measuring the travel times of waves going through the medium, and can be applied in global seismology, exploration geophysics and medical imaging. Recent progress in boundary rigidity and lens rigidity in Riemannian geometry provides the theoretical foundation for a novel numerical algorithm for recovering Riemannian metrics, which is directly related to the medium velocity field. Efficiency, accuracy and stability are enhanced by a new adaptive strategy for the phase-space method which can handle broken geodesics and inclusions in an unknown medium.

An innovative phase-space algorithm for travel-time tomography by using the Stefanov–Uhlmann identity for boundary rigidity problem is developed, which combines the advantages of both Lagrangian and Eulerian methods. The key idea of this approach is to utilize those geodesics that match the measurements well under the current approximating metric. As a result of the improved approximation of the metric, more geodesics will match the measurements better and will be used in the next step. By continuing this process, an increasing number of geodesics will be used so that one can recover the metrics in larger and larger regions of the domain. This adaptive approach improves stability by using only those more accurate geodesics at each step. It also improves efficiency by gradually involving more and more unknowns in a stable fashion. This is in the spirit of the layer stripping algorithm. Initially short geodesics that are usually close to the boundary are used to provide a good estimate of the metric in a boundary layer. Then, longer geodesics are used and the boundary layer of good estimate expands further into the interior. The crucial point is that one does not have to specify the layers, which is impossible without knowing the underlying metric. Instead the technique of data matching is used to automatically pick geodesics sequentially and the hybrid phase space method can recover the underlying metric in the neighborhood of the picked geodesics in physical space. This adaptive phase space method is applied to transmission and reflection tomography where broken geodesics have to be taken into account, and produces very good results.


Eric Chung is an assistant professor at the Chinese University of Hong Kong. Jianliang Qian is an associate professor at Michigan State University. Gunther Uhlmann and Hongkai Zhao are professors in the Department of Mathematics at University of California at Irvine.

Eric Chung

Jianliang Qian

Gunther Uhlmann

Hongkai Zhao





Unique recovery of inhomogeneity from transmission eigenvalues

Consider scattering of electromagnetic waves or sound waves from an inhomogeneous medium, where the inhomogeneity is confined to a spherical region and the medium properties vary only in the radial direction. A natural question is whether the inhomogeneity at any point in the spherical ball can be determined from an infinite set of constant values. The so-called transmission eigenvalues form such a discrete set of constant values and in various physical problems they can be obtained from far-field measurements and also from near-field measurements.

Transmission eigenvalues are in general an infinite set of complex constants, but some or even all may be real constants, depending on the physical problem. Given a physical wave scattering problem from an inhomogeneous medium confined to a spherical ball, there are various important problems to consider. Do transmission eigenvalues exist? Do they determine the properties of the inhomogeneous medium? Is such a determination unique, and if not what additional information may be used for a unique determination? Can the unique determination be made by using not all transmission eigenvalues but by using only a certain subset of them?

We consider the determination of the medium properties of a spherically-symmetric inhomogeneity confined to a ball by using only those transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. These eigenvalues are called ‘special transmission eigenvalues’, and they associate a ‘multiplicity’ for each special transmission eigenvalue. The uniqueness aspect of determining the properties of the inhomogeneous medium is analyzed when the special transmission eigenvalues and their multiplicities are known. This work shows that under certain conditions the data set uniquely determines the inhomogeneity. Also in a special case, the unique determination is possible provided one extra piece of information is available. We conjecture that, without those certain conditions, the unique determination is not possible.


Tuncay Aktosun is a professor of mathematics in the Department of Mathematics at University of Texas at Arlington. Drossos Gintides and Vassilis Papanicolaou are both professors of mathematics at the National Technical University of Athens.

Tuncay Aktosun

Drossos Gintides

Vassilis Papanicolaou

This experience was very positive because several people and experts in the field contacted us about the work. Some of them used our results to go further in their own research as well. So, we think that such an initiative (i.e. Insight papers) is constructive and useful.





Reconstruction of missing data in social networks based on temporal patterns of interactions

Dynamic activity involving social networks often has distinctive temporal patterns that can be exploited in situations involving incomplete information. Even when activity is highly stochastic, localized excitations in parts of the network can help identify actors in cases of unknown origin. Pinpointing the source of unknown activity in large social networks is a combinatorially complex problem that can be more easily computed via a non-convex constrained optimization.

Human event activity is often highly clustered in time, and the level of activity is typically based upon the number of recent events. As a case study of this phenomenon, the authors consider a well-known Los Angeles gang network in which the repeat activity between nodes of the network is modeled by a temporal Hawkes process. The nodes of the network represent individual street gangs and the activities are violent crimes between the gangs, some of which are unsolved. The goal is to correctly identify the gang affiliated with the unsolved crimes. The authors construct an energy functional – inspired by the true probabilistic likelihood associated with the Hawkes process – that depends quadratically on the ‘probability’ that an unsolved crime was committed by a specific gang. They maximize this functional under an ,2-constraint using gradient flow. This problem is well posed, and generally has a unique optimal solution. The algorithm performs almost identically to a combinatoric approach for small datasets, but runs in a fraction of the time; for large datasets, the combinatoric approach is computationally infeasible. For artificial datasets with properties similar to those of this gang network, the algorithm places the correct gang within the top three gangs of likelihood approximately 80% of the time.


Alexey Stomakhin is a doctoral student in applied mathematics at UCLA. Martin Short is adjunct assistant professor of mathematics at UCLA. Andrea Bertozzi is professor of mathematics at UCLA.

