Fractional Brownian vector fields

Multiscale Modeling and Simulation

Volumn 8, Issue 5, 2010, Pages 1645-1670

  Tafti, Pouya Dehghani ^a   Unser, Michael ^a  

^a EPFL   (Switzerland)

Author keywords

Fractional Brownian motion; Gel'fand Vilenkin stochastic analysis; Generalized random processes; Helmholtz decomposition; Invariance; Random vector fields; Self similarity

Indexed keywords

FRACTIONAL BROWNIAN MOTION; GENERALIZED RANDOM PROCESSES; HELMHOLTZ DECOMPOSITION; RANDOM VECTORS; SELF-SIMILARITIES;

BROWNIAN MOVEMENT; GELS; HELMHOLTZ EQUATION; INVARIANCE; STOCHASTIC SYSTEMS; WHITE NOISE;

VECTORS;

EID: 79251515004     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/090752638     Document Type: Article

References (64)

1. M. ARIGOVINDAN, Variational Reconstruction of Vector and Scalar Images from Nonuniform Samples, Ph. D. thesis, EPFL, Biomedical Imaging Group, Lausanne, Switzerland, http://bigwww.epfl.ch/publications/arigovindan0502.html (2005).
2. M. AVELLANEDA AND A. J. MAJDA, Simple examples with features of renormalization for turbulent transport, Philos. Trans. Roy. Soc. London Ser. A, 346 (1994), pp. 205-233.
3. A. BENASSI, S. JAFFARD, AND D. ROUX, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana, 13 (1997), pp. 19-90.
4. T. BLU AND M. UNSER, Self-similarity: Part II-Optimal estimation of fractal processes, IEEE Trans. Signal Process., 55 (2007), pp. 1364-1378.
5. G. BOURDAUD, Localisation et multiplicateurs des espaces de Sobolev homogenes, Manuscripta Math., 60 (1988), pp. 93-130.
6. B. CABRAL AND L. LEEDOM, Imaging vector fields using line integral convolution, in SIG-GRAPH'93: Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, ACM, New York, 1993, pp. 263-270.
7. R. CARMONA, Renormalization theories for incompressible flows with Gaussian statistics, in Monte Carlo Simulations in Oceanography: Proceedings of the 9th 'Aha Huliko'a Hawaiian Winter Workshop, University of Hawaii at Manoa, Honolulu, HI, 1997, pp. 81-89.
8. A. CRAYA, Contribution à l'analyse de la turbulence associée à des vitesses moyennes, Publ. Sci. Tech. Ministère de l'Air (Paris) 345, SDIT, Paris, 1958.
9. E. DERIAZ, M. FARGE, AND K. SCHNEIDER, Craya decomposition using compactly supported biorthogonal wavelets, Appl. Comput. Harmon. Anal., 28 (2010), pp. 267-284.
10. E. DERIAZ AND V. PERRIER, Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets, Appl. Comput. Harmon. Anal., 26 (2009), pp. 249-269.
11. R. L. DOBRUSHIN, Gaussian and their subordinated self-similar random generalized fields, Ann. Probab., 7 (1979), pp. 1-28.
12. F. W. ELLIOTT, JR., AND A. J. MAJDA, A wavelet Monte Carlo method for turbulent diffusion with many spatial scales, J. Comput. Phys., 113 (1994), pp. 82-111.
13. M. FARGE, Wavelet transforms and their applications to turbulence, in Annual Review of Fluid Mechanics 24, Annual Reviews, Palo Alto, CA, 1992, pp. 395-457.
14. M. FARGE, N. KEVLAHAN, V. PERRIER, AND É. GOIRAND, Wavelets and turbulence, Proc. IEEE, 84 (1996), pp. 639-669.
15. D. J. FIELD, Scale-invariance and self-similar "wavelet" transforms: An analysis of natural scenes and mammalian visual systems, in Wavelets, Fractals, and Fourier Transforms, M. Farge, J. C. R. Hunt, and J. C. Vassilicos, eds., Clarendon Press, Oxford, UK, 1993.
16. P. FLANDRIN, Wavelet analysis and synthesis of fractional Brownian motion, IEEE Trans. Inform. Theory, 38 (1992), pp. 910-916.
