ISSN:
 1930-8337

eISSN:
 1930-8345

All Issues

Volume 11, 2017

Volume 10, 2016

Volume 9, 2015

Volume 8, 2014

Volume 7, 2013

Volume 6, 2012

Volume 5, 2011

Volume 4, 2010

Volume 3, 2009

Volume 2, 2008

Volume 1, 2007

Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing.

This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 6 issues a year in February, April, June, August, October and December.
  • Publishes online only.
  • Indexed in Science Citation Index, ISI Alerting Services, CompuMath Citation Index, Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science and Zentralblatt MATH.
  • Archived in Portico and CLOCKSS.
  • IPI is a publication of the American Institute of Mathematical Sciences. All rights reserved.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

Select all articles

Export/Reference:

Stability for a magnetic Schrödinger operator on a Riemann surface with boundary
Joel Andersson  and  Leo Tzou 
2018, 12(1) : 1-28 doi: 10.3934/ipi.2018001 +[Abstract](70) +[HTML](38) +[PDF](538.76KB)
Abstract:

We consider a magnetic Schrödinger operator \begin{document} $(\nabla^X)^*\nabla^X+q$ \end{document} on a compact Riemann surface with boundary and prove a \begin{document} $\log\log$ \end{document}-type stability estimate in terms of Cauchy data for the electric potential and magnetic field under the assumption that they satisfy appropriate a priori bounds. We also give a similar stability result for the holonomy of the connection 1-form \begin{document} $X$ \end{document}.

ROI reconstruction from truncated cone-beam projections
Robert Azencott  , Bernhard G. Bodmann  , Tasadduk Chowdhury  , Demetrio Labate  , Anando Sen  and  Daniel Vera 
2018, 12(1) : 29-57 doi: 10.3934/ipi.2018002 +[Abstract](172) +[HTML](39) +[PDF](4980.26KB)
Abstract:

Region-of-Interest (ROI) tomography aims at reconstructing a region of interest \begin{document} $C$ \end{document} inside a body using only x-ray projections intersecting \begin{document} $C$ \end{document} and it is useful to reduce overall radiation exposure when only a small specific region of a body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve \begin{document} $Γ$ \end{document} in \begin{document} $\mathbb R^3$ \end{document} verifying the classical Tuy condition. In this generic situation, the non-trucated cone-beam transform of smooth density functions \begin{document} $f$ \end{document} admits an explicit inverse \begin{document} $Z$ \end{document} as originally shown by Grangeat. However \begin{document} $Z$ \end{document} cannot directly reconstruct \begin{document} $f$ \end{document} from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities \begin{document} $f$ \end{document} in \begin{document} $L^{∞}(B)$ \end{document} where \begin{document} $B$ \end{document} is a bounded ball in \begin{document} $\mathbb R^3$ \end{document}, our method iterates an operator \begin{document} $U$ \end{document} combining ROI-truncated projections, inversion by the operator \begin{document} $Z$ \end{document} and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI \begin{document} $C \subset B$ \end{document}, given \begin{document} $ε >0$ \end{document}, we prove that if \begin{document} $C$ \end{document} is sufficiently large our iterative reconstruction algorithm converges at exponential speed to an \begin{document} $ε$ \end{document}-accurate approximation of \begin{document} $f$ \end{document} in \begin{document} $L^{∞}$ \end{document}. The accuracy depends on the regularity of \begin{document} $f$ \end{document} quantified by its Sobolev norm in \begin{document} $W^5(B)$ \end{document}. Our result guarantees the existence of a critical ROI radius ensuring the convergence of our ROI reconstruction algorithm to an \begin{document} $ε$ \end{document}-accurate approximation of \begin{document} $f$ \end{document}. We have numerically verified these theoretical results using simulated acquisition of ROI-truncated cone-beam projection data for multiple acquisition geometries. Numerical experiments indicate that the critical ROI radius is fairly small with respect to the support region \begin{document} $B$ \end{document}.

Generalized stability estimates in inverse transport theory
Guillaume Bal  and  Alexandre Jollivet 
2018, 12(1) : 59-90 doi: 10.3934/ipi.2018003 +[Abstract](58) +[HTML](39) +[PDF](543.55KB)
Abstract:

Inverse transport theory concerns the reconstruction of the absorption and scattering coefficients in a transport equation from knowledge of the albedo operator, which models all possible boundary measurements. Uniqueness and stability results are well known and are typically obtained for errors of the albedo operator measured in the \begin{document} $L^1$ \end{document} sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in \begin{document} $O(1)$ \end{document} errors of the albedo operator and hence in \begin{document} $O(1)$ \end{document} error predictions on the reconstruction of the coefficients, which are not useful.

