Hybrid spline frames

Authors:
Say Song Goh, Tim N. T. Goodman and S. L. Lee

Journal:
Math. Comp. **78** (2009), 1537-1551

MSC (2000):
Primary 65D07, 41A15; Secondary 42C40, 42C30

DOI:
https://doi.org/10.1090/S0025-5718-09-02192-9

Published electronically:
January 21, 2009

MathSciNet review:
2501062

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using their unitary extension principle, Ron and Shen have constructed a normalized tight frame for consisting of spline functions with uniform knots. This paper constructs a normalized tight frame for comprising spline functions with knots on a hybrid of uniform and geometric mesh. The construction is motivated by applications in adaptive approximation using spline functions on a hybrid mesh that admits a natural dyadic multiresolution approximation of based on dilation and translation.

**1.**A. Basu, I. Cheng, Y. Pan,*Foveated Online 3D Visualization,*16th International Conference on Pattern Recognition (ICPR02), Canada, Volume 3, pp. 30 - 44.**2.**M. Bolduc and M. D. Levine,*A Real-time Foveated Sensor with Overlapping Receptive Fields,*Real-Time Imaging, 3(1977), 195 - 212 .**3.**P. J. Burt,*Smart sensing within a pyramid vision machine*, Proc. IEEE, 76 (1988), 1006 - 1015.**4.**E. C. Chang, S. Mallat and C. Yap,*Wavelet foveation*, Appl. Comput. Harmon. Anal., 9 (2000), 312 - 335. MR**1793421 (2002g:94003)****5.**O. Christensen,*An Introduction to Frames and Riesz Bases*, Birkhaüser, Boston, 2003. MR**1946982 (2003k:42001)****6.**C. K. Chui and W. He,*Compactly supported tight frames associated with refinable functions*, Appl. Comput. Harmon. Anal., 8 (2000), 293 - 319. MR**1754930 (2001h:42049)****7.**C. K. Chui, W. He and J. Stöckler,*Compactly supported tight and sibling frames with maximum vanishing moments*, Appl. Comput. Harmon. Anal., 13 (2002), 224 - 262. MR**1942743 (2004a:94011)****8.**C. K. Chui, W. He and J. Stöckler,*Nonstationary tight wavelet frames, I: bounded intervals*, Appl. Comput. Harmon. Anal., 17 (2004), 141 - 197. MR**2082157 (2005f:42082)****9.**C. K. Chui, W. He and J. Stöckler,*Nonstationary tight wavelet frames, II: unbounded intervals*, Appl. Comput. Harmon. Anal., 18 (2005), 25 - 66. MR**2110512 (2005j:42026)****10.**C. K. Chui and J. Z. Wang,*On compactly supported spline wavelets and a duality principle,*Trans. Amer. Math. Soc., 330 (1992), 903 - 915. MR**1076613 (92f:41020)****11.**I. Daubechies, B. Han, A. Ron and Z. Shen,*Framelets: MRA-based constructions of wavelet frames*, Appl. Comput. Harmon. Anal., 14 (2003), 1 - 46. MR**1971300 (2004a:42046)****12.**X. Gao, T. N. T. Goodman and S. L. Lee,*Foveated splines and wavelets*, Appl. Comput. Harmon. Anal. (to appear)**13.**K. Jetter and D.-X. Zhou,*Order of linear approximation from shift-invariant spaces,*Constr. Approx., 11 (1995), 423 - 438. MR**1367171 (96i:41024)****14.**A. Khodakovsky, P. Schröder, W. Sweldens, Progressive geometry compression, in: Proc. ACM SIGGRAPH, 2000, pp. 271-278.**15.**Y. Kuniyoshi, K. Nobuyuki, S. Rougeaux, T, Suehiro,*Active Stereo Vision System with Foveated Wide Angle Lens,*Proceedings of 2nd Asian Conference on Computer Vision, Singapore, 1995, pp. 191 - 200.**16.**M. Lounsbery, T.D. DeRose, J. Warren, Multiresolution analysis for surfaces of arbitrary topology type, ACM Trans. Graphics 16 (1997), 34-73.**17.**S. Mallat,*Foveal detection and approximation for singularities*, Appl. Comput. Harmon. Anal., 14 (2003), 133 - 180. MR**1981205 (2004c:42072)****18.**C. A. Micchelli,*Cardinal -splines*, in*Studies in Spline Functions and Approximation Theory*, Academic Press, New York, 1976, pp. 203 - 250. MR**0481767 (58:1866)****19.**A. Ron and Z. Shen,*Affine systems in : the analysis of the analysis operator*, J. Funct. Anal., 148 (1997), 408 - 447. MR**1469348 (99g:42043)****20.**I. J. Schoenberg,*Cardinal interpolation and spline functions*, J. Approx. Theory, 2 (1969), 167 - 206. MR**0257616 (41:2266)****21.**I. J. Schoenberg,*Cardinal Spline Interpolation*, CBMS-NSF Regional Conference Series in Applied Mathematics 12, SIAM, 1973. MR**0420078 (54:8095)**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65D07,
41A15,
42C40,
42C30

Retrieve articles in all journals with MSC (2000): 65D07, 41A15, 42C40, 42C30

Additional Information

**Say Song Goh**

Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

Email:
matgohss@nus.edu.sg

**Tim N. T. Goodman**

Affiliation:
Department of Mathematics, The University of Dundee, Dundee DD1 4HN, Scotland, United Kingdom

Email:
tgoodman@maths.dundee.ac.uk

**S. L. Lee**

Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

Email:
matleesl@nus.edu.sg

DOI:
https://doi.org/10.1090/S0025-5718-09-02192-9

Keywords:
Uniform splines,
geometric splines,
hybrid splines,
multiresolution,
tight frames

Received by editor(s):
September 26, 2006

Received by editor(s) in revised form:
March 15, 2008

Published electronically:
January 21, 2009

Additional Notes:
This research was partially supported by the Wavelets and Information Processing Programme of the Centre for Wavelets, Approximation and Information Processing, National University of Singapore, under a grant from DSTA

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.