The size of the particle image in pixels is an important factor in the final quality of the fit. Obviously, if the particle image is smaller than one pixel, then at most only four pixels can possibly contain information about said particle image and thus the fit is almost arbitrary (no sub-pixel resolution can be expected). As the size grows both the time needed to perform each fit1 and the chance of overlap increases.
For this test, all particle images had a magnitude equivalent to “bright” (as described in sec- tion12.5), which means that at most one pixel on each particle image had a value of 255.
1The computational time is of course dependent on the size of the kernel, which in turn is about the same as the size of the particle image.
For reference, enlarged samples of the images generated can be seen in section A.1 of Graff [2007d].
The case of perfect, noiseless images will be examined first. In this situation, the discretization error is evident, as was seen above. Scatter and average error plots show that the smaller the particle image, the more dependency betweenxandy error that appears.
Figure 12.4-1: Scatter plot of the error in the sub-pixel coordinate of the center for at most 15,000 recovered particle images for the case of no noise in the image and particle images of radius 1.0.
−0.1494 −0.0896 −0.0299 0.0299 0.0896 0.1494
−0.1494
−0.0896
−0.0299 0.0299 0.0896 0.1494
Scatter plot for no noise, particle images of radius 1.0 pixels
x error in pixels (−2σ:2σ)
y error in pixels (−2σ:2σ)
The scatter plot of the error for 1-pixel-radius particle images of figure12.4-1, for example, shows some very interesting patterns that are a consequence of the particle image being too small relative to the pixels. Dependency artifacts then increase a bit as the particle image size increases, which can be seen as spikes jutting out from the central distribution in the scatter plots—but as the particle image size increases, so does the magnitude of the error.
The 1.5-pixel-radius mark is also a transition for the behavior of the average xandy errors per image.
Between figures12.4-3,12.4-4, and12.4-5, it can be seen that the average error drops sharply, but that by the time the particle images are 2 pixels in radius, there is a difference between the average error inxandy. At this point, it is impossible to separate the contributions of the optimization itself, the pseudo-random position of the particle images, and the discretized domain. The discretization effect can be clearly seen in the scatter plot of figure12.4-6.
Note also that the average error remains relatively constant for the small particle images, but in figure12.4-5it is clear that it decreases as the images are moved closer and closer to the apertures.
This shows that at this particle image size, particle image overlap is already affecting the quality of the fit. The translation tests plot of section12.3is a more pure measurement of the fitting accuracy
Figure 12.4-2: Scatter plot of the error in the sub-pixel coordinate of the center for at most 15,000 recovered particle images for the case of no noise in the image and particle images of radius 4.0.
−0.2482 −0.1489 −0.0496 0.0496 0.1489 0.2482
−0.2482
−0.1489
−0.0496 0.0496 0.1489 0.2482
Scatter plot for no noise, particle images of radius 4.0 pixels
x error in pixels (−2σ:2σ)
y error in pixels (−2σ:2σ)
Figure 12.4-3: Plot of average magnitude of the pixel error in the sub-pixel coordinate of the center of recovered particle images for the case of no noise in the image and particle images of radius 1.0.
0 50 100 150
0.036 0.037 0.038 0.039 0.04 0.041 0.042 0.043 0.044
Image
Error magnitude (pixels)
Radius 1.0p, no noise
x y
Figure 12.4-4: Plot of average magnitude of the pixel error in the sub-pixel coordinate of the center of recovered particle images for the case of no noise in the image and particle images of radius 1.5.
0 50 100 150
8 8.5 9 9.5 10 10.5 11 11.5
x 10−3
Image
Error magnitude (pixels)
Radius 1.5p, no noise
x y
Figure 12.4-5: Plot of average magnitude of the pixel error in the sub-pixel coordinate of the center of recovered particle images for the case of no noise in the image and particle images of radius 2.0.
0 50 100 150
4 5 6 7 8 9 10 11 12 13
x 10−3
Image
Error magnitude (pixels)
Radius 2.0p, no noise
x y
Figure 12.4-6: Scatter plot of the error in the sub-pixel coordinate of the center for at most 15,000 recovered particle images for the case of no noise in the image and particle images of radius 2.0.
−0.0701 −0.0421 −0.014 0.014 0.0421 0.0701
−0.0701
−0.0421
−0.014 0.014 0.0421 0.0701
Scatter plot for no noise, particle images of radius 2.0 pixels
x error in pixels (−2σ:2σ)
y error in pixels (−2σ:2σ)
because there is no overlap in those particle images. The data presented in this section through the scatter plots and average error plots, however, is more realistic, since they more closely approximate experimental situations, so the evidence presented by those plots can be more readily applied to estimating experimental conditions.
The precision of the Gaussian fit is not the only performance benchmark. As shown in figure12.4- 7, the 0.5-pixel-radius particle image is completely inadequate since a substantial portion of them are lost. Moreover the histograms of figure12.3-2 show that the error for those that are recovered is substantial, which should be expected since no sub-pixel accuracy is possible with particle images that do not affect more than a single pixel.
