Computing Cross-Correlation Matrix, and multiplying Result by Correlated Vars

loesung · March 16, 2022, 22:47

Hello friends,

In the context of turbulent flows, I want to compute a spatially cross-correlation matrix of fourier-transformed velocity field (in one direction ), using either python or matlab, and then do the explicit multiplication

$\,\,\,\,\,\,\,\,\,\,\sqrt{r}\cdot R(r,r',t,t')\cdot \sqrt{r{'}}R \,\,\,\,\,\,\,\,\,\,\,\,\, (\dagger)$
Matlab has the function in the signals toolbox xcorr(), but I am not sure how to explictly do the above calculation with the and parameters multiplying the result of xcorr.
The problem in using matlab's xcorr(), is that there is not explicit access to the variables, so if I were to calculate just xcorr(var1,var2), then I don't see how to multiply the result by and - perhaps I can achieve this with some sort of weighted xcorr? But I am not sure that's what I need.

Another question is,
Since I am correlating in space and time, do I need to use something like xcorr2() (or its python equivalent)?
If I am correlating two vectors of length say 3, then xcorr returns a vector of length . I am also confused on this point, why the return value is this length

LuckyTran · March 17, 2022, 14:07

I simply don't understand what you are trying to calculate in 1) to contribute towards 1-3.

4) xcorr2 does perform correlation in both directions of the matrix. But I'm not sure if this gives you what you want in 1). If your data was arranged in two spatial dimensions, xcorr2 would give you the two-dimensional spatial correlation function. If your data is arranged in space-time, it gives you a spatial correlation and and a temporal correlation. But is this what you want?

5). A pair of 3-element vectors has 5 possible shifts. -2,-1,0,+1, and +2. The generalized cross-correlation therefore is a 5-element vector. What length were you expecting? If you do not permit zero padding and expected a single scalar output, then the element in the middle of the output contains the zero shift. +3, -3, and greater shifts would result in either a periodic shift or a null output depending on your definition of cross-correlation. In either case, there is no output needed for shifts exceeding -2 or +2 and so the output is no more than a 5 element vector.

loesung · March 17, 2022, 14:19

I stumbled across a book, "Particle Image Velocimetry: a Practical Guide" after posting this question, which clarified everything (in the Chapter 5: Image Evaluation Methods for PIV)

March 16, 2022, 22:47	Computing Cross-Correlation Matrix, and multiplying Result by Correlated Vars	#1
loesung New Member Join Date: Mar 2022 Posts: 23 Rep Power: 4	Hello friends, In the context of turbulent flows, I want to compute a spatially cross-correlation matrix of fourier-transformed velocity field (in one direction ), using either python or matlab, and then do the explicit multiplication $\,\,\,\,\,\,\,\,\,\,\sqrt{r}\cdot R(r,r',t,t')\cdot \sqrt{r{'}}R \,\,\,\,\,\,\,\,\,\,\,\,\, (\dagger)$ Matlab has the function in the signals toolbox xcorr(), but I am not sure how to explictly do the above calculation with the and parameters multiplying the result of xcorr. The problem in using matlab's xcorr(), is that there is not explicit access to the variables, so if I were to calculate just xcorr(var1,var2), then I don't see how to multiply the result by and - perhaps I can achieve this with some sort of weighted xcorr? But I am not sure that's what I need. Another question is, Since I am correlating in space and time, do I need to use something like xcorr2() (or its python equivalent)? If I am correlating two vectors of length say 3, then xcorr returns a vector of length . I am also confused on this point, why the return value is this length

March 17, 2022, 14:07		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,743 Rep Power: 66	I simply don't understand what you are trying to calculate in 1) to contribute towards 1-3. 4) xcorr2 does perform correlation in both directions of the matrix. But I'm not sure if this gives you what you want in 1). If your data was arranged in two spatial dimensions, xcorr2 would give you the two-dimensional spatial correlation function. If your data is arranged in space-time, it gives you a spatial correlation and and a temporal correlation. But is this what you want? 5). A pair of 3-element vectors has 5 possible shifts. -2,-1,0,+1, and +2. The generalized cross-correlation therefore is a 5-element vector. What length were you expecting? If you do not permit zero padding and expected a single scalar output, then the element in the middle of the output contains the zero shift. +3, -3, and greater shifts would result in either a periodic shift or a null output depending on your definition of cross-correlation. In either case, there is no output needed for shifts exceeding -2 or +2 and so the output is no more than a 5 element vector.

March 17, 2022, 14:19		#3
loesung New Member Join Date: Mar 2022 Posts: 23 Rep Power: 4	I stumbled across a book, "Particle Image Velocimetry: a Practical Guide" after posting this question, which clarified everything (in the Chapter 5: Image Evaluation Methods for PIV)