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Inverse Problems and Imaging includes 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, stochastic and statistical methods. The field of applications include 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.
IPI will be online only and publish bimonthly in 2017 in February, April, June, August, October and December.
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TOP 10 Most Read Articles in IPI, February 2017
1 
4DCT reconstruction with unified spatialtemporal patchbased regularization
Volume 9, Number 2, Pages: 447  467, 2015
Daniil Kazantsev,
William M. Thompson,
William R. B. Lionheart,
Geert Van Eyndhoven,
Anders P. Kaestner,
Katherine J. Dobson,
Philip J. Withers
and Peter D. Lee
Abstract
References
Full Text
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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 spatialtemporal 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 stateoftheart spatialtemporal 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 nonlocal patches which are most likely to belong to one intensity class. This modification significantly improves the signaltonoise 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 tradeoff between temporal and spatial resolution scenarios.

2 
Coordinate descent optimization for l^{1} minimization with application to compressed sensing; a greedy algorithm
Volume 3, Number 3, Pages: 487  503, 2009
Yingying Li
and Stanley Osher
Abstract
Full Text
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We propose a fast algorithm for solving the Basis Pursuit problem, min_{u}
$\{u_1\: \Au=f\}$, which has application to compressed sensing.
We design an efficient method for solving the related unconstrained problem min_{u} $E(u) = u_1 + \lambda \Auf\^2_2$ 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.

3 
Iterative choice of the optimal regularization parameter in TV image restoration
Volume 9, Number 4, Pages: 1171  1191, 2015
Alina Toma,
Bruno Sixou
and Françoise Peyrin
Abstract
References
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We present iterative methods for choosing the optimal regularization parameter for linear inverse problems with Total Variation regularization. This approach is based on the Morozov discrepancy principle or on a damped version of this principle and on an approximating model function for the data term. The theoretical convergence of the method of choice of the regularization parameter is demonstrated.
The choice of the optimal parameter is refined with a Newton method. The efficiency of the method is illustrated on deconvolution and superresolution experiments on different types of images. Results are provided for different levels of blur, noise and loss of spatial resolution. The damped Morozov discrepancy principle often outerperforms the approaches based on the classical Morozov principle and on the Unbiased Predictive Risk Estimator. Moreover, the proposed methods are fast schemes to select the best parameter for TV regularization.

4 
Application of mixed formulations of quasireversibility to solve illposed problems for heat and wave equations: The 1D case
Volume 9, Number 4, Pages: 971  1002, 2015
Eliane Bécache,
Laurent Bourgeois,
Lucas Franceschini
and Jérémi Dardé
Abstract
References
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In this paper we address some illposed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data.
The main objective is to introduce some variational mixed formulations of quasireversibility which enable us to solve these illposed problems by using some classical Lagrange finite elements.
The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasireversibility method.
Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations.

5 
Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem
Volume 9, Number 4, Pages: 1025  1049, 2015
Fioralba Cakoni,
Houssem Haddar
and Isaac Harris
Abstract
References
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We consider the interior transmission problem associated with the scattering by an inhomogeneous (possibly anisotropic) highly oscillating periodic media. We show that, under appropriate assumptions, the solution of the interior transmission problem converges to the solution of a homogenized problem as the period goes to zero. Furthermore, we prove that the associated real transmission eigenvalues converge to transmission eigenvalues of the homogenized problem. Finally we show how to use the first transmission eigenvalue of the period media, which is measurable from the scattering data, to obtain information about constant effective material properties of the periodic media. The convergence results presented here are not optimal. Such results with rate of convergence involve the analysis of the boundary correction and will be subject of a forthcoming paper.

