The ghost particles constitute a severe source of error in the tomographic reconstruction. Even though a model to understand and quantify the ghost particles formation is of crucial importance, more interesting questions arise regarding their influence on the velocity measurement in terms of bias and random errors. For
Chapter 2 – Tomographic PIV
example, Elsinga et al (2006a) documented an experiment on the wake behind a cylinder in which the ghost particles outnumbered the actual particles; the velocity measurement was still satisfactory nonetheless.
In Sec. 2.2.1 a model to estimate the number of ghost particles is presented. The model is slightly different from the one proposed by Elsinga et al (2006b) in its derivation, formulation and functional dependencies, and conceptually more similar to that by Wieneke (2008). In Sec. 2.2.2 the focus is on the effect of the ghost particles on the velocity measurement.
2.2.1 Estimate of the number of ghost particles
A ghost particle is formed in every occurrence of intersection of all the lines of sight in which a particle image is present. Consider, for example, an illuminated volume of size and a particle image of the first camera. The particle can be located anywhere along the line of sight, whose length is approximately (without affecting the generality of the problem the effect of the viewing angle is neglected). The possible candidates for the matching of a single particle of the first camera on the second camera are those included in a strip with length equal to the projection of the line of sight on the second camera (it can be estimated by multiplying for the average magnification and dividing by the pixel pitch ) and width equal to the particle image diameter . The number of candidates for the matching for each particle image of the first cameras in a 2 camera system can be statistically determined by multiplying this area for the particle image density. Normalizing with the number of true particles one obtains:
(2.5)
where is the particle diameter obtained by (1.2), and (1.14) is applied.
Two projections are enough to define for each candidate a trial position in the 3D space. The trial position is projected onto the third camera to find the possible matchings. In this case the search area is a circle with diameter equal to the particle image diameter. The number of spurious matchings is statistically determined by multiplying the particle image density for the search area, which is equal to
the source density . This leads to the general formula for a system:
(2.6)
The term in curly brackets is the depth of the volume in voxels (provided that the resolution ratio between voxels and pixels is equal to 1), while the term in square brackets is the diameter of the particle image.
Fig. 2.5 Reoccurrence of ghost particles in the two exposures. Solid lines and dark blue particles indicate
the first exposure; dashed lines and light blue particles relate to the second exposure. Dark and light red circles indicate ghost particles in the first and the second exposure, respectively. Left: ghost particle reoccurring in both the exposure. Right: ghost particle appearing only in the first exposure due to the presence of velocity gradients (Elsinga et al 2011).
2.2.2 The role of the ghost particles in the velocity measurement
The intuition might suggest that in Tomo-PIV the role of the ghost particles is marginal with respect to 3D-PTV, since MART iteratively damps their intensity; furthermore, cross-correlation is much more solid in terms of spurious matchings than particle tracking. These considerations are supported by the evidence that Tomo-PIV is able to work with particle image density more than 10 times larger than 3D-PTV. In many application reported in Tab. 1.1 the ratio (2.6) is close or even larger than 1. However, the contribution of the ghost particles to the cross- correlation maps is not exclusively dependent on their number per se. Elsinga et al (2011) observed that in some conditions ghost particles pairs between the exposures might form, and give an undesirable contribution to the cross- correlation map.
The process of ghost particles pairing is sketched in Fig. 2.5. Consider a set of 4 particles observed by 4 cameras, and suppose that the respective lines of sight intersect in a common point, determining the creation of a ghost particle. If the displacement along the direction normal to the viewing direction is nearly the same for the set of particles, a ghost particle will be formed in the second exposure by the same group of true particles. As a matter of fact, the ghost particle reoccurs in both the exposures in nearby locations, and its displacement is approximately the average displacement of the set of true particles. On the other hand, if the displacement is significantly different (more than one particle image diameter) for at least one particle of the set (i.e. if there is a significant velocity gradient) the ghost particles will not reoccur in the second exposure (see Fig. 2.5, right).
Chapter 2 – Tomographic PIV
Elsinga et al (2011) draw two importance consequences:
the coherent motion of the ghost particles might lead to velocity detection even outside of the illuminated volume. This problem can be avoided by identifying the illuminated region with the self-calibration technique (see Sec. 2.1.4) or by summing the particle intensities in the planes to determine the laser profile; the velocity vectors outside of this region are rejected;
since the ghost particles displacement is an average over a set of particles whose reciprocal separation might be much larger than the interrogation spot size, the effect is a smoothing (say, a modulation) of the velocity field.
The second consequence is much more severe. Some of the most recent studies tackle the problem without focusing on the pairing, i.e. the bias and random errors due to the ghost particles are challenged by increasing the reconstruction accuracy (see for example Petra et al 2009, Novara et al 2010, Novara & Scarano 2012a, de Silva et al 2013) or by using hybrid PIV-PTV approaches (Novara & Scarano 2012b, Wieneke 2013). More recently, Discetti et al (2012) proposed a low cost Tomo-PIV setup, consisting of two independent tomographic systems with cheap single- shutter cameras. Since the ghost particles distribution is related to the camera orientations, this system actually provides reconstructed distributions in which the ghost particles do not reoccur coherently in the two exposures even in absence of velocity gradients. More details are provided in Chapter 6.