# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

Open Access Articles

2020, 14(2): 267-290 doi: 10.3934/ipi.2020012 +[Abstract](320) +[HTML](194) +[PDF](2820.82KB)
Abstract:

Low rank approximation has been extensively studied in the past. It is most suitable to reproduce rectangular like structures in the data. In this work we introduce a generalization using "shifted" rank-\begin{document}$1$\end{document} matrices to approximate \begin{document}$\mathit{\boldsymbol{{A}}}\in \mathbb{C}^{M\times N}$\end{document}. These matrices are of the form \begin{document}$S_{\mathit{\boldsymbol{{\lambda}}}}(\mathit{\boldsymbol{{u}}}\mathit{\boldsymbol{{v}}}^*)$\end{document} where \begin{document}$\mathit{\boldsymbol{{u}}}\in \mathbb{C}^M$\end{document}, \begin{document}$\mathit{\boldsymbol{{v}}}\in \mathbb{C}^N$\end{document} and \begin{document}$\mathit{\boldsymbol{{\lambda}}}\in \mathbb{Z}^N$\end{document}. The operator \begin{document}$S_{\mathit{\boldsymbol{{\lambda}}}}$\end{document} circularly shifts the \begin{document}$k$\end{document}-th column of \begin{document}$\mathit{\boldsymbol{{u}}}\mathit{\boldsymbol{{v}}}^*$\end{document} by \begin{document}$\lambda_k$\end{document}.

These kind of shifts naturally appear in applications, where an object \begin{document}$\mathit{\boldsymbol{{u}}}$\end{document} is observed in \begin{document}$N$\end{document} measurements at different positions indicated by the shift \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document}. The vector \begin{document}$\mathit{\boldsymbol{{v}}}$\end{document} gives the observation intensity. This model holds for seismic waves that are recorded at \begin{document}$N$\end{document} sensors at different times \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document}. Other examples are a car that moves through a video changing its position \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document} in each of the \begin{document}$N$\end{document} frames, or non-destructive testing based on ultrasonic waves that are reflected by defects inside the material.

The main difficulty of the above stated problem lies in finding a suitable shift vector \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document}. Once the shift is known, a simple singular value decomposition can be applied to reconstruct \begin{document}$\mathit{\boldsymbol{{u}}}$\end{document} and \begin{document}$\mathit{\boldsymbol{{v}}}$\end{document}. We propose a greedy method to reconstruct \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document}. By using the formulation of the problem in Fourier domain, a shifted rank-\begin{document}$1$\end{document} approximation can be calculated in \begin{document}$O(NM\log M)$\end{document}. Convergence to a locally optimal solution is guaranteed. Furthermore, we give a heuristic initial guess strategy that shows good results in the numerical experiments.

We validate our approach in several numerical experiments on different kinds of data. We compare the technique to shift-invariant dictionary learning algorithms. Furthermore, we provide examples from application including object segmentation in non-destructive testing and seismic exploration as well as object tracking in video processing.

2019, 13(5): 959-981 doi: 10.3934/ipi.2019043 +[Abstract](934) +[HTML](309) +[PDF](321.87KB)
Abstract:

An inverse obstacle problem for the wave governed by the wave equation in a two layered medium is considered under the framework of the time domain enclosure method. The wave is generated by an initial data supported on a closed ball in the upper half-space, and observed on the same ball over a finite time interval. The unknown obstacle is penetrable and embedded in the lower half-space. It is assumed that the propagation speed of the wave in the upper half-space is greater than that of the wave in the lower half-space, which is excluded in the previous study: Ikehata and Kawashita, Inverse Problems and Imaging 12 (2018), no.5, 1173-1198. In the present case, when the reflected waves from the obstacle enter the upper half-space, the total reflection phenomena occur, which give singularities to the integral representation of the fundamental solution for the reduced transmission problem in the background medium. This fact makes the problem more complicated. However, it is shown that these waves do not have any influence on the leading profile of the indicator function of the time domain enclosure method.

