Spectral theory for random closed sets and estimating the covariance via frequency space

Advances in Applied Probability

Volumn 35, Issue 3, 2003, Pages 603-613

  Koch, Karsten ^a   Ohser, Joachim ^b   Schladitz, Katja ^b  

^a PHILIPPS UNIVERSITY MARBURG   (Germany)

^b FRAUNHOFER INSTITUTE FOR INDUSTRIAL MATHEMATICS ITWM   (Germany)

Author keywords

Bartlett spectrum; Fast Fourier transform; Power spectrum; Random set

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; COMPUTER SIMULATION; FAST FOURIER TRANSFORMS; GEOMETRY; MAXIMUM LIKELIHOOD ESTIMATION; SET THEORY; SPECTRUM ANALYSIS; THEOREM PROVING;

BARLETT SPECTRUM; COVARIANCE; FREQUENCY SPACE; POWER SPECTRUM; RANDOM CLOSED SETS; SPECTRAL THEORY;

RANDOM PROCESSES;

EID: 0041378175     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1239/aap/1059486820     Document Type: Article

References (22)

1. Bartlett, M. (1963). The spectral analysis of point processes. J. R. Statist. Soc. B 29, 264-296.
2. Bartlett, M. (1964). The spectral analysis of two-dimensional point processes. Biometrika 51, 299-311.
3. Berke, A., Neite, G., Riehemann, W. and Nembach, E. (1987). Characterization of periodic composites by laser-beam diffraction. J. Appl. Phys. 61, 1263-1267.
4. Böhm, S., Heinrich, L. and Schmidt, V. (2002). Kernel estimation of the spectral density of stationary random closed sets. Preprint.
5. Brigham, E. O. (1995). FFT: Schnelle Fourier-Transformation, 6th edn. Oldenbourg, München.
6. Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
7. Debye, P., Anderson, H. R. and Brumberger, H. (1957). Scattering by an inhomogeneous solid. II. The correlation function and its application. J. Appl. Phys. 28, 679-683.
8. Dreseler, B. and Schempp, W. (1980). Einführung in die Harmonische Analyse. Teubner, Stuttgart.
9. Frank, J. (1980). The role of correlation techniques in computer image processing. In Computer Processing of Electron Microscope Images, ed. P. Hawkes, Springer, Berlin, pp. 187-222.
10. Frigo, M. and Johnson, S. G. (1998). FFTW 2.1.3 (The fastest Fourier transform of the West). Available at http://www.fftw.org/.
11. Henry, N. F. M. and Lonsdale, K. (1969). International Tables for X-ray Crystallography, Vol. 1. Kynoch, Birmingham.
12. Koch, K. (2002). Spektralanalyse zufälliger abgeschlossener Mengen. Masters Thesis, University of Siegen.
13. Manolakis, D. and Proakis, J. (1996). Digital Signal Processing, 3rd edn. Prentice-Hall, Englewood Cliffs, NJ.
14. Marcotte, D. (1996). Fast variogram computation with FFT. Comput. Geosci. 22, 1175-1186.
15. Mugglestone, M. and Renshaw, E. (1996). A practical guide to the spectral analysis of spatial point processes. Comput. Statist. Data Anal. 21, 43-65.
16. Ohser, J. and Mücklich F. (2000). Statistical Analysis of Microstructures in Materials Science. John Wiley, Chichester.
17. Renshaw, E. and Ford, E. (1983). The interpretation of process from pattern using two-dimensional spectral analysis: methods and problems of interpretation. Appl. Statist. 32, 51-63.
18. Ripley, B. (1981). Spatial Statistics. John Wiley, New York.
19. Sasvari, Z. (2000). Positive Definite and Definitizable Functions. Akademie Verlag, Berlin.
20. Schneider, R. and Weil, W. (2000) Stochastische Geometrie. Teubner, Stuttgart.
21. Stoyan, D. (2002). Random systems of hard particles: models and statistics. Chinese J. Stereology Image Anal. 7, 1-14.
22. Vurpillot, F., Da Costa, G., Menand, A. and Blavette, D. (2001). Structural analysis in three-dimensional atom probe: a Fourier transform approach. J. Microscopy 203, 295-302.