How PIV Works
FFT Based Cross Correlation Computations
While modern computers are incredibly fast, the direct correlation method is still slow on average computers. However, multiplication of fast Fourier transformed signals are a common and expedient way to solve these issues and are here proved relevant to identifying the correlation plane magnitudes $(C(r,s))$. To begin, the cross correlation of a function $f(t)$ and $g(t)$ is equivalent to the convolution of the complex conjugate of $f(-t)$ and $g(t)$. This is useful, as the Fourier transform of the complex conjugate (bar accent) of $f(-t)$ is the complex conjugate of the Fourier Transform of the complex conjugate of $f(t)$ as described by equation 2.
Applying this understanding, we identify the convolution (*) of $f(t)$ and $g(t)$ in equation 3.
To find our peak value of the cross correlation then, we transform back to state space with the inverse Fourier transform. Equation 4 then summarizes the result.
Reminding ourselves why we underwent this process (turning our direct cross correlation multiplications into an expression relying on multiplication of Fourier transforms), we recognize that if $A_1$ is our $f(t)$ and $A_2$ is our $g(t)$, Equation 4 will provide $C(r,s)$ with a potentially faster computing process. Equation 5 provides the substitution result.
Then to identify the r,s which maximize the correlation plane $C(r,s)$, we simply need to write equation 5 in terms of fast Fourier transforms (FFT). Equation 6 then provides a pseudo-code version of what is needed to calculate the cross correlation of the displacement of the particles in the first region $(A_1)$ to the second region $(A_2)$.
As expected, this method is much less computationally expensive than the direct correlation, and typical PIV algorithms employ the FFT to identify the cross correlation of the interrogation regions.
References:
[1] B. L. Smith and D. R. Neal, “Particle Image Velocimetry,” Part. Image Velocim., p. 27, 2016.
Author: Jack Elliott
Date Published: June, 2022