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Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 48, Iss. 20 — Jul. 10, 2009
  • pp: 3894–3902

Abel inversion of deflectometric data: comparison of accuracy and noise propagation of existing techniques

Pankaj S. Kolhe and Ajay K. Agrawal  »View Author Affiliations

Applied Optics, Vol. 48, Issue 20, pp. 3894-3902 (2009)

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Abel inverse integral to obtain local field distributions from path-integrated measurements in an axisymmetric medium is an ill-posed problem with the integrant diverging at the lower integration limit. Existing methods to evaluate this integral can be broadly categorized as numerical integration techniques, semianalytical techniques, and least-squares whole-curve-fit techniques. In this study, Simpson’s 1 / 3 rd rule (a numerical integration technique), one-point and two-point formulas (semianalytical techniques), and the Guass–Hermite product polynomial method (a least-squares whole-curve-fit technique) are compared for accuracy and error propagation in Abel inversion of deflectometric data. For data acquired at equally spaced radial intervals, the deconvolved field can be expressed as a linear combination (weighted sum) of measured data. This approach permits use of the uncertainty analysis principle to compute error propagation by the integration algorithm. Least-squares curve-fit techniques should be avoided because of poor inversion accuracy with large propagation of measurement error. The two-point formula is recommended to achieve high inversion accuracy with minimum error propagation.

© 2009 Optical Society of America

OCIS Codes
(000.2190) General : Experimental physics
(100.6950) Image processing : Tomographic image processing
(120.5820) Instrumentation, measurement, and metrology : Scattering measurements
(280.1740) Remote sensing and sensors : Combustion diagnostics
(280.2490) Remote sensing and sensors : Flow diagnostics

ToC Category:
Instrumentation, Measurement, and Metrology

Original Manuscript: April 6, 2009
Revised Manuscript: June 1, 2009
Manuscript Accepted: June 15, 2009
Published: July 1, 2009

Pankaj S. Kolhe and Ajay K. Agrawal, "Abel inversion of deflectometric data: comparison of accuracy and noise propagation of existing techniques," Appl. Opt. 48, 3894-3902 (2009)

