## 2 Materials and Methods
### Propagation-Based PCI  
A simple formulation for the normalized, propagated intensity at the detector can be derived based on the so-called transport-of-intensity (TIE) equation [2; 3; 27]:  
\[\frac{I(\mathbf{r},z)}{I(\mathbf{r},0)}=1-\frac{z}{k}\
abla^{2}\phi(\mathbf{r}) \tag{1}\]  
where \(z\) is the propagation distance, \(k=2\pi/\lambda\) is the wave-number and \(\phi(\mathbf{r})\) is the phase shift inflicted by the sample. The propagation distance in Equation (1) has to be replaced by \(z_{eff}=z/M\) for cone-beam geometry and \(\mathbf{r}\), representing the detection plane coordinates perpendicular to the optical axis, has to be also scaled by \(M\), the magnification, according to the Fresnel scaling theorem [2]. The TIE gives satisfactory results in the so-called direct contrast regime, i.e., for moderate propagation distances and for most practical cases for laboratory PCI. The simplest form of TIE-based phase retrieval is obtained by inversion of Equation (1) via Fourier transform (\(\mathcal{F}\)) as  
\[\phi(\mathbf{r})=-\frac{k}{z}\mathcal{F}^{-\infty}\left[\frac{\mathcal{F}[I( \mathbf{r},z)/I(\mathbf{r},0)-1]}{\mathbf{k}^{2}}\right] \tag{2}\]  
\(\mathbf{k}\) is the Fourier-space equivalent of \(\mathbf{r}\). Equation (2) is the phase retrieval step of the original [PERSON] algorithm [28], which was integrated together with the ramp filter into the tomographic reconstruction. As explained above, the phase retrieval can be also performed prior to reconstruction on the projection images as is done mostly in practice, which is also implemented in our reconstruction package. The contact image I(\(\mathbf{r}\), 0) is experimentally challenging and not practical to measure. For pure phase objects or negligible attenuation it represents a normalization by an open-beam (no object) image, \(I_{o}\), instead of \(I(\mathbf{r},0)\) in Equation (2). Finite object attenuation is typically taken into account adding an empirical regularization parameter, \(\alpha\), to the denominator of Equation (2) also to compensate the singularity at zero frequency (\(\mathbf{k}=0\)) and stabilize the reconstruction. This leads to:  
\[\phi(\mathbf{r})=-\frac{k}{z}\mathcal{F}^{-1}\left[\frac{\mathcal{F}[I( \mathbf{r},z)/I_{o}-1]}{\alpha+\mathbf{k}^{2}}\right] \tag{3}\]  
Equation (3) is the modified Bronnikov algorithm (MBA) [29]. [PERSON] et al. [30] proposed an alternative approach the so-called Bronnikov-Aided-Correction (BAC), which is a two-step algorithm. In the first step, an approximation for the phase distribution, \(\phi(\mathbf{r})\) is obtained by applying Equation (3). Then the contact image is estimated by expressing it with the help of Equation (1) as  
\[I(\mathbf{r},0)=\frac{I(\mathbf{r},z)}{1-\gamma\
abla^{2}\phi(\mathbf{r})} \tag{4}\]  
where \(z/k\) was replaced by a control parameter \(\gamma\) that is typically determined empirically together with \(\alpha\)[2]. A common strategy for determining \(\alpha\) is to examine in \(\phi(\mathbf{r})\) obtained in the first step profiles through edges and material boundaries and decrease \(\alpha\) starting from larger values until edge effects are eliminated. Then adjust the strength of the phase correction in the second step increasing \(\gamma\) starting from small values and its final value is determined by visual inspection. We also apply this strategy with our reconstruction package and enable the user to choose the \(\alpha\) and \(\gamma\) parameters (see an illustration of it in Figures 2 and 4).Then the contact image, \(I(\mathbf{r},0)\), is used further in the conventional attenuation CT reconstruction pipeline. Besides the BAC algorithm, we implemented in our package the likely most often applied single-distance phase retrieval algorithm by [PERSON] et al. [31].