119 research outputs found
Choose your path wisely: gradient descent in a Bregman distance framework
We propose an extension of a special form of gradient descent --- in the
literature known as linearised Bregman iteration -- to a larger class of
non-convex functions. We replace the classical (squared) two norm metric in the
gradient descent setting with a generalised Bregman distance, based on a
proper, convex and lower semi-continuous function. The algorithm's global
convergence is proven for functions that satisfy the Kurdyka-\L ojasiewicz
property. Examples illustrate that features of different scale are being
introduced throughout the iteration, transitioning from coarse to fine. This
coarse-to-fine approach with respect to scale allows to recover solutions of
non-convex optimisation problems that are superior to those obtained with
conventional gradient descent, or even projected and proximal gradient descent.
The effectiveness of the linearised Bregman iteration in combination with early
stopping is illustrated for the applications of parallel magnetic resonance
imaging, blind deconvolution as well as image classification with neural
networks
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
Sparse Parallel MRI Based on Accelerated Operator Splitting Schemes
Recently, the sparsity which is implicit in MR images has been successfully exploited for fast MR imaging with incomplete acquisitions. In this paper, two novel algorithms are proposed to solve the sparse parallel MR imaging problem, which consists of l1 regularization and fidelity terms. The two algorithms combine forward-backward operator splitting and Barzilai-Borwein schemes. Theoretically, the presented algorithms overcome the nondifferentiable property in l1 regularization term. Meanwhile, they are able to treat a general matrix operator that may not be diagonalized by fast Fourier transform and to ensure that a well-conditioned optimization system of equations is simply solved. In addition, we build connections between the proposed algorithms and the state-of-the-art existing methods and prove their convergence with a constant stepsize in Appendix. Numerical results and comparisons with the advanced methods demonstrate the efficiency of proposed algorithms
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