84 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 1, JANUARY 2011 Sufficient Statistics as a Generalization of Binning in Spectral X-ray Imaging Adam S. Wang*, Student Member, IEEE, and Norbert J. Pelc Abstract—It is well known that the energy dependence of X-ray attenuation can be used to characterize materials. Yet, even with energy discriminating photon counting X-ray detectors, it is still unclear how to best form energy dependent measurements for spectral imaging. Common ideas include binning photon counts based on their energies and detectors with both photon counting and energy integrating electronics. These approaches can be generalized to energy weighted measurements, which we prove can form a sufficient statistic for spectral X-ray imaging if the weights used, which we term -weights, are basis attenuation functions that can also be used for material decomposition. To study the performance of these different methods, we evaluate the Cramér-Rao lower bound (CRLB) of material estimates in the presence of quantum noise. We found that the choice of binning and weighting schemes can greatly affect the performance of material decomposition. Even with optimized thresholds, binning condenses information but incurs penalties to decomposition precision and is not robust to changes in the source spectrum or object size, although this can be mitigated by adding more bins or removing photons of certain energies from the spectrum. On the other hand, because -weighted measurements form a sufficient statistic for spectral imaging, the CRLB of the material decomposition estimates is identical to the quantum noise limited performance of a system with complete energy information of all photons. Finally, we show that -weights lead to increased conspicuity over other methods in a simulated calcium contrast experiment. Index Terms—Cramér-Rao lower bound, dual energy, energy weighting, photon counting, spectral imaging, sufficient statistic. I. INTRODUCTION I T is well known that a material’s X-ray attenuation is dependent on the energies of the X-rays and that this dependence can be used for material selective imaging [1], [2]. Therefore, there is information about a measured object contained not only in the total number of photons transmitted through the object but also in the energy of each of these photons. How this information is collected greatly affects the ability of a system to efficiently utilize the energy dependence in what is known as spectral X-ray imaging. Early methods for spectral imaging include the use of dual kVp techniques that Manuscript received May 23, 2010; revised July 15, 2010; accepted July 17, 2010. Date of publication August 03, 2010; date of current version December 30, 2010. This work was supported in part by GE Healthcare and in part by the Lucas Foundation. Asterisk indicates corresponding author. *A. S. Wang is with the Departments of Electrical Engineering and Radiology, Stanford University, Stanford, CA 94305 USA (e-mail: adamwang@stanford.edu). N. J. Pelc is with the Departments of Radiology, Bioengineering, and Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail: pelc@stanford.edu). Digital Object Identifier 10.1109/TMI.2010.2061862 require two different exposures—one at a lower energy than the other—or the use of dual layer detectors. Both methods inherently suffer from spectral overlap, which degrades the spectral imaging performance [3]–[7]. Energy discriminating detectors offer the possibility of eliminating spectral overlap by directly partitioning the transmitted spectrum into sepa- rate measurements [8]–[10]. Our project began with work to understand and optimize the partition of the spectrum into bins and led to the observation that weighted measurements, a generalization of binning, can achieve universally optimal performance. Here, we evaluate several methods for forming energy dependent measurements in the context of ideal photon counting detectors with energy discriminating capabilities. The work led us to the discovery of a simple and elegant sufficient statistic for spectral X-ray imaging. II. BACKGROUND For now, we assume that the materials being measured have no K-edges within the detected X-ray spectrum; this will be relaxed later. Spectral imaging in the diagnostic energy range is often referred to as dual-energy imaging, which is based on the observation that X-rays in this energy range primarily in- teract by two physical mechanisms—photoelectric absorption and Compton scattering—and that in the absence of K-edges the energy dependence of these mechanisms is independent of the material [1]. Thus, materials without K-edges within the de- tected spectrum behave as if they were some linear combination of material independent basis functions (e.g., photoelectric ab- sorption as a function of energy and Compton scattering as a function of energy). In other words, in the energy range of in- terest, material 1 will have a linear attenuation coefficient at en- ergy that can be expressed as (1) where and depend on the material, and and are the attenuation functions attributable to the photoelectric and Compton interactions, respectively. Moreover, because all ma- terials without K-edges attenuate X-rays based on the same un- derlying principles, any material can be expressed as a linear combination of any other two materials. For materials 1, 2, and 3, constants and can be found such that (2) Thus, any object can be described as an amount of mate- rial 1 and an amount of material 2. To effectively estimate 0278-0062/$26.00 © 2010 IEEE