Alexey Stomakhin

Martin Short

Andrea Bertozzi

Best basis choice in geophysical tomography

The authors have introduced a new algorithm called the regularized functional matching pursuit (RFMP). It enhances current matching pursuits for application to ill-posed inverse problems on a 3D ball as occurs in geophysics and medical imaging. One main feature is the availability of a very large and overcomplete set of trial functions for expanding the unknown solution. This so-called dictionary can for example consist of global trial functions (such as orthogonal polynomials) and localized trial functions (such as spline or wavelet basis functions with wide and narrow ‘hats’) to cover phenomena of different spatial extent. The resulting expansion of the approximate solution consists of all kinds of trial functions but much fewer functions than the dictionary contains, and is sparse in this respect. As a numerical example, the static gravitational potential of the Earth is regionally

Doreen Fischer






Recovering hidden sparse signals using Bayes’ formula

Compressed sensing is useful where the hidden signal has a sparse representation in a certain basis. Starting with the seminal 2004 article by Daubechies, Defrise and De Mol, powerful deterministic methods have been introduced for computing sparse representations, especially in wavelet bases. The most useful feature of Bayesian inversion is the modular way in which measurement information and a priori knowledge can be combined using the likelihood and prior probability distributions. Bayesian inversion is useful in a wide variety of applications, including medical imaging, process monitoring, underground prospecting and nondestructive testing.

The present paper combines sparsity-promoting algorithms and the Bayesian inversion framework. Especially, the return on investment in computational resources is studied. Namely, in traditional numerical analysis it is standard practise to show that when the computational grid is refined, the increased computational burden is rewarded by a more accurate approximation of the underlying continuum solution. This aspect of numerical computation has not been studied much in the field of inverse problems. We show how sparsity-promoting, wavelet-based Bayesian inversion methods are discretization-invariant, or yield a better approximation of the same continuum estimate when the grid is refined.

Also, the paper addresses the classical problem of choosing values for free parameters in computational inversion. Assuming that we know a priori the sparsity level of the hidden signal, we can enforce that sparsity in the reconstruction as well. This approach is analogous to the L-curve method in that several reconstructions are computed with various parameter values. However, the novel sparsity-based approach seems to work even in cases when the L-curve method fails.


Matti Lassas and Samuli Siltanen are mathematicians working at the University of Helsinki, Finland. Ville Kolehmainen and Kati Niinimäki are physicists working at the University of Eastern Finland. All authors are with the Finnish Centre of Excellence in Inverse Problems Research and share a passion for developing medical imaging algorithms.

Matti Lassas

Samuli Siltanen

Ville Kolehmainen

Kati Niinimäki

inverted over South America. As a novelty, the resolution is essentially improved, since the algorithm automatically sets the majority of the centres of the localized basis functions to the Andes and the Caribbean, where more complicated structures occur in the solution. Moreover, RFMP is used to detect seasonal water mass transports in the Amazon area from satellite data. The project will be continued by combining gravitational and seismic data to recover deep structures of the Earth and by applying the RFMP to medical imaging.


The Geomathematics Group Siegen is a part of the Department of Mathematics of the University of Siegen, Germany. Doreen Fischer is a postdoctoral researcher in the group, and Volker Michel is a full professor who founded and heads the group.

Volker Michel






Reconstruction of the interfaces using complex geometrical optics solutions

In many situations we have access to the waves, scattered by the interface of a scatterer, only on a surface located away from the scatterer. The goal is then to read the collected scattered waves and try to provide information about the scatterer. Direct methods used to solve this problem are based on the use of at least two special type solutions. The first are the Green-type solutions on which the probe and the sampling methods are based. The second are the so-called complex geometrical optics solutions (CGO solutions). Originally, these solutions were constructed, by Sylvester and Uhlmann, to solve the Calderon problem. Later, using these special solutions as incident waves, Ikehata proposed the enclosure method to reconstruct the interfaces. The main idea is that the energy of the corresponding scattered waves is confined on one side of the level curves (or surfaces) of the phase of the CGO solutions. Using this property, we can enclose the obstacle by these curves (or surfaces) and hence reconstruct relevant features of the scatterers as its convex hull, for instance. A particularity of this method is that these features are reconstructed via direct formulas. This work analyzes mathematically this last method for the acoustic model. In particular, the authors have removed some unnatural geometrical assumptions on the scatterers imposed in the literature. The most relevant contribution is to have proposed a quite general way of proving the necessary estimates for justifying this method in more general settings. Indeed, this approach can be applied to the full Maxwell and elasticity models.


Mourad Sini is a senior fellow of the Austrian Academy of Sciences and works at the Radon Institute for Computational and Applied Mathematics (RICAM). Kazuki Yoshida is a postdoctoral researcher at Hokkaido University, Japan.

Mourad Sini

Kazuki Yoshida

Time reversal-defocusing may not be a bad thing

Time reversal (TR) is based on the invariance of the wave equation for a lossless medium to change in the sign of the time variable. By retransmitting the TR version of the time-dispersed received signal propagated from a probe source (PS) to a source-receiver array (SRA), one can re-acquire the probing pulse at the PS location. However, if some environmental variations occur between the two transmissions, the retrofocusing signal will be defocused. In most TR applications, defocusing is to be avoided, but in some cases, defocusing may not be a bad thing. Typical ocean environments show strong temporal variability, which is a major obstacle to adapting model-based sonar processing frameworks. The uncertainty of the ocean environment propagates through a complete chain for sonar performance: environment, acoustics, processing and operator. A conventional inversion approach in underwater acoustics is to find a model environment in which a forward sound propagation model can produce a replica with a good match to the observed data, which may lead to large errors because of the model mismatch. Based on the TR principle, one can re-acquire the probing pulse at the PS location by retransmitting

Jianlong Li

Wen Xu






Liling Jin

the TR version of the time-dispersed received signal propagated from a PS to an SRA. If environmental variations occur between the two transmissions, the retrofocusing signal will be defocused. We exploited the defocused data without producing a replica to invert the environmental variations between the two transmissions. By evaluating the degree of the SSP variation via retransmission at any time interval, there are potential applications in guiding when to update the SSP for matched field source localization, and when to update equalizers in underwater communications. The difficulties are in estimating the mode functions at the PS and SRA locations and analysing the effect of coupled modes with nonlinear waves on the TR inversion.


