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Topological reconstruction of sub-cellular motion with Ensemble Kalman velocimetry

  • * Corresponding author: vmaroula@utk.edu

    * Corresponding author: vmaroula@utk.edu
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  • Microscopy imaging of plant cells allows the elaborate analysis of sub-cellular motions of organelles. The large video data set can be efficiently analyzed by automated algorithms. We develop a novel, data-oriented algorithm, which can track organelle movements and reconstruct their trajectories on stacks of image data. Our method proceeds with three steps: (ⅰ) identification, (ⅱ) localization, and (ⅲ) linking. This method combines topological data analysis and Ensemble Kalman Filtering, and does not assume a specific motion model. Application of this method on simulated data sets shows an agreement with ground truth. We also successfully test our method on real microscopy data.

    Mathematics Subject Classification: Primary: 62M20, 62R40; Secondary: 62H35.


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  • Figure 1.  The motion of organelles, during an experiment starting at $ t_1 = 0 $ ending at $ t_N = T $, is identified at discrete times $ t_n $ (dots). For simplicity, space is represented with one dimension, although real datasets are two dimensional. The black dots represent the locations of organelles at different time levels. $ \tilde {\mathcal{R}} $ is the set contains the locations of all black dots

    Figure 2.  Here, $ \bar x^j $ is the position of an organelle producing the images shown (gray), $ \bar f_{n, +} $ and $ \bar f_{n, -} $ illustrate 1-level forward displacement and backward displacement of $ \bar x^j $, respectively. For clarity, the image produced by the organelle are shown as multi-peaked and space as 1D

    Figure 3.  The relations of forward fields and backward fields are indicated here. (a) shows the approach depiction of forward displacement fields, (b) shows the approach depiction of backward displacement fields. For clarity, time marches forward in (a) and backward in (b)

    Figure 4.  (a) shows $ \tilde {\mathcal{R}} $ as black dots and $ {\mathcal{R}} $ as gray lines; (b) shows $ {\mathcal{R}} $, $ {\mathcal{T}}_n $ in Eq. (12) and $ P_ {\mathcal{R}}^{-1}( {\mathcal{T}}_n) $ as blue segments; (c) shows $ {\mathcal{R}} $, $ {\mathcal{T}}_n $, $ P_ {\mathcal{R}}^{-1}( {\mathcal{T}}_n) $, $ \tilde R $ and reconstructed discrete trajectories. For visualization purpose, space is shown in 1D

    Figure 5.  Case I: The frame size is 320 by 320 pixels. Trajectories of 20 organelles are in red spanning from time $ t = 0\; $s to $ t = 99\; $s. Their motion is described by a diffusion process containing both a diffusion and a drift term. The starting distance of any two adjacent organelles at $ t = 0\; $s is 10 pixels

    Figure 6.  Case I: Four histograms of mean error of each frame. Each one compares estimated forward and backward displacement with ground truth along $ x $-axis and $ y $-axis, respectively

    Figure 7.  Case I: Linking result of all trajectories in red. The accuracy rate is 100%

    Figure 8.  Case I: Positions of organelles over time after adding perturbation $ U(-\epsilon, \epsilon) $ when $ \epsilon = 0, \ 1, \ 1.5, \ 2, \ 2.5, \ 3, \ 3.5, \ 4\; $pixels, respectively. If $ \epsilon $ increases, it is more difficult to detect trajectories, especially, when $ \epsilon = 4\; $ pixels, there are no clear patterns for all trajectories to be reconstructed

    Figure 9.  Case II: The left shows the rough detection result, the right show the locations after correction. The red dots in the left penal and blue pentagons in the right penal are the original locations before Bayesian identification. The red pentagons in the right panel are the fitted location after Bayesian identification

    Figure 10.  Case II: Estimated displacement fields of 17th frame using EnKF. Panel (a) shows the estimated displacement fields for the entire focal plane. Panel (b) shows the enlarged area of $ [140,230]\times[260,350] $ in Penal (a)

    Figure 11.  Case II: Trajectories reconstruction result

    Figure 12.  Case II: Four specific sets of trajectory reconstructions vs ground truth. Each panel shows reconstructions versus one true trajectory. The upper left is amplified from the area $ [290,380]\times[40,130] $ in Fig. 11; the upper right is amplified from the area $ [40,190]\times[190,340] $ in Fig. 11; the bottom left is amplified from the area $ [80,210]\times[200,330] $ in Fig. 11; the bottom right is amplified from the area $ [230,380]\times[200,350] $ in Fig. 11;

    Figure 13.  Case III: Panel (a) is the first frame of the video. Panel (b) exhibits all estimated trajectories in red. Panel (c) further shows each estimated trajectory in different colors

    Table 1.  Case I: Table of detection result

    $ \epsilon $ (in pixels) total $> 10\; s $ $ =100\% $ $ \geq 90\% $ $ \geq 50\% $
    $ \epsilon=1 $ 20 20 20 20 20
    $ \epsilon=1.5 $ 20 20 20 20 20
    $ \epsilon=2 $ 20 20 20 20 20
    $ \epsilon=2.5 $ 24 24 14 17 20
    $ \epsilon=3 $ 32 27 9 14 17
    $ \epsilon=3.5 $ 33 30 5 10 15
    $ \epsilon=4 $ 53 40 1 4 14
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