1 Efficient, non-iterative estimator for imaging contrast agents with spectral x-ray detectors Robert E. Alvarez Abstract —An estimator to image contrast agents and body materials with x-ray spectral measurements is de- scribed. The estimator is usable with the three or more basis functions that are required with high atomic number materials. The estimator variance is equal to the Cramèr- Rao lower bound (CRLB) and it is unbiased. Its parameters are computed from measurements of a calibration phantom with the clinical x-ray system and it is non-iterative. The estimator is compared with an iterative maximum likeli- hood estimator. Methods: The estimator first computes a linearized maximum likelihood estimate of the line in- tegrals of the basis set. Corrections for bias errors in the initial estimates are computed by interpolation with calibration phantom data. The final estimate is the initial estimate plus the correction. The estimator parameters are computed from measurements of a calibration phantom with the clinical x-ray system. The performance of the estimator is measured using a Monte Carlo simulation. Random photon counting with pulse height analysis data are generated. The mean squared errors of the estimates are compared to the CRLB. The random data are also processed with an iterative maximum likelihood estimator. Previous implementations of iterative estimators required advanced physics instruments not usually available in clinical institutions. Results: The estimator mean squared error (MSE) is essentially equal to the CRLB. The estimator outputs are close to those of the iterative estimator. The computation time is approximately 180 times shorter than the iterative implementation. Conclusion: The estimator is efficient and has advantages over alternate approaches such as iterative estimators. Key Words: spectral CT, dual energy, energy selective, Cramèr-Rao lower bound, Monte Carlo, maximum likeli- hood estimator, iterative estimator I. INTRODUCTION The state of the art of photon counting detectors is advancing rapidly and their use in clinical systems may be possible in the near future[1], [2]. These detectors have the capability to measure the energy of the individual photons with pulse height analysis (PHA). Since each PHA energy bin can be considered to provide a separate spectral measurement, they have the potential to provide many more spectra than were previously available and methods to process this information are required. This paper describes an estimator that uses these multiple x-ray spectrum measurements to extract energy dependent information optimally. The estimator is efficient over a large range of object thicknesses with variance essentially equal to the Cramèr-Rao lower bound (CRLB) and with bias much smaller than the noise standard devi- ation. The estimator is non-iterative and the parameters required to implement it can be computed from mea- surements of a calibration phantom with the clinical x-ray system. The rationale for the estimator is described and its use is justified by its empirical performance measured with a Monte Carlo simulation. The estimator implements a key step in the Alvarez- Macovski method[3]. With this method, the x-ray attenu- ation coefficient is approximated as a linear combination of basis functions of energy multiplied by coefficients that are independent of energy. The estimator uses measure- ments of the number of photons transmitted through the object with different spectra to compute the line integrals of the basis set coefficients. The vector of these line integrals will be referred to as the A-vector here. The number of basis functions, the dimensionality, determines the available information and the minimum number of spectra required to extract it[4]. A two function basis set is sufficient to approximate the attenuation coefficients of biological materials[3]. However, a three or higher dimension basis set is needed if an externally administered high atomic number contrast agent is used. In this case, measurements with three or more effective spectra are required. A previous paper[5] described a two dimension estimator[6], [7]. Because of the two dimension limita- tion, this implementation could not be used with contrast agents. In this paper, a new estimator is described that is usable with three or more basis functions as well as two functions. The rationale for the estimator is based on the near linearity of the logarithm of the number of photons trans- mitted through the object as a function of the A-vector. Assuming a linear system and a multivariate normal dis- tribution, the linear maximum likelihood estimator (linear MLE) is the minimum variance unbiased estimator[8]. Because the nonlinearity is small, the variance of the linear estimates is close to the CRLB of the actual, nonlinear system. The nonlinearity, however, leads to bias of the linear estimates that needs to be minimized for quantitative applications such as computed tomography (CT). This is done by correcting the initial estimates based on known calibration data. The final estimate is the initial estimate plus the correction. Since the correction values are computed from a table of A-vector values, the estimator is called the A-table estimator. A method to compute the parameters of the estimator from measurements of a calibration phantom with the clinical x-ray system is described. The design of the calibrator and ways to compute the estimator parameters