Jianlong Li and Wen Xu are members of the Department of Information Science and Electronic Engineering, Zhejiang University, China. Liling Jin is finishing her PhD supervised by Wen Xu in the same department.

Complex transmission eigenvalues for real-valued refractive indices

Transmission eigenvalue problems are a new class of eigenvalue problems that have recently arisen in the study of inverse scattering problems for acoustic and electromagnetic waves associated with real-valued refractive indices. These eigenvalues are of interest because they can be determined from scattering data and they carry information about the index of refraction of the medium. In particular, the first (real) transmission eigenvalue provides an ‘average’ value of the index of refraction, thus providing a simple ‘qualitative’ approach to the inverse scattering problem of determining the index of refraction from scattering data. Although it has been known for some time that transmission eigenvalue problems are not self-adjoint, until now only real eigenvalues have been shown to exist and only the properties of real eigenvalues have been studied. We have shown that for the simple case of a spherically stratified medium, there exist in general an infinite number of complex eigenvalues and that these eigenvalues lie in a strip parallel to the real axis. These results suggest that complex eigenvalues also exist for media that are not spherically stratified and that some of these eigenvalues may lie near the real axis and hence can be determined from scattering data. It also presents the intriguing question as to what information these complex eigenvalues provide about the scattering object. This article now places these questions on the agenda for future investigation by the inverse scattering community. As is often the case with non-self-adjoint eigenvalue problems, the analysis used in our investigation relies heavily on the theory of entire functions of a complex variable and thus opens a new opportunity for experts in this area to find new worlds to conquer in the field of inverse problems.


Yuk-J Leung and David Colton are, respectively, Associate Professor and Unidel Professor of Mathematics at the University of Delaware. Yuk-J Leung works on classical function theory while David Colton has worked on inverse problems and many other areas in applied analysis.

Yuk-J Leung

David Colton






Enhanced approximate cloaking by SH and FSH lining

A region is cloaked if its contents, together with the cloak, are invisible to electromagnetic waves. The route to cloaking that has received the most attention is that of transformation optics, made possible by taking advantage of the transformation rules for the material parameters of optics: the index of refraction for scalar optics, governed by the Helmholtz equation, and the electric permittivity and magnetic permeability for vector optics, as described by Maxwell’s equations. The ideal cloak uses singular ‘blow-up-a-point’ transformations, resulting in singular cloaking structures that pose a great challenge, both theoretically and practically.

In order to avoid the singular structure, a regularized ‘blow-up-a-small-region’ transformation is used for the construction and, instead of perfect invisibility, we consider approximate invisibility by showing that the wave scattering due to the cloaking construction is as small as possible. However, there are cloak-busting inclusions that defy any attempt to achieve a near-cloak via such a regularized construction. To defeat the cloak busts, certain mechanisms have to be incorporated into the construction. In our paper, two such mechanisms have been developed for the acoustical cloaking. The first, called an SH construction, incorporates a sound-hard layer between the cloaking region and cloaked region. Physically, this amounts to imposing a homogeneous Neumann boundary condition on the cloaking interface. The homogeneous Neumann boundary condition has been shown to hold for the singular ideal cloak by Greenleaf et al, hence, the SH construction is a ‘true’ approximate cloak in this sense. Indeed, the SH construction is shown to produce significantly improved accuracy of approximation, compared to earlier proposals in the literature. However, the sound-hard layer is an ideal state of acoustic materials. From a practical viewpoint, we develop a scheme that can produce the same performance as the SH construction by replacing the sound-hard layer with a high-density loss layer with well-chosen material parameters. It is also shown that the latter construction can be regarded as a finite realization of the former, and so is called an FSH construction.

In addition to the high-order accuracy of approximation, the FSH construction produces low infrared radiation since low-loss materials are used. In our ongoing research, we have found that the FSH construction is robust for qualitative wave detection and excels in cloaking active content, hence it has potential as a practical recipe for real cloaking construction.


Hongpeng Sun is a PhD student at the Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. Jingzhi Li is an associate professor of applied mathematics at the South University of Science and Technology of China. Hongyu Liu is an assistant professor in the Department of Mathematics and Statistics, University of North Carolina at Charlotte, USA.

Hongpeng Sun

Jingzhi Li

Hongyu Liu





Locating small conductivity inhomogeneities with two electrodes

Electrical impedance tomography (EIT) aims to deduce information about the internal conductivity distribution of a physical body from boundary measurements of current and voltage. One essential application of EIT is the detection of embedded inclusions in a known background. If the inhomogeneities are small, an efficient technique is to write an asymptotic expansion for the considered measurements with respect to the size of the inclusions and only consider the leading order term in the reconstruction process.

