Cramér–Rao lower bound of basis image noise in multiple-energy x-ray imaging

Abstract

We present an analytical method to compute the basis image noise in the context of multi-energy x-ray imaging based on the Cramér-Rao lower bound (CRLB). The proposed formalism extends the original idea of Alvarez and Macovski (1976 Phys. Med. Biol. 21 733) to estimate the noise in the photo-effect and Compton-effect basis images in the case of dual-energy imaging. It includes an arbitrary number of independent, spectrally distinct attenuation measurements and also goes beyond the two-dimensional decomposition of the attenuation, including, e.g., a contrast agent as a third basis material. To illustrate our method, we consider three simple applications. The first application is to study the influence of the exact values for the energy thresholds on the basis image noise for a binned photon-counting system. The second application relates to the same detector system as the first and is an investigation of the dependence of the basis image noise on the energy resolution of the detector system. Finally, the third application provides an example for the case of an energy-integrating detector: the aim is to optimize the front-scintillator layer thickness of a dual-crystal detector for dual-energy imaging. The CRLB is used to minimize the noise of a photo-effect/Compton-effect basis material decomposition.

Allowable Forward Model Misspecification for Accurate Basis Decomposition in a Silicon Detector Based Spectral CT

“…where (13) and the bias is obtained by solution of the maximum likelihood problem given the observed bin counts (14) where according to (3) [7]. For the bias calculation the expected value of counts in each bin was used, i.e., we apply the approximation .…”

“…Due to the channel variation of the gain (in units mV/keV) and the offset (in units mV), these reference voltages will correspond to different deposited energies in each channel (which can be thought of as a detector element) (7) In an energy weighted image, such channel inhomogeneities would result in ring artifacts if not properly compensated for [18] and [19]. When material basis decomposition is applied to the sinogram data, the inhomogeneities require that different forward equation parameters are used for each channel , according to (7). A problem of misspecification might occur however since and are only known up to certain precision (and their values are known to drift with temperature, albeit in a predictable fashion) [10].…”

“…A second approach is to directly determine the line integrals of from the sinogram data using a maximum likelihood approach and then reconstruct 's using the inverse Radon transform. Such an approach for multibin systems has been presented by Rößl and Herrmann [7] and successfully applied to real data on a prototype system by Rößl and Proksa [8].…”

“…The path connects the X-ray source with the detector element denotes the energy distribution of the X-ray spectrum, the detection efficiency. Following the notation of Rößl and Herrmann [7] denotes the probability that an X-ray photon with energy deposits charge corresponding to an energy in the detector and is therefore a bin function, giving the fraction of photons with energy that are deposited in the th bin. The parameters of the forward model are thus and , where is the number of energy bins.…”

Non-invasive material discrimination using spectral x-ray radiography

“…This selection of bin thresholds is based on previous analyses. 26,27 RESULTS AND DISCUSSION Figure 2 shows simulated transmission data, d, recorded by the spectral detector for the object in Figure 1, summed over all energy bins. The slight profile of the plutonium wedge can be seen in the middle of the image.…”