## 3 Phase Contrast Imaging Techniques on Conventional Systems
### Modulation-Based Imaging (Also Known as Speckle)  
Random phase modulations techniques were developed with X-rays in 2012 [25; 26] at synchrotrons, using the \"speckle phenomenon\" to create a random intensity pattern for which modifications upon introduction of the sample would give its phase information. In optics, speckles are random granular patterns that are produced by a coherent beam when deflected by an element with a rough surface (such as sandpaper at the synchrotron). With conventional sources with low or no coherence, higher Z materials have been used. The experimental set-up of modulation-based imaging (MoBI) is shown in Figure 10.  
The experimental set-up only requires the sample, an X-ray imaging detector and a randomly structured membrane with structures a few pixels in size. The principle of MoBI is to follow the deflection of the beam's rays due to the object. To do so, a first image reference, \(I_{r}\), is acquired with a random mask placed between the source and the detector, generating a random intensity modulations pattern, as shown on Figure 11. Then, a sample image, \(I_{s}\), is acquired when an object is added in the path of the X-ray beam. MoBI is based on the tracking of the local distortions of the random pattern observed when comparing \(I_{s}\) to \(I_{r}\). This comparison allows one to retrieve the attenuation, refraction maps and dark-field signal. As mentioned before, the refraction is proportional to the phase derivative. The phase image (\(\phi\)) can then be retrieved by integrating the refraction maps. In order to improve the quality of the result, it is possible to take several pairs of acquisitions (with and without the sample), moving the membrane between each pair of acquisitions. Figure 10: Modulation-based imaging set-up. The beam is modulated by a randomly structured membrane before reaching the sample. Then it propagates to the detector distorted by the membrane and the sample. Several images can be acquired with various positions of the membrane. Figure 9: Mesh-Based imaging phase retrieval principle. Simulated sample image with a phantom and a grid. The Fourier transform of this image contains the Fourier transform of the sample convoluted to a Dirac comb. The harmonics contain various pieces of information about the sample. This set-up, besides its simplicity of implementation, has the following main advantages:  
* No field of view limitation other than the detector (the random mask can be easy to manufacture). * No resolution limitation other than the optical system. * Being radiation-dose-efficient because no absorbing element is used between the sample and the detector, meaning that all photons passing through the sample eventually contribute to the image formation. For conventional systems, because the speckle phenomenon is difficult to produce, a new kind of membrane has to be invented using high-Z atomic elements (3D printed or powders deposition) to produce random intensity modulations [57]. Contrary to coded aperture systems [58] or EI that use binary masks (either letting the beam through unmodulated, or blocking it completely), the MoBI masks introduce an intensity modulation encoding each pixel of the image, meaning that sub-pixel precision is easier to obtain. Despite being easier to manufacture than coded aperture masks, and being easier to use (no need of precise alignment), due to the randomly introduced modulations being isotropic, it does not suffer from any frequency problems related to preferential direction detection as in mesh-based imaging and is sensitive to any directional structure. To summarize, the experimental complexity of PCI and dark-field is translated in MoBI to the numerical processing side. This practical simplicity holds promise for the feasibility of a rotation gantry. Then, through an image analysis method, one can track the modulations displacements caused by the refraction of the sample directly in two directions. One conventional source phase can be retrieved using a system of linear equations [57].