5,775 research outputs found
Multiscale Adaptive Representation of Signals: I. The Basic Framework
We introduce a framework for designing multi-scale, adaptive, shift-invariant
frames and bi-frames for representing signals. The new framework, called
AdaFrame, improves over dictionary learning-based techniques in terms of
computational efficiency at inference time. It improves classical multi-scale
basis such as wavelet frames in terms of coding efficiency. It provides an
attractive alternative to dictionary learning-based techniques for low level
signal processing tasks, such as compression and denoising, as well as high
level tasks, such as feature extraction for object recognition. Connections
with deep convolutional networks are also discussed. In particular, the
proposed framework reveals a drawback in the commonly used approach for
visualizing the activations of the intermediate layers in convolutional
networks, and suggests a natural alternative
Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing
For the solution of operator equations, Stevenson introduced a definition of
frames, where a Hilbert space and its dual are {\em not} identified. This means
that the Riesz isomorphism is not used as an identification, which, for
example, does not make sense for the Sobolev spaces and
. In this article, we are going to revisit the concept of
Stevenson frames and introduce it for Banach spaces. This is equivalent to
-Banach frames. It is known that, if such a system exists, by defining
a new inner product and using the Riesz isomorphism, the Banach space is
isomorphic to a Hilbert space. In this article, we deal with the contrasting
setting, where and are not identified, and
equivalent norms are distinguished, and show that in this setting the
investigation of -Banach frames make sense.Comment: 23 pages; accepted for publication in 'Numerical Functional Analysis
and Optimization
A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators
This chapter offers a detailed survey on intrinsically localized frames and
the corresponding matrix representation of operators. We re-investigate the
properties of localized frames and the associated Banach spaces in full detail.
We investigate the representation of operators using localized frames in a
Galerkin-type scheme. We show how the boundedness and the invertibility of
matrices and operators are linked and give some sufficient and necessary
conditions for the boundedness of operators between the associated Banach
spaces.Comment: 32 page
Adaptive Low-Rank Methods for Problems on Sobolev Spaces with Error Control in
Low-rank tensor methods for the approximate solution of second-order elliptic
partial differential equations in high dimensions have recently attracted
significant attention. A critical issue is to rigorously bound the error of
such approximations, not with respect to a fixed finite dimensional discrete
background problem, but with respect to the exact solution of the continuous
problem. While the energy norm offers a natural error measure corresponding to
the underlying operator considered as an isomorphism from the energy space onto
its dual, this norm requires a careful treatment in its interplay with the
tensor structure of the problem. In this paper we build on our previous work on
energy norm-convergent subspace-based tensor schemes contriving, however, a
modified formulation which now enforces convergence only in . In order to
still be able to exploit the mapping properties of elliptic operators, a
crucial ingredient of our approach is the development and analysis of a
suitable asymmetric preconditioning scheme. We provide estimates for the
computational complexity of the resulting method in terms of the solution error
and study the practical performance of the scheme in numerical experiments. In
both regards, we find that controlling solution errors in this weaker norm
leads to substantial simplifications and to a reduction of the actual numerical
work required for a certain error tolerance.Comment: 26 pages, 7 figure
Left-invariant evolutions of wavelet transforms on the Similitude Group
Enhancement of multiple-scale elongated structures in noisy image data is
relevant for many biomedical applications but commonly used PDE-based
enhancement techniques often fail at crossings in an image. To get an overview
of how an image is composed of local multiple-scale elongated structures we
construct a multiple scale orientation score, which is a continuous wavelet
transform on the similitude group, SIM(2). Our unitary transform maps the space
of images onto a reproducing kernel space defined on SIM(2), allowing us to
robustly relate Euclidean (and scaling) invariant operators on images to
left-invariant operators on the corresponding continuous wavelet transform.
Rather than often used wavelet (soft-)thresholding techniques, we employ the
group structure in the wavelet domain to arrive at left-invariant evolutions
and flows (diffusion), for contextual crossing preserving enhancement of
multiple scale elongated structures in noisy images. We present experiments
that display benefits of our work compared to recent PDE techniques acting
directly on the images and to our previous work on left-invariant diffusions on
orientation scores defined on Euclidean motion group.Comment: 40 page
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