If the investigated object is cylindrical and the boundary data are to be collected with only two electrodes, the following idea may be applicable: fix the position of one electrode, sweep the other electrode around the object maintaining one unit of current, and record the corresponding voltage difference as a function of the position of the moving electrode. After subtracting the analogous measurement with a homogeneous object, one arrives at the so-called sweep data. Under idealized circumstances, the sweep data extend as a complex analytic function to the exterior of the inhomogeneities. The leading order term of the asymptotic expansion for the sweep data in the size of the inclusions turns out to be a meromorphic function that has poles (only) inside the inhomogeneities. This makes it possible to adopt the following idea for the localization of the inclusions: compute a suitable Laurent–Padé approximate for the sweep data, i.e., a meromorphic function that approximates the sweep data in a certain way, and declare the poles of this approximate as the estimates of the inclusion locations. By considering the corresponding residues, it is also possible to extract information about the sizes and conductivity contrasts of the inhomogeneities. Although designed for ideal sweep data corresponding to point-like electrodes and a two-dimensional object, the reconstruction algorithm also provides information about conductivity inhomogeneities if applied to measurements simulated using a realistic electrode model and a 3D cylindrical body.


Nuutti Hyvönen and Otto Seiskari work at the Department of Mathematics and Systems Analysis of the Aalto University, Helsinki, Finland. Hyvönen is an Academy Research Fellow and Seiskari is his graduate student.

Nuutti Hyvönen

Otto Seiskari

Did you know?The Insights collection received an average of 473 views per month in 2013

Did you know?Insights articles are free to read on our journal homepage, and contain a link to the full article





An extended-DORT method and its application in a cavity configuration

Determining characteristics of an unknown object from the way it scatters incoming radiation is of great interest in applied science. If a limited amount of information is required, such as the footprint of an unknown target, several qualitative imagery schemes have been proposed. Among them, the decomposition of the time reversal operator (DORT) method works as a detection technique using an array of antennas functioning in transmit–receive mode, well adapted to identify point-like scatterers. We present an extension of this method for extended scatterers and compare it with previous methods. 2D examples, based on measured data sets acquired at Institut Fresnel with a microwave scanner consisting of an array of antennas in a metallic circular cavity, show that the method is computationally efficient and robust to noise.

The multi-incidence multi-receiver transfer matrix is the main piece of information being processed using DORT. Indeed, this transfer matrix can be diagonalized to derive the eigenstructure of the medium. When the medium contains several point-like scatterers, the number of significant singular values provides a direct insight into the number of well-resolved scatterers. It was shown that the singular vectors are proportional to the reflectivities of each scatterer. Furthermore, each singular vector provides phase and amplitude information that the antennas should jointly emit in order to focus on a given point-like scatterer.

Existing sampling methods, such as the linear sampling, the factorization method and MUSIC, use the same diagonalization procedure as the first step, but differ in the weighting coefficients. By adjusting the weighting coefficients, it is possible to focus either inside the scatterer support, on its boundary or at some locations where there are definitely no scatterers. Unfortunately, these methods rely on an adequate choice of external hyperparameter, which can be difficult to determine, in particular when dealing with real data, even if theoretical rules may apply. If they are incorrectly selected, it can dramatically modify the estimated shape of the scatterers. We propose a natural weighting function, exempt from any kind of tuning parameter, which can be seen as an extension of the classical DORT method to extended scatterers. In particular, it analyzes convergence properties and verifies its robustness with respect to noise by confronting it with experimental scattered fields. The electromagnetic fields scattered by various types of cylindrical metallic targets have been measured in a microwave scanner composed of an array of antennas radiating at fixed frequency in water while positioned inside a circular metallic casing.


Hervé Tortel, Amélie Litman and Jean-Michel Geffrin work at the Institute Fresnel, which is linked with Aix-Marseille University, CNRS and Ecole Centrale Marseille. Hervé Tortel and Amélie Litman are assistant professors, working on modelling and inversion algorithms. Jean-Michel Geffrin is a CNRS Research Engineer, working on microwave scattering measurement setups. This work was carried out during the PhD of Xiao-Yun Zhang, who is now working at the Science and Technology on Electronic Information Control Laboratory, Chengdu, China.

Hervé Tortel

Amélie Litman

Jean-Michel Geffrin





Wave-equation-based linearized seismic inversion

The inverse problem of seismology addresses the question of how to reconstruct material properties in the interior of the earth from seismic data measured at the surface of the earth. In the often-used acoustic approximation of seismic wave propagation, these properties are simply the velocity of the acoustic waves and the bulk density of the rocks. In this framework, the seismic data are solutions of the acoustic wave equation evaluated at the surface of the earth. The seismic inverse problem is therefore known as the inverse coefficient problem for the acoustic wave equation in the mathematical geophysics literature. A fruitful approach towards this problem is to make the ansatz that velocity and density functions can be decomposed as the sum of smooth and non-smooth terms, in such a way that the smooth terms explain the kinematics of propagating waves, whereas the non-smooth terms lead to scattering of waves. The inverse problem can be split similarly into a nonlinear one, estimating the smooth terms, and a linear one, estimating the non-smooth, or singular terms. The latter is equivalent to constructing an image of the singularities in the earth.

This paper discusses a number of new results in linearized seismic inversion. It starts from the Born–Kirchhoff scattering formalism and observes that, in the common case of simple jump discontinuities in velocity and density functions over smooth surfaces, the Kirchhoff integral can be seen as a Fourier integral operator mapping a reflection coefficient, which depends singularly on the three position variables and smoothly on two angular variables, to data. The two angular variables are the reflection angle and reflection azimuth describing the reflection of a plane wave at the surface of discontinuity. The first result of the paper is that this integral operator can be inverted explicitly and that the inverse can be constructed completely in terms of fundamental solutions of the acoustic wave equation. From the recovered angle-azimuth dependent reflection coefficients one can easily reconstruct the perturbations in velocity and density, so this also provides an explicit wave equation based solution of the linearized seismic inverse problem. The construction of the inverse uses so-called space-shift extended images as an alternative to angle-dependent reflection coefficients and leads to a modified Kirchhoff operator acting on these. This modified Kirchhoff operator can be fully expressed in terms of fundamental solutions of the wave equation, as opposed to the standard one, which depends on reflection angles, calculated by tracing seismic rays. The paper concludes with the construction of (partial) left and right inverses of this modified Kirchhoff operator.


