Point Spread Function Estimation in X-Ray Imaging with Partially Collapsed Gibbs Sampling

Abstract

The point spread function (PSF) of a translation invariant imaging system is its impulse response, which cannot always be measured directly. This is the case in high energy X-ray radiography, and it must be estimated from images of calibration objects indirectly related to the impulse response. When the PSF is assumed to have radial symmetry, it can be estimated from an image of an opaque straight edge. We use a non-parametric Bayesian approach, where the prior probability density for the PSF is modeled as a Gaussian Markov random field and radial symmetry is incorporated in a novel way. Markov Chain Monte Carlo posterior estimation is carried out by adapting a recently developed improvement to the Gibbs sampling algorithm, referred to as partially collapsed Gibbs sampling. Moreover, the algorithm we present is proven to satisfy invariance with respect to the target density. Finally, we demonstrate the efficacy of these methods on radiographic data obtained from a high-energy X-ray diagnostic system at the

Efficient Marginalization-Based MCMC Methods for Hierarchical Bayesian Inverse Problems

“…Gibbs sampling [3] and low-rank independence sampling [8] have been proposed for this type of problem, but the statistical efficiency of the resulting MCMC chains produced by either approach decreases as the dimension of the state increases. We discuss this phenomenon in detail and consider a solution using marginalization-based methods [22,33,45]. Marginalization can be just as computationally prohibitive for sufficiently large-scale problems.…”

“…This is because the Gaussian sample required for each iteration can be prohibitively expensive to compute, and because the statistical efficiency of the Markov chain degrades as the dimension of the state increases. The latter problem can be mitigated using marginalization-based techniques, such as those found in [22,33,45], but these can be computationally prohibitive as well. In this paper, we combine the low-rank techniques of [8] with the marginalization approach of [45].…”

An approach to characterizing spatial aspects of image system blur

“…The edge of the shown optically thick object with aperture provides an indirect measurement of the impulse response of the imaging system. A point-spread-function (PSF) is estimated using the techniques outlined in [14]. Distances between the X-ray source, the optically thick object being imaged, the scintillator, and the CCD array are in green.…”

“…See Figure 2 for an example of a checkerboard radiograph (top), with an averaged lineout of the rows highlighted in blue (bottom). Profiles from the checkerboard are used to estimate the spot size from a resulting integral equation at each edge [14]. Relatedly, in [15,18] the PSF is estimated along a single edge of a calibration-test object by means of parametric forms -a common approach used in a variety of applications (e.g., [5,24]).…”

“…Relatedly, in [15,18] the PSF is estimated along a single edge of a calibration-test object by means of parametric forms -a common approach used in a variety of applications (e.g., [5,24]). These methods are often nonlinear (exacerbating difficulties in the estimation and quantification of uncertainty), and parametric forms are often not adequate to capture an accurate representation of the blur that is the result of many components that act in aggregate [14].…”