TY - JOUR
T1 - Quasi-interpolatory refinable functions and construction of biorthogonal wavelet systems
AU - Kim, Hong Oh
AU - Kim, Rae Young
AU - Lee, Yeon Ju
AU - Yoon, Jungho
N1 - Funding Information:
Acknowledgements We are grateful to the anonymous referees for their valuable corrections and suggestions. R. Kim was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-331-C00014). H. Kim and J. Yoon were supported by the grant R01-2006-000-10424-0 from Korea Science and Engineering Foundation in Ministry of Science & Technology.
PY - 2010
Y1 - 2010
N2 - 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. 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.
AB - 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. 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.
KW - Biorthogonal wavelet
KW - Coifman wavelet
KW - Linear independence
KW - Multiresolution analysis
KW - Quasi-interpolation
KW - Refinable function
KW - Regularity
KW - Subdivision
UR - http://www.scopus.com/inward/record.url?scp=77956429185&partnerID=8YFLogxK
U2 - 10.1007/s10444-009-9129-4
DO - 10.1007/s10444-009-9129-4
M3 - Article
AN - SCOPUS:77956429185
VL - 33
SP - 255
EP - 283
JO - Advances in Computational Mathematics
JF - Advances in Computational Mathematics
SN - 1019-7168
IS - 3
ER -