81 research outputs found
Denoising using projections onto the epigraph set of convex cost functions
A new denoising algorithm based on orthogonal projections onto the epigraph set of a convex cost function is presented. In this algorithm, the dimension of the minimization problem is lifted by one and feasibility sets corresponding to the cost function using the epigraph concept are defined. As the utilized cost function is a convex function in RN, the corresponding epigraph set is also a convex set in RN+1. The denoising algorithm starts with an arbitrary initial estimate in RN+1. At each step of the iterative denoising, an orthogonal projection is performed onto one of the constraint sets associated with the cost function in a sequential manner. The method provides globally optimal solutions for total-variation, ℓ1, ℓ2, and entropic cost functions.1 © 2014 IEEE
Image restoration and reconstruction using projections onto epigraph set of convex cost fuchtions
Cataloged from PDF version of article.This thesis focuses on image restoration and reconstruction problems. These
inverse problems are solved using a convex optimization algorithm based on orthogonal
Projections onto the Epigraph Set of a Convex Cost functions (PESC).
In order to solve the convex minimization problem, the dimension of the problem
is lifted by one and then using the epigraph concept the feasibility sets corresponding
to the cost function are defined. Since the cost function is a convex
function in R
N , the corresponding epigraph set is also a convex set in R
N+1. The
convex optimization algorithm starts with an arbitrary initial estimate in R
N+1
and at each step of the iterative algorithm, an orthogonal projection is performed
onto one of the constraint sets associated with the cost function in a sequential
manner. The PESC algorithm provides globally optimal solutions for different
functions such as total variation, `1-norm, `2-norm, and entropic cost functions.
Denoising, deconvolution and compressive sensing are among the applications of
PESC algorithm. The Projection onto Epigraph Set of Total Variation function
(PES-TV) is used in 2-D applications and for 1-D applications Projection onto
Epigraph Set of `1-norm cost function (PES-`1) is utilized.
In PES-`1 algorithm, first the observation signal is decomposed using wavelet
or pyramidal decomposition. Both wavelet denoising and denoising methods using
the concept of sparsity are based on soft-thresholding. In sparsity-based denoising
methods, it is assumed that the original signal is sparse in some transform domain
such as Fourier, DCT, and/or wavelet domain and transform domain coefficients
of the noisy signal are soft-thresholded to reduce noise. Here, the relationship between
the standard soft-thresholding based denoising methods and sparsity-based
wavelet denoising methods is described. A deterministic soft-threshold estimation
method using the epigraph set of `1-norm cost function is presented. It is
demonstrated that the size of the `1-ball can be determined using linear algebra.
The size of the `1-ball in turn determines the soft-threshold. The PESC, PES-TV
and PES-`1 algorithms, are described in detail in this thesis. Extensive simulation
results are presented. PESC based inverse restoration and reconstruction
algorithm is compared to the state of the art methods in the literature.Tofighi, MohammadM.S
Phase and TV Based Convex Sets for Blind Deconvolution of Microscopic Images
In this article, two closed and convex sets for blind deconvolution problem
are proposed. Most blurring functions in microscopy are symmetric with respect
to the origin. Therefore, they do not modify the phase of the Fourier transform
(FT) of the original image. As a result blurred image and the original image
have the same FT phase. Therefore, the set of images with a prescribed FT phase
can be used as a constraint set in blind deconvolution problems. Another convex
set that can be used during the image reconstruction process is the epigraph
set of Total Variation (TV) function. This set does not need a prescribed upper
bound on the total variation of the image. The upper bound is automatically
adjusted according to the current image of the restoration process. Both of
these two closed and convex sets can be used as a part of any blind
deconvolution algorithm. Simulation examples are presented.Comment: Submitted to IEEE Selected Topics in Signal Processin
A Non-Local Structure Tensor Based Approach for Multicomponent Image Recovery Problems
Non-Local Total Variation (NLTV) has emerged as a useful tool in variational
methods for image recovery problems. In this paper, we extend the NLTV-based
regularization to multicomponent images by taking advantage of the Structure
Tensor (ST) resulting from the gradient of a multicomponent image. The proposed
approach allows us to penalize the non-local variations, jointly for the
different components, through various matrix norms with .
To facilitate the choice of the hyper-parameters, we adopt a constrained convex
optimization approach in which we minimize the data fidelity term subject to a
constraint involving the ST-NLTV regularization. The resulting convex
optimization problem is solved with a novel epigraphical projection method.
This formulation can be efficiently implemented thanks to the flexibility
offered by recent primal-dual proximal algorithms. Experiments are carried out
for multispectral and hyperspectral images. The results demonstrate the
interest of introducing a non-local structure tensor regularization and show
that the proposed approach leads to significant improvements in terms of
convergence speed over current state-of-the-art methods
Sublabel-Accurate Relaxation of Nonconvex Energies
We propose a novel spatially continuous framework for convex relaxations
based on functional lifting. Our method can be interpreted as a
sublabel-accurate solution to multilabel problems. We show that previously
proposed functional lifting methods optimize an energy which is linear between
two labels and hence require (often infinitely) many labels for a faithful
approximation. In contrast, the proposed formulation is based on a piecewise
convex approximation and therefore needs far fewer labels. In comparison to
recent MRF-based approaches, our method is formulated in a spatially continuous
setting and shows less grid bias. Moreover, in a local sense, our formulation
is the tightest possible convex relaxation. It is easy to implement and allows
an efficient primal-dual optimization on GPUs. We show the effectiveness of our
approach on several computer vision problems
Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
This paper develops a general framework for solving a variety of convex cone
problems that frequently arise in signal processing, machine learning,
statistics, and other fields. The approach works as follows: first, determine a
conic formulation of the problem; second, determine its dual; third, apply
smoothing; and fourth, solve using an optimal first-order method. A merit of
this approach is its flexibility: for example, all compressed sensing problems
can be solved via this approach. These include models with objective
functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or
a combination thereof. In addition, the paper also introduces a number of
technical contributions such as a novel continuation scheme, a novel approach
for controlling the step size, and some new results showing that the smooth and
unsmoothed problems are sometimes formally equivalent. Combined with our
framework, these lead to novel, stable and computationally efficient
algorithms. For instance, our general implementation is competitive with
state-of-the-art methods for solving intensively studied problems such as the
LASSO. Further, numerical experiments show that one can solve the Dantzig
selector problem, for which no efficient large-scale solvers exist, in a few
hundred iterations. Finally, the paper is accompanied with a software release.
This software is not a single, monolithic solver; rather, it is a suite of
programs and routines designed to serve as building blocks for constructing
complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This
version has updated reference
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