6 
Stabilized BFGS approximate Kalman filter
Volume 9, Number 4, Pages: 1003  1024, 2015
Alexander Bibov,
Heikki Haario
and Antti Solonen
Abstract
References
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The Kalman filter (KF) and Extended Kalman filter (EKF) are wellknown tools for assimilating data and model predictions. The filters require storage and multiplication of $n\times n$ and $n\times m$ matrices and inversion of $m\times m$ matrices, where $n$ is the dimension of the state space and $m$ is dimension of the observation space. Therefore, implementation of KF or EKF becomes impractical when dimensions increase. The earlier works provide optimizationbased approximative lowmemory approaches that enable filtering in high dimensions.
However, these versions ignore numerical issues that deteriorate performance of the approximations: accumulating errors may cause the covariance approximations to lose nonnegative definiteness, and approximative inversion of large closetosingular covariances gets tedious. Here we introduce a formulation that avoids these problems. We employ LBFGS formula to get lowmemory representations of the large matrices that appear in EKF, but inject a stabilizing correction to ensure that the resulting approximative representations remain nonnegative definite. The correction applies to any symmetric covariance approximation, and can be seen as a generalization of the Joseph covariance update.
We prove that the stabilizing correction enhances convergence rate of the covariance approximations.
Moreover, we generalize the idea by the means of NewtonSchultz matrix inversion formulae, which allows to employ them and their generalizations as stabilizing corrections.

7 
Template matching via $l_1$ minimization and its application to hyperspectral data
Volume 5, Number 1, Pages: 19  35, 2011
Zhaohui Guo
and Stanley Osher
Abstract
References
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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
searchandrescue 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),
generalizedlikelihood ratio test (GLRT) and matched filter. We
demonstrate that this algorithm achieves excellent results with both
high speed and accuracy by using Bregman iteration.

8 
Video stabilization of atmospheric turbulence distortion
Volume 7, Number 3, Pages: 839  861, 2013
Yifei Lou,
Sung Ha Kang,
Stefano Soatto
and Andrea L. Bertozzi
Abstract
References
Full Text
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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 wellposedness for the stabilizing PDE and the
linear stability of the numerical scheme.

9 
Locally sparse reconstruction using the $l^{1,\infty}$norm
Volume 9, Number 4, Pages: 1093  1137, 2015
Pia Heins,
Michael Moeller
and Martin Burger
Abstract
References
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This paper discusses the incorporation of local sparsity information, e.g. in
each pixel of an image, via minimization of the $\ell^{1,\infty}$norm. We
discuss the basic properties of this norm when used as a regularization
functional and associated optimization problems, for which we derive equivalent
reformulations either more amenable to theory or to numerical computation.
Further focus of the analysis is put on the locally 1sparse case, which is
well motivated by some biomedical imaging applications.
Our computational approaches are based on alternating direction methods of
multipliers (ADMM) and appropriate splittings with augmented Lagrangians. Those
are tested for a model scenario related to dynamic positron emission
tomography (PET), which is a functional imaging technique in nuclear medicine.
The results of this paper provide insight into the potential
impact of regularization with the $\ell^{1,\infty}$norm for
local sparsity in appropriate settings. However, it also
indicates several shortcomings, possibly related to the nontightness of the
functional as a relaxation of the $\ell^{0,\infty}$norm.

10 
A new KohnVogelius type formulation for inverse source problems
Volume 9, Number 4, Pages: 1051  1067, 2015
Xiaoliang Cheng,
Rongfang Gong
and Weimin Han
Abstract
References
Full Text
Related Articles
In this paper we propose a KohnVogelius type formulation
for an inverse source problem of partial differential equations.
The unknown source term is to be determined from both Dirichlet and
Neumann boundary conditions. We introduce two different boundary value problems,
which depend on two different positive real numbers $\alpha$ and $\beta$,
and both of them incorporate the Dirichlet and Neumann data into a single Robin
boundary condition. This allows noise in both boundary data.
By using the KohnVogelius type Tikhonov regularization, data to be fitted is transferred
from boundary into the whole domain, making the problem resolution more robust.
More importantly, with the formulation proposed here, satisfactory reconstruction could be achieved
for rather small regularization parameter through choosing properly the values of $\alpha$ and $\beta$.
This is a desirable property to have since a smaller regularization
parameter implies a more accurate approximation of the regularized problem to the original one.
The proposed method is studied theoretically.
Two numerical examples are provided to show the usefulness of the proposed method.

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