This blog post will show you how to measure zero-velocity regions with cross-correlation. You’ll learn what cross-correlation is, how it works, and practical procedures for measuring zero-velocity regions.
Definitions of Cross-Correlation Coefficient And Velocity Estimation
The cross-correlation coefficient is a measure of how well two signals match. In the case of two images, we can use this to determine how similar they are. It can also be used to find the velocity of an object using cross-correlation and Fourier analysis.
The cross-correlation function between two signals (or images) is defined as follows:
- C(t)=\sum_{n=0}^{N}\left(X(t+n)-X(t)\right)Y(n)
A Practical Procedure for Measurement of Zero Velocity Regions
The procedure for finding zero-velocity regions is as follows:
- Import two images into PIVlab and align them by their centers of mass. The image on the left will be shifted by half its size in x and y, as shown below (Figure 1).
- Plot cross correlation between input and shifted images at regular intervals along both axes (Figure 2).
- Find the highest peak in this plot; this corresponds to a zero velocity region with no relative motion between particles in either direction (Figure 3).
A Practical Example of Zero-Velocity Measurement With Taylor-Green Flow Using PIVlab
In this example, we will use a Taylor-Green flow to measure velocity in a zero-velocity region. We will then use PIVlab to measure the velocity with the cross-correlation function and compare it with our measurements obtained from Taylor-green flow.
The first step is to create an experiment in PIVlab with 3 planes of focus: plane 1 contains all particles, plane 2 contains only particles above the plane 1 plane (so they are in focus), and plane 3 contains only those below plane 2 (so they are out of focus). This way, we can visualize both particles inside and outside our zero velocity region using one visualization method (cross-correlation) instead of two separate methods (one for each plane).
Use Cross-Correlation to Measure the Relative Velocity Between Two Images
Cross-correlation can be used to measure the relative velocity between two images. The peak correlation coefficient position shows the average velocity of an object in the input signal, and multiple correlation peaks indicate zero velocity region along the object’s axis.
To use cross-correlation, you need two images (or signals) filtered with a Gaussian filter with the same standard deviation but different centers: one as an input and one as a reference. Then you will have to apply them to your data set separately and measure their cross-correlation function (CCF).
Correlation – a Measure of How Well Two Signals (Vectors) Match
Correlation is a measure of how well two signals match. It’s calculated by taking the dot product of two vectors and normalizing it by their standard deviations, then squaring the result:
corr(x,y) = |x||y|*((x-mean(x))(y-mean(y)))^2 / (std(x)*std(y))
Cross-Correlation – Correlation Measure Between an Input Signal And Shifted Version of The Input
Cross-correlation is a measure of how well two signals (vectors) match. It measures the degree to which two signals are correlated, or in other words, how much they resemble each other. If you shift one signal by some amount and then correlate it with another shifted version of itself, what you get is the cross-correlation between those two vectors:
$$X_{1} * X_{2} = X_{1}^T * X_{2}^T$$ where $X_1$ and $X_2$ are two vectors whose elements are defined by their respective values at time points t=0 through T-1.
The formula above tells us that we can compute the cross-correlation between two-time series simply by taking their transposes (which turns them into column matrices) and multiplying them together elementwise. This operation amounts to inner computing products between each pair of columns from both input sequences!
The Peak Correlation Coefficient Position Shows The Average Velocity of an Object in the Input Signal
This is also known as “zero velocity region” or “velocity mute.”
To get this information, we need to use cross-correlation because it allows us to compare two signals and determine how similar they are. Cross-correlation works by taking a small chunk of data from one signal (the ‘signal X’), multiplying it by another chunk of data from another signal (the ‘signal Y’), adding them together and then dividing by two:
\begin{equation}c(n)=sx(n) \times sy(n)+sy(n) \times sx(n)/2\end{equation} Where n = 0..N-1
Multiple Correlation Peaks Indicate Zero Velocity Region Along The Object’s Axis
The peak correlation coefficient position shows the average velocity of an object in the input signal. It measures how well two signals (vectors) match or how close they are to be aligned with each other. In physics and mathematics, cross-correlation is a measure used to quantify the degree of a linear relationship between two random variables or stochastic processes derived from different observations or measurements taken at different times. Cross-correlation has several applications, including image processing, seismology, sonar and radar systems.
Conclusion
A zero-velocity measurement helps understand the behavior of a flow in a particular region. This can be used to improve the performance of your experiments, as you can quickly identify areas with no motion and focus on them. For example, if an object moves at constant velocity inside a tank filled with water, the cross-correlation will show very high peak values along this axis. This peak’s position corresponds to the object’s average velocity within that particular frame (assuming it is not moving at all).
