How PIV Works
Sub-Pixel Estimation
One limitation of finding our vector values with only the maximum peak of our correlation plane $C(r,s)$ is that this would limit the range of possible output values to a maximum resolution of 1 pixel (i.e., the resolution of our correlation plane). Fortunately, there are several ways to increase our vector output resolution by first making an educated guess about the overall shape of the correlation plane, fitting the curve of this guess to the correlation plane points which are identified by $C(r,s)$, and finally finding the peak of our fitted curve(s).
The 2, 3-Point Gaussian Method
While the overall peak shape is difficult to determine, a common approximation is that the correlation peak follows a Gaussian (or normal) distribution. If we assume this correlation peak shape, then we can fit a Gaussian distribution to this correlation peak using the values nearest our maximum $C(r,s)$. To do this computation quickly, we will further assume that the $r$ and $s$ directions are each a 1-dimensional Gaussian distribution. With this assumption, we will fit our curve in the $r$ direction independently, then the $s$ direction independently. To illustrate this idea, we take an image pair as shown in Fig. 1, generated with a uniform movement of no $x$-direction displacement and +5.5 pixels $y$ direction displacement. Note: these results are in image coordinates, and the $y$ is flipped from conventional orientation.
Figure 1. A synthetic interrogation region pair with $x$ and $y$ displacement of 0 and 5.5 pixels respectively.
The correlation plane $C(r,s)$ for this image pair is demonstrated by Figure 2.
Figure 2. The correlation plane for the interrogation region pair in Figure 1.
Visually inspecting Figure 2, we recognize that the maximum value of our correlation plane is at $r = 5$, $s = 0$. This result is good, and close to the true value of $r = 5.5$, $s = 0$. However, we can do better. If we ‘cut’ the correlation plane at each of these full-pixel maximum locations, we can estimate the 1 dimensional curves in each direction. Figure 3 demonstrates this concept, where the correlation plane values along $s = 0$ and $r = 5$ are highlighted in red and yellow respectively.
Figure 3. The $C(r,s)$ values to consider the 1-Dimensional Gaussian curve fits, identified by the whole-pixel $C(r,s)$ maxima of $r = 5$, $s = 0$.
To fit these now 1 dimensional curves, we isolate the values along each ‘cut’. To demonstrate our resulting curve fit in the $r$ direction (the one we are hoping will get us closer to the correct sub-pixel value of 5.5), we hold $s$ constant at the maximum full-pixel value (0) and plotted the $C(r,s = 0)$ values in Figure 4.
Figure 4. The $C(r,s)$ values to consider the 1-Dimensional Gaussian curve fits, identified by the whole-pixel $C(r,s)$ maxima of $r = 5$, $s = 0$.
To make a quick fit and avoid allowing the noise around the peak to influence our curve fitting, we apply a Gaussian fit to the three highest points along this ‘cut’ (black) centered around our full-pixel maximum, denoted $i$. Figure 5 demonstrates the resulting Gaussian curve fit of these three points, and change in $r$ from the full-pixel value of $I = 0$ is the curve peak value, here $I = +0.48$.
Figure 5. The Gaussian curve fit for the $C(r,s)$ points neighboring the full-pixel peak in the $r$ direction. The maximum is at $i = 0.48$, indicating that 0.48 should be added to the full-pixel maximum $r$.
As we can see from Figure 5, the curve fit provides an estimate of +0.48 pixels from our previous peak. We add this result to the $r$ location of our $C(r,s)$ maximum, to find that the peak is approximately at 5.48 pixels. Remember, the true displacement was $r = 5.5$ pixels. What an improvement! Through these sub-pixel estimations, we can significantly increase the resolution of our PIV output relative to the single pixel resolution before. Empirically, these efforts lead to a general improvement on the scale of 0.1-pixel resolution, and the uncertainty of this output may increase or decrease according to other image parameters, which may reduce the validity of the assumption that the underlying peak curve is Gaussian.
To deploy this method quickly, the correction from our original peak (indexed $I$, $C(r,s)$ value $R$) through a three point Gaussian is readily identified through Equation 1.
Author: Jack Elliott
Date Published: June, 2022