628 research outputs found
Nonhomogeneous Wavelet Systems in High Dimensions
It is of interest to study a wavelet system with a minimum number of
generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in
[11] that for any real-valued expansive matrix M, a homogeneous
orthonormal M-wavelet basis can be generated by a single wavelet function. On
the other hand, it has been demonstrated in [21] that nonhomogeneous wavelet
systems, though much less studied in the literature, play a fundamental role in
wavelet analysis and naturally link many aspects of wavelet analysis together.
In this paper, we are interested in nonhomogeneous wavelet systems in high
dimensions with a minimum number of generators. As we shall see in this paper,
a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system
with almost all properties preserved. We also show that a nonredundant
nonhomogeneous wavelet system is naturally connected to refinable structures
and has a fixed number of wavelet generators. Consequently, it is often
impossible for a nonhomogeneous orthonormal wavelet basis to have a single
wavelet generator. However, for redundant nonhomogeneous wavelet systems, we
show that for any real-valued expansive matrix M, we can always
construct a nonhomogeneous smooth tight M-wavelet frame in with a
single wavelet generator whose Fourier transform is a compactly supported
function. Moreover, such nonhomogeneous tight wavelet frames are
associated with filter banks and can be modified to achieve directionality in
high dimensions. Our analysis of nonhomogeneous wavelet systems employs a
notion of frequency-based nonhomogeneous wavelet systems in the distribution
space. Such a notion allows us to separate the perfect reconstruction property
of a wavelet system from its stability in function spaces
Exact reconstruction with directional wavelets on the sphere
A new formalism is derived for the analysis and exact reconstruction of
band-limited signals on the sphere with directional wavelets. It represents an
evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999)
and Wiaux et al. (2005). The translations of the wavelets at any point on the
sphere and their proper rotations are still defined through the continuous
three-dimensional rotations. The dilations of the wavelets are directly defined
in harmonic space through a new kernel dilation, which is a modification of an
existing harmonic dilation. A family of factorized steerable functions with
compact harmonic support which are suitable for this kernel dilation is firstly
identified. A scale discretized wavelet formalism is then derived, relying on
this dilation. The discrete nature of the analysis scales allows the exact
reconstruction of band-limited signals. A corresponding exact multi-resolution
algorithm is finally described and an implementation is tested. The formalism
is of interest notably for the denoising or the deconvolution of signals on the
sphere with a sparse expansion in wavelets. In astrophysics, it finds a
particular application for the identification of localized directional features
in the cosmic microwave background (CMB) data, such as the imprint of
topological defects, in particular cosmic strings, and for their reconstruction
after separation from the other signal components.Comment: 22 pages, 2 figures. Version 2 matches version accepted for
publication in MNRAS. Version 3 (identical to version 2) posted for code
release announcement - "Steerable scale discretised wavelets on the sphere" -
S2DW code available for download at
http://www.mrao.cam.ac.uk/~jdm57/software.htm
Convex Optimization In Identification Of Stable Non-Linear State Space Models
A new framework for nonlinear system identification is presented in terms of
optimal fitting of stable nonlinear state space equations to input/output/state
data, with a performance objective defined as a measure of robustness of the
simulation error with respect to equation errors. Basic definitions and
analytical results are presented. The utility of the method is illustrated on a
simple simulation example as well as experimental recordings from a live
neuron.Comment: 9 pages, 2 figure, elaboration of same-title paper in 49th IEEE
Conference on Decision and Contro
A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity
The richness of natural images makes the quest for optimal representations in
image processing and computer vision challenging. The latter observation has
not prevented the design of image representations, which trade off between
efficiency and complexity, while achieving accurate rendering of smooth regions
as well as reproducing faithful contours and textures. The most recent ones,
proposed in the past decade, share an hybrid heritage highlighting the
multiscale and oriented nature of edges and patterns in images. This paper
presents a panorama of the aforementioned literature on decompositions in
multiscale, multi-orientation bases or dictionaries. They typically exhibit
redundancy to improve sparsity in the transformed domain and sometimes its
invariance with respect to simple geometric deformations (translation,
rotation). Oriented multiscale dictionaries extend traditional wavelet
processing and may offer rotation invariance. Highly redundant dictionaries
require specific algorithms to simplify the search for an efficient (sparse)
representation. We also discuss the extension of multiscale geometric
decompositions to non-Euclidean domains such as the sphere or arbitrary meshed
surfaces. The etymology of panorama suggests an overview, based on a choice of
partially overlapping "pictures". We hope that this paper will contribute to
the appreciation and apprehension of a stream of current research directions in
image understanding.Comment: 65 pages, 33 figures, 303 reference
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
- …