Abstract
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.
Highlights
In her celebrated paper 1, Daubechies constructed a family of compactly supported orthonormal scaling functions and their corresponding orthonormal wavelets
Based on SUT of twodimensional CQF, a parameterized method is presented for constructing real-valued two-dimensional orthogonal wavelet filters
Suppose that {bα}α∈Z2 is a two-dimensional CQF, {dαk}α∈Z2 k 1, 2, 3 satisfy 3.6 and 3.7, {bα}α∈Z2 is a two-dimensional CQF obtained by SUT of {bα}α∈Z2, {dαk}α∈Z2, which are derived by SUT of {dαk}α∈Z2 k 1, 2, 3, satisfy 3.6 and 3.7 simultaneously
Summary
In her celebrated paper 1 , Daubechies constructed a family of compactly supported orthonormal scaling functions and their corresponding orthonormal wavelets. Lai 13 proposed a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function in the multivariate setting. A transformation that we refer to as Shift Unitary Transform SUT of Conjugate Quadrature Filter CQF is proposed In terms of this transformation, we present a parametrization method for constructing two-dimensional orthogonal wavelet filters. Based on SUT of twodimensional CQF, a parameterized method is presented for constructing real-valued two-dimensional orthogonal wavelet filters.
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