## 3 Phase Contrast Imaging Techniques on Conventional Systems
### Propagation-Based Imaging  
Propagation-based imaging (PBI) is the most used experimental set-up in synchrotrons around the world. Its success is no doubt due to its very simple experimental set-up, the rapidity of the acquisition and the simple numerical phase retrieval processing. As presented in Figure 2, the key features of this method are the coherence of the source and the propagation distance between the sample and the detector. Apart from that, the set-up is similar to one of conventional radiography. The only difference with a standard clinical X-ray imaging system is the fact that the detector is separated from the sample by a distance that depends on the desired spatial resolution. The principle is that, thanks to the spatial coherence of the source, upon propagation, the waves distorted in the sample will create interference patterns that are more and more prevalent as the distance between the sample and the detector increases. Those interference fringes appear as white and black lines contouring the sample; it is what we call \"edge enhancement\". This interference pattern, appearing as intensity variations in the acquisitions, is proportional to the phase Laplacian of the wavesTo retrieve the phase information, various methods have been developed and optimized around the world, but the fundamental one is the so-called \"Paganin method\" [27]. It is based on the transport of intensity equation (TIE) in the paraxial approximation (see Equation (11)). \[-k\frac{\partial I(x,y)}{\partial z}=\
abla_{\perp}\cdot[I(x,y)\
abla\phi(x,y)] \tag{11}\]  
\(x\) and \(y\) are the spatial coordinates in the plane perpendicular to the propagation axis \(z\). An extensive development of this equation, including the derivation of the equation and its use with different approaches, can be found in the very well written review by [PERSON] et al. [28]. The resolution of this equation to extract the phase information proposed by [PERSON] et al. in [27] assumes a single material sample. This approximation allows one to assume constant \(\delta\) and \(\beta\) to solve the equation from only one acquisition. This assumption implies that the phase is proportional to the thickness \(t\): \(\phi=\delta t\). The equation to extract the thickness from the TIE then becomes:  
\[t(x,y)=-\frac{1}{\mu}log\left(\mathcal{F}^{-1}\Bigg{\{}\frac{\mathcal{F}\{I_{s }(x,y)/I_{r}(x,y)\}}{1-(z_{2}\delta/\mu)(k_{x}^{2}+k_{y}^{2})}\Bigg{\}}\right) \tag{12}\]  
where \(I_{s}\) is the sample image acquired with the sample, and \(I_{r}\) is the reference image corresponding here to the white field without the sample, used for normalization. Due to the strong single material hypothesis, the retrieved image does not give a quantitative measurement; however, it still gives a very good contrast between various materials. In order to get free from this assumption and get quantitative results, a variant of this method was invented with several propagation distances \(z\) (between the sample and the detector as shown in Figure 3) and based on the contrast transfer function (CTF) [29]:  
\[\phi(x,y)=\frac{1}{I_{r}}\mathcal{F}^{-1}\Bigg{\{}\frac{\sum_{z}\Big{[}I_{s}^{ (z)}-I_{r}-G^{(z)}(\mathbf{f})\mathcal{F}\{\
abla_{\perp}(\phi(x,y)\
abla_{ \perp}I_{r})\}\Big{]}H^{(z)}(\mathbf{f})}{\sum_{z}\big{(}H^{(z)}(\mathbf{f}) \big{)}^{2}}\Bigg{\}} \tag{13}\]  
where \(\mathbf{f}=(f_{x},f_{y})\) is the Fourier space vector. Note that the variables used in Fourier space are sometimes the frequency noted \(\mathbf{f}\) or \(\mathbf{u}\) or the angular frequency \(\mathbf{k}=2\pi\mathbf{f}\). As much as possible, we try to harmonize the notation. It was chosen to keep the original article notation when possible: \(G^{(z)}(f)=\frac{\lambda z}{2\pi}cos(\pi\lambda zf^{2})\) and \(H^{(z)}(f)=2 sin(\pi\lambda zf^{2})\). In this method, the equation is solved iteratively, neglecting the term containing the phase for initialization. Figure 2: Propagation-based imaging set-up.