Fons ten Kroode heads the seismic imaging R&D group at Royal Dutch Shell.

Fons ten Kroode





A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators

In signal analysis, we often have some a priori knowledge about the underlying structure of the signal that we need to exploit. Using this structure, we need to determine a certain number of parameters from the given signal measurements.

In many situations one is faced with the problem of determining parameters for a certain signal structure from the measured data. One well-known method for tackling this problem is the Prony method which indeed serves as the basic concept for a series of reconstruction methods, such as for example, MUSIC, ESPRIT, the matrix pencil method or the annihilating filter method. In this work, we provide a universalized approach to the Prony method that applies to a very general underlying signal structure. Let us assume that we can measure a certain small amount of data, and we know that there exists an underlying structure of the required signal that can be described via a finite linear combination of functions from a given huge function system. The task is now to determine the function parameters from the given data. More precisely, knowing that the given data y(k) can be described as a linear combination of M = #J functions fTj(x) from a function system, we have to determine the non-zero coefficients cj and the active functions fTj(x) of the system, i.e., the corresponding indices Tj . Observe that the function system may even have an infinite number of different functions. A prototype of such a problem is to determine the complex parameters cjj and Tj of a structured function of the form from the data y(k), k = 0 ,..., 2M − 1. In our paper, this original Prony method is strongly generalized to the case where fTj are allowed to be eigenfunctions of a linear operator. The virtue of our general view of the problem is that all properties, for example the stability results, the amount of required input data or the behaviour for erroneous input data, directly apply to all special Prony-like methods once they have been shown for the universal approach.


Thomas Peter is a PhD student at the Institute of Numerical and Applied Mathematics at the University of Göttingen. Gerlind Plonka is Professor for Applied Mathematics at the Institute of Numerical and Applied Mathematics at the University of Göttingen. She heads the research group for mathematical signal and image processing.

Thomas Peter

Gerlind Plonka

Convergence rates in ,1-regularization if the sparsity assumption fails

Over the last decade, sparsity has evolved as an extremely popular concept, introducing prior knowledge about the solution of inverse problems. The key idea is to find a basis system such that typical solutions can be represented in a sparse way, i.e., as linear combinations with only a few non-zero entries. In practice, however, one in fact finds basis systems such that typical solutions are only approximately sparse, i.e., many coefficients in the linear expansion are very small, but not zero. Analysing this situation carefully poses interesting mathematical challenges in regularization theory, since classical results always depend on the number of non-zero entries, deteriorating if this number goes to infinity.

Martin Burger






Jens Flemming

Bernd Hofmann

We start to analyse a situation of particular interest, namely the derivation of error estimates for the standard variational formulation of solving inverse problems with sparsity priors, consisting of a quadratic data fidelity and ,1-regularization, which is a convex relaxation of the function counting the number of nonzero entries. We replace the assumption of a sparse solution by the assumption of a solution with few large coefficients and the remaining part decaying to zero. For this case we try to develop error estimates between the exact solution and the one recovered from the above model in the case of noisy data. The modelling of our prior knowledge in a functional analytic setting has a surprising consequence: the source conditions usually used are not applicable, since they rely on a finite number of non-zero entries. On the other hand it is possible to obtain a formulation in terms of variational inequality conditions, which are the basis of further analysis. This yields an example where variational inequalities can go beyond classical conditions.


Martin Burger is Professor for Applied Mathematics at the University of Münster. He heads the Mathematical Imaging Group. Jens Flemming is a postdoctoral researcher in the Inverse Problems group at Chemnitz University of Technology. Bernd Hofmann is Professor for Analysis and heads the Chemnitz Inverse Problems Group.

The factorization method for inverse acoustic scattering in a layered medium

The factorization method (FM) is a sampling method for solving inverse problems where a geometric parameter has to be determined. This method is simple and fast but, more importantly, makes no explicit use of boundary conditions or topological properties of the underlying scatterer. From the mathematical point of view, the FM provides a criterion for sampling points which is both necessary and sufficient. The background of a scatterer is not homogeneous and often appears as a layered homogeneous medium. A medium of this type that is a nested body consisting of a finite number of homogeneous layers occurs in various applications such as radar, remote sensing, geophysics, and nondestructive testing. We use the FM to recover both the interface between layers and the imbedded obstacle. The validity of the FM for the imbedded obstacle is proven with the help of a mixed reciprocity principle and an application of the scattering operator. Due to the established mixed reciprocity principle, knowledge of the Green function for the background medium is no longer required, which makes the method attractive from a computational point of view. The paper is only concerned with sound-soft obstacles, but the analysis can easily be extended for sound-hard obstacles, or obstacles with separated sound-soft and sound-hard parts.


Andreas Kirsch is a professor of mathematics at Karlsruhe Institute of Technology. Oleksandr Bondarenko is his current PhD student. Xiaodong Liu is an assistant professor of applied mathematics at the Chinese Academy of Sciences.

Andreas Kirsch

Oleksandr Bondarenko

Xiaodong Liu






A new algorithm for discrete tomography

X-ray tomography is a widely used imaging technique that allows one to produce internal images in materials science or medical applications. An x-ray beam is transmitted through the sample and the projection is recorded on a detector at different angles. The most widely used reconstruction algorithm of the original image from these measurements approximates the inversion of the tomographic inverse linear problem. However, it provides satisfying results only for a well-determined system, with a large number of projections.

