Quantifying the Importance of the Statistical Assumption in Statistical X-ray CT Image Reconstruction

Abstract

Statistical image reconstruction (SIR) is a promising approach to reducing radiation dose in clinical computerized tomography (CT) scans. Clinical CT scanners use energy-integrating detectors. The CT signal follows a compound Poisson distribution, its probability density function (PDF) does not have an analytical form hence cannot be used in an SIR method. The goal of this work is to quantify the effects of using an approximate statistical assumption in SIR methods for clinical CT applications. We apply a pseudo-Ideal Observer (pIO) to simulated CT projection data of the fanbeam geometry at different dose levels. The simulation models the polychromatic X-ray tube spectrum, the effects of the bowtie filter, and the energy-integrating detectors. The pIO uses a pseudo likelihood function (pLF) to calculate the pseudo likelihood ratio, which is the decision variable used by the pIO in a binary detection task. The pLF is an approximation to the true LF of the underlying data. The pIO has inferior performance than the IO unless the pLF coincides with the LF; this performance difference quantifies the closeness between the pseudo likelihood and the exact one. Using lesion detectability in a signal known exactly, background known exactly binary detection task as a figure-of-merit, our results show that at down to 0.1% of a reference tube current level I0, the pIO that uses a Poisson approximation, or a matched variance Gaussian approximation in either the transmission or the line integral domain, achieves 99% the performance of the IO. The constant variance Gaussian approximation has only 70%-80% of the IO performance. At tube currents lower than 0.1% I0, the performance difference is more substantial. We conclude that at current clinical dose levels, it is important to account for the mean-dependent variance in CT projection data in SIR problem formulation, the exact PDF of the CT signal is not as important.

Blind X-Ray CT Image Reconstruction From Polychromatic Poisson Measurements

“…The Hessian expression in (23) implies that L A .I/ in (22b) is strongly convex if the design matrix A has full rank. Combining the convexity results in Lemmas 1 and 2 yields the biconvexity region for the NLL L.˛; I/ in (16).…”

“…To establish convexity of the NLL (18), we enforce monotonicity of the mass-attenuation spectrum ι.Ä/ in low-and high-Ä regions and also assume that the mid-Ä region has higher spectrum than the low-Ä region. Note that we do not require here that ι.Ä/ satisfy the basis-function expansion (10a); however, (10a) will be needed to establish the biconvexity of the NLL in (16). Hence, we define the three Ä regions using the spline parameters (13e) as well as an additional integer constant…”

“…We mimic real X-ray CT system calibration by scaling projection matrixâ nd spectrum Ã. "/ so that the maximum and minimum of the noiseless measurements .E E n / N nD1 are 2 16 and 20, respectively. Here, the scale ofˆcorresponds to the real size that each image pixel represents, and the scale of Ã.…”

“…We develop a blind sparse density-map reconstruction scheme from measurements corrupted by Poisson noise, where the signal sparsity in the density-map domain is enforced using a totalvariation (TV) norm penalty. The Poisson noise model is appropriate for measurements from photon-counting detectors and a good approximation for the more precise compound Poisson distribution for measurements from energy-integrating detectors [16,17].…”

Blind polychromatic X-ray CT reconstruction from poisson measurements

“…The Poisson noise model is appropriate for measurements from photon-counting detectors and a good approximation for the more precise compound-Poisson distribution for measurements from energy-integrating detectors [11,12].…”