17. I. M. GEL'FAND AND G. E. SHILOV, Generalized Functions. Volume I: Properties and Operations, Academic Press, New York, 1964.
18. I. M. GEL'FAND AND N. YA. VILENKIN, Generalized Functions. Volume IV: Applications of Harmonic Analysis, Academic Press, New York, 1964.
19. T. HIDA, Harmonic analysis on the space of generalized functions, Theory Probab. Appl., 15 (1970), pp. 117-122.
20. L. HÖRMANDER, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Grundlehren Math. Wiss. 256, Springer-Verlag, Berlin, 1983.
21. H. E. HURST, Long-term storage capacity of reservoirs, Trans. Amer. Soc. Civil Engineers, 116 (1951), pp. 770-808.
22. J.-P. KAHANE, Some Random Series of Functions, 2nd ed., Cambridge University Press, Cambridge, UK, 1985.
23. M. YA. KELBERT, N. N. LEONENKO, AND M. D. RUIZ-MEDINA, Fractional random fields associated with stochastic fractional heat equations, Adv. in Appl. Probab., 37 (2005), pp. 108-133.
24. M. S. KESHNER, 1/f noise, Proc. IEEE, 70 (1982), pp. 212-218.
25. V. KLYATSKIN, W. WOYCZYNSKI, AND D. GURARIE, Diffusing passive tracers in random incompressible flows: Statistical topography aspects, J. Statist. Phys., 84 (1996), pp. 797-836.
26. A. N. KOLMOGOROV, A note on the papers of R. A. Minlos and V. Sazonov, Theory Probab. Appl., 4 (1959), pp. 221-223.
27. W. E. LELAND, W. WILLINGER, M. S. TAQQU, AND D. V. WILSON, On the self-similar nature of ethernet traffic, SIGCOMM Comput. Commun. Rev., 25 (1995), pp. 202-213.
28. N. LEONENKO, Limit Theorems for Random Fields with Singular Spectrum, Math. Appl. 465, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
29. P. LÉVY, Le mouvement brownien, Mémor. Sci. Math. 126, Gauthier-Villars, Paris, 1954.
30. P. LÉVY, Processus stochastiques et mouvement brownien, 2ème ed., Gauthier-Villars Paris, 1965; reprinted, Éditions Jacques Gabay, Sceaux, France, 1992.
31. S. MALLAT, A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell., 11 (1989), pp. 674-693.
32. S. MALLAT, A Wavelet Tour of Signal Processing, 2nd ed., Academic Press, San Diego, CA, 1999.
33. B. B. MANDELBROT, Gaussian self-affinity and fractals, in Globality, The Earth, 1/f Noise, and R/S, Springer-Verlag, New York, 2002.
34. E. MASRY, The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion, IEEE Trans. Inform. Theory, 39 (1993), pp. 260-264.
35. Y. MEYER, F. SELLAN, AND M. TAQQU, Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion, J. Fourier Anal. Appl., 5 (1999), pp. 465-494.
36. R. A. MINLOS, Generalized random processes and their extension to a measure, in Selected Transl. Math. Statist. and Prob., Vol. 3, AMS, Providence, RI, 1963, pp. 291-313.
37. A. S. MONIN AND A. M. YAGLOM, Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. II, MIT Press, Cambridge, MA, 1975;
38. reprinted, Dover, Mineola, NY, 2007.
39. S. A. ORSZAG, Numerical methods for the simulation of turbulence, Phys. Fluids, 12 (1969), Suppl. II, pp. 250-257.
40. A. P. PENTLAND, Fractal-based description of natural scenes, IEEE Trans. Pattern Anal. Mach. Intell., 6 (1984), pp. 661-674.
41. B. PESQUET-POPESCU AND J. LÉVY-VÉHEL, Stochastic fractal models for image processing, IEEE Signal Process. Mag., 19 (2002), pp. 48-62.
42. A. PIETSCH, Nuclear Locally Convex Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 66, Springer-Verlag, New York, 1972.