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  1. N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153-157 (1826). [CrossRef]
  2. R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198-217 (1956). [CrossRef]
  3. U. Buck, “Inversion of molecular scattering data,” Rev. Mod. Phys. 46, 369-389 (1974). [CrossRef]
  4. C. J. Cremers and R. C. Birkebak, “Application of the Abel integral equation to spectrographic data,” Appl. Opt. 5, 1057-1064 (1966). [CrossRef] [PubMed]
  5. M. P. Freeman and S. Katz, “Determination of radial distribution of brightness in a cylindrical luminous medium with self-absorption,” J. Opt. Soc. Am. 50, 826-830 (1960). [CrossRef]
  6. P. Andanson, B. Cheminat, and A. M. Halbique, “Numerical solution of the Abel integral equation: application to plasma spectroscopy,” J. Phys. D Appl. Phys. 11, 209-215 (1978). [CrossRef]
  7. K. Bockasten, “Transformation of observed radiances into radial distribution of the emission of a plasma,” J. Opt. Soc. Am. 51, 943-947 (1961). [CrossRef]
  8. K. M. Green, M. C. Borras, P. P. Woskov, G. J. Flores III, K. Hadidi, and P. Thomas, “Electronic excitation temperature profiles in an air microwave plasma torch,” IEEE Trans. Plasma Sci. 29, 399-406 (2001). [CrossRef]
  9. V. Dribinski, A. Ossadtchi, V. A. Mandeleshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634-2642 (2002). [CrossRef]
  10. G. N. Minerbo and M. E. Levy, “Inversion of Abel's integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. 6, 598-616 (1969). [CrossRef]
  11. M. Deutsch and I. Beniaminy, “Inversion of Abel's integral equation for experimental data,” J. Appl. Phys. 54, 137-143(1983). [CrossRef]
  12. M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. 27, 1956-1959 (1988). [CrossRef] [PubMed]
  13. W. J. Glantschnig and A. Holliday, “Mass fraction profiling based on x-ray tomography and its application to characterizing porous silica boules,” Appl. Opt. 26, 983-989 (1987). [CrossRef] [PubMed]
  14. K. Tatekura, “Determination of the index profile of optical fibers from transverse interferograms using Fourier theory,” Appl. Opt. 22, 460 (1983). [CrossRef] [PubMed]
  15. E. Ampem-Lassen, S. T. Huntington, N. M. Dragomir, K. A. Nugent, and A. Roberts, “Refractive index profiling of axially symmetric optical fibers: a new technique,” Opt. Express 13, 3277-3282 (2005). [CrossRef] [PubMed]
  16. D. C. Hammond, Jr., “Deconvolution technique for line-of-sight optical measurements in axisymmetric sprays,” Appl. Opt. 20, 493-499 (1981). [CrossRef] [PubMed]
  17. E. Keren, E. Bar-Ziv, I. Glatt, and O. Kafri, “Measurements of temperature distribution of flames by moire deflectometry,” Appl. Opt. 20, 4263-4266 (1981). [CrossRef] [PubMed]
  18. J. R. Camacho, F. N. Beg, and P. Lee1, “Comparison of sensitivities of Moire deflectometry and interferometry to measure electron densities in z-pinch plasmas,” J. Phys. D. 40, 2026-2032 (2007). [CrossRef]
  19. P. V. Farrell and D. L. Hofeldt, “Temperature measurement in gases using speckle photography,” Appl. Opt. 23, 1055-1059(1984). [CrossRef] [PubMed]
  20. P. S. Greenberg, R. B. Klimek, and D. R. Buchele, “Quantitative rainbow schlieren deflectometry,” Appl. Opt. 34, 3810-3822 (1995). [CrossRef] [PubMed]
  21. K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, and D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89-95 (1998). [CrossRef]
  22. A. K. Agrawal, K. N. Alammar, and S. R. Gollahalli, “Application of rainbow schlieren deflectometry to measure temperature and oxygen concentration in a laminar jet diffusion flame,” Exp. Fluids 32, 689-691 (2002).
  23. K. S. Pasumarthi and A. K. Agrawal, “Schlieren measurements and analysis of concentration field in self-excited helium jets,” Phys. Fluids 15, 3683-3692 (2003). [CrossRef]
  24. B. S. Yildirim and A. K. Agrawal, “Full-field concentration measurements of self-excited oscillations in momentum-dominated helium jets,” Exp. Fluids 38, 161-173(2005). [CrossRef]
  25. T. Wong and A. K. Agrawal, “Quantitative measurements in an unsteady flame using high-speed rainbow schlieren deflectometry,” Meas. Sci. Technol. 17, 1503-1510 (2006). [CrossRef]
  26. R. Gorenflo and Y. Kovetz, “Solution of an Abel-type integral equation in the presence of noise by quadratic programming,” Numer. Math. 8, 392-406 (1966). [CrossRef]
  27. R. Gorenflo and S. Vessella, “Abel integral equations: analysis and applications,” in Lecture Notes in Mathematics, A. Dold, B. Eckmann, and F. Takens, eds. (Springer-Verlag, 1980).
  28. K. E. Atkinson, “The numerical solution of an Abel integral equation by a product trapezoidal method,” SIAM J. Numer. Anal. 11, 97-101 (1974). [CrossRef]
  29. S. Guerona and M. Deutsch, “A fast Abel inversion algorithm,” J. Appl. Phys. 75, 4313-4318 (1994). [CrossRef]
  30. C. J. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods,” Appl. Opt. 31, 1146-1152 (1992). [CrossRef] [PubMed]
  31. L. A. Vasil'ev, Schlieren Methods (Israel Program for Scientific Translations, 1971).
  32. O. H. Nestor and H. N. Olson, “Numerical methods for reducing line and surface probe data,” SIAM Rev. 2, 200-207(1960). [CrossRef]
  33. W. Frie, “Zur auswertung der Abelschen integralgleichung,” Ann. Phys. 465, 332-339 (1963). [CrossRef]
  34. R. Rubinstein and P. S. Greenberg, “Rapid inversion of angular deflection data for certain axisymmetric refractive index distributions,” Appl. Opt. 33, 1141-1144 (1994). [CrossRef] [PubMed]
  35. A. T. Ramsey and M. Diesso, “Abel inversions: error propagation and inversion reliability,” Rev. Sci. Instrum. 70, 380-383 (1999). [CrossRef]
  36. L. M. Smith, “Nonstationary noise effects in the Abel inversion,” IEEE Trans. Inf. Theory 34, 158-161 (1988). [CrossRef]
  37. R. K. Paul, J. T. Andrews, K. Bose, and P. K. Barhai, “Reconstruction errors in Abel inversion,” Plasma Devices Oper. 13, 281-290 (2005). [CrossRef]
  38. A. K. Shenoy, A. K. Agrawal, and S. R. Gollahalli, “Quantitative evaluation of flow computations by rainbow schlieren deflectometry,” AIAA J. 36, 1953-1960 (1998). [CrossRef]
  39. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).
  40. H. W. Coleman and W. G. Steele, Jr., Experimentation and Uncertainty Analysis for Engineers, 2nd ed. (Wiley, 1999).

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