• We start with preliminary comparisons of the PM and GPM strategies by trying on the respective spatial filters. We incorporate further filtering to imitate different phases of imaging, to raised emulate how spatial frequencies within the pattern are captured within the uncooked picture after which seem within the remaining retrieved pattern picture. These two strategies are additionally utilized to simulated projection photographs to measure the ensuing spatial decision of the imaging system.

### Evaluation of system switch capabilities

The PM, Eq. (1), is finally a tailor-made spatial frequency filter, derived beneath the transport of depth equation, used to blur a picture such that the section distinction seen at materials boundaries is unfold to reconstruct the pattern thickness. On condition that the one distinction between the PM and GPM strategies is the form of this spatial frequency filter, this turns into our first level of comparability. From Eqs. (1) and (3), we are able to outline switch capabilities for the appliance of every filter to uncooked photographs as

start{aligned} {mathscr{H}}_{textual content{PM}}(k_{x}, k_{y})&= frac{1}{1 + frac{delta Delta }{mu }(k_{x}^2 + k_{y}^2)}, finish{aligned}

(5)

start{aligned} {mathscr{H}}_{textual content{GPM}}(k_{x}, k_{y})&= frac{1}{1 – frac{2 delta Delta }{mu W^2}[cos (W k_{x}) + cos (W k_{y}) – 2]}, finish{aligned}

(6)

given as Eqs. (20) and (19) in Paganin et al.18, the place ({mathscr{H}}_{textual content{PM}}(k_{x}, k_{y})) represents the amplification utilized to every spatial frequency amplitude by the PM, and ({mathscr{H}}_{textual content{GPM}}(k_{x}, k_{y})) the amplification utilized by the GPM. From right here, a easy comparability between Eqs. (5) and (6) could be carried out by taking their ratio (Eq. (21) in2), (R(k_x,k_y) = {mathscr{H}}_{textual content{GPM}}(k_{x}, k_{y})/{mathscr{H}}_{textual content{PM}}(k_{x}, k_{y})) and plotting the end result on a normalized axis. Determine 1a does this for a variety of values of the mixed parameter (delta Delta /mu W^2), displaying the fractional distinction between the GPM and PM as a shaded area bounded by the (|k_y| cup |k_x|= 0) (higher certain) and (|k_x| = |k_y|) (decrease certain) traces. For the (|k_x| = |k_y|) traces, seen because the decrease certain of the Fig. 1a plots, we produce the one-dimensional spatial frequency axis by means of (k_r = Wsqrt{k_{x}^2 + k_{y}^2}), resulting in values above (pi) the place ((k_x, k_y)) prolong into the corners of a sq. picture. The shaded areas assist to reveal the new-found asymmetry of the GPM filter, correcting the PM algorithm to ship a extra uniform remedy of all edges in real-space, no matter their orientation. A extra useful technique to reveal quantitative variations between every algorithm is thru the fractional distinction, outlined as

start{aligned} {mathscr{D}}(k_{x}, k_{y})&= frac{{mathscr{H}}_{textual content{GPM}}(k_{x}, k_{y}) – {mathscr{H}}_{textual content{PM}}(k_{x}, k_{y})}{{mathscr{H}}_{textual content{PM}}(k_{x}, k_{y})}, finish{aligned}

(7)

which converges towards zero when the PM and GPM filters match, versus the ratio of Eqs. (5) and (6) which converges to 1. Determine 1b plots Eq. (7) throughout the (k_x) or (k_y) axis line ((|k_y| cup |k_x| = 0)). We see once more that the bigger frequencies expertise the best distinction between the 2 algorithms, resulting in elevated spatial decision within the GPM because of the better proportion of excessive spatial frequencies, and that they’re successfully indistinguishable on the decrease spatial frequencies. Nevertheless, Eqs. (5)–(7) solely signify the switch operate of the post-image processing, whereas to raised replicate experimental circumstances we should additionally account for the blurring impact of the detector imaging system and optical system (i.e. all blurring results other than the propagation and section retrieval). To do that we introduce the distinction switch operate ({mathscr{G}}(k_{x}, k_{y}, Gamma )). We describe the real-space detector PSF as an azimuthally symmetric Gaussian, in order that ({mathscr{G}}(k_{x}, k_{y}, Gamma )) is given by the Fourier remodel

start{aligned} {mathscr{G}}(k_{x}, k_{y}, Gamma )&= {mathscr{F}}left[ exp left( -frac{x^2+y^2}{Gamma ^2/2.355^2}right) right] finish{aligned}

(8)

start{aligned} = exp (-Gamma ^2(4pi ^2k_{x}^2 + 4pi ^2k_{y}^2)/2.355^2), finish{aligned}

(9)

the place (Gamma) is the a full width at half most (FWHM) in actual house, measured in pixels, and the components of (4pi ^2) within the exponential replicate the DFT normalization conference. Combining the imaging system switch operate, Eq. (9), with the section retrieval switch capabilities, Eqs. (5)–(6), offers the whole switch capabilities (bar{mathscr{H}}) as

start{aligned} bar{mathscr{H}}_{textual content{PM}}(k_{x}, k_{y})&= {mathscr{H}}_{textual content{PM}}(k_{x}, k_{y}){mathscr{G}}(k_{x}, k_{y}, Gamma ), finish{aligned}

(10)

start{aligned} bar{mathscr{H}}_{textual content{GPM}}(k_{x}, k_{y})&= {mathscr{H}}_{textual content{GPM}}(k_{x}, k_{y}){mathscr{G}}(k_{x}, k_{y}, Gamma ), finish{aligned}