2019, 13(2): 263-284 doi: 10.3934/ipi.2019014 +[Abstract](1996) +[HTML](509) +[PDF](497.96KB)
Abstract:

An inverse obstacle scattering problem for the electromagnetic wave governed by the Maxwell system over a finite time interval is considered. It is assumed that the wave satisfies the Leontovich boundary condition on the surface of an unknown obstacle. The condition is described by using an unknown positive function on the surface of the obstacle which is called the surface admittance. The wave is generated at the initial time by a volumetric current source supported on a very small ball placed outside the obstacle and only the electric component of the wave is observed on the same ball over a finite time interval. It is shown that from the observed data one can extract information about the value of the surface admittance and the curvatures at the points on the surface nearest to the center of the ball. This shows that a single shot contains a meaningful information about the quantitative state of the surface of the obstacle.

2019, 13(2): 285-307 doi: 10.3934/ipi.2019015 +[Abstract](2921) +[HTML](647) +[PDF](1075.82KB)
Abstract:

EIT is a non-linear ill-posed inverse problem which requires sophisticated regularisation techniques to achieve good results. In this paper we consider the use of structural information in the form of edge directions coming from an auxiliary image of the same object being reconstructed. In order to allow for cases where the auxiliary image does not provide complete information we consider in addition a sparsity regularization for the edges appearing in the EIT image. The combination of these approaches is conveniently described through the parallel level sets approach. We present an overview of previous methods for structural regularisation and then provide a variational setting for our approach and explain the numerical implementation. We present results on simulations and experimental data for different cases with accurate and inaccurate prior information. The results demonstrate that the structural prior information improves the reconstruction accuracy, even in cases when there is reasonable uncertainty in the prior about the location of the edges or only partial edge information is available.

2018, 12(5): 1173-1198 doi: 10.3934/ipi.2018049 +[Abstract](2711) +[HTML](729) +[PDF](469.0KB)
Abstract:

An inverse obstacle problem for the wave equation in a two layered medium is considered. It is assumed that the unknown obstacle is penetrable and embedded in the lower half-space. The wave as a solution of the wave equation is generated by an initial data whose support is in the upper half-space and observed at the same place as the support over a finite time interval. From the observed wave an indicator function in the time domain enclosure method is constructed. It is shown that, one can find some information about the geometry of the obstacle together with the qualitative property in the asymptotic behavior of the indicator function.

2017, 11(6): 1071-1090 doi: 10.3934/ipi.2017049 +[Abstract](2396) +[HTML](1164) +[PDF](2271.0KB)
Abstract:

In this paper, we investigate the relations between the Radon and weighted divergent beam and cone transforms. Novel inversion formulas are derived for the latter two. The weighted cone transform arises, for instance, in image reconstruction from the data obtained by Compton cameras, which have promising applications in various fields, including biomedical and homeland security imaging and gamma ray astronomy. The inversion formulas are applicable for a wide variety of detector geometries in any dimension. The results of numerical implementation of some of the formulas in dimensions two and three are also provided.

2017, 11(1): 99-123 doi: 10.3934/ipi.2017006 +[Abstract](2027) +[HTML](881) +[PDF](528.2KB)
Abstract:

An inverse obstacle scattering problem for the wave governed by the Maxwell system in the time domain, in particular, over a finite time interval is considered. It is assumed that the electric field \begin{document}$\boldsymbol{E}$\end{document} and magnetic field \begin{document}$\boldsymbol{ H}$\end{document} which are solutions of the Maxwell system are generated only by a current density at the initial time located not far a way from an unknown obstacle. The obstacle is embedded in a medium like air which has constant electric permittivity \begin{document}$ε$\end{document} and magnetic permeability \begin{document}$μ$\end{document}. It is assumed that the fields on the surface of the obstacle satisfy the Leontovich boundary condition \begin{document}$\boldsymbol{ ν}×\boldsymbol{H}-λ\,\boldsymbol{ ν}×(\boldsymbol{ E}×\boldsymbol{ ν})=\boldsymbol{ 0}$\end{document} with admittance \begin{document}$λ$\end{document} an unknown positive function and \begin{document}$\boldsymbol{ ν}$\end{document} the unit outward normal. The observation data are given by the electric field observed at the same place as the support of the current density over a finite time interval. It is shown that an indicator function computed from the electric fields corresponding two current densities enables us to know: the distance of the center of the common spherical support of the current densities to the obstacle; whether the value of the admittance \begin{document}$λ$\end{document} is greater or less than the special value \begin{document}$\sqrt{ε/μ}$\end{document}.