This paper revisit such stability estimates by introducing a more forgiving metric on the measurements errors, namely the \begin{document} $1-$ \end{document}Wasserstein distances, which penalize blurring or misalignment by an amount proportional to the width of the blurring kernel or to the amount of misalignment. We obtain new stability estimates in this setting.

We also consider the effect of errors, still measured in the \begin{document} $1-$ \end{document} Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allows us to consider discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.

Superconductive and insulating inclusions for linear and non-linear conductivity equations
Tommi Brander  , Joonas Ilmavirta  and  Manas Kar 
2018, 12(1) : 91-123 doi: 10.3934/ipi.2018004 +[Abstract](89) +[HTML](47) +[PDF](583.43KB)
Abstract:

We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear \begin{document} $p$ \end{document}-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation \begin{document} $\operatorname{div}(σ\lvert\nabla u\rvert^{p-2}\nabla u) = 0$ \end{document} where the measurable conductivity \begin{document} $σ\colonΩ\to[0,∞]$ \end{document} is zero or infinity in large sets and \begin{document} $1<p<∞$ \end{document}.

Assessment of the effect of tissue motion in diffusion MRI: Derivation of new apparent diffusion coefficient formula
Elie Bretin  , Imen Mekkaoui  and  Jérôme Pousin 
2018, 12(1) : 125-152 doi: 10.3934/ipi.2018005 +[Abstract](51) +[HTML](40) +[PDF](1660.87KB)
Abstract:

We investigate in this paper the diffusion magnetic resonance imaging (MRI) in deformable organs such as the living heart. The difficulty comes from the hight sensitivity of diffusion measurement to tissue motion. Commonly in literature, the diffusion MRI signal is given by the complex magnetization of water molecules described by the Bloch-Torrey equation. When dealing with deformable organs, the Bloch-Torrey equation is no longer valid. Our main contribution is then to introduce a new mathematical description of the Bloch-Torrey equation in deforming media. In particular, some numerical simulations are presented to quantify the influence of cardiac motion on the estimation of diffusion. Moreover, based on a scaling argument and on an asymptotic model for the complex magnetization, we derive a new apparent diffusion coefficient formula. Finally, some numerical experiments illustrate the potential of this new version which gives a better reconstruction of the diffusion than using the classical one.

Recovery of block sparse signals under the conditions on block RIC and ROC by BOMP and BOMMP
Wengu Chen  and  Huanmin Ge 
2018, 12(1) : 153-174 doi: 10.3934/ipi.2018006 +[Abstract](55) +[HTML](65) +[PDF](444.47KB)
Abstract:

In this paper, we consider the block orthogonal matching pursuit (BOMP) algorithm and the block orthogonal multi-matching pursuit (BOMMP) algorithm respectively to recover block sparse signals from an underdetermined system of linear equations. We first introduce the notion of block restricted orthogonality constant (ROC), which is a generalization of the standard restricted orthogonality constant, and establish respectively the sufficient conditions in terms of the block RIC and ROC to ensure the exact and stable recovery of any block sparse signals in both noiseless and noisy cases through the BOMP and BOMMP algorithm. We finally show that the sufficient condition on the block RIC and ROC is sharp for the BOMP algorithm.

On the parameter estimation problem of magnetic resonance advection imaging
Simon Hubmer  , Andreas Neubauer  , Ronny Ramlau  and  Henning U. Voss 
2018, 12(1) : 175-204 doi: 10.3934/ipi.2018007 +[Abstract](157) +[HTML](47) +[PDF](5194.5KB)
Abstract:

We present a reconstruction method for estimating the pulse-wave velocity in the brain from dynamic MRI data. The method is based on solving an inverse problem involving an advection equation. A space-time discretization is used and the resulting largescale inverse problem is solved using an accelerated Landweber type gradient method incorporating sparsity constraints and utilizing a wavelet embedding. Numerical example problems and a real-world data test show a significant improvement over the results obtained by the previously used method.

