3.3 Tests of the SIFT algorithm
3.3.1 Tests on simulated images
In order to test the accuracy of positioning and sizing and it’s dependence on noise, simulations were carried out of particle images in 3D and 2D. Similar to [65], simulated images were produced by drawing uniformly illuminated white spheres on a black background, using a ten times higher resolution than it is used in the later processing. The resulting image is then down-sampled by the factor ten to yield the actually wanted particle sizes. This allows for the production of images of arbitrary sized spheres located at sub-pixel/voxel positions. If desired, a simple noise source is simulated by adding random values to the pixels/voxels: After drawing the spheres, the background is0and the spheres are exactly1(maximum value). For the noise, values are obtained from a Poisson distribution with mean (and variance)10and are then divided by 20. This yields a clearly visible noise, which is by eye comparable to real images from confocal microscopy (see Fig.3.9).
3.3 Tests of the SIFT algorithm
Figure 3.9: Screenshot of the particle detection software detecting particles of increasing size in a simulated image with noise. Ring colors correspond to a detection in octave 1-red, 2-light blue, 3-yellow.
Radius determination of particles on a grid
One of the key features of the SIFT algorithm is the ability to determine the size of the features. For a test of the sizing capabilities, 49 spheres of increasing size were put on a7 × 7grid, as shown in Figure3.9.
Results are presented in Figure3.10, illustrating that the sizing error is at maximum5%. Sizes of small particles have a larger error and are more likely to be underestimated. However, particles with radii below 3px are usually too small for a safe detection anyway. In the 3D tests the number of DoGs per octave is𝑛 = 3, while it is𝑛= 4in the 2D case, which slightly reduces the sizing error. In the right panel of Figure3.10one can see that an additional Gaussian pre-filtering of the input data artificially increases the measured radius of small particles. With a pre-filtering using a of width of𝜎= 3.0, particles larger than𝑟 > 10px are not affected at all, but smaller particles appear to be more than10 % bigger. That only particles at radii smaller than5px are severely affected by the pre-filtering is good news. In a real confocal image the point spread function (PSF), which is characteristic to the microscopy setup, produces a similar blurring effect.
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∆∆zz=∆=0.3x/0.125 ∆x∆z=0.3/0.125 ∆x + noise
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ideal+ noiseGaussian (σ=3.0)
Figure 3.10: Sizing accuracy in 3D and 2D images. For 3D, a worse resolution in𝑧direction has not a big influence on the sizing accuracy (comparison of blue and green curves). Noise primarily affects small particle sizes (red curves) and a premature Gaussian filtering (cyan curve) only leads to an overestimation of the radius of small particles, big particles are not affected.
Figure 3.11: Screenshot of the particle detection software detecting particles of different radii in a simulated image with noise at a filling fraction of53 %. Ring colors correspond to the different octaves, where the particles were found.
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ideal+ noiseFigure 3.12:Sizing accuracy in dense 3D and 2D images of randomly positioned spheres. Averages from 60 particles are shown for each of the 25 different particle sizes in the simulation. Error bars correspond to the standard deviation in the radius estimation of those 60 particles. 3D volume fraction isΦ = 24 % and 2D area fraction isΦ2D= 53 %
Radius determination of particles on random positions
A more realistic test is to choose the particle positions randomly avoiding an overlap of neighbouring particles. Figure3.11shows a screenshot of the software detecting particles in a noisy 2D picture at a filling fraction of53 %.
While the particles on the grid in Figure3.9were well separated, now the particle distribution is much denser. As mentioned above, this leads to an underestimation of the radii: The position of a DoG mini-mum moves to smaller scales due to the contribution of the minima of nearby neighbour particles. There-fore, the estimated size is reduced by about5 %in a 3D image at24 %volume fraction and even by about 15 %in a 2D image at53 %area fraction of the particles (see Fig.3.12). In the 2D simulation the discrep-ancy is larger for two reasons: the higher filling fraction and the fact that fewer pixel values contribute to the size measurement compared to the many voxels in the 3D case. This statement is also supported by the larger error bars in 2D, as seen in the right panel of Figure3.12.
Positioning accuracy in dense particle distributions
For a realistic test of the positioning accuracy images with 1500 particles were simulated at comparatively high filling fractions of24 %and30 %in 3D and53 %in 2D. A number of 25 different particle radii were chosen, i.e. 60 particles for each radius. The SIFT algorithm was applied to the images and measured
3.3 Tests of the SIFT algorithm
Figure 3.13: Positioning accuracy in dense 3D and 2D images. Standard deviations of the deviations of measured positions compared to the positions used as input to the simulation. One point gives the average error in the𝑥,𝑦and𝑧coordinates of 60 particles with the radius given by the abscissa. Result are presented for ideal images (blue) and for noisy images (green). 3D volume fraction isΦ = 24 %and 2D area fraction isΦ2D= 53 %
positions were compared to the input positions of the simulations. The deviations of measured compared to simulated positions are determined for all directionsΔ𝑥, Δ𝑦andΔ𝑧, to obtain the distribution of the positioning errors. This is done individually for each of the 25 radii. The standard deviations of those error distributions are plotted in Figure3.13. One receives a curve that reveals the dependence of the average positioning error on the particle size. While in 3D the average error is always below5 %, in noisy 2D pictures the deviations are on average between10 %and15 %of a pixel size. It is understandable that the noise, described in detail in the beginning of this section, has a big influence; it almost doubles the average error.
As seen in Figure3.13, there is no significant dependence of the positioning accuracy on the particle size.
Therefore it makes sense to have a look at the actual error distribution. Figure3.14shows the histogram of the deviations of all 1500 particles together. Distributions for both, ideal and noisy pictures resemble Gaussian curves. Even with noise (green curves) the maximum error is clearly below0.15px in 3D and clearly below0.3px in 2D.
Figure 3.14: Distribution of the positioning error in dense 3D and 2D images with and without noise.
The histograms are taken from the displacements of measured compared to simulated positions in all directionsΔ𝑥,Δ𝑦,Δ𝑧. In all four simulations the number of particles is 1500, the volume fraction is 30 %in 3D and the area fraction is53 %in 2D.
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Gs=−1 raw Gs−1
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Figure 3.15:Positioning accuracy in dense 3D and 2D images with noise (see Figure3.13). Results using 𝐺𝑠=−1for the final refinement step (this work’s standard, blue) are compared to the usage of the raw data (green) and the Gaussian filtered images one scale distance below the local minima𝐺𝑠−1(proposed by Leocmach and Tanaka [65], red).
It was mentioned above that in contradiction to Leocmach and Tanaka [65], the final refinement step was not performed in the Gaussian blurred image𝐺𝑠−1 (with𝑠being the index of the DoG, where the local minimum was found) but in the first blurred image of the first octave𝐺𝑠=−1. As Figure3.15impressively shows, the blurring of𝐺𝑠−1 is too much, at least in the case of high filling fractions. The average error becomes larger with the particle size due to the increased blurring and it is even more than half a pixel size for larger particles. A comparison of using𝐺𝑠=−1to using the raw data shows no big difference, only for larger particles the error is slightly smaller with the blurred image, which is why𝐺𝑠=−1 was used in the data analysis throughout this work.