N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Dover, New York, 1975).

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).

[CrossRef]

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, the Netherlands, 1995).

N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Dover, New York, 1975).

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, New York, 1990), pp. 381–384.

G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, the Netherlands, 1995).

T. Kreis, Holographic Interferometry. Principles and Methods (Akademie, Berlin, 1996), Vol. 1.

K. G. Larkin, D. Bone, 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).

[CrossRef]

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).

[CrossRef]

A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).

[CrossRef]

J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).

[CrossRef]

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

K. G. Larkin, D. Bone, 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).

[CrossRef]

J. L. Marroquin, J. E. Figueroa, M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).

[CrossRef]

M. A. Stuff, J. N. Cederquist, “Coordinate transformations realizable with multiple holographic optical elements,” J. Opt. Soc. Am. A 7, 977–981 (1990).

[CrossRef]

Z. Jaroszewicz, A. Kolodziejczyk, D. Mouriz, S. Bara, “Analytic design of computer-generated Fourier-transform holograms for plane curves reconstruction,” J. Opt. Soc. Am. A 8, 559–565 (1991).

[CrossRef]

A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).

[CrossRef]

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly will introduce significant errors.

[CrossRef]

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

According to Bracewell (Ref. 4, p. 83), Lerch’s theorem states that if two functions f(x, y)and g(x, y)have the same Fourier transform, then f(x, y)-g(x, y)is a null function.

N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Dover, New York, 1975).

We shall refrain from calling this function the 2-D analytic signal at present because there are several conflicting definitions of analyticity 13in multiple dimensions. The alternative term “monogenic” does not seem appropriate because the word now has another widespread use in molecular genetics.

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

The Gaussian curvature Cat a critical point (zero gradient) is equal to the Hessian; C=ψ2,0ψ0,2-ψ1,12(1+ψ1,02+ψ0,12)3/2=H(1+ψ1,02+ψ0,12)3/2,ψ1,0=ψ0,1=0 ⇒ C=H.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

A zero gradient of the orientation implies a zero Hessian. The converse is not true, as there are surfaces, such as the cone, with a zero Hessian and a nonzero orientation gradient.

J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).

G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, the Netherlands, 1995).

T. Kreis, Holographic Interferometry. Principles and Methods (Akademie, Berlin, 1996), Vol. 1.

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, New York, 1990), pp. 381–384.

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).

[CrossRef]