In order to reduce the acquisition time or the total dose absorbed by the patient, tomography users need other reconstruction algorithms that incorporate known properties of the original image. For example, images with a discrete set of absorption values and few interfaces are frequently encountered in materials science. Using such prior information can make up for the lack of measurements and result in a high-precision reconstruction, an idea popularized by the field of compressed sensing. We have proposed a novel message-passing algorithm for discrete tomography that allows for an efficient computation of pixel expectations under the Bayesian posterior distribution. The algorithm works on two levels: it is iteratively sending probabilistic messages between the different light rays and different sets of messages within each light ray in order to satisfy the observed values of measurements. The resulting algorithm is fast and entirely distributed. We showed that for binary images an accurate reconstruction is obtained for highly undersampled measures, and, equally importantly, the performance is robust to Gaussian measurement noise.

Our algorithm also outperforms a classical optimization algorithm using total-variation minimization, which is the convex relaxation of our image prior. This opens the way to a new class of algorithms for practical applications of discrete tomographic reconstruction.


Florent Krzakala is an associate professor at the ESPCI Paristech. Marc Mézard is the director of the Ecole Normale Supérieure, Paris. Lenka Zdeborová is CNRS associate scientist at the Institute of Theoretical Physics in CEA, France. Emmanuelle Gouillart is an applied researcher at the joint unit of CNRS/Saint-Gobain.

Marc Mézard

Lenka Zdeborová

Florent Krzakala

Emmanuelle Gouillart





Reconstruction of piecewise constant fluorophore concentrations in optical tomography

In fluorescence diffuse optical tomography (FDOT) a series of light sources is placed on the surface of the sample and the emitted fluorescence light is measured from different projections. Conventional image reconstruction in fluorescence tomography is based on the computation of the so-called sensitivity, which is the linearization of the forward problem with respect to small changes in fluorophore parameters. Several standard regularization techniques, such as Tikhonov regularization, have been applied to regularize the identification process. In numerous applications, a priori information on a localization of the fluorophore inside the biological specimen is available. Mathematically, this corresponds to assuming piecewise constant fluorophore concentrations.

In the piecewise constant framework, we are looking for the support of the fluorophore concentrations. The problem becomes a shape optimization problem to identify the shape of the interfaces between the unknown regions and the background. For the numerical solution, shape sensitivity calculus, as developed by Sokolowski and Zolesio, may be employed. The constant concentration values in the subregions are also determined through standard minimization techniques in finite dimensions. The question of initialization of the reconstruction in shape optimization is important because an inaccurate initialization may yield a slow convergence or some subregions may be overlooked. To this end, topological sensitivity was introduced in shape optimization and recently applied for inverse problems.

The topological derivative measures the change in the objective functional when a small subregion with a fluorophore concentration different from the background is introduced. It is a powerful technique either to quickly detect small inclusions or to initialize a reconstruction algorithm for a larger inclusion. Shape sensitivity techniques may be used later on to locally improve the shape of the identified distributions.

In this paper we focus on the topological sensitivity part of this procedure. The computation of the topological sensitivity for FDOT involves using asymptotic analysis techniques for a coupled system of elliptic PDEs. In contrast to typical problems of shape optimization, different levels of topological differentiation must be computed, which leads to algorithmical challenges. Numerical results are performed and show a very accurate reconstruction compared with a single step algorithm based on a Tikhonov-type regularization.


Antoine Laurain heads the MATHEON research project ‘Inverse problems’ at the Technical University of Berlin, Germany. Michael Hintermüller is a MATHEON Research Professor and W3-Professor in Applied Mathematics at the Humboldt University of Berlin and is leading the START project at the Karl-Franzens University of Graz, Austria. Manuel Freiberger is a former research fellow at the University of Graz. Hermann Scharfetter is Professor at the Institute of Medical Engineering of the Technical University of Graz.

Michael Hintermüller

Manuel Freiberger

Antoine Laurain

Hermann Scharfetter





An exact inversion formula for cone beam vector tomography

We derive a novel, exact inversion formula for the cone beam transform (CBT) of vector fields. We consider reconstructing the solenoidal part of a smooth vector field f supported in the open unit ball knowing the cone beam data of f. The latter is the set of integrals of f dotted with a unit vector along a line for all lines intersecting a given curve in space. The CBT is the mathematical model of vector tomography, where a flow field f is reconstructed from ultrasound Doppler or time-of-flight measurements with sources located on a curve. Vector tomography is applied in medical diagnosis, oceanography, industry and plasma physics.

The main assumption about the source trajectory is that any plane intersecting the support of f intersects the source trajectory at least at three points. Our formula represents an analogue of the known inversion formulas for the scalar cone beam transform. In the scalar case, these formulas have been known since the 1980s. However, no inversion formulas were known in the vector case. The theoretical foundation of our formula is the analogue of the Grangeat relation which relates the CBT of f and the Radon transform (RT) of f. This formula was found recently by Kazantsev and Schuster. The first part of the formula, denoted f

1, is related to the tangential part of the RT of f. The second part, denoted f

2, is related to the normal part of the RT of the solenoidal part of f. The latter is computed by adding both parts: f

sol = f1 + f 2. Interestingly, the computation of f1 is in the filtered-backprojection (FBP) form, while the computation of f

2 is not in the FBP form. It is interesting to see whether the formula can be modified so that computation of both parts is in the FBP form. In the future we plan to test the new formula on simulated and, if available, real data.


Alexander Katsevich is a professor at the University of Central Florida. His research is in the development of image reconstruction algorithms for computer tomography. Thomas Schuster is a professor at the Department of Mathematics at the University of Saarland, Germany. His research has a special focus on vector and tensor tomography.

Thomas Schuster

Alexander Katsevich

My experience of being an Insights author has been really exciting. I felt very honoured to be selected for this programme and I believe that it represents a useful tool for authors to publicize their papers. I really hope that I’ll be able to write something else that might be considered for this interesting initiative.





