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A variational method for designing adaptive bandlimited wavelets

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Abstract

This paper proposes a new method of estimating an orthogonal bounded spectrum wavelet from a given signal. The method is based on minimizing the distance of the wavelet and the given signal in the sense of the metric of the L 2 space. In this method, the amplitude and phase of the mother wavelet are optimized simultaneously. The Lagrange multipliers technique is applied to consider the constraints due to the bounded spectrum wavelet and orthogonality conditions. The variational method reduced the optimal matching problem to the solution of a set of functional equations for the amplitude and phase of the optimal-matched bandlimited wavelet spectrum. Continuous functional equations are written with respect to Fourier coefficients of phase of transfer function of the quadrate low-pass filter at the sampled frequencies, and asset of discrete algebraic equations allows us to design the wavelet directly from the signal of interest. To demonstrate the performance of the presented method in this paper, it is employed to determine the matched wavelets of some specified signals.

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Abbreviations

\({\phi }\) :

scaling function for an orthogonal multiresolution analysis

\({\hat{{\phi}}(\omega)}\) :

spectrum of the scaling function

H(ω):

function of low-pass filter

G(ω):

functions of high-pass filter

\({\psi }\) :

mother wavelet for an orthogonal multiresolution analysis

\({\hat{{\psi}}(\omega)}\) :

spectrum of the mother wavelet

\({\omega }\) :

frequency

\({\theta_\psi (\omega)}\) :

phase of mother wavelet spectrum

\({\theta_\phi (\omega)}\) :

phase of scaling function spectrum

λ i (ω):

Lagrange multipliers

β:

Lagrange multiplier

a :

scaling factor

q 1,q 2 :

real auxiliary variables

δ f J :

variational derivatives of J with respect to arbitrary value f

θ H (ω):

phase of H(ω)

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Authors and Affiliations

  1. Sharif University of Technology, Tehran, Iran

    A. R. Bahrampour & S. Mohammad Ali Mirzaee

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  1. A. R. Bahrampour
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  2. S. Mohammad Ali Mirzaee
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Correspondence to S. Mohammad Ali Mirzaee.

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Bahrampour, A.R., Mohammad Ali Mirzaee, S. A variational method for designing adaptive bandlimited wavelets. SIViP 3, 363–374 (2009). https://doi.org/10.1007/s11760-008-0077-0

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  • DOI: https://doi.org/10.1007/s11760-008-0077-0

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