## Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform

JOSA A, Vol. 18, Issue 8, pp. 1871-1881 (2001)

http://dx.doi.org/10.1364/JOSAA.18.001871

Enhanced HTML Acrobat PDF (597 KB)

### Abstract

Utilizing the asymptotic method of stationary phase, I derive expressions for the Fourier transform of a two-dimensional fringe pattern. The method assumes that both the amplitude and the phase of the fringe pattern are well-behaved differentiable functions. Applying the limits in two distinct ways, I show, first, that the spiral phase (or vortex) transform approaches the ideal quadrature transform asymptotically and, second, that the approximation errors increase with the relative curvature of the fringes. The results confirm the validity of the recently proposed spiral phase transform method for the direct demodulation of closed fringe patterns.

© 2001 Optical Society of America

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(100.2650) Image processing : Fringe analysis

(100.5090) Image processing : Phase-only filters

(110.6980) Imaging systems : Transforms

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

**History**

Original Manuscript: August 24, 2000

Revised Manuscript: January 16, 2001

Manuscript Accepted: January 16, 2001

Published: August 1, 2001

**Citation**

Kieran G. Larkin, "Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform," J. Opt. Soc. Am. A **18**, 1871-1881 (2001)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-18-8-1871

Sort: Year | Journal | Reset

### References

- 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]
- 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]
- M. A. Stuff, J. N. Cederquist, “Coordinate transformations realizable with multiple holographic optical elements,” J. Opt. Soc. Am. A 7, 977–981 (1990). [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.
- 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]
- A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966). [CrossRef]
- M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982). [CrossRef]
- D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986). [CrossRef] [PubMed]
- J. L. Marroquin, J. E. Figueroa, M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997). [CrossRef]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.