Kalman-based methods for the solution of inverse problems

The ensemble Kalman filter (EnKF) is a data assimilation technique that uses an ensemble of states and parameters that is sequentially updated to blend the model and data available at a given time. The implementation of the EnKF is computationally inexpensive and derivative-free, so the model is easily incorporated in a non-intrusive black-box fashion enabling the user to leverage already available legacy codes. Since its introduction by Evensen in 1994, the EnKF has had enormous impact in the fields of hydrology, meteorology and oceanography.

Motivated by successful applications of the EnKF to solve parameter estimation problems in dynamical models, we propose a novel application of an iterative ensemble Kalman method for the solution of a wider class of inverse problems, including static models with no dynamics. We consider the inverse problem of finding parameters given noisy observations of the forward operator applied to the unknown parameters. Of particular interest is the case where the forward operator arises from physical systems and involves the solution of PDEs. A central aspect of EnKF implementations is the selection of the prior ensemble, which, in the standard approach, is typically chosen as random draws from a prior distribution. In the context of the proposed framework, the prior ensemble becomes an essential design parameter that, in turn, determines the subspace where the estimate is sought. More precisely, we prove that the proposed Kalman method possesses an invariant subspace property that ensures the estimator of the unknown parameters belongs to the subspace spanned by the prior ensemble. In other words, the proposed Kalman method can be thought of as a parameter identification method in the aforementioned subspace. It is therefore natural to compare the proposed Kalman method with regularized least-squares techniques on subspace spanned by the prior ensemble, and with the best approximation in that subspace, as this provides a lower bound on achievable accuracy.

We test the proposed framework for a wide class of inverse problems in our recent paper. The proposed Kalman method is applied to estimate log-permeability from head measurements in a single-phase groundwater flow model. In addition, we use the Kalman method to estimate the initial condition of the velocity field in the Navier–Stokes equation from measurements of the velocity at later times. All our experiments suggest that, when properly stopped, the EnKF provides a computationally efficient and easy to implement regularization technique that can be used for generic inverse problems and can constrain the solution to be consistent with some prior knowledge.


Marco Iglesias and Kody Law are research fellows in the Mathematics Institute at the University of Warwick. Andrew Stuart is a professor of applied mathematics at the University of Warwick.

Kody Law

Andrew Stuart

Marco Iglesias





Blind deconvolution for astronomical imaging

Blind deconvolution is the problem of image deblurring when the blur is unknown. By assuming that the detected image is (a) given by the convolution between the unknown object f and the unknown point spread function (PSF) h, and (b) corrupted with Poisson noise, the blind deconvolution problem reduces to the minimization of the Kullback–Leibler (KL) divergence on both f and h. A faithful reconstruction of the unknowns strongly depends on the available a priori information on both the object and the PSF.

In this paper, we focus on astronomical imaging by assuming that an adaptive optics (AO) system is used to compensate for atmospheric blur and that a parameter characteristic of this correction, the so-called Strehl ratio (SR), is approximately known. More generally, this parameter is characteristic of any aberrated system, such as for instance the Hubble Space Telescope (HST) before the COSTAR correction. We propose to solve the resulting constrained minimization problem by means of an inexact alternating minimization method, whose global convergence to stationary points of the objective function has recently been proved in a general setting. The method is iterative and each iteration, also called outer iterations, consists of alternating an update of the object and the PSF by means of a fixed number of iterations, also called inner iterations, of the scaled gradient projection (SGP) method. We illustrate its effectiveness by means of numerical experiments whose results indicate that the method, pushed to convergence, is very promising in the reconstruction of non-dense stellar clusters.


Marco Prato is a researcher at the Department of Physics, Computer Science and Mathematics, University of Modena and Reggio Emilia, Italy. Andrea La Camera and Mario Bertero are, respectively, a postdoctoral researcher and a professor at the Department of Computer Science, Biomedical Engineering, Robotics and Systems Engineering (DIBRIS), University of Genova, Italy. Silvia Bonettini is a researcher at the Department of Mathematics and Computer Science, University of Ferrara, Italy.

Andrea La Camera

Mario Bertero

Marco Prato

Silvia Bonettini





Quantitative photoacoustic tomography using the radiative transfer equation

In PAT, short pulses of laser light irradiate the tissue and are absorbed by specific chromophores. This absorption leads to a small increase in local temperature and pressure, which subsequently relaxes and propagates an acoustic wave that may be detected by an acoustic detector array at the tissue surface. A photoacoustic image is formed by taking these surface measurements of the times-of-arrival of the acoustic waves and reconstructing the initial acoustic pressure distribution which resulted from the optical excitation. This is a well-posed inverse problem that has largely been solved. However, photoacoustic images are not directly representative of underlying tissue constituents or physiology because they are weighted by the (unknown) light distribution in the tissue. Photoacoustic imaging would become a much more powerful tool if it were possible to obtain quantitative images of specific chromophores within the tissue. This is the aim of quantitative photoacoustic tomography, which seeks to determine quantitative estimates of either the optical absorption coefficient or specific chromophore concentration distributions from a given photoacoustic image or set of images at multiple wavelengths. This optical inverse problem is nonlinear due to the dependence of the light influence on optical absorption, which is ill-posed due to the diffusive nature of light propagation in turbid media and large scale due to the high resolution of inherently three-dimensional photoacoustic images.

