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Principles of Tomography

  • Pradipta Kumar Panigrahi
  • Krishnamurthy Muralidhar
Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

Abstract

Most physical systems involving heat and mass transfer have 3D variation of temperature and species concentration within the apparatus. When schlieren or shadowgraph imaging is employed, one obtains a depth-averaged view of the 3D variation. The image data is often called a path integral or a projection of the thermal (or concentration) field. Tomography is a procedure for recovering the 3D information of the field variable from a collection of projections. The projection data is recorded at various angles by turning the experimental apparatus or the light beam, specifically the source-detector axis. This chapter presents tomography in the form of algorithms, of which CBP, ART, and MART are a few. Tomographic algorithms are known to be sensitive to noise in projection data. Issues such as sensitivity and the impact of having limited data are discussed. The algorithms are validated using simulated data as well as from physical experiments of crystal growth and jet interactions. Finally, the use of POD as a tool for dealing with unsteady data is presented.

Keywords

CBP ART MART Entropy Sensitivity Extrapolation scheme 

References

  1. 1.
    Anderson AH, Kak AC (1984) Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm. Ultrason Imaging 6:81–94Google Scholar
  2. 2.
    Bahl S, Liburdy JA (1991) Three dimensional image reconstruction using interferometric data from a limited field of view with noise. Appl Opt 30(29):4218–4226CrossRefGoogle Scholar
  3. 3.
    Bahl S, Liburdy JA (1991) Measurement of local convective heat transfer coefficients using three-dimensional interferometry. Int J Heat Mass Transf 34:949–960CrossRefGoogle Scholar
  4. 4.
    Censor Y (1983) Finite series-expansion reconstruction methods. Proc IEEE 71(3):409–419CrossRefGoogle Scholar
  5. 5.
    Faris GW, Byer RL (1988) Three dimensional beam deflection optical tomography of a supersonic jet. Appl Opt 27(24):5202–5212CrossRefGoogle Scholar
  6. 6.
    Gilbert PFC (1972) Iterative methods for three-dimensional reconstruction of an object from its projections. J Theor Biol 36:105–117CrossRefGoogle Scholar
  7. 7.
    Gordon R, Bender R, Herman GT (1970) Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J Theor Biol 29:471–481CrossRefGoogle Scholar
  8. 8.
    Gordon R, Herman GT (1974) Three dimensional reconstructions from projections: a review of algorithms. Int Rev Crystallogr 38:111–151CrossRefGoogle Scholar
  9. 9.
    Gull SF, Newton TJ (1986) Maximum entropy tomography. Appl Opt 25:156–160CrossRefGoogle Scholar
  10. 10.
    Herman GT (1980) Image reconstruction from projections. Academic Press, New YorkMATHGoogle Scholar
  11. 11.
    Kaczmarz MS (1937) Angenaherte auflosung von systemen linearer gleichungen. Bull Acad Polonaise Sci Lett Classe Sci Math Natur Serier A35:355–357Google Scholar
  12. 12.
    Lanen TAWM (1990) Digital holographic interferometry in flow research. Opt Commn 79:386–396CrossRefGoogle Scholar
  13. 13.
    Liu TC, Merzkirch W, Oberste-Lehn K (1989) Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density. Exp Fluids 7:157–163CrossRefGoogle Scholar
  14. 14.
    Mayinger F (eds) (1994) Optical measurements: techniques and applications. Springer, BerlinGoogle Scholar
  15. 15.
    Mewes D, Friedrich M, Haarde W, Ostendorf W (1990) Tomographic measurement techniques for process engineering studies. In: Cheremisinoff NP (ed) Handbook of heat and mass transfer, Chapter 24, vol 3Google Scholar
  16. 16.
    Michael YC, Yang KT (1992) Three-dimensional mach-zehnder interferometric tomography of the rayleigh-benard problem. J Heat Transf Trans ASME 114:622–629CrossRefGoogle Scholar
  17. 17.
    Mishra D, Muralidhar K, Munshi P (1998) Performance evaluation of fringe thinning algorithms for interferometric tomography. Opt Lasers Eng 30:229–249CrossRefGoogle Scholar