Scattering problems for perturbations of the multidimensional biharmonic operator
Teemu Tyni  and  Valery Serov 
2018, 12(1) : 205-227 doi: 10.3934/ipi.2018008 +[Abstract](96) +[HTML](58) +[PDF](420.82KB)
Abstract:

Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by first and zero order perturbations, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weighted Sobolev space \begin{document}$H_{-δ}^2 $\end{document}. The main result of this paper is the proof of Saito's formula, which can be used to prove a uniqueness theorem for the inverse scattering problem. The proof of Saito's formula is based on norm estimates for the resolvent of the direct operator in \begin{document}$H_{-δ}^1 $\end{document}.

Parametrices for the light ray transform on Minkowski spacetime
Yiran Wang 
2018, 12(1) : 229-237 doi: 10.3934/ipi.2018009 +[Abstract](44) +[HTML](36) +[PDF](353.3KB)
Abstract:

We consider restricted light ray transforms arising from an inverse problem of finding cosmic strings. We construct a relative left parametrix for the transform on two tensors, which recovers the space-like and some light-like singularities of the two tensor.

A scaled gradient method for digital tomographic image reconstruction
Jianjun Zhang  , Yunyi Hu  and  James G. Nagy 
2018, 12(1) : 239-259 doi: 10.3934/ipi.2018010 +[Abstract](120) +[HTML](45) +[PDF](779.26KB)
Abstract:

Digital tomographic image reconstruction uses multiple x-ray projections obtained along a range of different incident angles to reconstruct a 3D representation of an object. For example, computed tomography (CT) generally refers to the situation when a full set of angles are used (e.g., 360 degrees) while tomosynthesis refers to the case when only a limited (e.g., 30 degrees) angular range is used. In either case, most existing reconstruction algorithms assume that the x-ray source is monoenergetic. This results in a simplified linear forward model, which is easy to solve but can result in artifacts in the reconstructed images. It has been shown that these artifacts can be reduced by using a more accurate polyenergetic assumption for the x-ray source, but the polyenergetic model requires solving a large-scale nonlinear inverse problem. In addition to reducing artifacts, a full polyenergetic model can be used to extract additional information about the materials of the object; that is, to provide a mechanism for quantitative imaging. In this paper, we develop an approach to solve the nonlinear image reconstruction problem by incorporating total variation (TV) regularization. The corresponding optimization problem is then solved by using a scaled gradient descent method. The proposed algorithm is based on KKT conditions and Nesterov's acceleration strategy. Experimental results on reconstructed polyenergetic image data illustrate the effectiveness of this proposed approach.

