Publication: General n-dimensional quadrature transform and its application to interferogram demodulation
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Advisors (or tutors)
Optical Society of America
Quadrature operators are useful for obtaining the modulating phase 0 in, interferometry and temporal signals in electrical communications. In carrier-frequency interferometry and electrical communications, one uses the Hilbert transform to obtain the quadrature of the signal. In these, cases the Hilbert transform gives the desired quadrature because the modulating phase is monotonically increasing. We propose an n-dimensional quadrature operator that transforms cos(φ) into -sin(φ) regardless of the-frequency spectrum of the signal. With the quadrature of the phase-modulated signal, one can easily calculate the value of φ over all the domain of interest. Our quadrature operator is composed of-two n-dimensional vector fields: One is related to the gradient of the image normalized with respect to local frequency magnitude, and the other is related to the sign of the local frequency of the signal: The inner product of these two vector fields gives us the desired quadrature signal. This quadrature operator is derived in the image space by use of differential vector calculus and in the frequency domain by use of a n-dimensional generalization of the Hilbert transform. A robust numerical algorithm is given to find the modulating phase of two-dimensional single-image closed-fringe interferograms by use of the ideas put forward.
© 2003 Optical Society of America. J. L. Marroquín and M. Servín were partially supported by grants 34575A and 33429-E from Consejo Nacional de Ciencia y Tecnología, México.
1. T. Kreis, ‘‘Digital holgraphic interference-phase measurement using the Fourier-transform method’’, J. Opt. Soc. Am. A 3, 847–855 (1986). 2. M. Servín, J. L. Marroquín, and F. J. Cuevas, ‘‘Fringe-following regularized phase tracker for demodulation of closed-fringe interferogram’’, J. Opt. Soc. Am. A 18, 689–695 (2001). 3. J. L. Marroquín, M. Servín, and R. Rodríguez-Vera, ‘‘Adap-tive quadrature filters and the recovery of phase from fringe pattern images’’, J. Opt. Soc. Am. A 14, 1742 1753(1997). 4. J. L. Marroquín, R. Rodríguez-Vera, and M. Servín, ‘‘Local phase from local orientation by solution of a sequence of linear systems’’, J. Opt. Soc. Am. A 15, 1536 1543(1998). 5. K. G. Larkin, D. J. Bone, and M. A. Oldfield, ‘‘Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform’’, J. Opt. Soc. Am. A 18, 1862–1870 (2001). 6. M. Takeda, H. Ina, and S. Kobayashi, ‘‘Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry’’, J. Opt. Soc. Am. 72, 156–160 (1982). 7. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 2000). 8. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980). 9. E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces (Princeton U. Press, Princeton, N.J., (1971). 10. J. A. Quiroga, M. Servín, and F. J. Cuevas, ‘‘Modulo 2π fringe orientation angle estimation by phase unwrapping with a regularized phase-tracking algorithm’’, J. Opt. Soc. Am. A 19, 1524–1531 (2002). 11. D. Richards,Advanced Mathematical Methods with Maple (Cambridge U. Press, Cambridge, UK., 2002). 12. D. C. Ghiglia and L. A. Romero, ‘‘Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods’’, J. Opt. Soc. Am. A 11, 107–117 (1994). 13. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory Algorithms, and Software (Wiley, New York, 1998). 14. M. Servín, F. J. Cuevas, D. Malacara, and J. L. Marroquín, ‘‘Phase unwrapping through demodulation using the regularized phase-tracking technique’’, Appl. Opt. 38, 1934–1940 (1999). 15. M. Schwartz, Information Transmission Modulation and Noise(McGraw Hill, New York, 1980).