1,233 research outputs found
Coxeter Groups and Wavelet Sets
A traditional wavelet is a special case of a vector in a separable Hilbert
space that generates a basis under the action of a system of unitary operators
defined in terms of translation and dilation operations. A
Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on
foldable figures, which tesselate the embedding space by reflections in their
bounding hyperplanes instead of by translations along a lattice. Although both
theories look different at their onset, there exist connections and
communalities which are exhibited in this semi-expository paper. In particular,
there is a natural notion of a dilation-reflection wavelet set. We prove that
dilation-reflection wavelet sets exist for arbitrary expansive matrix
dilations, paralleling the traditional dilation-translation wavelet theory.
There are certain measurable sets which can serve simultaneously as
dilation-translation wavelet sets and dilation-reflection wavelet sets,
although the orthonormal structures generated in the two theories are
considerably different
Nonlinear Approximation Using Gaussian Kernels
It is well-known that non-linear approximation has an advantage over linear
schemes in the sense that it provides comparable approximation rates to those
of the linear schemes, but to a larger class of approximands. This was
established for spline approximations and for wavelet approximations, and more
recently by DeVore and Ron for homogeneous radial basis function (surface
spline) approximations. However, no such results are known for the Gaussian
function, the preferred kernel in machine learning and several engineering
problems. We introduce and analyze in this paper a new algorithm for
approximating functions using translates of Gaussian functions with varying
tension parameters. At heart it employs the strategy for nonlinear
approximation of DeVore and Ron, but it selects kernels by a method that is not
straightforward. The crux of the difficulty lies in the necessity to vary the
tension parameter in the Gaussian function spatially according to local
information about the approximand: error analysis of Gaussian approximation
schemes with varying tension are, by and large, an elusive target for
approximators. We show that our algorithm is suitably optimal in the sense that
it provides approximation rates similar to other established nonlinear
methodologies like spline and wavelet approximations. As expected and desired,
the approximation rates can be as high as needed and are essentially saturated
only by the smoothness of the approximand.Comment: 15 Pages; corrected typos; to appear in J. Funct. Ana
Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type
Coorbit space theory is an abstract approach to function spaces and their
atomic decompositions. The original theory developed by Feichtinger and
Gr{\"o}chenig in the late 1980ies heavily uses integrable representations of
locally compact groups. Their theory covers, in particular, homogeneous
Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the
recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces
cannot be covered by their group theoretical approach. Later it was recognized
by Fornasier and the first named author that one may replace coherent states
related to the group representation by more general abstract continuous frames.
In the first part of the present paper we significantly extend this abstract
generalized coorbit space theory to treat a wider variety of coorbit spaces. A
unified approach towards atomic decompositions and Banach frames with new
results for general coorbit spaces is presented. In the second part we apply
the abstract setting to a specific framework and study coorbits of what we call
Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel
spaces of various types of interest as coorbits. We obtain several old and new
wavelet characterizations based on precise smoothness, decay, and vanishing
moment assumptions of the respective wavelet. As main examples we obtain
results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal
spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed
smoothness, and even mixtures of the mentioned ones. Due to the generality of
our approach, there are many more examples of interest where the abstract
coorbit space theory is applicable.Comment: to appear in Journal of Functional Analysi
Nested quasicrystalline discretisations of the line
One-dimensional cut-and-project point sets obtained from the square lattice
in the plane are considered from a unifying point of view and in the
perspective of aperiodic wavelet constructions. We successively examine their
geometrical aspects, combinatorial properties from the point of view of the
theory of languages, and self-similarity with algebraic scaling factor
. We explain the relation of the cut-and-project sets to non-standard
numeration systems based on . We finally examine the substitutivity, a
weakened version of substitution invariance, which provides us with an
algorithm for symbolic generation of cut-and-project sequences
An Algebraic Perspective on Multivariate Tight Wavelet Frames. II
Continuing our recent work we study polynomial masks of multivariate tight
wavelet frames from two additional and complementary points of view: convexity
and system theory. We consider such polynomial masks that are derived by means
of the unitary extension principle from a single polynomial. We show that the
set of such polynomials is convex and reveal its extremal points as polynomials
that satisfy the quadrature mirror filter condition. Multiplicative structure
of such polynomial sets allows us to improve the known upper bounds on the
number of frame generators derived from box splines. In the univariate and
bivariate settings, the polynomial masks of a tight wavelet frame can be
interpreted as the transfer function of a conservative multivariate linear
system. Recent advances in system theory enable us to develop a more effective
method for tight frame constructions. Employing an example by S. W. Drury, we
show that for dimension greater than 2 such transfer function representations
of the corresponding polynomial masks do not always exist. However, for wavelet
masks derived from multivariate polynomials with non-negative coefficients, we
determine explicit transfer function representations. We illustrate our results
with several examples
Standardised convolutional filtering for radiomics
The Image Biomarker Standardisation Initiative (IBSI) aims to improve
reproducibility of radiomics studies by standardising the computational process
of extracting image biomarkers (features) from images. We have previously
established reference values for 169 commonly used features, created a standard
radiomics image processing scheme, and developed reporting guidelines for
radiomic studies. However, several aspects are not standardised.
Here we present a preliminary version of a reference manual on the use of
convolutional image filters in radiomics. Filters, such as wavelets or
Laplacian of Gaussian filters, play an important part in emphasising specific
image characteristics such as edges and blobs. Features derived from filter
response maps have been found to be poorly reproducible. This reference manual
forms the basis of ongoing work on standardising convolutional filters in
radiomics, and will be updated as this work progresses.Comment: 62 pages. For additional information see https://theibsi.github.io
Iterated function systems and permutation representations of the Cuntz algebra
We study a class of representations of the Cuntz algebras O_N, N=2,3,...,
acting on L^2(T) where T=R/2\pi Z. The representations arise in wavelet theory,
but are of independent interest. We find and describe the decomposition into
irreducibles, and show how the O_N-irreducibles decompose when restricted to
the subalgebra UHF_N\subset O_N of gauge-invariant elements; and we show that
the whole structure is accounted for by arithmetic and combinatorial properties
of the integers Z. We have general results on a class of representations of O_N
on Hilbert space H such that the generators S_i as operators permute the
elements in some orthonormal basis for H. We then use this to extend our
results from L^2(T) to L^2(T^d), d>1 ; even to L^2(\mathbf{T}) where \mathbf{T}
is some fractal version of the torus which carries more of the algebraic
information encoded in our representations.Comment: 84 pages, 11 figures, AMS-LaTeX v1.2b, full-resolution figures
available at ftp://ftp.math.uiowa.edu/pub/jorgen/PermRepCuntzAlg in eps files
with the same names as the low-resolution figures included her
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