## Journals

- Advances in Mathematics of Communications
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- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
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- Mathematical Foundations of Computing
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- Electronic Research Announcements
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- AIMS Mathematics

IPI

The paper addresses the generalization of the half-quadratic minimization method for the restoration of
images having values in a complete, connected Riemannian manifold.
We recall the half-quadratic minimization method using the notation of the $c$-transform
and adapt the algorithm to our special variational setting. We prove the convergence of the method for
Hadamard spaces.
Extensive numerical examples for images with values on spheres, in the rotation group $SO(3)$, and
in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm.
In particular, the method with $SO(3)$-valued data
shows promising results for the restoration of images obtained
from Electron Backscattered Diffraction which are of interest in material science.

IPI

In this paper, we consider tomographic reconstruction for axially symmetric
objects from a single radiograph formed by fan-beam X-rays.
All contemporary methods are based on the assumption
that the density is piecewise constant or linear.
From a practical viewpoint, this is quite a restrictive approximation.
The method we propose is based on
high-order total variation regularization.
Its main advantage is to reduce
the staircase effect
while keeping sharp edges and enable the recovery of smoothly varying regions.
The optimization problem is solved using
the augmented Lagrangian method which has
been recently applied in image processing.
Furthermore, we use a
one-dimensional (1D) technique for fan-beam X-rays to
approximate 2D tomographic reconstruction for cone-beam X-rays. For
the 2D problem, we treat the cone beam as fan beam located at
parallel planes perpendicular to the symmetric axis. Then the density of the whole
object is recovered layer by layer.
Numerical
results in 1D show that the proposed method has improved the
preservation of edge location and the accuracy of the
density level when compared with several other contemporary methods.
The 2D numerical tests show that cylindrical symmetric objects can be recovered
rather accurately by our high-order regularization model.

keywords:
high-order total variation
,
augmented Lagrangian method.
,
radiograph
,
Tomography
,
Abel
inversion

IPI

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.

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

keywords:

IPI

Real images usually have two layers, namely, cartoons (the piecewise
smooth part of the image) and textures (the oscillating pattern
part of the image). Both these two layers have sparse
approximations under some tight frame systems such as framelet,
translation invariant wavelet, curvelet, and local DCTs. In this
paper, we solve image inpainting problems by using two separate
tight frame systems which can sparsely represent cartoons and
textures respectively. Different from existing schemes in the
literature which are either analysis-based or synthesis-based
sparsity priors, our minimization formulation balances these two
priors. We also derive iterative algorithms to find their
solutions and prove their convergence. Numerical simulation examples
are given to demonstrate the applicability and usefulness of our
proposed algorithms in image inpainting.

IPI

We propose iterative thresholding algorithms based on the
iterated Tikhonov method for image deblurring problems.
Our method is similar in idea to the modified linearized Bregman algorithm (MLBA) so
is easy to implement. In order to obtain good restorations, MLBA requires an accurate estimate of the regularization parameter $\alpha$ which
is hard to get in real applications.
Based on previous results in iterated Tikhonov method, we design
two nonstationary iterative thresholding algorithms which give near
optimal results without estimating $\alpha$. One of them is based
on the iterative soft thresholding algorithm and the other is based
on MLBA. We show that the nonstationary methods,
if converge, will converge to the same minimizers of
the stationary variants.
Numerical results
show that the accuracy and convergence of our nonstationary methods are very robust
with respect to the changes in the parameters and the restoration
results are comparable to those of MLBA with optimal $\alpha$.

IPI

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 L1-shaped data-fidelity 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 above-mentioned 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 candidates---the 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 outlier-free 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.

## Year of publication

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