Nonlinear optimization techniques, whereby the difference between the measured image and a corresponding numerical model is iteratively minimized, have previously been used successfully to recover quantitative estimates of the optical properties underlying a simulated photoacoustic image. However, these methods typically use the diffusion approximation to the radiative transfer equation (RTE) to model the transportation of light within the tissue, which is not sufficiently accurate for PAT. In this article, the full RTE is used to model the light propagation. The acoustic propagation and image reconstruction are also included in the simulation of the measured data; the presented results therefore tackle both the acoustic and optical inverse problems, as would be required in practice. The article also highlights the importance of using memory-efficient minimization techniques for quantitative photoacoustics; PAT images may contain several million unknowns, and so the storage of large Jacobian or Hessian matrices required to execute some available minimization schemes will be unfeasible. Here we show that a gradient-based minimization scheme, which avoids the storage of such large matrices, can be used together with the full RTE to successfully recover both quantitative estimates of both optical absorption and scattering coefficients simultaneously from simulated photoacoustic images.


Teedah Saratoon is a PhD student in the Department of Medical Physics and Bioengineering at University College London (UCL). Tanja Tarvainen is a research fellow at the University of Eastern Finland and the Centre for Medical Image Computing at UCL. Ben Cox is a senior lecturer in the Photoacoustic Imaging Group at UCL. Simon Arridge is a professor of imaging science in the Centre for Medical Image Computing at UCL.

Tanja Tarvainen

Ben Cox

Teedah Saratoon

Simon Arridge



Inverse Problems


Inverse Problems

Editorial boardEditor-in-ChiefA K Louis Universität des Saarlandes, Saarbrücken, Germany

Editorial BoardG Alessandrini Università di Trieste, Italy S R Arridge University College London, UK M Bonnet École Nationale Superiéure de Techniques Avancées (ENSTA), France M Burger Universität Münster, Germany M Cheney Rensselaer Polytechnic Institute, Troy, NY, USA J Cheng Fudan University, Shanghai, People’s Republic of China D L Colton University of Delaware, Newark, USA T M Habashy Schlumberger-Doll Research, Cambridge, MA, USA B Hofmann Technische Universität Chemnitz, Germany V Isakov Wichita State University, KS, USA

M Jiang Peking University, People’s Republic of China B Kaltenbacher Karl-Franzens-Universität Graz, Austria P Lamm Michigan State University, East Lansing, USA R Novikov École Polytechnique, Palaiseau, France E T Quinto Tufts University, MA, USA A Rieder Karlsruhe Institut für Technologie (KIT), Germany W Rundell Texas A&M University, College Station, USA O Scherzer Universität Innsbruck, Austria T Schuster Universität des Saarlandes, Saarbrücken, Germany L Tenorio Colorado School of Mines, Golden, USA

International Advisory PanelG Bal Columbia University, NY, USA M Bertero Università di Genova, Italy L Borcea Rice University, Houston, TX, USA D Calvetti Case Western Reserve University, OH, USA O Dorn Universidad Carlos II de Madrid, Spain H Engl University of Vienna, Austria F A Grünbaum University of California, Berkeley, USA E Haber Emory University, Atlanta, GA, USA S I Kabanikhin Novosibirsk State University, Russia

J P Kaipio University of Kuopio, Finland D Lesselier Laboratoire des Signaux et Systèmes (CNRS-SUPELEC-UPS), Gif-sur-Yvette, France J R McLaughlin Rensselaer Polytechnic Institute, Troy, NY, USA F Natterer Universität Münster, Germany P C Sabatier Université des Sciences et Techniques du Languedoc, Montpellier, France J Sylvester University of Washington, Seattle, USA W W Symes Rice University, Houston, TX, USA G Uhlmann University of Washington, Seattle, USA M Yamamoto University of Tokyo, Japan


Inverse Problems

How to submit your researchHere is our guide to successfully submitting your article to Inverse Problems.

Article requirementsArticles in Inverse Problems must meet the highest scientific quality standards, both in terms of originality and significance, and the research results should make substantial advances in the field. Authors and referees are asked to take into account that Inverse Problems is an interdisciplinary journal when considering what is significant and worthy of publication. In particular, they are asked to bear in mind the following points.

• Papers should have carefully written introductions, which briefly summarize and explain the results so that readers from other related disciplines can understand them.

• Theoretical and mathematical papers should have an introduction clearly stating the potential applications of the problem and the results.

• Experimental and computational papers should be presented clearly for the whole inverse problems community.

• Papers are typically between 16 and 25 pages in length. However, we do not have a strict upper or lower page limit on submission. If your submission exceeds 30 pages we will request that the reviewers explicitly comment on whether the length is justified by the scientific content and whether the work could be shortened without detriment.

Open accessInverse Problems offers an open access option called hybrid open access. Read the IOP Publishing open access policy (http://iopscience.iop.org/info/page/openaccess) to find out more. You may also post your accepted manuscript (before editing and typesetting by the publisher) on your personal website or institutional repositories, subject to conditions.

SubmissionAll of our journals operate a fast online submission system. To submit to Inverse Problems simply visit iopscience.org/ip/submit. This will take you through to our online submission pages, where you will need to follow the steps described.

To help early career researchers prepare their papers for publication, we have published a digital brochure ‘Introductory guide for authors’, available at iopscience.org/author-guide.

iopscience.org/ip

ISSN 0266-5611

InverseProblems



Inverse Problems

http://iopscience.iop.org/info/page/openaccess


http://cms.iopscience.org/3205ba4f-d278-11e1-9b7a-4d5160a0f0b4/index.html

We would like to thank all of our authors, referees, board members and supporters

across the world for their vital contribution to the work and progress of Inverse Problems.

IOP PublishingTemple Circus, Temple Way, Bristol BS1 6HG, UK

E-mail [email protected] Web iopscience.org/ip

iopscience.org/ip

ISSN 0266-5611

InverseProblems



mailto:ip%40iop.org?subject=


insights collection – summaries of the excellent ......in inverse problems to the wider research...

Documents