Large scale cell tracking using an approximated Sinkhorn algorithm

Large-scale cell tracking using an approximated
Sinkhorn-Knopp algorithm
Submitted to the
School of Engineering Science,
Information Science Department
Osaka University
February 2021
Parth Shirish Nandedkar

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2020 Bachelor’s Thesis
Large-scale cell tracking using an approximated Sinkhorn-Knopp algorithm
Parth Shirish Nandedkar
Abstract
The popular field of Cell Tracking has a lot of recent developments, with the influx
of many different deep learning methods. However, not only are cells far more
numerous, the different kinds of organisms have cells with entirely different
characteristics. Apart from these practical considerations, the issues of blurry
intercellular boundaries and mitosis are also faced. It is for these reasons that much of
biological research on tracking the behaviour of cells still involves manual annotation of
cells in each photographed slide.
In this research, we focus on vastly speeding up the tracking of cells, basing our
experiments on the data for the multi-cellular pattern formation of over a hundred
thousand cells of a microorganism, the Dictyostelium discoideum species of amoeba, in
neurologically diseased animal subjects. The method follows the traditional two-phase
(detection phase and association phase) approach to cell tracking. However, due to the
sheer number of cells, faster, but less precise bipartite graph matching or linear
programming (optimal transport) approaches were tested and a final candidate - a
modified and greatly speeded-up Sinkhorn-Knopp algorithm, the Sparse Sinkhorn-
Knopp algorithm - was reached.
The proposed algorithm allows researchers to automate the annotation of cells, and
in the case of the Dictyostelium discoideum cell data, allows for the association of over
10,000 detected cells into continuous cell tracks in a mere 25 seconds per slide of data,
in 𝑂(𝑛) time and space complexity for 𝑛 cells.
Keywords: Cell tracking, Multi-cellular patterns, Hungarian algorithm, Bipartite
Matching, Optimal Transport, Sparse matrices, Sinkhorn-Knopp algorithm

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Contents
1 Introduction
2 Related work
2.1 Deep Learning-based approaches
2.1.1 Motion and Position Map (MPM) method
2.1.2 CellNucleiTracker
2.2 Non-Deep Learning based approaches
2.2.1 CancerCellTracker
2.2.2 Macrophage tracker (Hungarian algorithm)
2.2.3 Candidate cell region detection-based association
2.2.4 Probabilistic event detection of stem cells
2.2.5 Tracking using Particle filters and Time-to-Event analysis
2.3 General multi object tracking methods
2.3.1 DeepMOT
2.3.2 Multi-object tracking using dynamical graph matching
3 Proposed method
3.1 Pipeline
3.2 Detection Phase
3.3 Tracklets generation
3.3.1 Features and cost matrix
3.3.2 Nearest Neighbor matching
3.3.3 Hungarian matching
3.3.4 Sinkhorn-Knopp matching
3.3.5 Sparse Sinkhorn-Knopp matching
3.4 Tracklets association

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4 Experiments
4.1 Details on the dataset
4.2 Examples
4.3 Protocol
4.4 Tracking accuracy
4.5 Computational cost compared to pure Sinkhorn-Knopp algorithm
5 Conclusions
Acknowledgements
Bibliography

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Chapter 1
Introduction
There have been many advances in the neurological study of diseased or malnourished
neural tissue. One of these advances is based on modelling the neurons with the
Dictyostelium discoideum species of amoeba. The lifecycle of D. discoideum is relatively
short, which allows for timely viewing of all stages. This makes it a model organism to
study cellular and biochemical processes in neural tissues. However, due to its ever-
changing nature and the large number of cells, the identification of cells fluorescent
imaging data for these is still done manually. The goal of this research thesis is to
automate this process of identifying, numbering and associating (joining the tracks of)
each cell across multiple frames of fluorescent imaging data.
When malnourished, thousands of dictyostelium cells collectively migrate and form
patterns as shown in Figure 1.1 below. In particular, only a few ‘leader’ cells release a
small amount of Cyclic adenosine monophosphate (cAMP) as a signal to the remaining
‘follower’ cells which then leads to this migration. Such collective cell migration
contributes to many organismal phenomena in neural tissues, including morphogenesis,
wound healing, and cancer invasion [1, 2], making the timely analysis of the movement
of a 100,000 or more cells imperative for the study of cell-cell communication and
desieases like brain cancer. The provided data has over a 100,000 dictyostelium cells
laced with two different fluorescent protiens allowing for two different modes of
fluorescent imaging of the amoeba.

