Noise and analyzer-crystal angular position analysis for analyzer-based phase-contrast imaging

Abstract

The analyzer-based phase-contrast x-ray imaging (ABI) method is emerging as a potential alternative to conventional radiography. Like many of the modern imaging techniques, ABI is a computed imaging method (meaning that images are calculated from raw data). ABI can simultaneously generate a number of planar parametric images containing information about absorption, refraction, and scattering properties of an object. These images are estimated from raw data acquired by measuring (sampling) the angular intensity profile of the x-ray beam passed through the object at different angular positions of the analyzer crystal. The noise in the estimated ABI parametric images depends upon imaging conditions like the source intensity (flux), measurements angular positions, object properties, and the estimation method. In this paper, we use the Cramér-Rao lower bound (CRLB) to quantify the noise properties in parametric images and to investigate the effect of source intensity, different analyzer-crystal angular positions and object properties on this bound, assuming a fixed radiation dose delivered to an object. The CRLB is the minimum bound for the variance of an unbiased estimator and defines the best noise performance that one can obtain regardless of which estimation method is used to estimate ABI parametric images. The main result of this paper is that the variance (hence the noise) in parametric images is directly proportional to the source intensity and only a limited number of analyzer-crystal angular measurements (eleven for uniform and three for optimal non-uniform) are required to get the best parametric images. The following angular measurements only spread the total dose to the measurements without improving or worsening CRLB, but the added measurements may improve parametric images by reducing estimation bias. Next, using CRLB we evaluate the multiple-image radiography, diffraction enhanced imaging and scatter diffraction enhanced imaging estimation techniques, though the proposed methodology can be used to evaluate any other ABI parametric image estimation technique.

Figure 1

Schematic diagram of ABI imaging system (not drawn to scale). In this paper, we investigate the effect of the analyzer position on the noise performance.

Figure 2

MIR projections of a human thumb. The images are acquired with the beam energy of 40keV at 25 analyzer positions ranging from −4.8 to 4.8 μrad with 0.4 μrad increments (see experimental sections for acquisition details).

Figure 3

Example of a Rocking Curve for Si (333) Crystal at beam energy of (a) 40 keV (FWHM = 1.4μrad); (b) 60keV (FWHM = 1.0μrad).

Figure 4

Examples of information functions for parameter plotted on the logarithmic scale. Each function indicates the relative value of obtaining an angular measurement at a given angular analyzer position θ_l in regard to estimation of one of the object parameters. (a) For thumb sample imaged at 40 keV, (b) for breast data imaged at 60keV, in c) and d) we show information functions for extreme object parameters.

Figure 5

CRLB for an unbiased estimator vs. number of angular measurements using a uniform set of analyzer angular positions. (a) For thumb sample imaged at 40 keV, (b) for breast data imaged at 60keV.

Figure 6

CRLB for an unbiased estimator vs. number of angular measurements for both optimal and uniform set of analyzer angular positions. (a) For thumb sample imaged at 40 keV, (b) for breast data imaged at 60keV.

Figure 7

Analyzer angular positions in optimal and uniform set of analyzer angular positions for different number of angular measurements. (a) For thumb sample imaged at 40 keV, (b) for breast data imaged at 60keV.

Figure 8

Variance, MSE, squared bias, and CRLB vs. number of angular measurements for optimized set of analyzer angular positions. (a) For thumb sample imaged at 40 keV, (b) for breast data imaged at 60keV.

Figure 9

Estimated MIR images using uniform set of analyzer angular positions with different number of angular measurements. Observe that there is little or no improvement after L>11.