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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.
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, April 2017
1 
Wavelet inpainting by nonlocal total variation
Volume 4, Number 1, Pages: 191  210, 2010
Xiaoqun Zhang
and Tony F. Chan
Abstract
Full Text
Related Articles
Wavelet inpainting problem consists of filling in missed data in the wavelet domain. In [17], Chan, Shen, and Zhou proposed an efficient method to recover piecewise constant or smooth images by combining total variation regularization and wavelet representations. In this paper, we extend it to nonlocal total variation regularization in order to recover textures and local geometry structures simultaneously. Moreover, we apply an efficient algorithm framework for both local and nonlocal regularizers. Extensive
experimental results on a variety of loss scenarios and natural
images validate the performance of this approach.

2 
Correlation priors
Volume 5, Number 1, Pages: 167  184, 2011
Lassi Roininen,
Markku S. Lehtinen,
Sari Lasanen,
Mikko Orispää
and Markku Markkanen
Abstract
References
Full Text
Related Articles
We propose a new class of Gaussian priors, correlation priors. In contrast to some wellknown smoothness priors, they have stationary covariances. The correlation priors are given in a parametric form with two parameters: correlation power and correlation length. The first parameter is connected with our prior information on the variance of the unknown. The second parameter is our prior belief on how fast the correlation of the unknown approaches zero. Roughly speaking, the correlation length is the distance beyond which two points of the unknown may be considered independent.
The prior distribution is constructed to be essentially independent of the discretization so that the a posteriori distribution will be essentially independent of the discretization grid. The covariance of a discrete correlation prior may be formed by combining the Fisher information of a discrete white noise and differentorder difference priors. This is interpreted as a combination of virtual measurements of the unknown. Closedform expressions for the continuous limits are calculated. Also, boundary correction terms for correlation priors on finite intervals are given.
A numerical example, deconvolution with a Gaussian kernel and a correlation prior, is computed.

3 
Identification of soundsoft 3D obstacles from phaseless data
Volume 4, Number 1, Pages: 131  149, 2010
Olha Ivanyshyn
and Rainer Kress
Abstract
Full Text
Related Articles
The inverse problem for timeharmonic acoustic wave scattering to recover a soundsoft obstacle from a
given incident field and the far field pattern of the scattered field is considered. We split this problem into two
subproblems; first to reconstruct the shape from the modulus of the data and this is followed by
employing the full far field
pattern in a few measurement points to find the location of the obstacle.
We extend a nonlinear integral equation approach for shape reconstruction from the modulus of the far field
data [6] to the threedimensional case. It is known, see [13], that the location
of the obstacle cannot be reconstructed from only the modulus of the far field pattern since it is invariant under
translations. However, employing the underlying invariance relation
and using only few far field measurements in the backscattering direction we
propose a novel approach for the localization of the obstacle.
The efficient implementation of the method is described and the feasibility of the approach is illustrated by numerical
examples.

4 
Variational denoising of diffusion weighted MRI
Volume 3, Number 4, Pages: 625  648, 2009
Tim McGraw,
Baba Vemuri,
Evren Özarslan,
Yunmei Chen
and Thomas Mareci
Abstract
Full Text
Related Articles
In this paper, we present a novel variational formulation for
restoring high angular resolution diffusion imaging (HARDI) data. The
restoration formulation involves smoothing signal measurements over
the spherical domain and across the 3D image lattice. The
regularization across the lattice is achieved using a total
variation (TV) norm based scheme, while the finite element method
(FEM) was employed to smooth the data on the sphere at each lattice
point using first and second order smoothness constraints. Examples
are presented to show the performance of the
HARDI data restoration scheme and its effect on fiber direction
computation on synthetic data, as well as on real data sets
collected from excised rat brain and spinal cord.

5 
On uniqueness in the inverse conductivity problem with local data
Volume 1, Number 1, Pages: 95  105, 2007
Victor Isakov
Abstract
Full Text
Related Articles
We show that the DirichlettoNeumann map given on an arbitrary part of the boundary of a threedimensional domain with zero Dirichlet (or Neumann) data on the remaining (spherical or plane) part of the boundary
uniquely determines conductivity or potential coefficients. This is the first uniqueness result for the Calderon problem with zero data on unaccessible part of the boundary. Proofs use some modification of the method of complex geometrical solutions due to CalderonSylvesterUhlmann.

