There are a few sources of errors it is worth keeping in mind when you measure velocities of particles from single images.
If we consider a particle that, in the exposure time, moves its centre as plotted in figure 4.2(left). The particle has a finite size, the surface it actually covers during that time is something like figure 4.2(right). Even though our images are over-exposed, so the particles appear larger than they actually are, we still have a limited resolution, so the digital images we sample will look like the examples shown in figure 4.3.
Figure 4.2: Particle trace. Left: The track of the particle. Right: The resulting (negative) image together with the track.
Figure: Particle trace. Examples of the image resulting from the track shown in figure 4.2. Sampled at three different resolutions (negative images).
Figure: Particle trace. The rightmost image from figure 4.3 overlayed with the velocity vector that corresponds to the diameter of the image and with the actual locations of the particle at the two ends of the trace. The diameter underestimates the extend of the image by pixel along each axis at each end of the image, and the image extends a particle diameter further than the actual movement along each axis.
Figure 4.5: Three real images of particle traces. The left and centre ones are single particles, and the rightmost images shows two particles that got so close together that the tracing software can not distinguish them. These images also show how the actual vertical resolution on the images is two pixels, due to the interaction between the camera, the VCR, and the frame grabber. (We managed to use this to our advantage in the measurements on the Faraday experiment.) The pixels are squares.
I define the location of a particle as the centre of intensity in the part of the image that is identified as the particle. The position of each pixel that makes up the image of the particle is weighed with the intensity of that pixel.
The definition of the end points of the track of a particle I have chosen to use is the two pixels in the image of the particle track which are furthest away from each other. This will usually give an error in the positions close to the difference between the particle diameter and the pixel size. - The particle diameter takes you slightly too far away, and the pixel size brings you closer unless you use the outer corners and not the centres of the pixels. The particle radius varies between and 1 pixel radius in the soap film measurements.