You may try to invent some algorithm based on the k-nearest neighbor

To simplify it at the extreme level, you look for each pixel on image A the nearest pixel in image B with the same image level, that will be your displacement vector which, divided by the time interval, will give the velocity you are looking for.

Obviously this will result in a huge amount of error, so you need to stabilize it.

For example, instead of simply looking for pixel with same graylevel, you may look for similar patches on the two images. I.e., not only pixel P on image A has the same level of pixel Q in image B, but also neighboring pixels of  P in image A have similar graylevels of neighboring pixels of Q in image B.

There is literally an infinity of methods to measure the dissimilarity between two patches, you should use the one fits you best. I would suggest you to decompose each patch on a series of Laguerre-Gauss filter and look at the coefficients.

I hope these hints help you, feel free to ask more if you need more help.

Hello, Mr. Zhang.

I have attempted to derive a method to resolve this problem. Please find it in the PDF attached. This is no way the final or definitely correct way, but I hope this helps get a broad estimate.

Giuseppe and Saumita, thank you very much. 

To Giuseppe's answer: here, the saturation is not image level, it is the volume fraction of gas in the pores within a pixel.  

So the flow velocity is not simply a displacement from one point to another point. We must consider fraction changes. Another problem is the fractions always change. It is difficult to find any patches.  

Yesterday, I thought this problem more. I think it is maybe difficult to solve the problem by a pure image processing way.  

Your suggestions about k-nearest neighbor and Laguerre-Gauss filter are valuable for some of my other problems. Thank you.

To Saumitra's answer:  thank you for your idea and deriving an equation.  

Since the system has  two fluids migration in a porous rock,   the D in your equation, may be similar to the permeability k in Darcy's law based  PDE. As the (relative)  permeability changes when fluid saturation changes, the D is also a function of saturation. 

Another problem is that the local flux for a pixel,  is not only related to the saturation changes near the pixel, but also all saturation before the pixel (1D case, along flow direction). For example, the saturation of a pixel may be not change so much within T2-T1, but some fluid pass it to other pixels,  so the flux pass it is large. 

But anyway, I think your idea is valuable; I will  a test. Thank you very much! 

Mr. Zhang,

I wasn't aware that the problem involves flow in porous media. In that case, I doubt if Fick's equations are valid. In case you happen to try them out, could you please let me know if they work? Thanks!

As far as I understand that your image evolution is driven by a physical model that you know up to a certain extend.

In other words, the image at time t+dt can be predicted from the image at time t (and maybe you also need it at time t-dt, t-2dt, ...) and maybe from some unknow parameters.

If that is the case you can formulate your problem in the image assimilation framework.

In other words, you write down the equations of the time evolution of your system and you search for all unknown parameters by minimizing the discrepancy with observation. It is a standard variational problem, which works well in many situations (though it is quite slow).

I hope this helps

Mr Papari,

This sounds very interesting. In order to understand this better, could you please provide a few references where such a method has been used?

Mr Joshi

I made a test assuming the flow is diffusion describable.  Please see the attched figure.As our discussion,  the estimated gas flux, is only sensitive to the plume front, where has apparent saturation changes; and cannot reveal the real flux at other regions. 

Thanks. 

Mr Papari

Thank you for your suggestion. I have searched " image assimilation" technique. It is interesting and may helpful for my problem. 

Thank you!

Mr. Zhang,

This image really helps understand the shortfall of the diffusion assumption - as you pointed out, pixels where the concentration doesn't change are shown to have very low flux, while that is clearly not the case.

Thank you for the post. I never received a notification for it, else I would have seen it earlier.

Hello Yi Zhang,

here you find a powerpoint on data assimilation, with good references inside

http://www.dtcenter.org/com-GSI/users/docs/presentations/2010_tutorial/L1-0628-Fundamental_DA_TomAuligne.pdf

I hope it helps!

Thank you Mr. Papari!

The physics-based optical flow method is applicable to your problem.  The physics-based optical flow equation should be derived from the relevant fluid transport equation and imaging process by using the projection method. Thus, the physical and mathematical meanings of the optical flow can be clearly defined. For X-ray images that are transmitting images, it has been done (see T. Liu and L. Shen, “Fluid flow and optical flow” Journal of Fluid Mechanics, Vol. 614, No. 11, pp. 253-291, 2008). 

Regarding to the intensity change, it is indeed a problem for both PIV and optical flow method.  If the time interval between two images is sufficiently smaller than the timescale of the intensity change, the intensity change could be small enough for optical flow computation.  In addition, the intensity change could be corrected or compensated by pre-processing before optical flow computation, which is a normal procedure in our studies.   

Prof. Liu

Thank you very much for this information. I will read your paper and test the recommended method.  Is there any open code of physics-based optical flow for beginners?  For my study, the physics is two-phase flow in porous media.  I didn't see any previous case yet.

You can email me at [email protected] to discuss some details.