In a real defocusing camera, the position of particles in space is found by first searching an image for particle images and then searching through the particle images to find the ones that match the pattern of the aperture layout. This requires that thex, ycoordinate of the particle images be known to good accuracy because this error propagates directly into the calculatedX, Y, Zposition and then velocity of the particle. A real sensor is a “discretized” space in that it is made of pixels. They are the smallest image element and nothing smaller than a pixel can be resolved. In determining the sub-pixel position of a particle image, a two-dimensional Gaussian is fitted over the particle images in a least-squares sense; the position of the peak of the resulting Gaussian is taken as the location of the particle image. Thus it is critical that particle images occupy more than one pixel on the sensor to have any hope of exacting its position. Of course if the particle images are too big then overcrowding is an issue. Simulations show that the accuracy in determining the sub-pixel location of particle images peaks with particle images whose radius (measured as the distance to the point where the intensity drops to 1/e2 of maximum) is 2 pixels1.
For a given seeding particle size, the two principal factors that affect the size of the image is the “sharpness” of the lens and the sensor planar resolution. For the purposes of this discussion, we defineplanar resolution as the number of pixels per millimeter of field of view at the reference plane. By the definition of magnification, the field of view dimensionW at the reference plane of a
sensor of dimensionwis
M = w
W ⇒W = w M
If s is the dimension of one pixel, then w = N s where N is the number of pixels along that dimension, and the planar resolution will be
R=N/W =NM w =
M
s (6.2-1)
The sharpness of the lens is critical because if a particle is treated as a small sphere whose image must cover four pixels in diameter then for a given defocusing camera the minimum particle size would be 4/R. In the case of the Emilio Camera, this would correspond to just under 400 microns. In actuality, particle images will be larger than this geometric estimate; the lens will blur the image slightly. Once the effects of diffraction from the aperture are considered, given enough light particles much smaller than this prediction can be imaged. In the case of the Emilio Camera, tests indicate that particles about 100 microns in diameter yield good images at f/22 when illuminated by a 200 mJ-per-pulse laser expanded to illuminate the intended probe volume (see figure 3.2-5). In effect then it is bestnot to have an extremely sharp lens as any physical blurring device independent of
Z coordinate (such as diffraction) is useful in expanding the particle images over enough pixels so that its sub-pixel position can be measured more accurately.
Typical blurring or soft-focus filters available for photography are inadequate because they are essentially a pseudo-random (or worse—a structured) pattern of micro-prisms used to scatter the rays before they enter the lens. They may work well with incoherent light and large entrance pupils, but when imaging microscopic particles illuminated by a laser at high f-number they make the particle images “jump” around in the image2 rather than move continuously, thus any advantage gained
by physical blurring (which is minimal in the case of these filters) is obliterated by the “increased discretization” of the image domain.
More sophisticated blurring filters exist—those which are meant to be mounted on the aperture plane. The advantage of having the blurring element at the aperture is that this is the only place in a lens where the rays from any point in space pass through the exact same location and thus would be altered equally. These are not cheap, and are very rare for 35-mm-format lenses since the lens would have to be disassembled to install the filter. Rarely lenses are available in “soft focus” versions that implement such filters; realistically these are not an option for DDPIV cameras because they are so rare. There is potential for a custom-designed blurring element to be made for installation together with the replacement aperture (see chapter9), such as those depicted inPalum[2001].
2This is exactly what these filters intend to do: move adjacent point sources of light around on the image so that
These difficulties in imaging and reconstructing the particle image location imply that a tolerance must be used when matching images together. This tolerance is called the pixel tolerance and corresponds to how many pixels away from the predicted pattern the particle images are allowed to be. The pixel tolerance must be increased to accommodate for errors resulting from the Gaussian fit (the quality of the particle images) and errors introduced by multi-plane dewarping. However it should also be minimized to eliminate as many ghost particles as possible. With the Ian Camera this tolerance is typically around 0.75 for actual experiments on seeded flows, whereas with the Emilio Camera a tolerance of 0.50 pixels shows good results.