
Exact section retrieval for propagationbased photographs utilizing discrete arithmetic
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 tailormade 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) in^{2}), (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 onedimensional 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 newfound asymmetry of the GPM filter, correcting the PM algorithm to ship a extra uniform remedy of all edges in realspace, 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 postimage 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 realspace 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 Xray 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 Xray 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 photoncounting detectors, will likely be finest suited to learn from the GPM algorithm, whereas for oblique Xray 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 azimuthallyaveraged 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 phaseretrieved picture. We then use the measured decision within the propagationbased section distinction picture, prephase 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 phaseretrieved 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 endon 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 approximation^{23} on a (occasions 5) upsampled 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 prepropagation to suppress artefacts arising from the pixelated boundaries of the round phantom, earlier than the second blurring filter was utilized postpropagation 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 propagation^{24}.
From Fig. 2b we see, for these lowZ 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.