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Medical Physics / Volume 39 / Issue 2 / RADIATION IMAGING PHYSICS

Med. Phys. 39, 654 (2012); http://dx.doi.org/10.1118/1.3675399 (4 pages)

XCOM intrinsic dimensionality for low-Z elements at diagnostic energies

Hans Bornefalk

Department of Physics, Royal Institute of Technology, SE-106 91 Stockholm, Sweden

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(Received 6 October 2011; accepted 13 December 2011; revised 13 December 2011; published online 12 January 2012)

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Purpose: To determine the intrinsic dimensionality of linear attenuation coefficients (LACs) from XCOM for elements with low atomic number (Z = 1–20) at diagnostic x-ray energies (25–120 keV). H0q, the hypothesis that the space of LACs is spanned by q bases, is tested for various q-values.
Methods: Principal component analysis is first applied and the LACs are projected onto the first q principal component bases. The residuals of the model values vs XCOM data are determined for all energies and atomic numbers. Heteroscedasticity invalidates the prerequisite of i.i.d. errors necessary for bootstrapping residuals. Instead wild bootstrap is applied, which, by not mixing residuals, allows the effect of the non-i.i.d residuals to be reflected in the result. Credible regions for the eigenvalues of the correlation matrix for the bootstrapped LAC data are determined. If subsequent credible regions for the eigenvalues overlap, the corresponding principal component is not considered to represent true data structure but noise. If this happens for eigenvalues l and l + 1, for any l ≤ q, H0q is rejected.
Results: The largest value of q for which H0q is nonrejectable at the 5%-level is q = 4. This indicates that the statistically significant intrinsic dimensionality of low-Z XCOM data at diagnostic energies is four.
Conclusions: The method presented allows determination of the statistically significant dimensionality of any noisy linear subspace. Knowledge of such significant dimensionality is of interest for any method making assumptions on intrinsic dimensionality and evaluating results on noisy reference data. For LACs, knowledge of the low-Z dimensionality might be relevant when parametrization schemes are tuned to XCOM data. For x-ray imaging techniques based on the basis decomposition method (Alvarez and Macovski, Phys. Med. Biol. 21, 733–744, 1976), an underlying dimensionality of two is commonly assigned to the LAC of human tissue at diagnostic energies. The finding of a higher statistically significant dimensionality thus raises the question whether a higher assumed model dimensionality (now feasible with the advent of multibin x-ray systems) might also be practically relevant, i.e., if better tissue characterization results can be obtained.

© 2012 American Association of Physicists in Medicine

Article Outline

  1. INTRODUCTION
  2. BACKGROUND
  3. METHODS
  4. RESULTS
  5. CONCLUSIONS
Digital Object Identifier
http://dx.doi.org/10.1118/1.3675399

KEYWORDS and PACS

Keywords

biological tissues, bootstrapping, diagnostic radiography, eigenvalues and eigenfunctions, photons, principal component analysis, XCOM, linear attenuation coefficients, linear subspace dimensionality, bootstrap

PACS

  • 87.59.B-

    Radiography

  • 02.50.-r

    Probability theory, stochastic processes, and statistics

PUBLICATION DATA

ISSN

0094-2405 (print)  

Publisher

$abstract.society.value is a Member of CrossRef American Association of Physicists in Medicine

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