64,046 research outputs found
Epigraphical splitting for solving constrained convex optimization problems with proximal tools
International audienceWe propose a proximal approach to deal with a class of convex variational problems involving nonlinear constraints. A large family of constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different blocks of the linearly-transformed signal. For such constraints, the associated projection operator generally does not have a simple form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. In particular, we focus on constraints involving q-norms with q ≥ 1, distance functions to a convex set, and L1,p-norms with p ∈ {2, +∞}. The proposed approach is validated in the context of image restoration by making use of constraints based on Non-Local Total Variation. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems. A second application to a pulse shape design problem is provided in order to illustrate the flexibility of the proposed approach
Automatic regularization parameter selection for the total variation mixed noise image restoration framework
Image restoration consists in recovering a high quality image estimate based only on
observations. This is considered an ill-posed inverse problem, which implies non-unique
unstable solutions. Regularization methods allow the introduction of constraints in such
problems and assure a stable and unique solution. One of these methods is Total Variation,
which has been broadly applied in signal processing tasks such as image denoising, image
deconvolution, and image inpainting for multiple noise scenarios. Total Variation features
a regularization parameter which defines the solution regularization impact, a crucial step
towards its high quality level. Therefore, an optimal selection of the regularization parameter
is required. Furthermore, while the classic Total Variation applies its constraint to the
entire image, there are multiple scenarios in which this approach is not the most adequate.
Defining different regularization levels to different image elements benefits such cases. In
this work, an optimal regularization parameter selection framework for Total Variation image
restoration is proposed. It covers two noise scenarios: Impulse noise and Impulse over
Gaussian Additive noise. A broad study of the state of the art, which covers noise estimation
algorithms, risk estimation methods, and Total Variation numerical solutions, is
included. In order to approach the optimal parameter estimation problem, several adaptations
are proposed in order to create a local-fashioned regularization which requires no
a-priori information about the noise level. Quality and performance results, which include
the work covered in two recently published articles, show the effectivity of the proposed
regularization parameter selection and a great improvement over the global regularization
framework, which attains a high quality reconstruction comparable with the state of the art
algorithms.Tesi
Variational models for multiplicative noise removal
학위논문 (박사)-- 서울대학교 대학원 자연과학대학 수리과학부, 2017. 8. 강명주.This dissertation discusses a variational partial differential equation (PDE) models for restoration of images corrupted by multiplicative Gamma noise. The two proposed models are suitable for heavy multiplicative noise which is often seen in applications. First, we propose a total variation (TV) based model with local constraints. The local constraint involves multiple local windows which is related a spatially adaptive regularization parameter (SARP). In addition, convergence analysis such as the existence and uniqueness of a solution is also provided. Second model is an extension of the first one using nonconvex version of the total generalized variation (TGV). The nonconvex TGV regularization enables to efficiently denoise smooth regions, without staircasing artifacts that appear on total variation regularization based models, and to conserve edges and details.1. Introduction 1
2. Previous works 6
2.1 Variational models for image denoising 6
2.2.1 Convex and nonconvex regularizers 6
2.2.2 Variational models for multiplicative noise removal 8
2.2 Proximal linearized alternating direction method of multipliers 10
3. Proposed models 13
3.1 Proposed model 1 :exp TV model with SARP 13
3.1.1 Derivation of our model 13
3.1.2 Proposed TV model with local constraints 16
3.1.3 A SARP algorithm for solving model (3.1.16) 27
3.1.4 Numerical results 32
3.2 Proposed model 2 :exp NTGV model with SARP 51
3.2.1 Proposed NTGV model 51
3.2.2 Updating rule for in (3.2.1) 52
3.2.3 Algorithm for solving the proposed model (3.2.1) 55
3.2.4 Numerical results 62
3.2.5 Selection of parameters 63
3.2.6 Image denoising 65
4. Conclusion 79Docto
Solving a variational image restoration model which involves L∞ constraints
In this paper, we seek a solution to linear inverse problems arising in image restoration in terms of a recently posed optimization problem which combines total variation minimization and wavelet-thresholding ideas. The resulting nonlinear programming task is solved via a dual Uzawa method in its general form, leading to an efficient and general algorithm which allows for very good structure-preserving reconstructions. Along with a theoretical study of the algorithm, the paper details some aspects of the implementation, discusses the numerical convergence and eventually displays a few images obtained for some difficult restoration tasks
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
Domain decomposition methods for compressed sensing
We present several domain decomposition algorithms for sequential and
parallel minimization of functionals formed by a discrepancy term with respect
to data and total variation constraints. The convergence properties of the
algorithms are analyzed. We provide several numerical experiments, showing the
successful application of the algorithms for the restoration 1D and 2D signals
in interpolation/inpainting problems respectively, and in a compressed sensing
problem, for recovering piecewise constant medical-type images from partial
Fourier ensembles.Comment: 4 page
- …