Precise phase retrieval for propagation-based images using discrete mathematics

Abstract

The ill-posed problem of phase retrieval in optics, using one or more intensity measurements, has a multitude of applications using electromagnetic or matter waves. Many phase retrieval algorithms are computed on pixel arrays using discrete Fourier transforms due to their high computational efficiency. However, the mathematics underpinning these algorithms is typically formulated using continuous mathematics, which can result in a loss of spatial resolution in the reconstructed images. Herein we investigate how phase retrieval algorithms for propagation-based phase-contrast X-ray imaging can be rederived using discrete mathematics and result in more precise retrieval for single- and multi-material objects and for spectral image decomposition. We validate this theory through experimental measurements of spatial resolution using computed tomography (CT) reconstructions of plastic phantoms and biological tissues, using detectors with a range of imaging system point spread functions (PSFs). We demonstrate that if the PSF substantially suppresses high spatial frequencies, the potential improvement from utilising the discrete derivation is limited. However, with detectors characterised by a single pixel PSF (e.g. direct, photon-counting X-ray detectors), a significant improvement in spatial resolution can be obtained, demonstrated here at up to 17%.

Figure 1

Comparative analysis of the PM and GPM transfer functions. (a) Displays the ratio of the phase retrieval transfer functions, Eqs. (6) and (5), for various values of $\delta \mathrm{\Delta }/\mu W^2$, using a horizontal and diagonal slice of the 2D filter to create a bounded region presenting the asymmetry of the GPM filter. (b) Plots the horizontal, ($k_x,k_y=0$), line of the fractional difference in transfer functions described by Eq. (7), used as a comparison to (c) the imaging system transfer function, Eq. (12), which incorporates a $\mathcal{G}(k_x,k_y,\mathrm{\Gamma })$ PSF, set as $\mathrm{\Gamma }=3.0$ FWHM which reduces the plot’s vertical scaling, as well as the phase retrieval transfer function. Finally, (d) directly displays the effect of varying the PSF width, $\mathrm{\Gamma }$, on the imaging system transfer function, for the case $\delta \mathrm{\Delta }/\mu W^2=10$.

Figure 2

(a) Azimuthally averaged imaging system Line Spread Functions (LSF)s of the circular phantom image showing the effect of phase retrieval on spatial resolution from the phase contrast (PC) and phase retrieved images using the PM and GPM algorithms for a sample composed of water. Underlying dashed curves represent Pearson VII fits used to measure the LSF width. The phase contrast PSF was rescaled vertically by a factor of 8 for plotting. (b) Plots the percentage improvement in resolution, according to Eq. (14) of the GPM, plotted against the initial resolution of the simulated object.

Figure 3

The effect of rebinning the raw phase contrast image ($\times n$) upon the spatial resolution of the phase-retrieved images, using the PM and GPM methods. (a) Plots the transfer function fractional difference, Eq. (12), along the $(k_x,k_y=0)$ line, for various levels of rebinning. Here the filter conditions were chosen to initially match the detector PSF and the conditions used in (b) which shows the Pearson VII fit reconstructions of imaging system PSF measurements conducted at the outer edge of a cylindrical PMMA phantom, shown in inset.

Figure 4

PSFs measured by azimuthally averaging the outer edge PMMA cylinders in CT recorded at 2 m propagation distance with $24\text{keV}$ beam energy using (a) a Advacam Modupix and (b) a LAMBDA photon-counting detector. Voxel size = $55\mathrm{\mu }\text{m}$. The PMMA cyclinder used in (a) contains an off-centre circular cavity while the cylinder in (b) is solid PMMA nearing the detector width in size.

Figure 5

CT slices of a rat lung reconstructed from phase retrieved projections using either the (a) PM or (c) GPM algorithms. (b) Shows an interleaved image of the two methods, with a blue band along the bottom in columns from the PM method and a red band along the top in columns from the GPM, with an inset region in (b) bounded by dashed green lines magnified in (d) for direct comparison of the two methods.

Figure 6

Pearson VII fit comparison of spatial resolution achieved through a PM and GPM application of the Beltran two-material phase retrieval algorithm, featuring a PMMA phantom with aluminium inset. Data was recorded using the LAMBDA detector at a propagation distance of $2\text{m}$, and spatial resolution analysis was performed using the aluminium-PMMA material interface.

Figure 7

PSF comparison of the dual-energy phase retrieval images showing reconstructed slices of electron density maps shown in the graph inset. PSFs have been fit with Pearson VII functions for quantifying the PSF FWHM. (a) Shows a spatial resolution improvement of $16(6)\%$ at $1\text{m}$ propagation distance, while (b) shows the same phantom at $2\text{m}$ propagation distance showing a $29(2)\%$ improvement, likely due to the higher SNR afforded by low-pass filtering in the phase retrieval algorithm.