830 research outputs found
Shift Unitary Transform for Constructing Two-Dimensional Wavelet Filters
Due to the difficulty for constructing two-dimensional wavelet filters, the commonly used wavelet filters are tensor-product of one-dimensional wavelet filters. In some applications, more perfect reconstruction filters should be provided. In this paper, we introduce a transformation which is referred to as Shift Unitary Transform (SUT) of Conjugate Quadrature Filter (CQF). In terms of this transformation, we propose a parametrization method for constructing two-dimensional orthogonal wavelet filters. It is proved that tensor-product wavelet filters are only special cases of this parametrization method. To show this, we introduce the SUT of one-dimensional CQF and present a complete parametrization of one-dimensional wavelet system. As a result, more ways are provided to randomly generate two-dimensional perfect reconstruction filters
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
Rigorous free fermion entanglement renormalization from wavelet theory
We construct entanglement renormalization schemes which provably approximate
the ground states of non-interacting fermion nearest-neighbor hopping
Hamiltonians on the one-dimensional discrete line and the two-dimensional
square lattice. These schemes give hierarchical quantum circuits which build up
the states from unentangled degrees of freedom. The circuits are based on pairs
of discrete wavelet transforms which are approximately related by a
"half-shift": translation by half a unit cell. The presence of the Fermi
surface in the two-dimensional model requires a special kind of circuit
architecture to properly capture the entanglement in the ground state. We show
how the error in the approximation can be controlled without ever performing a
variational optimization.Comment: 15 pages, 10 figures, one theore
Compactly supported wavelets and representations of the Cuntz relations, II
We show that compactly supported wavelets in L^2(R) of scale N may be
effectively parameterized with a finite set of spin vectors in C^N, and
conversely that every set of spin vectors corresponds to a wavelet. The
characterization is given in terms of irreducible representations of
orthogonality relations defined from multiresolution wavelet filters.Comment: 10 or 11 pages, SPIE Technical Conference, Wavelet Applications in
Signal and Image Processing VII
Projections and Dyadic Parseval Frame MRA Wavelets
A classical theorem attributed to Naimark states that, given a Parseval frame
in a Hilbert space , one can embed in
a larger Hilbert space so that the image of is the
projection of an orthonormal basis for . In the present work, we
revisit the notion of Parseval frame MRA wavelets from two papers of
Paluszy\'nski, \v{S}iki\'c, Weiss, and Xiao (PSWX) and produce an analog of
Naimark's theorem for these wavelets at the level of their scaling functions.
We aim to make this discussion as self-contained as possible and provide a
different point of view on Parseval frame MRA wavelets than that of PSWX.Comment: 19 page
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