43. M. S. PINSKER, The theory of curves in Hilbert space with stationary nth increments, Izv. Akad. Nauk SSSR. Ser. Mat., 19 (1955), pp. 319-344 (in Russian).
44. YU. V. PROHOROV, The method of characteristic functionals, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, University of California Press, Berkeley, CA, 1961, pp. 403-419; also available online from http://projecteuclid.org/euclid.bsmsp/1200512613.
45. W. RUDIN, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976.
46. M. D. RUIZ-MEDINA, V. V. ANH, AND J. M. ANGULO, Fractional generalized random fields of variable order, Stochastic Anal. Appl., 22 (2004), pp. 775-799.
47. P. SAGAUT AND C. CAMBON, Homogeneous Turbulence Dynamics, Cambridge University Press, Cambridge, UK, 2008.
48. G. SAMORODNITSKY AND M. S. TAQQU, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, 1994.
49. N. SARSHAR AND X. WU, On rate-distortion models for natural images and wavelet coding performance, IEEE Trans. Image Process., 16 (2007), pp. 1383-1394.
50. P. D. TAFTI, R. DELGADO-GONZALO, A. F. STALDER, AND M. UNSER, Fractal modelling and analysis of flow-field images, in Proceedings of the Seventh IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI 2010), Rotterdam, The Netherlands, 2010, pp. 49-52.
51. P. D. TAFTI AND M. UNSER, Self-similar random vector fields and their wavelet analysis, in Proceedings of the SPIE Conference on Mathematical Imaging: Wavelets XIII, M. Papadakis, V. K. Goyal, and D. Van De Ville, eds., Vol. 7446, San Diego, CA, 2009, pp. 74460Y-1-74460Y-8.
52. P. D. TAFTI, D. VAN DE VILLE, AND M. UNSER, Invariances, Laplacian-like wavelet bases, and the whitening of fractal processes, IEEE Trans. Image Process., 18 (2009), pp. 689-702.
53. M. TALAGRAND, Pettis integral and measure theory, Mem. Amer. Math. Soc., 51 (1984), no. 307.
54. M. S. TAQQU, Fractional Brownian motion and long-range dependence, in Theory and Applications of Long-Range Dependence, P. Doukhan, G. Oppenheim, and M. S. Taqqu, eds., Birkhäuser Boston, Boston, MA, 2003, pp. 5-38.
55. M. S. TAQQU, V. TEVEROVSKY, AND W. WILLINGER, Is network traffic self-similar or multifractal?, Fractals, 5 (1997), pp. 63-73. (Pubitemid 127526956)
56. M. UNSER AND T. BLU, Fractional splines and wavelets, SIAM Rev., 42 (2000), pp. 43-67.
57. M. UNSER AND T. BLU, Self-similarity: Part I-Splines and operators, IEEE Trans. Signal Process., 55 (2007), pp. 1352-1363.
58. M. UNSER AND P. D. TAFTI, Stochastic Models for Sparse and Piecewise-Smooth Signals, manuscript, in review.
59. N. N. VAKHANIA, Probability Distributions on Linear Spaces, North-Holland Ser. Probab. Appl. Math., North-Holland, New York, 1981.
60. B. VEDEL, Confinement of the infrared divergence for the Mumford process, Appl. Comput. Harmon. Anal., 21 (2006), pp. 305-323.
61. D. VEITCH AND P. ABRY, A wavelet-based joint estimator of the parameters of long-range dependence, IEEE Trans. Inform. Theory, 45 (1999), pp. 878-897.
62. J. A. VIECELLI AND E. H. CANFIELD, Functional representation of power-law random fields and time series, J. Comput. Phys., 95 (1991), pp. 29-39.
63. A. M. YAGLOM, Correlation Theory of Stationary and Related Random Functions. Volume I. Basic Results, Springer Ser. Statist., Springer-Verlag, New York, 1987.
64. A. M. YAGLOM AND M. S. PINSKER, Random processes with stationary increments of the nth order, Doklady Akad. Nauk SSSR (N. S.), 90 (1953), pp. 731-734.