A fuzzy-registration approach to track cell divisions in time-lapse fluorescence microscopy

Background Particle-tracking in 3D is an indispensable computational tool to extract critical information on dynamical processes from raw time-lapse imaging. This is particularly true with in vivo time-lapse fluorescence imaging in cell and developmental biology, where complex dynamics are observed at high temporal resolution. Common tracking algorithms used with time-lapse data in fluorescence microscopy typically assume a continuous signal where background, recognisable keypoints and independently moving objects of interest are permanently visible. Under these conditions, simple registration and identity management algorithms can track the objects of interest over time. In contrast, here we consider the case of transient signals and objects whose movements are constrained within a tissue, where standard algorithms fail to provide robust tracking. Results To optimize 3D tracking in these conditions, we propose the merging of registration and tracking tasks into a fuzzy registration algorithm to solve the identity management problem. We describe the design and application of such an algorithm, illustrated in the domain of plant biology, and make it available as an open-source software implementation. The algorithm is tested on mitotic events in 4D data-sets obtained with light-sheet fluorescence microscopy on growing Arabidopsis thaliana roots expressing CYCB::GFP. We validate the method by comparing the algorithm performance against both surrogate data and manual tracking. Conclusion This method fills a gap in existing tracking techniques, following mitotic events in challenging data-sets using transient fluorescent markers in unregistered images.


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It is generally understood that automated imaging and tracking methods can ex-4 ceed manual tracking of objects in large data-sets. These solutions are of growing 5 importance in life science applications, where time-lapse microscopy is a power-6 ful and popular tool for capturing dynamics at cellular and sub-cellular scales [1]. 7 Indeed, a wide range of microscopy [2] and computational approaches [3,4,5,6] 8 have been developed and adapted to particle tracking in biology. Developmental 9 biology has benefit enormously from automated in vivo tracking methods, ranging 10 from cell lineage tracing in both animals [7,8,9] and plants [10,11,12], to cell 11 shape tracking [13,14] and tracing of growing organs as in the case of plant roots 12 [11, 15,16,17,18].
one frame to what is found to be the same object in a later frame, so that linked 23 objects share the same identity label, or identifier. The set of linked identifications 24 of a given object through time is referred to as its lineage. 25 Tracking algorithms are greatly influenced by the structure of the data. Macro- 26 scopic objects may have complex morphologies that can be used to improve their 27 identification [22]. Objects smaller than imaging resolution, on the other hand, will 28 appear as featureless blobs; in these cases tracking relies more strongly on dynam-29 ical models alone [23]. For example, "multiple-hypothesis tracking" [24,25] takes 30 an exhaustive and deterministic approach to consider all possible lineage trees. A 31 similar approach has been previously applied to tracking plant root cell nuclei in 32 3D [11]. Stochastic algorithms have also been proposed [26], specifically to deal 33 with particularly noisy data-sets. Whenever persistent features, or reference points 34 known as keypoints, can be easily identified in the background, a common strategy 35 is to build a link of the same keypoint in two frames [27,28,29,30]. Keypoints 36 appearing in more than one frame are defined as inliers, while those appearing in 37 only one frame are treated as outliers. If inliers exist, a geometric transformation 38 (including for example translation, rotation, reflection, scaling) linking them in the 39 background can be determined and used as a first guess for tracking the objects of 40 interest in the foreground. 41 Unfortunately, tracking transient events in 3D is still challenging and imprecise 42 especially in plant biology where soft tissue rarely o↵ers usable reference points. 43 Here, we o↵er a novel solution to this problem, by exchanging the common method 44 of morphological analysis [20] for a fuzzy registration-tracking approach, where fuzzy 45 indicates a random sampling of points. As a case-study, we collected and analysed 46 a 4D data-set of cell division events in the Arabidopsis root meristem, throughout several consecutive days. This was achieved with time-lapse 3D scanning of grow-48 ing transgenic roots expressing the fluorescent reporter CYCB::GFP , through a 49 previously described light-sheet microscope setup [16]. The mitotic events are here 50 identified by the short-lived CYCB::GFP signal, sparsely distributed in a soft tissue 51 lacking fluorescent reference keypoints and generally di cult to identify due to low 52 signal-to-noise ratio. This represents a very common set of circumstances in time-53 lapse imaging of plant tissues. Due to its general strategy and features, we believe 54 that the methodology proposed might be readily applied to similar 4D data-sets 55 collected from other tissues. The identification of inliers is the key challenge in the kind of experimental data-60 set that we are discussing. When single-particle tracking methods require pre-61 registration, they may not be robust against lack of guaranteed inliers. On the 62 other hand, when single-particle tracking does not need pre-registration, it is typi-63 cally based on models of the objects' motion, which are not optimal with transient 64 objects and with low signal-to-noise ratio across extended image sequences. 65 We treated the position, appearance and disappearance of transient objects as 66 a random spatial process. The frame-to-frame displacement of objects embedded 67 in a tissue can be described as the e↵ect of two random variables: (1) large-scale 68 movements of the tissue within the field of view, due for example to its growth or 69 to the inability of the microscope to focus on a fixed point of the field of view over 70 time; (2) small-scale fluctuations of the objects of interests within the tissue.