On the other end of the spectrum, the large particle images take up so much room that the overlap becomes a considerable problem, as shown in figure12.4-8.
Adding some slight noise to the image greatly affects the results. With magnitude 5 noise, the histogram spreads over a width nearly 8 times larger than the case with no noise, and any bias due to discretization is overwhelmed, as is obvious from comparison of figures12.4-6and12.4-10.
The evidence can also be seen in the results of the systematic translation test of figures 12.3-1 and12.4-11.
For each increase in the magnitude of the noise, the error distribution increases in width by the same factor. The increased noise also diminishes the number of correctly identified particle images.
Not only do extra particle images begin to appear, but they contribute to the error of properly identified particle images in two ways: first, the fit is obviously affected, though if the particle image is large enough, the effect should be minimized. But there is also added uncertainty in identifying
Figure 12.4-7: Population statistics for recovered particle images for the case of no noise in the image and particle images of radius 0.5.
0 50 100 150
0 5 10 15 20 25 30 35
Image
Percent of total particle images
Radius 0.5p, no noise
Double Extra Lost
Figure 12.4-8: Population statistics for recovered particle images for the case of no noise in the image and particle images of radius 6.0.
0 50 100 150
0 5 10 15 20 25 30 35 40
Image
Percent of total particle images
Radius 6.0p, no noise
Double Extra Lost
Figure 12.4-9: Histogram of the pixel error in thexcoordinate of the sub-pixel coordinate of the center of recovered particle images for the case of noise of magnitude 5 in the image.
−0.1 −0.0714 −0.0429 −0.0143 0.0143 0.0429 0.0714 0.1
0 1.1 2.3 3.4 4.6
x error in pixels
Percent population
x error histogram vs. radius, noise mag. 5
0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 Particle image radius in pixels
Figure 12.4-10: Scatter plot of the error in the sub-pixel coordinate of the center for at most 15,000 recovered particle images for the case of noise of magnitude 5 in the image and particle images of radius 2.0.
−0.0777 −0.0466 −0.0155 0.0155 0.0466 0.0777
−0.0777
−0.0466
−0.0155 0.0155 0.0466 0.0777
Scatter plot for noise mag. 5, particle images of radius 2.0 pixels
x error in pixels (−2σ:2σ)
y error in pixels (−2σ:2σ)
Figure 12.4-11:Error in thexcoordinate of the sub-pixel center of the Gaussian fit in a systematic translation of 1/100 of a pixel per step for images with noise of magnitude 5.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
Actual x position (pixels)
Error in x position (pixels)
Translation test, x error, mag. 5 noise
0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 Par. img. radius (pixels)
the potential particle image peaks, about which the kernel for fitting is centered. The cumulative result is that at extremely high noise levels, there is little advantage to particle image size from the point of view of fit accuracy, as can be seen in figure12.4-12. (Note that a noise of magnitude 50 is well beyond what can be achieved during a real experiment.) There is, however, still an advantage from the point of view of the number of particle images recovered, as seen in figure12.4-13.
Figure 12.4-12: Histogram of the pixel error in thexcoordinate of the sub-pixel coordinate of the center of recovered particle images for the case of noise of magnitude 50 in the image.
−0.5 −0.3571 −0.2143 −0.0714 0.0714 0.2143 0.3571 0.5
0 0.6 1.2 1.8 2.4
x error in pixels
Percent population
x error histogram vs. radius, noise mag. 50
0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 Particle image radius in pixels
Figure 12.4-13: Percent lost particle images for the case of noise of magnitude 50 in the image for particle images of each radius.
0 50 100 150
0 10 20 30 40 50 60 70 80 90 100
Image
Percent of total particle images
Percent lost particle images for each radius, noise mag. 50
0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 Par. img. radius (pixels)
As will be seen in section12.5, the susceptibility of the accuracy to noise is much lower than that to image intensity. Between perfect images (figure12.4-14) and those with absurd amounts of noise (figure12.4-15), the 99th percentile grows by less than 50% for most particle image radii, though, as mentioned above, the increase in noise decreases the difference between the performance at different particle image radii.
Figure 12.4-14: Percentile of the radial pixel error magnitude in the sub-pixel coordinate of the center of recovered particle images for the case of no noise in the image.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 25 50 75 100
Error in pixels
Percent population
Percentile radial error magnitude vs. radius, no noise
0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 Par. img. radius (pix.)
Figure 12.4-15: Percentile of the radial pixel error magnitude in the sub-pixel coordinate of the center of recovered particle images for the case of noise of magnitude 50 in the image.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 25 50 75 100
Error in pixels
Percent population
Percentile radial error magnitude vs. radius, noise mag. 50
0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 Par. img. radius (pix.)