2016, 10(1): 131-163 doi: 10.3934/ipi.2016.10.131 +[Abstract](2168) +[PDF](574.5KB)
Abstract:
In this paper, a time domain enclosure method for an inverse obstacle scattering problem of electromagnetic wave is introduced. The wave as a solution of Maxwell's equations is generated by an applied volumetric current having an orientation and supported outside an unknown obstacle and observed on the same support over a finite time interval. It is assumed that the obstacle is a perfect conductor. Two types of analytical formulae which employ a single observed wave and explicitly contain information about the geometry of the obstacle are given. In particular, an effect of the orientation of the current is catched in one of two formulae. Two corollaries concerning with the detection of the points on the surface of the obstacle nearest to the centre of the current support and curvatures at the points are also given.
2015, 9(2): 447-467 doi: 10.3934/ipi.2015.9.447 +[Abstract](2512) +[PDF](10881.1KB)
Abstract:
In this paper, we consider a limited data reconstruction problem for temporarily evolving computed tomography (CT), where some regions are static during the whole scan and some are dynamic (intensely or slowly changing). When motion occurs during a tomographic experiment one would like to minimize the number of projections used and reconstruct the image iteratively. To ensure stability of the iterative method spatial and temporal constraints are highly desirable. Here, we present a novel spatial-temporal regularization approach where all time frames are reconstructed collectively as a unified function of space and time. Our method has two main differences from the state-of-the-art spatial-temporal regularization methods. Firstly, all available temporal information is used to improve the spatial resolution of each time frame. Secondly, our method does not treat spatial and temporal penalty terms separately but rather unifies them in one regularization term. Additionally we optimize the temporal smoothing part of the method by considering the non-local patches which are most likely to belong to one intensity class. This modification significantly improves the signal-to-noise ratio of the reconstructed images and reduces computational time. The proposed approach is used in combination with golden ratio sampling of the projection data which allows one to find a better trade-off between temporal and spatial resolution scenarios.
2015, 9(2): 431-446 doi: 10.3934/ipi.2015.9.431 +[Abstract](1938) +[PDF](505.7KB)
Abstract:
In X-ray computed tomography (CT) it is generally acknowledged that reconstruction methods exploiting image sparsity allow reconstruction from a significantly reduced number of projections. The use of such reconstruction methods is inspired by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT reconstruction, nor any relation between sparsity and a sufficient number of measurements for recovery, i.e., perfect reconstruction from noise-free data. We consider reconstruction through 1-norm minimization, as proposed in CS, from data obtained using a standard CT fan-beam sampling pattern. In empirical simulation studies we establish quantitatively a relation between the image sparsity and the sufficient number of measurements for recovery within image classes motivated by tomographic applications. We show empirically that the specific relation depends on the image class and in many cases exhibits a sharp phase transition as seen in CS, i.e., same-sparsity images require the same number of projections for recovery. Finally we demonstrate that the relation holds independently of image size and is robust to small amounts of additive Gaussian white noise.
2013, 7(3): i-ii doi: 10.3934/ipi.2013.7.3i +[Abstract](1376) +[PDF](345.2KB)
Abstract:
In 2012, there were two scientific conferences in honor of Professor Tony F. Chan's 60th birthday. The first one was The International Conference on Scientific Computing'', which took place in Hong Kong from January 4-7. The second one, The International Conference on the Frontier of Computational and Applied Mathematics'', was held at the Institute of Pure and Applied Mathematics (IPAM) of UCLA from June 8-10. Invitations were also sent out to conference speakers, participants, Professor Chan's former colleagues, collaborators and students, to solicit for original research papers. After the standard peer review processes, we have collected 23 papers in this special issue dedicated to Professor Chan to celebrate his contribution and leadership in the area of scientific computing and image processing.

