Theory

Current EI phase retrieval methods allow the extraction of three different sample properties: absorption, refraction and scattering. These can be linked to changes

in the detected intensity distribution in an intuitive way, as demonstrated in Fig. 4.9, in which the beams with and without the sample in place are compared, for a single detector pixel. Sample absorption leads to a reduction in the total area, while the shift in the centre position is caused by sample refraction, and sample scattering broadens the beam.

Figure 4.9: Beam distribution in an EI setup, shown for a single detector pixel. The presence of a sample leads to an attenuated, shifted and broadened intensity distribution. Adapted with permission from AIP Publishing: Applied Physics Letters, Ref [3], Copyright 2014.

The intensity recorded with the EI setup by a single detector pixel can be de- scribed by:

I(x) I0

= (C ∗ O)(x − ∆xR)T , (4.1)

where I0 is the intensity transmitted through the sample-mask aperture, C is the

IC describing the intensity change as a function of relative masks positioning x (see Fig. 4.11(b)), O is the sample’s scattering distribution, ∆xRis the beam shift

sample, and ∗ is the convolution operator [3]. Both C and O can be represented as a sum of Gaussian functions:

C(x) = N X n=1 An p2πσ2 n exp[−(x − µn)2/2σ2n] , (4.2) O(x) = M X m=1 Am p2πσ2 m exp[−(x − µm)2/2σm2] . (4.3)

Thus Eq. 4.1 can be rewritten as: I(x) I0 = TX m X n Amnexp  − (x − µmn) 2 2σ2 mn  , (4.4) where σ_mn2 = σ2_m+ σ_n2, Amn= AmAn(1/p2πσmn2 ) and µmn= µm+ µn.

Equation 4.4 can be analytically inverted to obtain solutions for absorption, re- fraction and scattering, in the case where three images are acquired in positions x1 = −x3, x2 = 0 with respect to the IC (“global retrieval”) [77]. When applied

to experimental data, this approach therefore assumes the same positions on the IC for every pixel over the entire FOV, not taking into account local variations due to misalignment of optical elements or masks imperfections. These variations are mostly compensated for by normalizing the raw data by flat-field images. As has been shown by Endrizzi et al. [78], Eq. 4.4 can also be used to retrieve sample absorption, refraction and scattering, without making any assumptions on system alignment. This is done by applying it on a pixel-by-pixel basis, where Eq. 4.4 is used as a model function for a non-linear curve-fitting, solved by the least-squares method: the intensity values from the IC scan are used to obtain reference values for each pixel, of the amplitude, mean, standard deviation and

offset of the curve. Then, for the same pixel, the intensity values from the sample images acquired at the three different positions of the sample-mask are used in the curve-fitting process. The parameters (amplitude, mean, standard deviation and offset) of the fitted curve are then compared to the reference values, resulting in the extraction of sample absorption, refraction and scattering. Therefore, owing to its pixel-by-pixel computation, this method (“local retrieval”) automatically takes system misalignment and mask defects into account and corrects for them. This has been applied to planar data, and was shown to correct for a significant degree of system misalignment [78].

While the algorithm described here is based on the local retrieval method, a fur- ther extension of the retrieval equation was required in order to adapt it for use on CT scans. Since sample properties are extracted from comparison with refer- ence parameters drawn from the IC scan, if the latter change over time due to system instabilities, the model should account for such changes. In particular, as shown below, it was found that the mean position of the IC varies over time. To accommodate for this variation, two correction steps were implemented: the first was real-time illumination tracking during the scan (as described below), while the second was incorporated into the retrieval algorithm and involved the use of information from background regions in the images. This additional informa- tion was used to estimate the new mean position of the IC (hence updating the reference value), and therefore added a degree of freedom to the fitting process. The correction term ∆µ, corresponding to the translational shift of the IC, was incorporated into Eq. 4.4 in the following way:

I(x) I0 = TX m X n Amnexp  − (x − µ 0 mn)2 2σ2 mn  , (4.5)

where µ0_mn = µm+ µ0n, µ0n = µn(ij)+ ∆µ, and i and j correspond to individual

pixel coordinates.

The illumination curve’s mean position can be calculated for each pixel as µn(ij)

from an initial IC scan. To find the translational shift of the IC over time, begin by defining a background area in the FOV. This background region could be any part of the FOV which is not covered by the sample throughout its rotation. However, since information from this region is used to generate correction terms for the sample data, it is beneficial that the background region is chosen as close as possible to the sample, typically next to it or above it. For each pixel in the background region, the new position of the illumination curve’s mean is estimated as µn im(ij) by means of a least-squares curve fitting using values from the three

images acquired at each dithering step. The shift ∆µ is then determined according to: ∆µ = P i P jµn im(ij)− µn(ij) Nij , (4.6)

where Nij is the total number of pixels in the defined background region. The

shift ∆µ is then applied in the phase retrieval to all pixels in the same dithering step image. Notably, in contrast to the global retrieval, here no flat-field images are needed for processing, as all the required pixel-wise information is drawn from the IC scan.

A diagram of the image reconstruction workflow using the modified local retrieval algorithm is shown in Fig. 4.10.

Figure 4.10: The image reconstruction workflow using the modified local retrieval algorithm.