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Figure 1.1: The formation of circular patterns in malnourished Dictyostelium cells.
Cells releasing cAMP are shown in red [15].
As this is a tell tracking problem, we will first discuss the basics method of cell
tracking. Cell tracking consists of two phases, the detection phase, and the Association
Phase [3]. The detection phase consists of identifying the bounding boxes of each cell
and assigning temporary IDs to them in each frame. Recently, trained CNNs and RNNs
are being used to detect the cells in each frame [4, 5, 6, 10, 17]. In the association
phase, these detection results associated in successive time-frames to form tracklets, in
a process called "Tracklets generation”. These tracklets are then associated wherever
possible, to form continuous overall trajectories for as many cells as possible in the
“Global Association” stage. This process is shown visually in Figure 1.2 below. The
global association stage always follows the detection phase and the tracklets generation
stage with a delay in order to gather as much data as required to associate a tracklet
ending in the current frame to a new tracklet starting in the next few frames.
Figure 1.2: The pipeline of most Cell Trackers [3]
The above method for tracking cells has many issues however, such as those shown
below. Figure 1.3 shows examples of these issues, with 1.3(a) showing the case of

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similarly shaped cells, leading many deep learning based methods and traditional object
tracking astray. Figure 1.3(b), a part of our target dataset, shows the random shape
changes many types of cells undergo from frame to frame. Figure 1.3(c), depicts the
extremely blurry intercellular boundaries in our target dataset, which renders many cell
segmentation algorithms futile. Lastly, Figure 1.3(d) is an example of Cell Mitosis
occurring between frames, which needs its own algorithm to track newly separated
cells, such as the Mitosis tracker developed by S. Huh [7]. The fifth and most important
issue for practical purposes is temporal efficiency or computational cost, which rules out
most existing cell tracking or multi object tracking (MOT) methods for this research.
Figure 1.3: (a) Example cell data, with each cell showing nearly identical shapes [23]. (b)
The cells in the centre and the bottom right vastly change their shape and size
respectively. (c) Cells stuck to each other with no clear intercellular boundaries in our
dataset. (d) Cell mitosis occurs for the bright cell in the centre.
At present, there are many different methods to track (associate) detected cells, and
can be broadly classified into deep and non-deep. Deep learning methods for cell
tracking have been developed over the last decade, and are based on detection using a
RNN (or Bi-RNN), a U-net (a specialized CNN for medical image segmentation) or a
siamese network [4, 5, 10, 17, 18] trained on sample images of the type of cell and fed
cell ID annotation data. However, upon application of these methods to our target
dataset, the shortcomings of all deep learning methods are clear. Adding annotation

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data for a 100,000 cells impractical. On top of this, these networks also almost
completely base their identification of cells upon existing characteristics of the specific
type of cell, which inevitably requires high resolution biomedical imaging. This is not
the case for our very low-resolution dataset as shown in Figure 1.1 or 1.3(b),(c). Since
cell imaging is low resolution when imaging such a vast number of cells, we can
conclude that deep learning methods will be inaccurate and largely overfitted. Apart
from these issues, neural networks may not be able to keep track of the constant cell
shape changes of the Dictyostelium discoideum amoeba, as discussed in detail in the
related work section.
In non-deep learning approaches, we have the basic K-nearest neighbour (KNN)
algorithm, bipartite graph matching algorithms, and the linear programming based
optimal transport algorithms [19 , 9, 12]. We use KNN as a baseline for testing the
other methods here, namely the Hungarian bipartite matching algorithm and the
Sinkhorn-Knopp optimal transport algorithm. The Hungarian algorithm, or the Kuhn-
Munkres algorithm [20] is a simple linear programming approach to solving the general
assignment problem provided the costs (based on similarity metrics) between every
combination of cells in two different groups of equal number of cells. The advantage
with this method is the ability to get a robust matching between two successive frames
of cells (current frame 𝑡 and previous frame 𝑡 − 1) without impractical amounts of
annotation or training. However, the naïve Hungarian algorithm has 𝑂(𝑛 )
computational cost (where n is the number of cells). This can be reduced to 𝑂(𝑘𝑛 ) by
limiting the scope of matching to only the K nearest neighbours (Hungarian-on-KNN,
or the sparse assignment problem [21]). This makes for a small 𝑂(𝑘𝑛) space cost matrix
and a decently fast match for a thousand cells (about 40 seconds per match for 2,000
cells), however it still takes more than 15 minutes to match two frames of 10,000 cells
each.
The Sinkhorn-Knopp algorithm [12, 13, 14] is also a linear programming based
approach, which, when provided the costs between cells in one frame and in another
frame, converts these costs into entropies, and maximizes the sum of these entropies,
closing in to an asymptotic solution, while never truly reaching the exact solution [13].
The advantages with this optimal transport algorithm are numerous, starting from
having no need of annotation or training, to the temporal and spatial cost [12]. It also
allows for specifying the weights of each object in the form of probability vectors, which

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in our case corresponds to the total brightness of the cell. When two cells visually
merge, the Sinkhorn algorithm detects that their brightness has been added, and
matches these merger cells to the merged cell. The Sinkhorn-Knopp algorithm is 𝑂(𝑛 )
in time and space (due to the matrix operations), making it a much better candidate
for speeding it up by tinkering with the process.
The algorithm proposed in this paper for solving the issue of computational cost for
large-scale cell tracking is the Sparse Sinkhorn-Knopp algorithm. This is a modification
of the Sinkhorn-Knopp algorithm, in which we generate a sparse cost matrix by
calculating costs only for a limited number of candidate cells matching certain criteria.
This takes advantage of the faster 𝑂(𝑘𝑛) sparse matrix multiplications, making the
computational cost of the match 𝑂(𝑘𝑛) for n cells. Also, another major advantage of
the proposed algorithm also 𝑂(𝑘𝑛) in spatial complexity since the sparse cost matrix
can be stored in any of the CSR, CSC, COO or LIL formats. This also tackles the
shape-changing nature of amoeba and neurons by not relying on learning cell shape and
structure. The issues of blurry intercellular boundaries and cell mitosis remain however,
making the global association of tracklets inaccurate for this method in its current
state.

