Wayne Lawton

Necessary and sufficient conditions for constructing orthonormal wavelet bases

This paper proves a previous conjecture of the author characterizing sequences h∈l2(Z) that yield orthonormal wavelet bases of L2(R) in terms of the multiplicity of the eigenvalue 1 of an operator associated to h. The proof utilizes a result of Cohen characterizing these sequences in terms of the real zeros of their Fourier transforms. The mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2(Z) into L2(R). Related conjectures are discussed.

An algorithm for matrix extension and wavelet construction

This paper gives a practical method of extending an n x r matrix P(z), r ≤ n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P(z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P(z) has rank r for each z ∈ T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.

Applications of complex valued wavelet transforms to subband decomposition

A method for constructing complex valued linear phase FIR conjugate quadrature filters and associated wavelet bases is described. Each filter is derived by replating certain zeros of a real valued FIR conjugate quadrature filter by their reciprocal conjugates. The derived filters have the same frequency response magnitudes as the original filters and their linear phase property permits the use of symmetrization in subband decomposition to avoid border discontinuities that result from signal periodization. Subband decomposition and reconstruction using both a length 6 filter associated with a Daubechies (1988) wavelet bases and a related length 6 complex valued linear phase filter are compared to illustrate the reduced border effects. >

Tight frames of compactly supported affine wavelets

This paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by Daubechies [Commun. Pure Appl. Math. 41, 909 (1988)]. For each integer N≥1, a family of wavelet functions ψ having support [0,2N−1] is constructed such that {ψjk(x)=2j/2ψ(2jx−k) kj,k∈Z} is a tight frame of L2(R), i.e., for every f∈L2(R), f=c∑jk 〈ψjk‖f〉ψjk for some c>0. This family is parametrized by an algebraic subset VN of R4N. Furthermore, for N≥2, a proper algebraic subset WN of VN is specified such that all points in VN outside of WN yield orthonormal bases. The relationship between these tight frames and the theory of group representations and coherent states is discussed.

Characterization of compactly supported refinable splines

AbstractWe prove that a compactly supported spline functionφ of degree k satisfies the scaling equation

Tubes in tubes: catheter navigation in blood vessels and its applications

\phi (x) = \sum _{n = 0}^N c(n)\phi (mx - n)

Construction of conjugate quadrature filters with specified zeros

for some integerm ≥ 2, if and only if