  18. 18.
    Mishra D, Muralidhar K, Munshi P (1999a) Interferometric study of rayleigh-benard convection using tomography with limited projection data. Exp Heat Transf 12(2):117–136CrossRefGoogle Scholar
  19. 19.
    Mishra D, Muralidhar K, Munshi P (1999c) A robust MART algorithm for tomographic applications. Numer Heat Transf B Fundam. 35(4):485–506CrossRefGoogle Scholar
  20. 20.
    Mishra D, Muralidhar K, Munshi P (1999d) Interferometric study of rayleigh-benard convection at intermediate rayleigh numbers. Fluid Dynamics Res 25(5):231–255CrossRefGoogle Scholar
  21. 21.
    Mukutmoni D, Yang KT (1995) Pattern selection for rayleigh-benard convection in intermediate aspect ratio boxes. Numer Heat Transf Part A 27:621–637CrossRefGoogle Scholar
  22. 22.
    Munshi P (1997) Application of computerized tomography for measurements in heat and mass transfer, proceedings of the 3rd ISHMT-ASME heat and mass transfer conference held at IIT Kanpur (India) during 29–31 December 1997. Narosa Publishers, New DelhiGoogle Scholar
  23. 23.
    Natterer F (1986) The mathematics of computerized tomography. Wiley, New YorkMATHGoogle Scholar
  24. 24.
    Ostendorf W, Mayinger F, Mewes D 1986 A tomographical method using holographic interferometry for the registration of three dimenisonal unsteady temperature profiles in laminar and turbulent flow, proceedings of the 8th international heat transfer conference, San Francisco, USA, pp 519–523Google Scholar
  25. 25.
    Sirovich L (1989) Chaotic dynamics of coherent structures. Physica D 37:126–145CrossRefMathSciNetGoogle Scholar
  26. 26.
    Snyder R, Hesselink L (1985) High speed optical tomography for flow visualization. Appl Opt 24:23Google Scholar
  27. 27.
    Snyder R (1988) Instantaneous three dimensional optical tomographic measurements of species concentration in a co-flowing jet, Report No. SUDAAR 567, Stanford University, USAGoogle Scholar
  28. 28.
    Soller C, Wenskus R, Middendorf P, Meier GEA, Obermeier F (1994) Interferometric tomography for flow visualization of density fields in supersonic jets and convective flow. Appl Opt 33(14):2921–2932CrossRefGoogle Scholar
  29. 29.
    Srivastava A, Singh D, Muralidhar K (2009) Reconstruction of time-dependent concentration gradients around a KDP crystal growing from its aqueous solution. J Crystal Growth 311:1166–1177Google Scholar
  30. 30.
    Subbarao PMV, Munshi P, Muralidhar K (1997) Performance evaluation of iterative tomographic algorithms applied to reconstruction of a three dimensional temperature field. Numer Heat Transf B Fundam 31(3):347–372CrossRefGoogle Scholar
  31. 31.
    Tanabe K (1971) Projection method for solving a singular system. Numer Math 17:302–214CrossRefMathSciNetGoogle Scholar
  32. 32.
    Tolpadi AK, Kuehn TH (1991) Measurement of three-dimensional temperature fields in conjugate conduction-convection problems using multidimensional interferometry. Int J Heat Mass Transfer 34(7):1733–1745CrossRefGoogle Scholar
  33. 33.
    Torniainen ED, Hinz A, Gouldin FC (1998) Tomographic analysis of unsteady. Reacting Flows AIAA J 36:1270–1278CrossRefGoogle Scholar
  34. 34.
    Velarde MG, Normand C (1980) Convection. Sci American 243(1):79–94Google Scholar
  35. 35.
    Verhoeven D (1993) Multiplicative algebraic computed tomography algorithms for the reconstruction of multidirectional interferometric data. Opt Eng 32:410–419CrossRefGoogle Scholar
  36. 36.
    Watt DW, Vest CM (1990) Turbulent flow visualization by interferometric integral imaging and computed tomography. Exp Fluids 8:301–311CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  • Pradipta Kumar Panigrahi
    • 1
  • Krishnamurthy Muralidhar
    • 2
  1. 1.Department of Mechanical EngineeringIndian Institute of Technology KanpurKanpurIndia
  2. 2.Department of Mechanical EngineeringIndian Institute of Technology KanpurKanpurIndia

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