Quadratic variational framework for signal design on the 2-sphere

IEEE Transactions on Signal Processing

Volumn 59, Issue 11, 2011, Pages 5243-5252

  Wei, Liying ^a   Kennedy, Rodney A ^b   Lamahewa, Tharaka A ^b  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Directional derivative; isotropic convolution; uncertainty principle; unit sphere

Indexed keywords

2-SPHERE; DIRECTIONAL DERIVATIVE; DIRECTIVITY; ENERGY CONCENTRATION; FILTER DESIGNS; FREDHOLM INTEGRAL EQUATIONS; ISOTROPIC CONVOLUTION; KEY EQUATION; MATRIX EQUATIONS; SIGNAL DESIGN; SPATIAL DOMAINS; SPATIAL RESOLUTION; SPECTRAL DOMAINS; TOTAL ENERGY; UNCERTAINTY PRINCIPLE; UNIT SPHERE; VARIATIONAL FRAMEWORK; WEIGHTING FUNCTIONS;

CONVOLUTION; DESIGN; INTEGRAL EQUATIONS; MATRIX ALGEBRA; STRAIN MEASUREMENT;

SPHERES;

EID: 80054077324     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2011.2162506     Document Type: Article

References (50)

1. D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-I," Bell Syst. Tech. J., vol. 40, pp. 43-63, Jan. 1961.
2. H. J. Landau and H. O. Pollack, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-II," Bell Syst. Tech. J., vol. 40, no. 1, pp. 65-84, Jan. 1961.
3. H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-III: The dimension of the space of essentially time- and band-limited signals," Bell Syst. Tech. J., vol. 41, pp. 1295-1336, Jul. 1962.
4. L. E. Franks, Signal Theory. Englewood Cliffs, NJ: Prentice-Hall, 1969.
5. G. Lachs, "Optimization of signal waveforms," IEEE Trans. Inf. Theory, vol. IT-9, pp. 95-97, Apr. 1963.
6. A.Wyner and H. Landau, "Optimum waveform signal sets with amplitude and energy constraints," IEEE Trans. Inf. Theory, vol. IT-30, no. 4, pp. 615-622, Jul. 1984.
7. A. Nuttall and F. Amoroso, "Minimum Gabor bandwidth of M orthogonal signals," IEEE Trans. Inf. Theory, vol. 14, no. 5, pp. 440-444, Jul. 1965.
8. A. Nuttall, "Minimum rms bandwidth of M time-limited signals with specified code or correlation matrix," IEEE Trans. Inf. Theory, vol. IT-14, no. 5, pp. 699-707, Sep. 1968.
9. A. Papoulis, "Minimum-bias windows for high-resolution spectral estimates," IEEE Trans. Inf. Theory, vol. IT-19, no. 1, pp. 9-12, Jan. 1973.
10. L. Xu, C. Bi, X. Chen, and J. Chen, "Resolution enhancement of nearfield acoustic holography by interpolation using band-limited signal restoration method," Chinese Sci. Bull., vol. 53, no. 20, pp. 3142-3150, Oct. 2008.
11. T. Bülow, "Spherical diffusion for surface smoothing and denoising," Comput. Inf. Sci. Dept., Univ. of Pennsylvania , Philadelphia, PA, Tech. Rep. (CIS), Dec. 2001.
12. T. Bülow, "Spherical diffusion for 3D surface smoothing," IEEE Trans. Pattern Anal. Mach. Intell., vol. 26, no. 12, pp. 1650-1654, Dec. 2004.
13. J. D. McEwen, M. P. Hobson, and A. N. Lasenby, "Optimal filters on the sphere," IEEE Trans. Signal Process., vol. 56, no. 8, pp. 3813-3823, Aug. 2008.
14. J. L. Sanz, D. Herranz, and E. M. Gonzalez, "Optimal detection of sources on a homogeneous and isotropic background," Astrophys. J., vol. 552, pp. 484-492, May 2001.
15. D. Herranz, J. L. Sanz, R. B. Barreiro, and E. M. Gonzalez, "Scaleadaptive filters for the detection/separation of compact sources," Astrophys. J., vol. 580, pp. 610-625, 2002.
16. M. Tegmark, D. H. Hartmann, M. S. Briggs, and C. A. Meegan, "The angular power spectrum of BASTE 3B gamma-ray bursts," Astrophys. J., vol. 468, p. 214, Sep. 1996.
17. E. Hivon, K. M. Gorski, C. B. Netterfield, B. P. Crill, S. Prunet, and F. Hansen, "Master of the CMB anisotropy power spectrum: A fast method for statistical analysis of large and complex CMB data sets," Astrophys. J., vol. 567, pp. 2-17, Mar. 2002.
18. S. Mitra, A. S. Sengupta, and T. Souradeep, "CMB power spectrum estimation using noncircular beams," Phys. Rev. D., vol. 70, p. 103002, 2004.
19. G. Faÿ, F. Guilloux, M. Betoule, J. F. Cardoso, J. Delabrouille, and M. L. Jeune, "CMB power spectrum estimation using wavelets," Phys. Rev. D., vol. 78, no. 8, p. 20, 2008.
20. A. S.Y. Poon, R.W. Broderson, andD. N. C. Tse, "Degrees of freedom in multiple-antenna channels: A signal space approach," IEEE Trans. Inf. Theory, vol. 51, no. 2, pp. 523-536, Feb. 2005.
21. R. A. Kennedy and T. D. Abhayapala, "Spatial concentration of wavefields: Towards spatial information content in arbitrary multipath scattering," in Proc. 4th Austr. Commun. Theory Workshop (AusCTW), Melbourne, Australia, Feb. 2003, pp. 38-45.
22. R. A. Kennedy, P. Sadeghi, T. D. Abhayapala, and H. M. Jones, "Intrinsic limits of dimensionality and richness in random multipath fields," IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2542-2556, Jun. 2007.
