title: Estimator for photon counting energy selective x-ray imaging with multibin pulse height analysis.
abstract: OBJECTIVE
  This paper describes a noniterative estimator for the energy dependent information from photon counting detectors with multibin pulse height analysis (PHA). The estimator uses the two function decomposition of the attenuation coefficient [R. E. Alvarez and A. Macovski, Phys. Med. Biol. 21, 733-744 (1976)] and its output is the line integrals of the basis set coefficients. The output noise variance and bias is compared to other noniterative estimators and to the Cramèr-Rao lower bound (CRLB).
  
  
  METHODS
  The estimator first computes an initial estimate from a linearized maximum likelihood estimator. The errors in the initial estimates are determined at a set of points from measurements on a calibration phantom. The errors at these known points are interpolated to create two-dimensional look up tables of corrections to the initial estimates. During image acquisition, the linearized maximum likelihood estimate for each data point is used as an input to the correction look up tables, and the final output is the sum of the estimate and the correction. The performance of the estimator is compared to generalizations of the polynomial and rational polynomial estimators for multibin data. The estimators are compared by the mean square error (MSE) and its components, the bias, and the variance of the output. The variance is also compared to the CRLB. The performance is simulated with two to five bins PHA data. The CRLB at a fixed object thickness is also computed as a function of the number of bins.
  
  
  RESULTS
  For two bin data, all the estimators' variances are equal to the CRLB. With three or more bins, only the proposed estimator achieves the CRLB while the others, which were not optimized for noise performance, have much larger output variance. The bias of the proposed estimator is equal to the polynomial estimator for calibration phantoms with 40 or more steps, that is, 1600 combinations of basis materials, but is larger than the rational polynomial bias. In all cases at the photon counts tested, the MSE is essentially equal to the variance, indicating that the bias errors are negligible compared to the variance.
  
  
  CONCLUSIONS
  The estimator provides a noniterative method to compute the energy dependent information from multibin PHA data that achieves the CRLB over a wide range of operating conditions and has low output bias. The estimator can be calibrated based on the measurements of a calibration phantom; so, it does not require measurements of the x-ray energy spectrum or the detector response functions.
authors.affiliation: Aprend Technology Mountain View, California 94043, USA. ralvarez@aprendtech.com
authors.forename: Robert E
authors.initials: RE
authors.lastname: Alvarez
mesh_terms: D000465:Algorithms; D003627:Data Interpretation, Statistical; D017785:Photons; D011829:Radiation Dosage; D011859:Radiography / Q000379:methods*; D011874:Radiometry / Q000379:methods*; D015203:Reproducibility of Results; D012680:Sensitivity and Specificity; D014965:X-Rays