id: 05772895
dt: b
an: 05772895
au: Lee, Kyoung-Yong
ti: Construction of Hilbert transform pairs of MRA tight frames and its
    application.
so: Dortmund: Univ. Dortmund, Fachbereich Mathematik (Diss.). 133~p. (2007).
py: 2007
pu: Dortmund: Univ. Dortmund, Fachbereich Mathematik (Diss.)
la: EN
cc: 
ut: 
ci: 
li: https://eldorado.tu-dortmund.de/handle/2003/24951
ab: Summary: Hilbert transform pairs of wavelets, biorthogonal wavelets and
    frames were found to be attractive in many applications. The Hilbert
    transform pairs are, however, hardly adoptable for applications, since
    their two-scale symbols are not trigonometric polynomials. Moreover,
    the symbols can not be implemented as FIR filters, nor rational IIR
    filters. That is the reason why approximations are constructed by many
    researchers in spite of the theoretical existence of the Hilbert
    transform pairs of wavelets, biorthogonal wavelets, and frames. But
    these conventional approaches have two drawbacks. Firstly, the wavelets
    and refinable functions do not have closed forms. Secondly, the
    symmetry, or “linear phase”, of the wavelets and refinable
    functions is an important constraint in many applications. But, the
    results, however, show that it is not easy to get symmetric Hilbert
    transform pairs of wavelets (or generators of frames).  In the first
    half of this thesis, we study the construction of Hilbert transform
    pair of multiresolution analysis (MRA) tight frames, which overcomes
    the drawbacks of the conventional constructions. Namely, our first
    research contributions are as follows: { indent4mm \item{$\bullet$} We
    show that for a given MRA tight frame $\{ψ_{j,k,\ell}\}$, the family
    $\{{\cal H}ψ_{j,k,\ell}\}$ is an MRA tight frame as well. Furthermore,
    we present a general method producing an MRA tight frame
    $\{Tψ_{j,k,\ell}\}$ from a given one, where $T$ is a linear operator
    including the Hilbert transform. \item{$\bullet$} For the sake of the
    application, we demonstrate an approximate Hilbert transform
    $\{Ψ_{j,k,\ell}\}$ such that $\widetildeΨ_j\approx {\cal H}ψ_j$ and
    $\widetildeΨ_j$ has closed form and almost symmetry. } In the second
    half of this thesis, we focus on the work of Zhao. He constructed the
    biorthogonal wavelet $\{Λψ,Λ^{-1}\widetildeψ\}$ for a given
    biorthogonal wavelet $\{ψ,\widetildeψ\}$ and applied it to the
    filtered backprojection algorithm of computed tomography. The
    $Λ$-operator is defined by $Λ=\cal{HD}$, where $\cal D$ is the
    differential operator. The $Λ$-operator appears in the inversion
    formula for the Radon transform and plays an important role in the
    filtered backprojection algorithm. His construction is based on the
    general method of generating a biorthogonal wavelet from a given one.
    The associated filters are described by IIR filters and they were
    approximated by FIR filters by truncation.  We generalize the result of
    Zhao to MRA bi-frames in association with the first two results of this
    thesis. Namely, the other main results of this thesis are as follows: {
    indent4mm \item{$\bullet$} We show that it is possible to construct the
    MRA bi-frame $(\{Λψ_{j,k,\ell}\},\{Λ^{-1}ψ_{j,k,\ell}\})$ for a
    given MRA tight frame $\{ψ_{j,k,\ell}\}$. In addition, we present a
    general method generating an MRA bi-frame
    $(\{Tψ_{j,k,\ell}\},\{T^{-1}\widetildeψ_{j,k,\ell}\})$ from a given
    $(\{ψ_{j,k,\ell}\},\{\widetildeψ_{j,k,\ell}\})$ where the linear
    operator $T$ (possibly unbounded) includes the Hilbert transform,
    differentiation/integration, and the $Λ$-operator. \item{$\bullet$}
    Using the second result of this thesis, we present an approximation of
    $(\{Λψ_{j,k,\ell}\},\{Λ^{-1}ψ_{j,k,\ell}\})$. } In addition to the
    approximation, we propose an approximation of the Ram-Lak filter. We
    expect that this result can be employed in the filtered backprojection
    algorithm of computed tomography.
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