Quasi-interpolatory refinable functions and construction of biorthogonal wavelet systems

Authors:
Hong Oh Kim;Rae Young Kim;Yeon Ju Lee;Jungho Yoon
Affiliations:
Department of Mathematical Sciences, KAIST, Daejeon, South Korea 305-701;Department of Mathematics, Yeungnam University, Gyeongsangbuk-do, South Korea 712-749;Department of Mathematical Sciences, KAIST, Daejeon, South Korea 305-701;Department of Mathematics, Ewha W. University, Seoul, South Korea 120-750
Venue:
Advances in Computational Mathematics
Year:
2010

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Abstract

We present a new family of compactly supported and symmetric biorthogonal wavelet systems. Each refinement mask in this family has tension parameter 驴. When 驴驴=驴0, it becomes the minimal length biorthogonal Coifman wavelet system (Wei et al., IEEE Trans Image Proc 7:1000---1013, 1998). Choosing 驴 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. 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.