2013, 7(3): 839-861 doi: 10.3934/ipi.2013.7.839 +[Abstract](3125) +[PDF](1592.7KB)
Abstract:
We present a method to enhance the quality of a video sequence captured through a turbulent atmospheric medium, and give an estimate of the radiance of the distant scene, represented as a latent image,'' which is assumed to be static throughout the video. Due to atmospheric turbulence, temporal averaging produces a blurred version of the scene's radiance. We propose a method combining Sobolev gradient and Laplacian to stabilize the video sequence, and a latent image is further found utilizing the lucky region" method. The video sequence is stabilized while keeping sharp details, and the latent image shows more consistent straight edges. We analyze the well-posedness for the stabilizing PDE and the linear stability of the numerical scheme.
2011, 5(1): 19-35 doi: 10.3934/ipi.2011.5.19 +[Abstract](2210) +[PDF](582.3KB)
Abstract:
Detecting and identifying targets or objects that are present in hyperspectral ground images are of great interest. Applications include land and environmental monitoring, mining, military, civil search-and-rescue operations, and so on. We propose and analyze an extremely simple and efficient idea for template matching based on $l_1$ minimization. The designed algorithm can be applied in hyperspectral classification and target detection. Synthetic image data and real hyperspectral image (HSI) data are used to assess the performance, with comparisons to other approaches, e.g. spectral angle map (SAM), adaptive coherence estimator (ACE), generalized-likelihood ratio test (GLRT) and matched filter. We demonstrate that this algorithm achieves excellent results with both high speed and accuracy by using Bregman iteration.
2010, 4(2): i-iii doi: 10.3934/ipi.2010.4.2i +[Abstract](1443) +[PDF](37.7KB)
Abstract:
Life expectancy in the developed and developing countries is constantly increasing. Medicine has benefited from novel biomarkers for screening and diagnosis. At least for a number of diseases, biomedical imaging is one of the most promising means of early diagnosis. Medical hardware manufacturer's progress has led to a new generation of measurements to understand the human anatomical and functional states. These measurements go beyond simple means of anatomical visualization (e.g. X-ray images) and therefore their interpretation becomes a scientific challenge for humans mostly because of the volume and flow of information as well as their nature. Computer-aided diagnosis develops mathematical models and their computational solutions to assist data interpretation in a clinical setting. In simple words, one would like to be able to provide a formal answer to a clinical question using the available measurements. The development of mathematical models for automatic clinical interpretation of multi-modalities is a great challenge.

2009, 3(3): 487-503 doi: 10.3934/ipi.2009.3.487 +[Abstract](2658) +[PDF](682.0KB)
Abstract:
We propose a fast algorithm for solving the Basis Pursuit problem, minu $\{|u|_1\: \Au=f\}$, which has application to compressed sensing. We design an efficient method for solving the related unconstrained problem minu $E(u) = |u|_1 + \lambda \||Au-f\||^2_2$ based on a greedy coordinate descent method. We claim that in combination with a Bregman iterative method, our algorithm will achieve a solution with speed and accuracy competitive with some of the leading methods for the basis pursuit problem.
2009, 3(2): i-i doi: 10.3934/ipi.2009.3.2i +[Abstract](1505) +[PDF](34.3KB)
Abstract:
This special issue is dedicated to Professors David Colton and Rainer Kress in honor of their contribution and leadership in the area of direct and inverse scattering theory for more then 30 years. The papers in this special issue were solicited from the invited speakers at the International Conference on Inverse Scattering Problems organized in honor of the 65th birthdays of David Colton and Rainer Kress held in the seaside resort of Sestry Levante, Italy, May 8-10, 2008.
As organizers of this conference and close collaborators of Professors Colton and Kress, we are very honored to have had the opportunity to facilitate this special scientific and social event. It was a particular occasion that gathered together long term colleagues, collaborators, former students and friends of Professors Colton and Kress. And now it gives us particular pleasure to be guest editors of this special issue of Inverse Problems and Imaging which is a collection of original research papers in the area of scattering theory and inverse problems. Much of the work presented here has been directly or indirectly influenced by these two scientists, offering the reader a glimpse of their significant impact in this research area.
We would like to thank all of those who have contributed a paper for this special issue. A special thanks goes to the Editor in Chief of Inverse Problems and Imaging, Lassi Päivärinta, for supporting and facilitating this publication. We would also like to thank all the participants of the Sestri Levante Conference who made such a successful, stimulating and pleasant event possible. Last (but definitely not least!) we would like to thank the sponsors of the conference: the European Office of Aerospace Research and Development of the United States Air Force Office of Scientific Research, the University of Genova, the University of Verona, the Istituto Nazionale di Alta Matematica - Gruppo Nazionale di Calcolo Scientifico, the University of Göttingen, the University of Delaware and INRIA Center of Saclay Ile de France.

2018  Impact Factor: 1.469