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The main purpose of the algorithm is to find the best large-scale transforma-72 tion that explains most of the object correspondence, despite the noisy small-scale 73 dynamics. The strategy adopted was to find correspondences between the random 74 process at time t and the random process at time t ⌧ where ⌧ is a lag variable. In 75 this work, we only considered a ⌧ = 1step, which is equivalent to approximating the 76 tissue as e↵ectively rigid at the given temporal scale. As mentioned above, objects 77 that appear in both frames are called inliers and those that do not appear in both 78 frames are called outliers and cannot be linked by a global frame-frame transforma-79 tion. For example, outliers can be either debris, i.e. any object which is determined 80 not to be an object of interest, or could be an object of interest that has just exited 81 or entered the new frame.

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The least-squares-loss method is generally described as the minimization of P |y 122ŷ | 2 , where y is a proposal vector andŷ is the target vector. In our case, the proposal distance > " between objects' centroids is assumed. The value of " was determined 129 from the data and is always slightly larger than the average object radius. When 130 considering distances between proposal objects and target objects, only distances 131 to the first nearest neighbour within a radial distance " were considered. Each 132 candidate transformation was applied to all objects u i (t) 2 U (t), i 2 {1, 2, ..., N}.

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Letũ i = (u i ). We used a cost function i . If the projected object had no nearest neighbour within a sphere of radius ", 136 the capped distance " + 1 was attributed. This is illustrated in Fig. 1.

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The cost function will rank transformations by how well they explain the move- We used computationally generated (surrogate) data to validate the tracking algo-158 rithm against a known spatial transformation. To be representative of the biological 159 images intended to be tracked, the surrogate data were generated respecting a min-160 imum object separation ". A number of parameters were introduced (Table 1)      A third independent validation of our method was performed with a "lag test". We counted the number of objects where identifiers were in agreement when performing the tracking using di↵erent lags ⌧ . For example, we may expect the following equivalence where, with an abuse of notation, ⌧ corresponds to the highest-scoring transforma-211 tion with lag ⌧ . As a representative example of this test, we considered 734 objects 212 in a sample sequence: 42 (or ⇡ 6%) showed disagreements between 1 and 2 and, 213 excluding these, 29 (or ⇡ 4%) showed disagreements between 1 and 3 in specific 214 frames ( Figure 5).  larger epsilon values (Fig. 11).   For this study we imaged and analyzed N = 3 roots. A single primary root was 268 grown and imaged on a custom-made light-sheet microscope setup, as previously 269 described [16]. In essence, the root was hydroponically grown in a perfusion 5ml  typically corresponded to light saturation and/or low signal-to-noise ratio, and was 290 due to a low number of mitotic events. In such instances additional thresholding 291 was applied to the data before wavelet denoising. When the noise exceeded 0.1, the signal-to-noise ratio was so low that the frame was marked as degenerate. Fig. 8a 293 contrasts di↵erent noise levels in sample images and Fig. 8b plots image properties 294 over time for a given experiment.

Object detection 317
We performed a "pre-segmentation" procedure to detect blob centroids. Segmenta-318 tion plays a central role in many image processing pipelines and typically involves 319 (i) thresholding and identifying background, (ii) using distance/gradient transfor-320 mations with peak detection to identify markers and (iii) routines such as watershed 321 [35] to segment blob labels. For our data, we have found it appropriate not to carry 322 out the final segmentation. Instead, we carried out the pre-segmentation steps from 323 thresholding to peak detection in detecting object centroids. Given the variabil-324 ity in the data over a large frame sequence, we have found a simple "annealing 325 thresholding" to be e↵ective. This simply increases a threshold iteratively so as to  Having applied the threshold filter, we identified centroids by (i) performing a di↵erence of Gaussians to emphasise blob-like objects, (ii) applying a maximum 334 filter and (iii) returning the coordinates of the local maxima (peaks) in the image.

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To allow for a fully automated routine that can cope with arbitrary datasets (in the scope of our light-sheet microscopy datasets) the emphasis in the centroid detection 337 stage has been to avoid spurious centroid detections at the risk of under-sampling, 338 while optimising for objects to be identified for at least two frames somewhere 339 during the peak of their light intensity arc.  Object position updates were treated as small Gaussian perturbations of a global transformation. If the position of an object is given as

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The current implementation contains significant "meta analysis" overhead. Availability of data and materials 461 The Python code [36] can be run against sample data hosted in a publicly accessible data repository [37]. The 462 free-to-use code, which has an MIT license, can be run on any operating system running Python. The README at 463 the project homepage provides instructions to run the code with the sample data. 464 Competing interests 465 The authors declare that they have no competing interests. 466 Ethics approval and consent to participate     aggressive Gaussian smoothing (C) is used to find the largest connected component in the narrow band of the data corresponding to the region of activity in the root tip (D).