The interior transmission problem
David Colton  , Lassi Päivärinta  and  John Sylvester 
2007, 1(1) : 13-28 doi: 10.3934/ipi.2007.1.13 +[Abstract](86) +[PDF](204.8KB) Cited By(104)
Fast dual minimization of the vectorial total variation norm and applications to color image processing
Xavier Bresson  and  Tony F. Chan 
2008, 2(4) : 455-484 doi: 10.3934/ipi.2008.2.455 +[Abstract](139) +[PDF](1915.3KB) Cited By(101)
Two-phase approach for deblurring images corrupted by impulse plus gaussian noise
Jian-Feng Cai  , Raymond H. Chan  and  Mila Nikolova 
2008, 2(2) : 187-204 doi: 10.3934/ipi.2008.2.187 +[Abstract](73) +[PDF](922.2KB) Cited By(66)
Iteratively solving linear inverse problems under general convex constraints
Ingrid Daubechies  , Gerd Teschke  and  Luminita Vese 
2007, 1(1) : 29-46 doi: 10.3934/ipi.2007.1.29 +[Abstract](63) +[PDF](270.3KB) Cited By(57)
Regularized D-bar method for the inverse conductivity problem
Kim Knudsen  , Matti Lassas  , Jennifer L. Mueller  and  Samuli Siltanen 
2009, 3(4) : 599-624 doi: 10.3934/ipi.2009.3.599 +[Abstract](73) +[PDF](451.7KB) Cited By(53)
On the existence of transmission eigenvalues
Andreas Kirsch 
2009, 3(2) : 155-172 doi: 10.3934/ipi.2009.3.155 +[Abstract](173) +[PDF](214.7KB) Cited By(51)
Augmented Lagrangian method for total variation restoration with non-quadratic fidelity
Chunlin Wu  , Juyong Zhang  and  Xue-Cheng Tai 
2011, 5(1) : 237-261 doi: 10.3934/ipi.2011.5.237 +[Abstract](127) +[PDF](2454.5KB) Cited By(48)
Non-local regularization of inverse problems
Gabriel Peyré  , Sébastien Bougleux  and  Laurent Cohen 
2011, 5(2) : 511-530 doi: 10.3934/ipi.2011.5.511 +[Abstract](71) +[PDF](1841.8KB) Cited By(41)
On uniqueness in the inverse conductivity problem with local data
Victor Isakov 
2007, 1(1) : 95-105 doi: 10.3934/ipi.2007.1.95 +[Abstract](198) +[PDF](156.4KB) Cited By(40)
Discretization-invariant Bayesian inversion and Besov space priors
Matti Lassas  , Eero Saksman  and  Samuli Siltanen 
2009, 3(1) : 87-122 doi: 10.3934/ipi.2009.3.87 +[Abstract](54) +[PDF](441.9KB) Cited By(37)
A scaled gradient method for digital tomographic image reconstruction
Jianjun Zhang  , Yunyi Hu  and  James G. Nagy 
2018, 12(1) : 239-259 doi: 10.3934/ipi.2018010 +[Abstract](120) +[HTML](45) +[PDF](779.26KB) PDF Downloads(28)
ROI reconstruction from truncated cone-beam projections
Robert Azencott  , Bernhard G. Bodmann  , Tasadduk Chowdhury  , Demetrio Labate  , Anando Sen  and  Daniel Vera 
2018, 12(1) : 29-57 doi: 10.3934/ipi.2018002 +[Abstract](172) +[HTML](39) +[PDF](4980.26KB) PDF Downloads(18)
Scattering problems for perturbations of the multidimensional biharmonic operator
Teemu Tyni  and  Valery Serov 
2018, 12(1) : 205-227 doi: 10.3934/ipi.2018008 +[Abstract](96) +[HTML](58) +[PDF](420.82KB) PDF Downloads(17)
Recovery of block sparse signals under the conditions on block RIC and ROC by BOMP and BOMMP
Wengu Chen  and  Huanmin Ge 
2018, 12(1) : 153-174 doi: 10.3934/ipi.2018006 +[Abstract](55) +[HTML](65) +[PDF](444.47KB) PDF Downloads(15)
On the parameter estimation problem of magnetic resonance advection imaging
Simon Hubmer  , Andreas Neubauer  , Ronny Ramlau  and  Henning U. Voss 
2018, 12(1) : 175-204 doi: 10.3934/ipi.2018007 +[Abstract](157) +[HTML](47) +[PDF](5194.5KB) PDF Downloads(14)
Superconductive and insulating inclusions for linear and non-linear conductivity equations
Tommi Brander  , Joonas Ilmavirta  and  Manas Kar 
2018, 12(1) : 91-123 doi: 10.3934/ipi.2018004 +[Abstract](89) +[HTML](47) +[PDF](583.43KB) PDF Downloads(14)
Generalized stability estimates in inverse transport theory
Guillaume Bal  and  Alexandre Jollivet 
2018, 12(1) : 59-90 doi: 10.3934/ipi.2018003 +[Abstract](58) +[HTML](39) +[PDF](543.55KB) PDF Downloads(12)
Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data
Chengxiang Wang  , Li Zeng  , Yumeng Guo  and  Lingli Zhang 
2017, 11(6) : 917-948 doi: 10.3934/ipi.2017043 +[Abstract](269) +[HTML](15) +[PDF](1923.8KB) PDF Downloads(11)
Stability for a magnetic Schrödinger operator on a Riemann surface with boundary
Joel Andersson  and  Leo Tzou 
2018, 12(1) : 1-28 doi: 10.3934/ipi.2018001 +[Abstract](70) +[HTML](38) +[PDF](538.76KB) PDF Downloads(10)
Coordinate descent optimization for l1 minimization with application to compressed sensing; a greedy algorithm
Yingying Li  and  Stanley Osher 
2009, 3(3) : 487-503 doi: 10.3934/ipi.2009.3.487 +[Abstract](110) +[PDF](682.0KB) PDF Downloads(8)

2016  Impact Factor: 1.094

Editors

Referees

Librarians

Email Alert

[Back to Top]