Comparison to Crocker and Grier’s method

The main reason to switch from the almost traditionally used algorithm introduced by Crocker and Grier [35] (CG) to the new SIFT-based algorithm is the ability to determine the particle radii with a high precision. This allows one to discriminate between small and big particles in the binary samples used within this work. In order to show how big the enhancement is, not only in the radius determination but also in the overall particle detection, results are shown here comparing the performance of both codes on one of the typical binary samples investigated in whit work: PMMA particles with diameters of2µm and1µm, colored with a long-lasting near-infrared dye in a mixture where the number density of small particles is 2 times the number density of big particles. The mixing ratio was determined by weighing the two components as dry powder and employing the radius of the dry particles measured in the electron microscope. A confocal micrograph of the sample is shown in Figure3.20. Obviously the small particles are a lot harder to detect, especially due to their lower luminosity.

For both algorithms the parameters, such as filter sizes and detection cut-offs were tuned with the aim to maximize the number of detected particles and to minimize the number of false detections (false posi-tives). As a first comparison a histogram of the so-called pixel bias is shown in Fig.3.21. It is computed as the remainder of the division of a single coordinate (here𝑥) by the according pixel/voxel dimension (usually the pixel size or the distance between𝑧 slices). Ideally, the histogram should be totally flat, indicating that the positions determined by the code are randomly distributed around the camera’s pixel positions. Figure3.21shows that with the CG method the histogram is not perfectly flat, there are more particles at a position sitting just in the middle between two pixels. For small particles there are even two additional peaks at𝑥= 0and𝑥= Δ𝑥= 0.124, which means that more particles than expected are seen directly at a pixel position. For the SIFT algorithm the histogram is flat, coordinates do not show any pixel bias. The reason for the better performance of SIFT is clearly the much better refinement step: With the knowledge of the particle radius all the pixels/voxels inside the spheres are used in the computation of the sub-pixel position.

In the framework of the CG algorithm one can estimate a particle’s radius by computing the radius of gyration from intensity values. All the pixel/voxel values inside a sphere with radius𝑤around the particle position contribute, and𝑤 is the same size that is also used in the filtering and refinement steps (see

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Figure 3.21: Histograms of the pixel bias, computed by taking the remainder of the division of a coor-dinate by the pixel size. CG (Crocker and Grier) and SIFT algorithms are compared for big and small particles seen in a binary sample (cf.3.20). Ideally the histogram is flat, which means that pixel positions and particle coordinates are totally uncorrelated. Here𝑥, 𝑦, 𝑧denotes the pixel/voxel position of the detected particle. It is clear that this calculation relies on the particles being separated well enough from their next nearest neighbours. A distance smaller than 𝑤must result in an overestimation of the radius. On the other hand, with𝑤being too small the radius could be underestimated. For strongly charged particles at low enough densities a good choice for 𝑤 that allows for the detection and sizing of all particles should be possible, but looking at Figure3.20this might not be the case for the sample discussed here.

In Figure3.22the resulting histograms from both algorithms are presented. Actually the radius estimation according to CG is not totally wrong, at least one can infer that there are two different particle sizes in the sample. The peaks around4px∼ 0.5µm and6.5px∼ 0.8µm indeed give a good initial estimation for the particle radii. However, the two populations cannot be separated well enough. At the red division line there is a considerable overlap of small and big particles. The histogram according to the new SIFT-based algorithm gives a lot more distinct peaks, the overlap of small and big populations is really zero. With radii of3.5px∼ 0.45µm and7px∼ 0.9µm the particle radii are a little underestimated but as expected the factor between big and small is indeed2.

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Figure 3.22: Histograms of the radius distribution of the binary mixture (see Figure3.20) as detected by the Crocker and Grier (CG) and by the new SIFT algorithm. An estimated separation line between small and big is printed in red.

3.3 Tests of the SIFT algorithm

Figure 3.23: Two-dimensional histograms of the distribution of radius and mass (left panel) and radius and DoG minimum depth (right panel), as detected by the two compared algorithms. Color code is chosen to be logarithmically increasing, such that a bin with no particle is dark blue, a bin with one particle is light blue and a bin with about 10000 particles is dark red. Separation lines between small and big particles are shown in red.

More information is given in Figure3.23where a second dimension is added to the histogram. For the CG algorithm it is the mass of the particle, which is defined by the denominator of Equation3.22. For the SIFT-based algorithm it is the depth of the minimum in the DoG space. While the additional dimension does not seem to help much in the discrimination of big and small particles in the CG case, for the SIFT case it proves to be very helpful. The CG mass of the big particles is about 10 times larger than that of the small particles. So it seems that the small particles are really at the detection limit for the CG algorithm.

For SIFT the average difference in the minimum depth is only a factor of two, thus the small particles are well in the detectable regime.

As mentioned above, one realizes that the discrimination between small and big particles is not good enough with the CG algorithm which is the main reason to go for the new SIFT-based method. Another very important issue is the detectability of small particles. As one can already see in the radius distribution in Figure3.22the CG method predicts the population of small and big particles to be of a very similar size, but as mentioned above the mixing ratio is 1:1 but 2:1. The SIFT-based algorithm gives the correct size distribution, the peak from the small particles is twice as high, implying that there are two times more small particles.

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Figure 3.24:Number densities of the two particle components of the sample as measured by the Crocker and Grier (CG) algorithm and the new SIFT-based method

In Figure 3.24this is illustrated further with the number densities of the two populations. The SIFT algorithm gives the correct mixing ratio while the CG method overestimates the number of big particles and does not detect all the small particles. The higher number of big particles might come from the overlap of the distributions at the separation line but the lack of small particles clearly points out not all the particles can be detected with the CG method. An interesting observation is also the fluctuation of the number densities. With the SIFT detection the number density of the big particles does fluctuate less than that of the small particles, which could have two reasons: The small particles are still hard to detect, so that not all of them are detected in every snapshot and/or their number really does fluctuate more, since they can move in and out of the field of view more quickly and more numerously. However, since the mixing ratio of the detected particles goes together with the value expected from the mixing of the dry powders, one can consider the SIFT results as being very reliable.