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XCOM intrinsic dimensionality for low‐Z elements at diagnostic energies
Author(s) -
Bornefalk Hans
Publication year - 2012
Publication title -
medical physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.473
H-Index - 180
eISSN - 2473-4209
pISSN - 0094-2405
DOI - 10.1118/1.3675399
Subject(s) - curse of dimensionality , physics , medical imaging , statistical physics , computer science , artificial intelligence
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).H 0 q, 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 ,H 0 qis rejected. Results: The largest value of q for whichH 0 qis 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.