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Title: Singular integrals, scale-space and wavelet transforms
Authors: Goh, S.S. 
Goodman, T.N.T.
Lee, S.L. 
Keywords: B-spline scale-space
Gaussian scale-space and wavelet transforms
Singular integral operators
Wavelet transforms
Wavelets and framelets
Issue Date: Dec-2013
Citation: Goh, S.S., Goodman, T.N.T., Lee, S.L. (2013-12). Singular integrals, scale-space and wavelet transforms. Journal of Approximation Theory 176 : 68-93. ScholarBank@NUS Repository.
Abstract: The Gaussian scale-space is a singular integral convolution operator with scaled Gaussian kernel. For a large class of singular integral convolution operators with differentiable kernels, a general method for constructing mother wavelets for continuous wavelet transforms is developed, and Calderón type inversion formulas, in both integral and semi-discrete forms, are derived for functions in Lp spaces. In the case of the Gaussian scale-space, the semi-discrete inversion formula can further be expressed as a sum of wavelet transforms with the even order derivatives of the Gaussian as mother wavelets. Similar results are obtained for B-spline scale-space, in which the high frequency component of a function between two consecutive dyadic scales can be represented as a finite linear combination of wavelet transforms with the derivatives of the B-spline or the spline framelets of Ron and Shen as mother wavelets. © 2013 Elsevier Inc.
Source Title: Journal of Approximation Theory
ISSN: 00219045
DOI: 10.1016/j.jat.2013.09.007
Appears in Collections:Staff Publications

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