# American Institute of Mathematical Sciences

June  2020, 2(2): 101-121. doi: 10.3934/fods.2020007

## Topological reconstruction of sub-cellular motion with Ensemble Kalman velocimetry

 1 Mathematics Dept., University of Tennessee, Knoxville, TN 37996, USA 2 Center for Biological Physics, Arizona State University, Tempe, AZ 85281, USA

* Corresponding author: vmaroula@utk.edu

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.

Citation: Le Yin, Ioannis Sgouralis, Vasileios Maroulas. Topological reconstruction of sub-cellular motion with Ensemble Kalman velocimetry. Foundations of Data Science, 2020, 2 (2) : 101-121. doi: 10.3934/fods.2020007
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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
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
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)
(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
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
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
Case I: Linking result of all trajectories in red. The accuracy rate is 100%
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
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
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)
Case II: Trajectories reconstruction result
; 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 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;
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
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
 $\epsilon$ (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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