Adaptive multivariate approximation using binary space partitions and geometric wavelets

SIAM Journal on Numerical Analysis

Volumn 43, Issue 2, 2005, Pages 707-732

  Dekel, S ^a   Leviatan, D ^b  

^a RealTimeImage   (Israel)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

Adaptive multivariate approximation; Binary space partitions; Geometric wavelets; Nonlinear approximation; Piecewise polynomial approximation

Indexed keywords

ADAPTIVE SYSTEMS; COMPUTATIONAL GEOMETRY; FUNCTION EVALUATION; MATHEMATICAL MODELS; WAVELET TRANSFORMS;

ADAPTIVE MULTIVARIATE APPROXIMATION; BINARY SPACE PARTITIONS; GEOMETRIC WAVELETS; NONLINEAR APPROXIMATION; PIECEWISE POLYNOMIAL APPROXIMATION;

APPROXIMATION THEORY;

EID: 20144369360     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/040604649     Document Type: Article

References (26)

1. p, p < 1, Proc. Amer. Math. Soc., 120 (1994), pp. 97-100.
2. E. CANDÈS AND D. DONOHO, New Tight Frames of Curvelets and Optimal Representations of Objects with Smooth Singularities, Technical report, Stanford, CA, 2002.
3. I. DAUBECHIES, Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Ser. in Appl. Math. 61, SIAM, Philadelphia, 1992.
4. S. DEKEL AND D. LEVIATAN, The Bramble - Hilbert lemma for convex domains, SIAM J. Math. Anal., 35 (2004), pp. 1203-1212.
5. S. DEKEL AND D. LEVIATAN, Whitney estimates for convex domains with applications to multivariate piecewise polynomial approximation, Found. Comput. Math., 4 (2004), pp. 345-368.
6. S. DEKEL, D. LEVIATAN, AND M. SHARIR, On bivariate smoothness spaces associated with nonlinear approximation, Constr. Approx., 20 (2004), pp. 625-646.
7. R. DEVORE, Nonlinear approximation, Acta Numer., 7 (1998), pp. 51-150.
8. R. DEVORE, B. JAWERTH, AND B. LUCIER, Image compression through wavelet transform coding, IEEE Trans. Inform. Theory, 38 (1992), pp. 719-746.
9. R. DEVORE AND G. LORENTZ, Constructive Approximation, Springer-Verlag, New York, 1991.
10. R. DEVORE AND V. POPOV, Interpolation of Besov spaces, Trans. Amer. Math. Soc., 305 (1988), pp. 397-414.
11. D. DONOHO, CART and best-ortho-basis: A connection, Ann. Statist., 25 (1997), pp. 1870-1911.
12. J. HERSHBERGER AND S. SURI, Binary space partitions for 3D subdivisions, in Proceedings of the 14th Annual ACM-SIAM Joint Symposium on Discrete Algorithms, Baltimore, MD, 2003, SIAM, Philadelphia, 2003, pp. 100-108.
13. B. KARAIVANOV AND P. PETRUSHEV, Nonlinear piecewise polynomial approximation beyond Besov spaces, Appl. Comput. Harmon. Anal., 15 (2003), pp. 177-223.
14. B. KARAIVANOV, P. PETRUSHEV, AND R. SHARPLEY, Algorithms for nonlinear piecewise polynomial approximation: Theoretical aspects, Trans. Amer. Math. Soc., 355 (2003), pp. 2585-2631.
15. M. S. PATERSON AND F. F. YAO, Efficient binary space partitions for hidden-surface removal and solid modeling, Discrete Comput. Geom., 5 (1990), pp. 485-503.
16. P. PETRUSHEV, Multivariate n-term rational and piecewise polynomial approximation, J. Approx. Theory, 121 (2003), pp. 158-197.
17. H. RADHA, M. VETTERLI, AND R. LEONARDI, Image compression using binary space partitioning trees, IEEE Trans. Image Process., 5 (1996), pp. 1610-1624.
18. A. SAID AND W. PEARLMAN, A new fast and efficient image codec based on set partitioning in hierarchical trees, IEEE Trans. Circuits Systems Video Technol., 6 (1996), pp. 243-250.
19. P. SALEMBIER AND L. GARRIDO, Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval, IEEE Trans. Image Process., 9 (2000), pp. 561-576.
20. M. SHAPIRO, An embedded hierarchical image coder using zerotrees of wavelet coefficients, IEEE Trans. Signal Process., 41 (1993), pp. 3445-3462.
21. R. SHUKLA, P. L. DRAGOTTI, M. N. Do, AND M. VETTERLI, Rate-distortion optimized tree structured compression algorithms for piecewise polynomial images, IEEE Trans. Image Process., 14 (2005), pp. 343-359.
22. J. L. STARCK, E. CANDÈS, AND D. L. DONOHO, The curvelet transform for image denoising, IEEE Trans. Image Process., 11 (2000), pp. 670-684.
23. D. TAUBMAN, High performance scalable image compression with EBCOT, IEEE Trans. Image Process., 9 (2000), pp. 1151-1170.
24. C. TÓTH, A note on binary plane partitions, in Proceedings of the 17th ACM Symposium on Computational Geometry, ACM, New York, 2001, pp. 151-156.
25. C. TÓTH, Binary space partitions for line segments with a limited number of directions, SIAM J. Comput., 32 (2003), pp. 307-325.
26. M. WAKIN, J. ROMBERG, H. CHOI, AND R. BARANIUK, Geometric methods for wavelet-based image compression, in Wavelets: Applications in Signal and Image Processing X, M. Unser, A. Aldroubi, and A. Laine, eds., SPIE, Bellingham, WA, 2003, pp. 507-520.