5,659 research outputs found
Stable image reconstruction using total variation minimization
This article presents near-optimal guarantees for accurate and robust image
recovery from under-sampled noisy measurements using total variation
minimization. In particular, we show that from O(slog(N)) nonadaptive linear
measurements, an image can be reconstructed to within the best s-term
approximation of its gradient up to a logarithmic factor, and this factor can
be removed by taking slightly more measurements. Along the way, we prove a
strengthened Sobolev inequality for functions lying in the null space of
suitably incoherent matrices.Comment: 25 page
Accelerated gradient methods for total-variation-based CT image reconstruction
Total-variation (TV)-based Computed Tomography (CT) image reconstruction has
shown experimentally to be capable of producing accurate reconstructions from
sparse-view data. In particular TV-based reconstruction is very well suited for
images with piecewise nearly constant regions. Computationally, however,
TV-based reconstruction is much more demanding, especially for 3D imaging, and
the reconstruction from clinical data sets is far from being close to
real-time. This is undesirable from a clinical perspective, and thus there is
an incentive to accelerate the solution of the underlying optimization problem.
The TV reconstruction can in principle be found by any optimization method, but
in practice the large-scale systems arising in CT image reconstruction preclude
the use of memory-demanding methods such as Newton's method. The simple
gradient method has much lower memory requirements, but exhibits slow
convergence. In the present work we consider the use of two accelerated
gradient-based methods, GPBB and UPN, for reducing the number of gradient
method iterations needed to achieve a high-accuracy TV solution in CT image
reconstruction. The former incorporates several heuristics from the
optimization literature such as Barzilai-Borwein (BB) step size selection and
nonmonotone line search. The latter uses a cleverly chosen sequence of
auxiliary points to achieve a better convergence rate. The methods are memory
efficient and equipped with a stopping criterion to ensure that the TV
reconstruction has indeed been found. An implementation of the methods (in C
with interface to Matlab) is available for download from
http://www2.imm.dtu.dk/~pch/TVReg/. We compare the proposed methods with the
standard gradient method, applied to a 3D test problem with synthetic few-view
data. We find experimentally that for realistic parameters the proposed methods
significantly outperform the gradient method.Comment: 4 pages, 2 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 Singular Value Thresholding Algorithm for Matrix Completion
This paper introduces a novel algorithm to approximate the matrix with minimum
nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood
as the convex relaxation of a rank minimization problem and arises in many important
applications as in the task of recovering a large matrix from a small subset of its entries (the famous
Netflix problem). Off-the-shelf algorithms such as interior point methods are not directly amenable
to large problems of this kind with over a million unknown entries. This paper develops a simple
first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in
which the optimal solution has low rank. The algorithm is iterative, produces a sequence of matrices
{X^k,Y^k}, and at each step mainly performs a soft-thresholding operation on the singular values
of the matrix Y^k. There are two remarkable features making this attractive for low-rank matrix
completion problems. The first is that the soft-thresholding operation is applied to a sparse matrix;
the second is that the rank of the iterates {X^k} is empirically nondecreasing. Both these facts allow
the algorithm to make use of very minimal storage space and keep the computational cost of each
iteration low. On the theoretical side, we provide a convergence analysis showing that the sequence
of iterates converges. On the practical side, we provide numerical examples in which 1,000 × 1,000
matrices are recovered in less than a minute on a modest desktop computer. We also demonstrate
that our approach is amenable to very large scale problems by recovering matrices of rank about
10 with nearly a billion unknowns from just about 0.4% of their sampled entries. Our methods are
connected with the recent literature on linearized Bregman iterations for ℓ_1 minimization, and we
develop a framework in which one can understand these algorithms in terms of well-known Lagrange
multiplier algorithms
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