## 3 Phase Contrast Imaging Techniques on Conventional Systems
### Mesh-Based Imaging  
In 2008, in the search of a technique requiring less exposure than GI and a simpler acquisition set-up, [PERSON] et al. [24] invented spatial harmonic imaging. It has rapidly gone from a 1D to a 2D differential phase contrast modality [51]. The experimental set-up shown in Figure 8 consists simply of a 2D grid (or mesh) placed close to the sample in the path of the beam. The used meshes are adapted to the resolution of the system in order to have a period of typically 3 to 4 pixels and are usually made of woven steel. Only two acquisitions are needed, one with only the grid in the path of the beam to collect the reference image (\(I_{r}\)), and one with the sample added in the path of the beam (\(I_{s}\)). Ideally, the grid should entirely absorb the rays and be as thin as possible. Figure 7: Edge illumination imaging set-up. A beam emitted toward the sample is first split into a thin line beamlet by a vertical grid G0. Those beamlets go through the sample before propagating to the detector plane, where a second grid G1 filters the beamlets that have been deflected by the sample. Several acquisitions are acquired with various positions of G0 and G1. The resulting sample image acquired multiplies the effect of the sample with the effect of the grid. Therefore, when going through Fourier transformation, the effects of the sample are convoluted with a 2D Dirac comb (Fourier transform of the mesh). Additionally, by isolating different harmonics of the Fourier transforms of the sample and reference images, it is possible to retrieve attenuation, scattering and differential phase contrast images of the samples. Figure 9 presents an example of a simulated sample image of a phantom with a grid and the associated Fourier transform. On the Fourier transform image, we can clearly see the different harmonics due to the mesh. Those harmonics are isolated by band pass filters, such as the blue and red circles drawn on the image, with radii (\(r\)) smaller than the separation width (SW) of the harmonics. After filtering the different (\(m\),\(n\)) harmonics, their real space images \(I_{sm,n}(x,y)\) can be retrieved. As the mesh is not composed of perfect delta Dirac functions, those images can be corrected by the same harmonic image of the reference image \(I_{r=m,n}(x,y)\) to obtain: \(I_{m,n}(x,y)=I_{sm,n}(x,y)/I_{r,m,n}(x,y)\). The information contained in that harmonic image can be written as:  
\[I_{m,n}(x,y)=I_{0}(x,y)S_{m,n}(x,y)\exp(i\psi_{m,n}(x,y)) \tag{14}\]  
where \(I_{0}(x,y)\) is the intensity transmitted through the object, \(S_{m,n}(x,y)\) is the real-valued scattering amplitude and \(\psi_{m,n}(x,y)=Mz_{2}\mathbf{g}_{m,n}\mathbf{\alpha}(x,y)\) is proportional to the phase derivative in the \(\mathbf{g}_{m,n}\) direction. Here, \(M\) is the magnification, \(z_{2}\) is the propagation distance between the sample and detector and \(\mathbf{\alpha}(x,y)\) is the refraction angle. From Equation (14), the sample transmission is obtained by taking the 0\({}^{th}\) harmonic \(I_{0}(x,y)=I_{0,0}(x,y)\). When taking the (0,1) and (1,0) harmonics, we get:  
\[I_{0,1}(x,y)/I_{0}=S_{0,1}(x,y)\exp(i\psi_{0,1}(x,y)) \tag{15}\]  
Then, \(S_{0,1}(x,y)\) is the norm of this complex ratio, and its angle gives \(\psi_{0,1}(x,y)\) = \(Mz_{2}\mathbf{g}_{0,1}\mathbf{\alpha}(x,y)=Mz_{2}\alpha_{x}(x,y)\), from which the refraction along the \(x\) axis \(\alpha_{x}(x,y)\) can be easily extracted. The same treatment on the (1, 0) harmonic gives the refraction angle along the \(y\) axis \(\alpha_{y}(x,y)\). A more complex, even though quite similar harmonic-based method, also allows us to retrieve the dark-field signal. This method was developed for conventional sources from the very beginning, first with simple 1D grids that gave only one phase derivative at a time [24; 52], but it was rapidly adapted to 2D phase derivative by using meshes instead of simple grids [51]. It was also very rapidly used for in vivo imaging [53].