Tests on a rigid sample

Tests on simulation data is a good way to check and verify the method, but for a realistic estimation of positioning errors it is crucial to look at real microscopy data. To do this, a rigid rigid sample was built of PMMA-particles with a size of about2.6µm in diameter, dyed with Rhodamine B. In shelf the particles were suspended in Decaline, for the fabrication of the sample one drop of the suspension was put on a cover glass (No 1.5) and mixed with several drops of Norrland Optical Adhesive 81, which has a refractive index of1.56, not very far away from decaline with∼ 1.49. After 15 minutes of UV curing the glue, a rigid sample was obtained. The micrograph in Figure3.16gives an impression of the particle distribution.

Using two different objectives (100x and 63x),100snapshots were recorded at time intervals of 20 sec-onds, each with a size of30µm in𝑧−direction starting in a depth of𝑧= 10µm. Particles were detected using the software and the tracking algorithm, which is described in the subsequent chapter, was used to join the positions to full tracks for each particle. About 300 tracks could be recorded and the positions were corrected by removing the total drift of the whole sample. Positioning errors are obtained by simply computing the deviation of the detected position from the average detected position of a particle. For an computation of errors in 2D, the whole processing was restricted to the usage of one single slice of the 3D images taken at𝑧= 15µm.

The histograms presented in Figure3.17show the error distributions for the 100x oil objective. With 𝜎_𝑥,𝑦 = 0.015µm the standard deviation from the mean position for𝑥and𝑦is about12 %of the pixel size, which is about three times larger then the average error in the simulations with artificial noise. For𝑧the

3.3 Tests of the SIFT algorithm

Figure 3.16:Micrograph from the rigid sample used for the positioning tests. Pixel size is0.125µm.

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deviation from mean [µm]

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frequency

3D

∆z=0.2µm

x, σ_x=0.015µm y, σ_y=0.015µm z, σ_z=0.030µm

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deviation from mean [µm]

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2D

∆x=∆y=0.125µm

x, σ_x=0.032µm y, σ_y=0.029µm

Figure 3.17:Positioning error distribution in all three directions obtained by recording 100 snapshots of a rigid sample. Images were recorded with a 100x oil objective withNA = 1.45, the voxel size is given byΔ𝑥,Δ𝑦,Δ𝑧. The field of view contained about 300 particles.

average error is even higher with𝜎 = 0.03µm being about15 % of the voxel size in𝑧direction. The small difference between𝑥and𝑦direction is a remnant from the drift, which was stronger in𝑥direction.

Comparing 3D and 2D one obtains the same result as in the simulations: The error is twice as large in 2D since fewer pixels contribute to the calculation of each particle’s position.

For the data in Figure3.17a 2x2 binning of the camera was used. This implies that all pixel values are actually the sum of 4 pixels, each with a size of0.0625µm. Using the full resolution actually makes no sense for three reasons: First, the detected wavelength is much larger than the size of single pixels, thus the Abbe limit comes into play. Second, the used camera would be way too slow in the 1x1 mode. Third, 4 times longer exposure time would be necessary to obtain the same luminosity and the same low noise level. Towards the other direction, going to an even worse resolution, tests were also performed in the 4x4 binning mode, which doubles the pixel size once more to0.25µm. Additionally, bigger stepsΔ𝑧in the𝑧direction upto0.5µm were tested as well. It turned out that the average positioning error expressed in absolute micrometer values was not larger in those tests. This leads to the assumption that the error is not mainly determined by camera noise but rather by the point spread function (PSF) of the objective.

The PSF blurs the image in all three directions and does it particularly strongly in the𝑧direction, so that the error is always larger for𝑧.

More tests were performed using the 63x water objective with a pixel resolution of0.2µm in the 2x2 binning mode. Figure3.18shows that the errors are a lot larger. This is not very astonishing since the

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deviation from mean [µm]

Figure 3.18:Positioning error distribution in all three directions obtained by recording 100 snapshots of a rigid sample. Images were recorded with a 63x water objective withNA = 1.3, the voxel size is given byΔ𝑥,Δ𝑦,Δ𝑧, the field of view contained about 300 particles.

objective does not fit well to the refractive index of the suspensions. However, at least for 3D, with 𝜎_𝑥,𝑦 ∼ 0.035µm the error in𝑥and𝑦 direction is still acceptable. Plateau values of mean square dis-placements (MSDs) in our suspensions are usually about0.02µm², which is still 15 times higher than 𝜎_𝑥,𝑦² ∼ 0.0012µm². Therefore, even with the actually inappropriate water objective, errors in the mea-surement of the MSD should stay below10 %.

Another test is to check whether the radius of a particle is always the same and how large is the error in the radius determination due to the camera noise. In Figure3.19e see the deviation of the radii from their mean value, obtained from 100 consecutive snapshots recorded with the 100x oil objective. With 0.022µm the average deviation, which is the error in this case, is very small compared to the mean radius of all particles of about 1.45µm, an error of only1.5%. Obviously, in a sample with moving particles this error should be larger, as the distance to the nearest neighbour changes from snapshot to snapshot.

To get a feeling for this error, particles were also tracked in a moving sample and the relative standard deviation of the radius of single particles was computed in a measurement of∼ 1500frames. For small particles (4 to 5px in diameter) the relative error for the radius was up to5 %, but for big particles (∼10px and more) it was again as low as1.5%.

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deviation from mean radius [µm]

Figure 3.19: Error distribution of the radius determination obtained by recording 100 snapshots of a rigid sample. The measured mean radius of all particles was1.45µm. Images were recorded with a 100x oil objective withNA = 1.45, the voxel size is given byΔ𝑥,Δ𝑦,Δ𝑧, the field of view contained about 300 particles.

3.3 Tests of the SIFT algorithm

Figure 3.20: Micrograph of a binary mixture of PMMA particles with diameters of1 and2µm at a volume fraction ofΦ = 28 %. Pixel size is0.125µm.