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PURPOSE: To derive fundamental limits on the effect of pulse pileup and quantum noise in photon counting detectors on the signal to noise ratio (SNR) and noise variance of energy selective x-ray imaging systems.METHODS: An idealized model of the response of counting detectors to pulse pileup is used. The model assumes a nonparalyzable response and delta function pulse shape. The model is used to derive analytical formulas for the noise and energy spectrum of the recorded photons with pulse pileup. These formulas are first verified with a Monte Carlo simulation. They are then used with a method introduced in a previous paper [R. E. Alvarez, "Near optimal energy selective x-ray imaging system performance with simple detectors," Med. Phys. 37, 822-841 (2010)] to compare the signal to noise ratio with pileup to the ideal SNR with perfect energy resolution. Detectors studied include photon counting detectors with pulse height analysis (PHA), detectors that simultaneously measure the number of photons and the integrated energy (NQ detector), and conventional energy integrating and photon counting detectors. The increase in the A-vector variance with dead time is also computed and compared to the Monte Carlo results. A formula for the covariance of the NQ detector is developed. The validity of the constant covariance approximation to the Cramèr-Rao lower bound (CRLB) for larger counts is tested.RESULTS: The SNR becomes smaller than the conventional energy integrating detector (Q) SNR for 0.52, 0.65, and 0.78 expected number photons per dead time for counting (N), two, and four bin PHA detectors, respectively. The NQ detector SNR is always larger than the N and Q SNR but only marginally so for larger dead times. Its noise variance increases by a factor of approximately 3 and 5 for the A1 and A2 components as the dead time parameter increases from 0 to 0.8 photons per dead time. With four bin PHA data, the increase in variance is approximately 2 and 4 times. The constant covariance approximation to the CRLB is valid for larger counts such as those used in medical imaging.CONCLUSIONS: The SNR decreases rapidly as dead time increases. This decrease places stringent limits on allowable dead times with the high count rates required for medical imaging systems. The probability distribution of the idealized data with pileup is shown to be accurately described as a multivariate normal for expected counts greater than those typically utilized in medical imaging systems. The constant covariance approximation to the CRLB is also shown to be valid in this case. A new formula for the covariance of the NQ detector with pileup is derived and validated. Source

Alvarez R.E.,Aprend Technology
Medical Physics

Purpose: To develop and test a method to quantify the effect of dimensionality on the noise in energy selective x-ray imaging. Methods: The Cramèr-Rao lower bound (CRLB), a universal lower limit of the covariance of any unbiased estimator, is used to quantify the noise. It is shown that increasing dimensionality always increases, or at best leaves the same, the variance. An analytic formula for the increase in variance in an energy selective x-ray system is derived. The formula is used to gain insight into the dependence of the increase in variance on the properties of the additional basis functions, the measurement noise covariance, and the source spectrum. The formula is also used with computer simulations to quantify the dependence of the additional variance on these factors. Simulated images of an object with three materials are used to demonstrate the trade-off of increased information with dimensionality and noise. The images are computed from energy selective data with a maximum likelihood estimator. Results: The increase in variance depends most importantly on the dimension and on the properties of the additional basis functions. With the attenuation coefficients of cortical bone, soft tissue, and adipose tissue as the basis functions, the increase in variance of the bone component from two to three dimensions is 1.4 × 103. With the soft tissue component, it is 2.7 × 104. If the attenuation coefficient of a high atomic number contrast agent is used as the third basis function, there is only a slight increase in the variance from two to three basis functions, 1.03 and 7.4 for the bone and soft tissue components, respectively. The changes in spectrum shape with beam hardening also have a substantial effect. They increase the variance by a factor of approximately 200 for the bone component and 220 for the soft tissue component as the soft tissue object thickness increases from 1 to 30 cm. Decreasing the energy resolution of the detectors increases the variance of the bone component markedly with three dimension processing, approximately a factor of 25 as the resolution decreases from 100 to 3 bins. The increase with two dimension processing for adipose tissue is a factor of two and with the contrast agent as the third material for two or three dimensions is also a factor of two for both components. The simulated images show that a maximum likelihood estimator can be used to process energy selective x-ray data to produce images with noise close to the CRLB. Conclusions: The method presented can be used to compute the effects of the object attenuation coefficients and the x-ray system properties on the relationship of dimensionality and noise in energy selective x-ray imaging systems. © 2013 American Association of Physicists in Medicine. Source

Purpose: 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 Cramr-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. © 2011 American Association of Physicists in Medicine. Source

Alvarez R.E.,Aprend Technology
IEEE Transactions on Medical Imaging

An estimator to image contrast agents and body materials with x-ray spectral measurements is described. The estimator is usable with the three or more basis functions that are required to represent the attenuation coefficient of 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 likelihood estimator. The estimator first computes a linearized maximum likelihood estimate of the line integrals of the basis set coefficients. Corrections for errors in the initial estimates are computed by interpolation with calibration phantom data. The final estimate is the initial estimate plus the correction. 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. The estimator mean squared error is essentially equal to the CRLB. The estimator outputs are close to those of the iterative estimator but the computation time is approximately 180 times shorter. The estimator is efficient and has advantages over alternate approaches such as iterative estimators. © 2015 IEEE. Source

Purpose: This article describes a method to achieve near optimal performance with low energy resolution detectors. Tapiovaara and Wagner [Phys. Med. Biol. 30, 519-529 (1985)] showed that an energy selective x-ray system using a broad spectrum source can produce images with a larger signal to noise ratio (SNR) than conventional systems using energy integrating or photon counting detectors. They showed that there is an upper limit to the SNR and that it can be achieved by measuring full spectrum information and then using an optimal energy dependent weighting. Methods: A performance measure is derived by applying statistical detection theory to an abstract vector space of the line integrals of the basis set coefficients of the two function approximation to the x-ray attenuation coefficient. The approach produces optimal results that utilize all the available energy dependent data. The method can be used with any energy selective detector and is applied not only to detectors using pulse height analysis (PHA) but also to a detector that simultaneously measures the total photon number and integrated energy, as discussed by Roessl [Med. Phys. 34, 959-966 (2007)]. A generalization of this detector that improves the performance is introduced. A method is described to compute images with the optimal SNR using projections in a "whitened" vector space transformed so the noise is uncorrelated and has unit variance in both coordinates. Material canceled images with optimal SNR can also be computed by projections in this space. Results: The performance measure is validated by showing that it provides the Tapiovaara-Wagner optimal results for a detector with full energy information and also a conventional detector. The performance with different types of detectors is compared to the ideal SNR as a function of x-ray tube voltage and subject thickness. A detector that combines two bin PHA with a simultaneous measurement of integrated photon energy provides near ideal performance across a wide range of operating conditions. Conclusions: Low energy resolution detectors can be used in energy selective x-ray imaging systems to produce images with near optimal performance. © 2010 American Association of Physicists in Medicine. Source

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