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-Purpose: A neural network estimator to process x-ray spectral measurementsfrom photon counting detectors with pileup. The estimator is used with anexpansion of the attenuation coefficient as a linear combination of functionsof energy multiplied by coefficients that depend on the material composition atpoints within the object [R.E. Alvarez and A. Macovski, Phys. Med. Biol., 1976,733-744]. The estimator computes the line integrals of the coefficients frommeasurements with different spectra. Neural network estimators are trained withmeasurements of a calibration phantom with the clinical x-ray system. Oneestimator uses low noise training data and another network is trained with datacomputed by adding random noise to the low noise data. The performance of theestimators is compared to each other and to the Cramer-Rao lower bound (CRLB).Methods: The estimator performance is measured using a Monte Carlo simulationwith an idealized model of a photon counting detector that includes only pileupand quantum noise. Transmitted x-ray spectra are computed for a calibrationphantom. The transmitted spectra are used to compute random data for photoncounting detectors with pileup. Detectors with small and large dead times areconsidered. Neural network training data with extremely low noise are computedby averaging the random detected data with pileup for a large numbers ofexposures of the phantom. Each exposure is equivalent to a projection image orone projection of a computed tomography scan. Training data with high noise arecomputed by using data from one exposure. Finally, training data are computedby adding random data to the low noise data. The added random data aremultivariate normal with zero mean and covariance equal to the samplecovariance of data for an object with properly chosen attenuation. To test theestimators, random data are computed for different thicknesses of three testobjects with different compositions. These are used as inputs to the neuralnetwork estimators. The mean squared errors (MSE), variance and square of thebias of the neural networks' outputs with the random object data are eachcompared to the CRLB. Results: The MSE for a network trained with low noisedata and added noise is close to the CRLB for both the low and high pileupcases. Networks trained with very low noise data have low bias but largevariance for both pileup cases. ralvarez@aprendtech.com Networks trained withhigh noise data have both large bias and large variance. Conclusion: With aproperly chosen level of added training data noise, a neural network estimatorfor photon counting data with pileup can have variance close to the CRLB withnegligible bias.

Near optimal neural network estimator for spectral x-ray photon counting data with pileup