Second-generation wavelet denoising methods for irregularly spaced data in two dimensions

Signal Processing

Volumn 86, Issue 7, 2006, Pages 1435-1450

  Delouille, Véronique ^a   Jansen, Maarten ^b,c,d   von Sachs, Rainer ^e  

^a ROYAL OBSERVATORY OF BELGIUM   (Belgium)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

^c RESEARCH FOUNDATION FLANDERS FWO   (Belgium)

^d UNIVERSITY OF LEUVEN   (Belgium)

^e UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

Biorthogonal; Irregular; Lifting; Relaxation; Scattered data; Semi orthogonalization; Two dimensional; Wavelet

Indexed keywords

COMPUTER SIMULATION; IMAGE RECONSTRUCTION; LAGRANGE MULTIPLIERS; LIGHT SCATTERING; POLYNOMIALS; SECOND HARMONIC GENERATION;

BIORTHOGONAL; IRREGULAR; LIFTING; RELAXATION; SCATTERED DATA; SEMI-ORTHOGONALIZATION; WAVELET;

SIGNAL TO NOISE RATIO;

EID: 33646139179     PISSN: 01651684     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sigpro.2005.08.005     Document Type: Article

References (35)

1. Donoho D.L., and Johnstone I.M. Minimax estimation via wavelet shrinkage. Ann. Statist. 26 (1998) 879-921
2. Donoho D.L. De-noising via soft-thresholding. IEEE Trans. Inform. Theory 41 (1995) 613-627
3. Donoho D.L., and Johnstone I.M. Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 (1994) 425-455
4. Donoho D.L., and Johnstone I.M. Adapting to unknown smoothness via wavelet shrinking. J. Amer. Statist. Assoc. 90 (1995) 1200-1224
5. Cai T., and Brown L.D. Wavelet shrinkage for nonequispaced samples. Ann. Statist. 26 5 (1998) 1783-1799
6. Hall P., and Turlach B.A. Interpolation methods for nonlinear wavelet regression with irregularly spaced design. Ann. Statist. 25 5 (1997) 1912-1925
7. Kovac A., and Silverman B.W. Extending the scope of wavelet regression methods by coefficient-dependent thresholding. J. Amer. Statist. Assoc. 95 (2000) 172-183
8. Delyon B., and Juditsky A. On the computation of wavelet coefficients. J. Approx. Theory 88 (1997) 47-79
9. Antoniadis A., and Pham D.T. Wavelet regression for random or irregular design. Comput. Statist. Data Anal. 28 (1998) 353-369
10. Delouille V., Simoens J., and von Sachs R. Smooth design-adapted wavelets for nonparametric stochastic regression. J. Amer. Statist. Assoc. 99 467 (2004) 643-658
11. Vanraes E., Jansen M., and Bultheel A. Stabilized wavelet transforms for nonequispaced data smoothing. Signal Processing 82 12 (2002) 1979-1990
12. Sweldens W. The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal. 29 2 (1997) 511-546
13. Carnicer J.M., Dahmen W., and Peña J.M. Local decomposition of refinable spaces and wavelets. Appl. Comput. Harmon. Anal. 3 (1996) 127-153
14. V. Delouille, J. Simoens, R. von Sachs, Design-adapted wavelet estimator for two-dimensional tensor product irregular designs, in: M. Unser, A. Aldroubi, A. Laine (Eds.), X Proceedings of SPIE, Wavelets: Applications in Signal and Image Processing, vol. 5207, 2003, pp. 880-891.
15. Estrin D., Govindan R., Heidemann J., and Kumar S. Next century challenges: scalable coordination in sensor networks. Proceedings of the ACM/IEEE MobiCom'99 (August 1999) 263-270
16. Delouille V., Neelamani R., and Baraniuk R.G. Robust distributed estimation in sensor networks using the embedded polygons algorithm. Third International Symposium on Information Processing in Sensor Networks. IPSN (April 2004) 405-413
17. M. Jansen, G. Nason, B. Silverman, Scattered data smoothing by empirical Bayesian shrinkage of second generation wavelet coefficients, in: M. Unser, A. Aldroubi (Eds.), IX Proceedings of the SPIE, Wavelets: Applications in Signal and Image Processing, vol. 4478, 2001, pp. 87-97.
18. Simoens J., and Vandewalle S. A stabilized lifting construction of wavelets on irregular meshes on the interval. SIAM J. Sci. Comput. 24 4 (2003) 1356-1378
19. Johnstone I.M., and Silverman B.W. Needles and straw in haystacks: empirical Bayes estimates of possibly sparse sequences. Ann. Statist. 32 4 (2004) 1594-1649
20. I.M. Johnstone, B.W. Silverman, Empirical Bayes selection of wavelet thresholds, Ann. Statist. 33 (2005), to appear.
21. D. Nychka, B. Bailey, S. Ellner, P. Haarland, M. O'Connell. FUNFITS: data analysis and statistical tools for estimating functions, Technical Report, North Carolina State University, 1998.
22. Hansen M., Kooperberg C., and Sardy S. Triogram models. J. Amer. Statist. Assoc. 93 (1998) 101-119
23. Dahmen W. Some remarks on multiscale transformations, stability and orthogonality. In: Laurent P.J., Le Méhauté A., and Schumaker L.L. (Eds). Curves and Surfaces II (1991), A.K. Peters, Boston 1-32
24. Daubechies I., and Sweldens W. Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl. 4 3 (1998) 245-267
25. Sweldens W. The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3 2 (1996) 186-200
26. Barber C.B., Dobkin D.P., and Huhdanpaa H.T. The quickhull algorithm for convex hulls. ACM Trans. Math. Software 22 4 (1996) 469-483
27. J. Krämer, Delaunay triangulation in two and three dimensions, Master's Thesis, Wilhelm Schickard Institute for Computer Science Graphical-Interactive Systems, University of Tübingen, 1995.
28. I. Guskov, Multivariate subdivision scheme and divided differences, Technical Report, Princeton University, 1998.
29. Guskov I., Sweldens W., and Schröder P. Multiresolution signal processing for meshes. Computer Graphics Proceedings (SIGGRAPH 99) (1999) 325-334
30. Spanier E.H. Algebraic Topology (1996), McGraw-Hill, New York
31. V. Delouille, M. Jansen, R. von Sachs, Second generation wavelet denoising for irregularly spaced data in two dimensions, Discussion Paper 0305, Institut de Statistique, UCL, Belgium, 2003.
32. W. Sweldens, P. Schröder, Building your own wavelets at home, in: Wavelets in Computer Graphics, ACM SIGGRAPH Course Notes, 1996, pp. 15-87.
33. Daubechies I., Guskov I., and Sweldens W. Regularity of irregular subdivision. Constr. Approx. 15 3 (1999) 381-426
34. M. Jansen, G. Nason, B. Silverman, Second generation wavelets and Empirical Bayes estimates for scattered data smoothing, Technical Report, Department of Mathematics, University of Bristol, Bristol, UK, 2001.
35. Jansen M., Malfait M., and Bultheel A. Generalized cross validation for wavelet thresholding. Signal Processing 56 1 (January 1997) 33-44