6 
Augmented Lagrangian method for total variation restoration with nonquadratic fidelity
Volume 5, Number 1, Pages: 237  261, 2011
Chunlin Wu,
Juyong Zhang
and XueCheng Tai
Abstract
References
Full Text
Related Articles
Recently augmented Lagrangian method has been successfully applied
to image restoration. We extend the method to total variation (TV)
restoration models with nonquadratic fidelities. We will first
introduce the method and present an iterative algorithm for TV
restoration with a quite general fidelity. In each iteration, three
subproblems need to be solved, two of which can be very efficiently
solved via Fast Fourier Transform (FFT) implementation or closed
form solution. In general the third subproblem need iterative
solvers. We then apply our method to TV restoration with $L^1$ and
KullbackLeibler (KL) fidelities, two common and important data
terms for deblurring images corrupted by impulsive noise and Poisson
noise, respectively. For these typical fidelities, we show that the
third subproblem also has closed form solution and thus can be
efficiently solved. In addition, convergence analysis of these
algorithms are given. Numerical experiments demonstrate the
efficiency of our method.

7 
Twophase approach for deblurring images corrupted by impulse plus gaussian noise
Volume 2, Number 2, Pages: 187  204, 2008
JianFeng Cai,
Raymond H. Chan
and Mila Nikolova
Abstract
References
Full Text
Related Articles
The restoration of blurred images corrupted with impulse noise is a
difficult problem which has been considered in a series of recent
papers. These papers tackle the problem by using variational methods
involving an L1shaped datafidelity term. Because of this term, the
relevant methods exhibit systematic errors at the corrupted pixel locations
and require a cumbersome optimization stage. In this work we
propose and justify a much simpler alternative approach which
overcomes the abovementioned systematic errors and leads to much
better results. Following a theoretical derivation based on a
simple model, we decouple the problem into two phases. First, we
identify the outlier candidatesthe pixels that are likely to be
corrupted by the impulse noise, and we remove them from our data set. In a
second phase, the image is deblurred and denoised simultaneously
using essentially the outlierfree data. The resultant optimization
stage is much simpler in comparison with the current full
variational methods and the outlier contamination is more accurately
corrected. The experiments show that we obtain a 2 to 6 dB improvement
in PSNR. We emphasize that our method can be adapted to deblur
images corrupted with mixed impulse plus Gaussian noise, and hence
it can address a much wider class of practical problems.

8 
A family of inversion formulas in thermoacoustic tomography
Volume 3, Number 4, Pages: 649  675, 2009
Linh V. Nguyen
Abstract
Full Text
Related Articles
We present a family of closed form inversion formulas in thermoacoustic tomography in the case of a constant sound speed. The formulas are presented in both timedomain and frequencydomain versions. As special cases, they imply most of the previously known filtered backprojection type formulas.

9 
The interior transmission problem
Volume 1, Number 1, Pages: 13  28, 2007
David Colton,
Lassi Päivärinta
and John Sylvester
Abstract
Full Text
Related Articles
The interior transmission problem is a boundary value problem that plays a basic role in inverse scattering theory but unfortunately does not seem to be included in any existing theory in partial differential equations.This paper presents old and new results for the interior transmission problem ,in particular its relation to inverse scattering theory and new results on the spectral theory associated with this class of boundary value problems.

10 
An efficient computational method for total variationpenalized Poisson likelihood estimation
Volume 2, Number 2, Pages: 167  185, 2008
Johnathan M. Bardsley
Abstract
References
Full Text
Related Articles
Approximating nonGaussian noise processes with Gaussian models is standard in data analysis. This is due in large part to the fact that Gaussian models yield parameter estimation problems of least squares form, which have been extensively studied both from the theoretical and computational points of view. In image processing applications, for example, data is often collected by a CCD camera, in which case the noise is a Guassian/Poisson mixture with the Poisson noise dominating for a sufficiently strong signal. Even so, the standard approach in such cases is to use a Gaussian approximation that leads to a negativelog likelihood function of weighted least squares type.
In the Bayesian pointofview taken in this paper, a negativelog prior (or regularization) function is added to the negativelog likelihood function, and the resulting function is minimized. We focus on the case where the negativelog prior is the wellknown total variation function and give a statistical interpretation. Regardless of whether the least squares or Poisson negativelog likelihood is used, the total variation term yields a minimization problem that is computationally challenging. The primary result of this work is the efficient computational method that is presented for the solution of such problems, together with its convergence analysis. With the computational method in hand, we then perform experiments that indicate that the Poisson negativelog likelihood yields a more computationally efficient method than does the use of the least squares function. We also present results that indicate that this may even be the case when the data noise is i.i.d. Gaussian, suggesting that regardless of noise statistics, using the Poisson negativelog likelihood can yield a more computationally tractable problem when total variation regularization is used.

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