Journal Article10.1007/S10589-012-9525-4
A cyclic projected gradient method
TL;DR: This paper investigates the incorporation of superstep cycles into the projected gradient method and shows for two problems in compressive sensing and image processing that the resulting simple cyclic projected gradient algorithm can numerically compare with various state-of-the-art first-order algorithms.
read more
Abstract: In recent years, convex optimization methods were successfully applied for various image processing tasks and a large number of first-order methods were designed to minimize the corresponding functionals. Interestingly, it was shown recently in Grewenig et al. (2010) that the simple idea of so-called "superstep cycles" leads to very efficient schemes for time-dependent (parabolic) image enhancement problems as well as for steady state (elliptic) image compression tasks. The "superstep cycles" approach is similar to the nonstationary (cyclic) Richardson method which has been around for over sixty years.
In this paper, we investigate the incorporation of superstep cycles into the projected gradient method. We show for two problems in compressive sensing and image processing, namely the LASSO approach and the Rudin-Osher-Fatemi model that the resulting simple cyclic projected gradient algorithm can numerically compare with various state-of-the-art first-order algorithms. However, due to the nonlinear projection within the algorithm convergence proofs even under restrictive assumptions on the linear operators appear to be hard. We demonstrate the difficulties by studying the simplest case of a two-cycle algorithm in ?2 with projections onto the Euclidean ball.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
First Order Algorithms in Variational Image Processing
TL;DR: The success of non-smooth variational models in image processing is heavily based on efficient algorithms, taking into account the specific structure of the models as sum of different convex terms, splitting algorithms are an appropriate choice.
111
FSI Schemes: Fast Semi-Iterative Solvers for PDEs and Optimisation Methods
David Hafner,Peter Ochs,Joachim Weickert,M. Reißel,Sven Grewenig +4 more
- 12 Sep 2016
TL;DR: A simple and highly efficient acceleration strategy is introduced, leading to so-called Fast Semi-Iterative (FSI) schemes that extrapolate the basic solver iteration with the previous iterate to derive suitable extrapolation parameters.
15
•Journal Article
Barzilai-borwein-like method for solving large-scale non-linear systems of equations
TL;DR: In this article, a derivative-free Barzilai-Borwein-like algorithm is developed for solving large-scale non-linear systems of equations, which is based on approximating the Jacobian matrix in a quasi-Newton manner using a scalar multiple of an identity matrix.
15
An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems
TL;DR: The optimally generalized steepest-descent algorithm (OGSDA) is proven to be convergent with very fast convergence speed, accurate and robust against noisy disturbance, which is confirmed by numerical tests of some well-known ill-posed linear problems and linear inverse problems.
8
Performance of First-Order Algorithms for TV Penalized Weighted Least-Squares Denoising Problem
Alex Sawatzky
- 30 Jun 2014
TL;DR: First-order state-of-the-art computational schemes are applied on a total variation penalized weighted least-squares denoising problem and their performance is evaluated on numerical examples simulating a Poisson noise perturbation.
References
Regression Shrinkage and Selection via the Lasso
TL;DR: A new method for estimation in linear models called the lasso, which minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant, is proposed.
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information
TL;DR: In this paper, the authors considered the model problem of reconstructing an object from incomplete frequency samples and showed that with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the lscr/sub 1/ minimization problem.
Nonlinear total variation based noise removal algorithms
TL;DR: In this article, a constrained optimization type of numerical algorithm for removing noise from images is presented, where the total variation of the image is minimized subject to constraints involving the statistics of the noise.
17.3K
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
Amir Beck,Marc Teboulle +1 more
TL;DR: A new fast iterative shrinkage-thresholding algorithm (FISTA) which preserves the computational simplicity of ISTA but with a global rate of convergence which is proven to be significantly better, both theoretically and practically.
14.3K
A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging
Antonin Chambolle,Thomas Pock +1 more
TL;DR: A first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure can achieve O(1/N2) convergence on problems, where the primal or the dual objective is uniformly convex, and it can show linear convergence, i.e. O(ωN) for some ω∈(0,1), on smooth problems.