Abstract

Quadrature operators are useful for obtaining the modulating phase ϕ 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

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2002

2001

1999

1998

1997

1994

1986

1982

Bone, D. J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 2000).

Cuevas, F. J.

Ghiglia, D. C.

Ina, H.

Kobayashi, S.

Kreis, T.

Larkin, K. G.

Malacara, D.

Marroquin, J. L.

Oldfield, M. A.

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory Algorithms, and Software (Wiley, New York, 1998).

Quiroga, J. A.

Richards, D.

D. Richards, Advanced Mathematical Methods with Maple (Cambridge U. Press, Cambridge, UK., 2002).

Rodriguez-Vera, R.

Romero, L. A.

Schwartz, M.

M. Schwartz, Information Transmission Modulation and Noise (McGraw Hill, New York, 1980).

Servin, M.

Stein, E.

E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces (Princeton U. Press, Princeton, N.J., 1971).

Takeda, M.

Weiss, G.

E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces (Princeton U. Press, Princeton, N.J., 1971).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 2000).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces (Princeton U. Press, Princeton, N.J., 1971).

D. Richards, Advanced Mathematical Methods with Maple (Cambridge U. Press, Cambridge, UK., 2002).

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory Algorithms, and Software (Wiley, New York, 1998).

M. Schwartz, Information Transmission Modulation and Noise (McGraw Hill, New York, 1980).

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