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Non-uniqueness and instability of ‘ankylography’

Abstract

Arising from K. S. Raines et al. Nature 463, 214–217 (2010)10.1038/nature08705

Raines et al.1 have proposed ‘ankylography’ as a three-dimensional tomographic modality with a single monochromatic beam1. They believe that the three-dimensional structure of an object is “in principle determined by the 2D spherical pattern” that “is sampled at a sufficiently fine scale on the Ewald sphere.” Although they restricted their approach to unspecified “certain circumstances”, they claim potential “broad applications in the physical and life sciences”. Nevertheless, the imaging principles and proposed methodology of ankylography are flawed, bringing into question its applications, which was also pointed out by others, such as in ref. 2.

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Figure 1: Numerical studies of ‘ankylography’ and ‘one-view divergent tomography’.

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References

  1. Raines, K. S. et al. Three-dimensional structure determination from a single view. Nature 463, 214–217 (2010)

    Article  ADS  CAS  Google Scholar 

  2. Thibault, P. Feasibility of 3D reconstruction from a single 2D diffraction measurement. Preprint at http://arXiv.org/abs/0909.1643v1 (2009)

  3. Crimmins, T. R. & Fienup, J. R. Uniqueness of phase retrieval for functions with sufficiently disconnected support. J. Opt. Soc. Am. 73, 218–221 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  4. Foley, J. T. & Butts, R. R. Uniqueness of phase retrieval from intensity measurements. J. Opt. Soc. Am. 71, 1008–1014 (1981)

    Article  ADS  Google Scholar 

  5. Hamaker, C. et al. The divergent beam x-ray transform. Rocky Mountain J. Math. 10, 253–284 (1980)

    Article  MathSciNet  Google Scholar 

  6. Ye, Y., Yu, H. & Wang, G. Interior tomography: mathematical analysis. In Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning, and Inverse Problems (eds Censor, Y., Jiang, M. & Wang, G. ) 543–561 (Medical Physics Publishing, 2010)

    Google Scholar 

  7. Inouye, T. Image reconstruction with limited angle projection data. IEEE Trans. Nucl. Sci. 26, 2665–2669 (1979)

    Article  ADS  Google Scholar 

  8. Fienup, J. R. Reconstruction of an object from modulus of its Fourier transform. Opt. Lett. 3, 27–29 (1978)

    Article  ADS  CAS  Google Scholar 

  9. Donoho, D. L. Compressed sensing. IEEE Trans. Inf. Theory 52, 1289–1306 (2006)

    Article  MathSciNet  Google Scholar 

  10. Yu, H. & Wang, G. Compressed sensing based interior tomography. Phys. Med. Biol. 54, 2791–2805 (2009)

    Article  Google Scholar 

  11. Anderson, N. G. et al. Spectroscopic (multi-energy) CT distinguishes iodine and barium contrast material in MICE. Eur. Radiol. 20, 2126–2134 (2010)

    Article  CAS  Google Scholar 

  12. Natterer, F. The Mathematics of Computerized Tomography. Chs II, IV and VI (John Wiley & Sons, 1986)

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Contributions

G.W. conceptualized and wrote the manuscript. G.W., H.Y. and W.C. identified non-uniqueness of ankylography in simple cases, and A.K. constructed the general argument. H.Y., W.C. and G.W. analysed stability of ankylography and simulated spectrography.

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Wang, G., Yu, H., Cong, W. et al. Non-uniqueness and instability of ‘ankylography’. Nature 480, E2–E3 (2011). https://doi.org/10.1038/nature10635

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