Deblurring of X-Ray Spectra Acquired with a Nal-Photomultiplier Detector by Constrained Least-Squares Deconvolution

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A constrained least-squares technique to correct diagnostic x-ray tube energy spectra for inherent blurring by scintillation detectors was developed. The measured detector response function to monoenergetic sources was used to construct a matrix that modeled the energy broadening in the crystal. This blurring operator, along with an estimate of statistical noise in the count data, comprised the a priori system knowledge required for application of the method. Tungsten anode spectra up to 90 kVp were acquired with a Nal-photomultiplier detector system at a source-to-detector distance of 30 cm. X-ray tube output was collimated at the detector by a 0.5 mm diameter pinhole collimator. Measured Nal spectra were compared to both published reference data and to spectra acquired in our laboratory with a Ge detector system. Application of the constrained least-squares technique involved first defining a criterion function that combined an assessment of the goodness of fit with a weighted measure of the smoothness of the solution. Minimization of this function resulted in the corrected spectrum. While it is not possible to recover the characteristic tungsten peaks, the success of our method in deconvolving the measured spectra was demonstrated by a significant improvement in agreement with reference data. To provide a measure of this agreement, a histogram of the differences between the two curves was generated. The full width at half maximum (FWHM) of the Gaussian distribution fit to the histogram was used to quantify the similarity between the spectra and the reference data, both before and after correction. As spectral agreement improves, the FWHM becomes smaller. We show that application of the constrained least-squares technique improved spectral matching by 20%–60%.