Could you give more information on which method (iterative, direct) you're using for phase retrieval or link to the paper you're trying to implement? 

The method that I'm trying to implement is on this paper in the link I attached. The method is direct but still I'm trying to understand it. All the other methods(HIO, OSS etc) require an initial guess.

Thanks for your response

http://onlinelibrary.wiley.com/doi/10.1046/j.1365-2818.2002.01010.x/full

That's the Paganin filter. For this method, the value you need to guess is the ratio of delta/beta. Other parameters given by your experiment (resolution, energy, sample-detector distance). You always can start from 1 to see what happens and increase to ~1000 to check the improvement of the contrast. To be more specific, you can check the values of delta and beta of your sample at the energy you did your experiment and use it as the initial guess. There're different implementations of this method like in python https://github.com/DiamondLightSource/Savu/blob/master/savu/plugins/filters/paganin_filter.py  or in imagej: https://imagej.nih.gov/ij/plugins/ankaphase/ankaphase-userguide.html.

If you want to use Matlab to apply the paganin filter then you could use the TomoTools GUI which you can download from https://github.com/RSBradley/TomoTools . You can download specific code for applying the filter too, see https://github.com/RSBradley/TomoTools/blob/master/addins/phase%20retrieval/tie_hom.m

Paganin's approach via the ImageJ plugin ANKAphase

https://imagej.nih.gov/ij/plugins/ankaphase/

has the advantage that it offers some image restoration in order to overcome the blurring introduce by this kind-of-low pass filter.

some examples on how to use it can be found here:

https://imagej.nih.gov/ij/plugins/ankaphase/ankaphase-jsr2011-authorprint.pdf

Thank you all your answers are very useful. The difficult thing is that I have to calculate delta and beta constants of air at 40 Celcius degrees and I haven't found it yet. I will keep trying thank you very much. 

A lot of X-ray constants are available online, i.e. if you are in the photon energy range of up to 30 keV then check http://henke.lbl.gov/optical_constants/

- otherwise tools like XOP can help you:

http://www.esrf.eu/Instrumentation/software/data-analysis/xop2.4

It seems that I cannot have the results I want. Maybe I cannot understand the algorithm. Using ANKA phase at pixel size I  put 0.0048 microns because i have a 512*512 pixel image representing a 2.5*2.5 microns frame. Is it correct and if not what should I put for pixel size?

Thanks in advance your help is of paramount importance to me

Can you perhaps provide some more details on your experimental setup? The field of view you describe is quite small: do you use perhaps microscopy? Paganins approach works basically for parallel-beam geometry. Also if the pixels areso small, i.e. 5 nm, perhaps you are already working rather in a holographic regime than in the near-field condition?

Sorry that was a mistake. The pixel size to me is 5 microns and the image is 2.5*2.5mm. I have used synchrotron data to obtain images of cavitating flow inside a diesel injector nozzle. The problem is that the cloud that I want to represent is very low contrasted and not totally closed sometimes. So I was thinking about using Paganin filter as a phase retrieval algorithm and then do the visualization better with the pixel with the same refractive index, i.e. the same phase, obtained by this algorithm. Do yo have something more to suggest?

Thanks again very much 

Paganin/phase-retrieval commonly gives you a  boost in contrast when applying it with tomographic reconstruction. In radiography only I never found it quite helpful. Your data sounds like dynamic data? I am not a real expert but I heard that in such cases approaches like "optical flow" are helpful.

What do you mean by 'dynamic data'? I have 16000 images of the nozzle as a plane with no rotation to create a tomography. This is not my purpose. My purpose to somehow manage to see the cloud of the cavitation in the nozzle by a kind of edge detection. But to do this, at first I have to enhance the low contrast of the image. That's why I thought about the paganin filter to use the obtained refractive index.

Dynamic data = movies as you describe above. Again, I would rather consider something like optical flow techniques. Alexey Ershov might be a good contact for you: https://www.researchgate.net/profile/Alexey_Ershov2/publications

Robert Bradley provided a very good implementation of the two retrieval methods. I would suggest you to run the code step by step for understanding. One hint, use padarray instead of padtovalue function.

The retireval method will help in improving the CNRs, also may reduce the spatial resolution.

Muhammad U. Ghani the phase retrieval method will help in improving the CNR, also may reduce the spatial resolution. can you suggest some papers on the relationship between the CNR and spatial resolution when phase retrieval is applied?

@Lin Zhang

Here are some papers that you can look into

https://scholar.google.com/scholar?cites=11884185817867836688&as_sdt=5,37&sciodt=0,37&hl=en#d=gs_qabs&u=%23p%3D7qKEhtAP8Z4J