Quasi-interpolatory refinable functions and construction of biorthogonal wavelet systems
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Quasi-interpolatory refinable functions and construction of biorthogonal wavelet systems Hong Oh Kim · Rae Young Kim · Yeon Ju Lee · Jungho Yoon
Received: 14 May 2008 / Accepted: 29 April 2009 / Published online: 29 May 2009 © Springer Science + Business Media, LLC 2009
Abstract We present a new family of compactly supported and symmetric biorthogonal wavelet systems. Each refinement mask in this family has tension parameter ω. When ω = 0, it becomes the minimal length biorthogonal Coifman wavelet system (Wei et al., IEEE Trans Image Proc 7:1000–1013, 1998). Choosing ω away from zero, we can get better smoothness of the refinable functions at the expense of slightly larger support. Though the construction of the new biorthogonal wavelet systems, in fact, starts from a new class of quasi-interpolatory subdivision schemes, we find that the refinement masks accidently coincide with the ones by Cohen et al. (Comm Pure Appl Math 45:485–560, 1992, §6.C) (or Daubechies 1992, §8.3.5), which are designed for the purpose of generating biorthogonal wavelets close to orthonormal cases.
Communicated by R. Q. Jia. H. O. Kim · Y. J. Lee Department of Mathematical Sciences, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, South Korea H. O. Kim e-mail: [email protected] Y. J. Lee e-mail: [email protected] R. Y. Kim Department of Mathematics, Yeungnam University, 214-1, Dae-dong, Gyeongsan-si, Gyeongsangbuk-do, 712-749, South Korea e-mail: [email protected] J. Yoon (B) Department of Mathematics, Ewha W. University, Seoul, 120-750, South Korea e-mail: [email protected]
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However, the corresponding mathematical analysis is yet to be provided. In this study, we highlight the connection between the quasi-interpolatory subdivision schemes and the masks by Cohen, Daubechies and Feauveau, and then we study the fundamental properties of the new biorthogonal wavelet systems such as regularity, stability, linear independence and accuracy. Keywords Subdivision · Coifman wavelet · Biorthogonal wavelet · Multiresolution analysis · Quasi-interpolation · Refinable function · Regularity · Linear independence Mathematics Subject Classifications (2000) 41A15 · 41A25 · 41A30 · 42A05
1 Introduction During the last decades, the theory of wavelets and multiresolution analysis has established itself firmly as one of the most successful methods for a broad range of signal processing applications. The construction of classical wavelets is now well-understood due to pioneer works such as [5–7]. Many properties, such as symmetry (or antisymmetry), vanishing moments, regularity and short support, are required for a practical use in application areas. It has been well-known that orthogonality and symmetry are conflicting properties for the design of compactly supported wavelets [7]. In order to maintain the symmetric properties of wavelet systems, the orthogonality constraint has been relaxed to semi-orthogonality or biorthogonality. In particular, spline functions have been a good source for wavelet constructions. We select so
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