319 research outputs found
Adaptive regularization for image reconstruction from subsampled data
Choices of regularization parameters are central to variational methods for image restoration. In this paper, a spatially adaptive (or distributed) regularization scheme is developed based on localized residuals, which properly balances the regularization weight between regions containing image details and homogeneous regions. Surrogate iterative methods are employed to handle given subsampled data in transformed domains, such as Fourier or wavelet data. In this respect, this work extends the spatially variant regularization technique previously established in [15], which depends on the fact that the given data are degraded images only. Numerical experiments for the reconstruction from partial Fourier data and for wavelet inpainting prove the efficiency of the newly proposed approach
Adaptive regularization for image reconstruction from subsampled data
Choices of regularization parameters are central to variational methods
for image restoration. In this paper, a spatially adaptive (or distributed)
regularization scheme is developed based on localized residuals, which
properly balances the regularization weight between regions containing image
details and homogeneous regions. Surrogate iterative methods are employed to
handle given subsampled data in transformed domains, such as Fourier or
wavelet data. In this respect, this work extends the spatially variant
regularization technique previously established in [15], which depends on the
fact that the given data are degraded images only. Numerical experiments for
the reconstruction from partial Fourier data and for wavelet inpainting prove
the efficiency of the newly proposed approach
Solving Adaptive Image Restoration Problems via a Modified Projection Algorithm
We introduce a new general TV regularizer, namely, generalized TV regularization, to study image denoising and nonblind image deblurring problems. In order to discuss the generalized TV image restoration with solution-driven adaptivity, we consider the existence and uniqueness of the solution for mixed quasi-variational inequality. Moreover, the convergence of a modified projection algorithm for solving mixed quasi-variational inequalities is also shown. The corresponding experimental results support our theoretical findings
First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite
universal and flexible tool, allowing for highly successful approaches on tasks
like denoising, deblurring, inpainting, segmentation, super-resolution,
disparity, and optical flow estimation. The overall structure of such
approaches is of the form ; where the functional is a data fidelity term also
depending on some input data and measuring the deviation of from such
and is a regularization functional. Moreover is a (often linear)
forward operator modeling the dependence of data on an underlying image, and
is a positive regularization parameter. While is often
smooth and (strictly) convex, the current practice almost exclusively uses
nonsmooth regularization functionals. The majority of successful techniques is
using nonsmooth and convex functionals like the total variation and
generalizations thereof or -norms of coefficients arising from scalar
products with some frame system. The efficient solution of such variational
problems in imaging demands for appropriate algorithms. Taking into account the
specific structure as a sum of two very different terms to be minimized,
splitting algorithms are a quite canonical choice. Consequently this field has
revived the interest in techniques like operator splittings or augmented
Lagrangians. Here we shall provide an overview of methods currently developed
and recent results as well as some computational studies providing a comparison
of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure
CT Image Reconstruction by Spatial-Radon Domain Data-Driven Tight Frame Regularization
This paper proposes a spatial-Radon domain CT image reconstruction model
based on data-driven tight frames (SRD-DDTF). The proposed SRD-DDTF model
combines the idea of joint image and Radon domain inpainting model of
\cite{Dong2013X} and that of the data-driven tight frames for image denoising
\cite{cai2014data}. It is different from existing models in that both CT image
and its corresponding high quality projection image are reconstructed
simultaneously using sparsity priors by tight frames that are adaptively
learned from the data to provide optimal sparse approximations. An alternative
minimization algorithm is designed to solve the proposed model which is
nonsmooth and nonconvex. Convergence analysis of the algorithm is provided.
Numerical experiments showed that the SRD-DDTF model is superior to the model
by \cite{Dong2013X} especially in recovering some subtle structures in the
images
A combined first and second order variational approach for image reconstruction
In this paper we study a variational problem in the space of functions of
bounded Hessian. Our model constitutes a straightforward higher-order extension
of the well known ROF functional (total variation minimisation) to which we add
a non-smooth second order regulariser. It combines convex functions of the
total variation and the total variation of the first derivatives. In what
follows, we prove existence and uniqueness of minimisers of the combined model
and present the numerical solution of the corresponding discretised problem by
employing the split Bregman method. The paper is furnished with applications of
our model to image denoising, deblurring as well as image inpainting. The
obtained numerical results are compared with results obtained from total
generalised variation (TGV), infimal convolution and Euler's elastica, three
other state of the art higher-order models. The numerical discussion confirms
that the proposed higher-order model competes with models of its kind in
avoiding the creation of undesirable artifacts and blocky-like structures in
the reconstructed images -- a known disadvantage of the ROF model -- while
being simple and efficiently numerically solvable.Comment: 34 pages, 89 figure
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
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