Reversing subdivision rules: Local linear conditions and observations on inner products

Journal of Computational and Applied Mathematics

Volumn 119, Issue 1-2, 2000, Pages 29-67

  Bartels, Richard H ^a   Samavati, Faramarz F ^b  

^a UNIVERSITY OF WATERLOO   (Canada)

^b SHAHID BEHESHTI UNIVERSITY   (Iran)

Author keywords

Curves; Least squares; Multiresolution; Subdivision; Surfaces; Wavelets

Indexed keywords

COMPUTATIONAL GEOMETRY; INTERPOLATION; LEAST SQUARES APPROXIMATIONS; MATRIX ALGEBRA;

SUBDIVISION RULES;

DIGITAL ARITHMETIC;

EID: 0033707642     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00370-8     Document Type: Article

References (39)

1. Aldroubi A., Abry P., Unser M. Construction of biorthogonal wavelets starting from any two multiresolutions. IEEE Trans. Signal Process. 46:1998;1130-1133.
2. 2 SIAM J. Math. Anal. 25:1994;1412-1432.
3. Carnicer J.M., Dahmen W., Peña J.M. Local decomposition of refinable spaces and wavelets. Appl. Comput. Harm. Anal. 3:1996;127-153.
4. A. Certain, J. Popović, T. DeRose, T. Duchamp, D. Salesin, W. Stuetzle, Interactive multiresolution surface viewing, in: H. Rushmeier (Ed.), SIGGRAPH '96 Conference Proceedings, ACM SIGGRAPH and Addison-Wesley, Reading, MA, 1996, pp. 91-98.
5. Chaikin G. An algorithm for high speed curve generation. Comput. Graphics Image Process. 3:1974;346-349.
6. Chui C.K. An Introduction to Wavelets, Wavelet Analysis and its Applications, Vol. 1. 1992;Academic Press, New York.
7. Cohen A. Biorthogonal wavelets. Chui C.K. Wavelets: A Tutorial in Theory and Applications. 1992;123-152 Academic Press, New York.
8. Cohen A. Wavelet bases, approximation theory and subdivision schemes. Cheney E.W., Chui C.K., Schumaker L.L. Approximation Theory VII. 1993;63-85 Academic Press, Boston, MA.
9. Cohen A., Daubechies I. On the instability of arbitrary biorthogonal wavelet packets. SIAM J. Math. Anal. 24:1993;1340-1354.
10. Cohen A., Daubechies I., Feauveau J. Bi-orthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45:1992;485-560.
11. Dahmen W., Micchelli C.A. Banded matrices with banded inverses II: locally finite decompositions of spline spaces. Constr. Approx. 9:1993;263-281.
12. Dahmen W., Micchelli C.A. Biorthogonal wavelet expansions. Constr. Approx. 13:1997;293-328.
13. Daubechies I. Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41:1998;909-996.
14. I. Daubechies, W. Sweldens, Factoring wavelet transforms into lifting steps, Technical Report, Bell Laboratories, Lucent Technologies, 1996.
15. DeVore R.A., Jawerth B., Lucier B.J. Surface compression. Comput. Aided Geom. Design. 9:1992;219-239.
16. R.A. DeVore, B.J. Lucier, Wavelets, Acta Numerica, Cambridge University Press, Cambridge, 1992, pp. 1-56.
17. Dyn N., Levin D., Gregory J. A 4-point interpolatory subdivision scheme for curve design. Comput. Aided Geom. Design. 4:1987;257-268.
18. Dyn N., Levin D., Gregory J.A. A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Graphics. 9:1990;160-169.
19. M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, W. Stuetzle, Multiresolution analysis of arbitrary meshes, in: R. Cook (Ed.), SIGGRAPH'95 Conference Proceedings ACM SIGGRAPH and Addison-Wesley, Reading, MA, 1995, pp. 173-182.
20. Faber G. Über stetige functionen. Math. Ann. 66:1909;81-94.
21. Finkelstein A., Salesin D.H. Multiresolution curves. Glassner A. Proceedings of SIGGRAPH'94. 1994;261-268 ACM Press, New York.
22. D.R. Forsey, R.H. Bartels, Hierarchical B-spline refinement, in: J. Dill (Ed.), Proceedings of SIGGRAPH '88, (Comput. Graphics 22 (1988) ) pp. 205-212.
23. Goldman R. Illicit expressions in vector algebra. ACM Trans. Graphics. 4:1985;223-243.
24. C. Herley, M. Vetterli, Biorthogonal bases of symmetric compactly supported wavelets, Wavelets, fractals, and Fourier transforms, Institute of Mathematics Applied Conference Series, New Series Vol. 43, Oxford University Press, New York, 1993, pp. 91-108.
25. Jawerth B., Sweldens W. An overview of wavelet based multiresolution analyses. SIAM Rev. 36:1994;377-412.
26. Karoui A., Vaillancourt A. Families of biorthogonal wavelets. Comput. Math. Appl. 28:1994;25-39.
27. J. Kautsky, R. Turcajová, Discrete biorthogonal wavelet transforms as block circulant matrices, ftp :// ftp.cs.flinders.edu.au / pub / wavelets / bio1.ps, 1994.
28. L. Kobbelt, S. Campagna, J. Vorsatz, H.-P. Seidel, Interactive multi-resolution modeling on arbitrary meshes, in: M. Cohen (Ed.), SIGGRAPH '98 Conference Proceedings, ACM SIGGRAPH and Addison-Wesley, Reading, MA, 1998, pp. 105-114.
29. Lawton A., Lee S.L., Shen Z. An algorithm for matrix extension and wavelet construction. Math. Comp. 65:1996;723-737.
30. Lounsbery M., DeRose T.D., Warren J. Multiresolution analysis for surfaces of arbitrary topological type. ACM Trans. Graphics. 16:1997;34-73.
31. Reissell L.-M. Wavelet multiresolution representation of curves and surfaces. Graphical Models and Image Processing. 58:1996;198-217.
32. Samavati F.F., Bartels R.H. Multiresolution curve and surface representation: reversing subdivision rules by least-squares data fitting. Comput. Graphics Forum. 18:1998;97-119.
33. P. Schröder, W. Sweldens, Spherical wavelets: efficiently representing functions on the sphere, in: R. Cook (Ed.), SIGGRAPH '95 Conference Proceedings, ACM SIGGRAPH and Addison-Wesley, Reading, MA, 1995, pp. 161-172.
34. Stollnitz E.J., DeRose T.D., Salesin D.H. Wavelets for Computer Graphics. 1996;Morgan Kaufmann Publishers, San Francisco.
35. W. Sweldens, Compactly supported wavelets which are biorthogonal with respect to a weighted inner product, Proceedings of the 14th IMACS World Congress and also available at http :// cm.bell-labs.com / who / wim / papers / papers.html#weight.
36. Sweldens W. The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal. 29:1998;511-546.
37. W. Sweldens, P. Schröder, Building your own wavelets at home, in: Wavelets in Computer Graphics, ACM SIGGRAPH Course Notes, 1996.
38. Warren J. Sparse filter banks for binary subdivision schemes. Goodman T., Martin R. The Mathematics of Surfaces VII. 1996;427-438 IMA Press, London.
39. D. Zorin, P. Schröder, W. Sweldens, Interpolating subdivision for meshes with arbitrary topology, in: H. Rushmeier (Ed.), SIGGRAPH '96 Conference Proceedings, ACM SIGGRAPH and Addison-Wesley, Reading, MA, 1996, pp. 189-192.