26 research outputs found
Shearlets and Optimally Sparse Approximations
Multivariate functions are typically governed by anisotropic features such as
edges in images or shock fronts in solutions of transport-dominated equations.
One major goal both for the purpose of compression as well as for an efficient
analysis is the provision of optimally sparse approximations of such functions.
Recently, cartoon-like images were introduced in 2D and 3D as a suitable model
class, and approximation properties were measured by considering the decay rate
of the error of the best -term approximation. Shearlet systems are to
date the only representation system, which provide optimally sparse
approximations of this model class in 2D as well as 3D. Even more, in contrast
to all other directional representation systems, a theory for compactly
supported shearlet frames was derived which moreover also satisfy this
optimality benchmark. This chapter shall serve as an introduction to and a
survey about sparse approximations of cartoon-like images by band-limited and
also compactly supported shearlet frames as well as a reference for the
state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data",
Birkh\"auser-Springe
Matrix-free interior point method for compressed sensing problems
We consider a class of optimization problems for sparse signal reconstruction
which arise in the field of Compressed Sensing (CS). A plethora of approaches
and solvers exist for such problems, for example GPSR, FPC AS, SPGL1, NestA,
\ell_{1}_\ell_{s}, PDCO to mention a few. Compressed Sensing applications
lead to very well conditioned optimization problems and therefore can be solved
easily by simple first-order methods. Interior point methods (IPMs) rely on the
Newton method hence they use the second-order information. They have numerous
advantageous features and one clear drawback: being the second-order approach
they need to solve linear equations and this operation has (in the general
dense case) an computational complexity. Attempts have been made to
specialize IPMs to sparse reconstruction problems and they have led to
interesting developments implemented in and PDCO softwares. We
go a few steps further. First, we use the matrix-free interior point method, an
approach which redesigns IPM to avoid the need to explicitly formulate (and
store) the Newton equation systems. Secondly, we exploit the special features
of the signal processing matrices within the matrix-free IPM. Two such features
are of particular interest: an excellent conditioning of these matrices and the
ability to perform inexpensive (low complexity) matrix-vector multiplications
with them. Computational experience with large scale one-dimensional signals
confirms that the new approach is efficient and offers an attractive
alternative to other state-of-the-art solvers