23. W. Freeden and M. Schreiner, "Non-orthogonal expansions on the sphere," Math. Methods Appl. Sci., vol. 18, pp. 83-120, 1995.
24. F. J. Narcowich and J. D. Ward, "Nonstationary wavelets on the -sphere for scattered data," Appl. Comput. Harmon. Anal., vol. 3, pp. 324-336, Oct. 1996.
25. A. Albertella, F. Sansò, and N. Sneeuw, "Band-limited functions on a bounded spherical domain: The Slepian problem on the sphere," J. Geodesy, vol. 77, pp. 436-447, 1999.
26. R. Pail, G. Plank, and W. D. Schuh, "Spatially restricted data distributions on the sphere: The method of orthonormalized functions and applications," J. Geodesy, vol. 75, pp. 44-56, 2001.
27. L. Miranian, "Slepian functions on the sphere, generalized Gaussian quadrature rule," Inverse Problems, vol. 20, pp. 877-892, 2004.
28. F. J. Simons, F. A. Dahlen, and M. A.Wieczorek, "Spatiospectral concentration on a sphere," SIAM Rev., vol. 48, pp. 504-536, 2006.
29. q-1," J. Comput. Appl. Math., vol. 199, pp. 68-79, Feb. 2007.
30. M. A. Wieczorek and F. J. Simons, "Minimum-variance multitaper spectral estimation on the sphere," J. Four. Anal. Appl., vol. 13, pp. 665-692, 2007.
31. F. Guilloux, G. Faÿ, and J. F. Cardoso, "Practical wavelet design on the sphere," Appl. Comput. Harmon. Anal., vol. 26, pp. 143-160, Mar. 2009.
32. L. Wei, R. A. Kennedy, and T. A. Lamahewa, "Signal concentration on unit sphere: An azimuthally moment weighting approach," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (IEEE ICASSP), Dallas, TX, Mar. 14-19, 2010, pp. 3698-3701.
33. L.Wei, R. A. Kennedy, and T. A. Lamahewa, "Signal concentration on unit sphere: A local -th moment zenithal energy concentration measure," in Proc. 11th Aust. Commun. Theory Workshop (AusCTW), Canberra, Australia, Feb. 2010, pp. 97-101.
34. L. Wei and R. A. Kennedy, "On spectral concentration of signals on the 2-sphere under a generalized moment weighting criterion," in Proc. Eur. Signal Process. Conf. (EUSIPCO),Aalborg,Denmark,Aug. 2010, p. 4.
35. M. K. Chung, K. M. Dalton, L. Shen, A. C. Evans, and R. J. Davidson, "Weighted Fourier series representation and its application to quantifying the amount of gray matter," IEEE Trans. Med. Imag., vol. 26, no. 4, pp. 566-581, Apr. 2007.
36. M. K. Chung, K.M. Dalton, and R. J. Davidson, "Tensor-based cortical surface morphometry via weighted spherical harmoic representation," IEEE Trans. Med.l Imag., vol. 27, no. 8, pp. 1143-1151, Aug. 2008.
37. E. W. Hobson, The Thoery of Spherical and Ellipsoidal Harmonics. New York: Chelsea, 1931.
38. R. A. Kennedy, T. A. Lamahewa, and L. Wei, "On azimuthally symmetric 2-sphere convolution," in Proc. 6th U.S./Aust. Joint Workshop on Defence Appl. Signal Process. (DASP), Sep. 26-30, 2009, p. 6.
39. K. Seon, "Smoothing of all-sky survey map with Fisher-von Mises function," J. Korean Phys. Soc., vol. 48, no. 3, pp. 331-334, Mar. 2006.
40. L. Debnath and P. Mikusinski, Introduction to Hilbert Spaces With Applications. San Diego, CA: Academic, 1990.
41. A. R. Edmonds, Angular Momentum in Quantum Mechanics. Princeton, NJ: Princeton Univ. Press, 1996.
42. F. A. Dahlen and F. J. Simons, "Spectral estimation on a sphere in geophysics and cosmology," Geophys. J. Int., vol. 174, pp. 774-807, Aug. 2008.
43. J. R. Driscoll, "Computing Fourier transforms and convolutions on the 2-sphere," Adv. Appl. Math., vol. 15, pp. 202-250, June 1994.
44. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing. NewYork: Cambridge Univ. Press, 1988.
45. B. Rafaely, "Plane-wave decomposition of the sound field on a sphere by spherical convolution," J. Acoust. Soc. Amer., vol. 116, no. 4, pp. 2149-2157, Oct. 2004.
46. B. Rafaely, "Analysis and design of spherical microphone arrays," IEEE Trans. Speech, Audio Process., vol. 13, no. 1, pp. 135-143, Jan. 2005.
47. M. Agmon, B. Rafaely, and J. Tabrikian, "Maximum directivity beamformer for spherical-aperture microphones," in Proc. IEEE Workshop Appl. Signal Process. Audio Acoust., New Paltz, NY, Oct. 18-21, 2009, pp. 153-156.
48. B. Rafaely, "Phase-mode versus delay-and-sum spherical microphone array processing," IEEE Signal Process. Lett., vol. 12, no. 10, pp. 713-716, Oct. 2005.
49. A. Koretz and B. Rafaely, "Dolph-Chebyshev beampattern design for spherical arrays," IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2417-2420, Jun. 2009.
50. S. F. Yan, H. H. Sun, U. P. Svensson, X. H. Ma, and J. M. Hovem, "Optimal modal beamforming for spherical microphone arrays," IEEE Trans. Audio Speech Lang. Process., vol. 19, no. 2, pp. 361-371, Feb. 2011.