(11)

permitting us to equally outline a brand new fractional distinction as

start{aligned} bar{mathscr{D}}(k_{x}, k_{y})&= frac{bar{mathscr{H}}_{textual content{GPM}}(k_{x}, k_{y}) – bar{mathscr{H}}_{textual content{PM}}(k_{x}, k_{y})}{{mathscr{H}}_{textual content{PM}}(k_{x}, k_{y})}. finish{aligned}

(12)

By incorporating the blurring results of the imaging and imaging system into Eq. (7), we are able to increase on the filter comparisons in Paganin et al.18 and reveal how imaging PSFs could restrict the relative spatial decision enchancment between the algorithms. Determine 1c plots Eq. (12) for a similar (delta Delta /mu W^2) values as in Fig. 1b, now together with a Gaussian PSF with FWHM of three pixels. We see that the amplitudes of all spatial frequencies are diminished relative to panels (a) and (b), significantly on the larger spatial frequencies, and the distinction between the GPM and PM is diminished general. This predicts that there could solely be a really small distinction in spatial decision between the PM and GPM strategies when the PSF (Gamma) is a couple of pixels extensive, as is typical of most oblique X-ray detectors. We additionally see that the most important distinction is now shifted to medium spatial frequencies on this instance. Determine 1d plots Eq. (12) for (delta Delta /mu W^2=10), whereas various the PSF measurement in pixels ((Gamma)). The ‘no convolution’ pattern shows the fractional distinction with out incorporating a PSF, ({mathscr{D}}(k_{x}, k_{y})), as a reference level. We observe that rising the PSF measurement decreases the amplitudes of excessive spatial frequencies, therefore seemingly reducing the potential enchancment to decision obtainable by way of the GPM. Nevertheless, for PSF widths round 1 pixel, typical of direct X-ray detectors, equivalent to photon counting detectors, the excessive spatial frequency amplitudes are nonetheless 20% elevated beneath the GPM algorithm. This leads us to suspect that direct detectors, equivalent to photon-counting detectors, will likely be finest suited to learn from the GPM algorithm, whereas for oblique X-ray detectors, which might possess PSF widths of two or extra pixels, the development could also be comparatively minor.

### Spatial decision enchancment in projection photographs

The earlier part confirmed how the form of the picture switch operate can range between the PM and GPM section retrieval algorithms on account of including life like PSFs to the system. Right here we quantify what impact the PSF has on spatial decision together with section retrieval by way of numerical simulation. We carried out the comparability by simulating the propagation of a wavefield by means of cylindrical phantoms utilizing the TIE (particulars under) till section distinction fringes had been produced. Subsequent, we utilized every section retrieval algorithm and created an azimuthally-averaged profile of the phantom edge, which was differentiated to create a line unfold operate (LSF) that may be measured to guage the spatial decision of the phase-retrieved picture. We then use the measured decision within the propagation-based section distinction picture, pre-phase retrieval, as a foundation for our decision comparability. Observe that, whereas LSFs are one dimensional, and PSFs are two dimensional, each are distinct measurements of spatial decision entities, and we’ll use the phrases interchangeably all through the paper. A typical LSF is properly approximated by a Pearson VII operate,

start{aligned} P(x)&= Aleft[ 1 + frac{4(x – x_0)^2}{Gamma ^2}left( 2^{frac{1}{m}} – 1right) right] ^{-m}, finish{aligned}

(13)

the place (x_0) is the height place, (Gamma) is the FWHM and m is the exponent that units the place on the spectrum between Lorentzian ((m=1)) and Gaussian (approximated by (m>10)) behaviour. Lastly, we use the FWHM values of the imaging system PSF measured within the phase-retrieved photographs from the PM ((Gamma _{textual content{PM}})) and GPM ((Gamma _{textual content{GPM}})) to calculate a fractional enchancment in spatial decision between the 2 algorithms,

start{aligned} Gamma _{textual content{I}}&= frac{Gamma _{textual content{PM}} – Gamma _{textual content{GPM}}}{Gamma _{textual content{PM}}}. finish{aligned}

(14)

Our simulations used end-on cylindrical phantoms composed of water (({0.998},{textual content{g}},{textual content{cm}}^{{-3}})) and polymethyl methacrylate (PMMA, ({1.19},{textual content{g}},{textual content{cm}}^{{-3}})) with a radius of 900.5 pixels, created on a (2048times 2048) pixel array with pixel measurement ({textual content{W}} = {25},upmu {textual content{m}}). The wavefield immediately after transmission by means of the phantom was constructed utilizing the projection approximation23 on a (occasions 5) up-sampled grid, assuming an object thickness of 6 mm, and propagated with the TIE till a single section distinction fringe grew to become seen within the wavefield depth (4 mm). A small Gaussian blurring filter ((Gamma = 1.0) pixel) was utilized to the thickness map pre-propagation to suppress artefacts arising from the pixelated boundaries of the round phantom, earlier than the second blurring filter was utilized post-propagation to simulate a detectors with various PSFs. Section retrieval was then carried out utilizing both the PM or GPM strategies. Determine 2a exhibits instance imaging system PSF measurements, earlier than and after section retrieval, incorporating a ({mathscr{G}}(k_{x}, k_{y}, Gamma )) element, simulated by means of a Gaussian blurring filter, utilized after the TIE propagation24.

From Fig. 2b we see, for these low-Z supplies, a (sim {6}%) enchancment within the decision when the section distinction PSF FHWM is the same as the pixel measurement. This profit reduces with rising detector PSF width. At 2 pixels extensive, a (sim {2}{%}) enchancment is seen, and solely a (sim {1}%) enchancment is seen at 3 pixels PSF FWHM. This reinforces that the advantage of the GPM technique is closely depending on the detector PSF and can present the best enchancment over the PM when the detector PSF width is equal to a single pixel or smaller.