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Chapter 2
Related work
There have been several published methods in order to deal with issues such as
inaccurate cell segmentation and cell mitosis, however as this research focuses on
mainly the time and space complexity encountered in forming cell tracks, most of the
related work below was scrapped in order to generate the sparse version of the
Sinkhorn-Knopp algorithm. All the tried and tested methods have been listed and
discussed below. Some of these methods are general multi-object trackers.
2.1 Deep learning-based approaches
2.1.1 Motion and Position Map(MPM) method
The MPM method is a recently developed cutting-edge cell detection method [4]. It
uses U-nets to both detect and associate cells, the tracklet formation being based on a
likelihood estimation formula which uses the cell’s motion data for the next few frames.
In the detection phase, each detected cell is first given an ID using a Likelihood Map
(LM) generated by using a U-net trained for cell detection of the specific type of cell.
As the first step to associate the cells across frames, tracklets must be generated.
For this purpose, Cell Motion Fields (CMFs) for different frame intervals (2 frames ,3
frames and so on) are generated as shown in Figure 2.1. The CMF gives us data on the
direction and magnitude of each cell’s motion. A predicted matching is made based on
the Motion and Position Map calculated from a formula with trainable weights.

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Figure 2.1: The Cell Motion Field (CMF) used for matching, the vectors show the
magnitude and direction of displacement in the duration of the interval [4]
The training of the MPM-net (the U-net performing the matches) consists of five
steps. We take two successive frames and find the relative cell position and motion,
each as annotated in Figure 2.2(a). Next, we calculate the ‘Cell Motion Vectors’(CMV)
which are the Motion vectors from the neighbor points around the annotated cell of ID
‘𝑖’ in frame ‘𝑡’ to the annotated cell in frame ‘𝑡 − 1’. This collection of vectors is shown
in blue in Figure 2.2(b). Next the ‘Cell Position Likelihood Map’ (CPLM) is calculated
which, in a similar fashion to random walk theory, is a 3-D Gaussian distribution
around the ‘𝑡 − 1’ position. as in Figure 2.2(c). The Motion and Position Map (MPM)
is formed from the above CMV and CPLM data using a formula which contains
trainable weights. Finally, a CNN (U-net) is trained on the 3D vector data of this
MPM using MSE loss which trains the weights in the formula. This is the ‘MPM-Net’.
The main advantage of this method is since the number of trainable weights is
minimal, the simplest of CNNs can function as MPM-Nets, and with great accuracy.
Also, since MPM represents both detection and association, it guarantees coherence,
i.e., if a cell is detected, an association of the cell is always produced. Other similar
CMF-based or multi-tasking learning based methods [5] have the tendency to be
incoherent, due to having independent detection and association phases.

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Figure 2.2: The calculation of the MPM from the CMV and the CPLM [4].
However, upon application of this method to a small sample target dataset, the
flaws of this approach in relation to the given problem will be met. The MPM-net,
while requiring fewer epochs and training samples to train than other deep learning
based methods, still needs large-scale annotations. Also, coherence between association
and detection phases does not solve the issues of shape-shifts and blurry boundaries in
the detection phase. On the contrary, the training and the results of the simpler MPM-
net architecture is highly susceptible to such detection lapses.
2.1.2 CellNucleiTracker
Another NN (U-net) based method [6], which is based on learning the features of the
cells, especially the location of the nuclei. As seen in Figure 2.3, this method was nearly
perfectly accurate for plant cells, and epithelial or circulatory tissue, which are largely
immobile cells. Also, the resolution of the frames is much higher, allowing the network
to learn the different features of each type of cell.

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Figure 2.3: The pretrained network of CellNucleiTracker showed nearly perfect
detection and segmentation for high resolution images of plant and animal tissue [6].
Since such conditions are not present in our dataset, this method was very inaccurate
for the Dictyostelium cell dataset, as seen in the results of the pre-trained network
(Figure 2.4).
Figure 2.4: The pretrained network could not handle the provided low resolution frames
of Dictyostelium amoeba cells

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2.2 Non-deep learning based approaches
2.2.1 CancerCellTracker
CancerCellTracker is an OpenSource project for solving the Fluo-H2DL-HeLa dataset
of the cell tracking challenge [8]. The dataset consists of cancerous cells with cell
mitosis constantly occurring. It uses the watershed algorithm to segment cells and
builds a feature vector for cell tracking including the information of position, shape,
spatial distribution and texture as discussed in Li et. al. [8]. It also has a cell family
history tracking particularly useful for cancerous cells.
Not requiring annotation and training, this method is also faster than the MPM and
DeepMOT methods discussed in this section, even when obtaining the results. However
identity switching is common in the results as in Figure 2.5, and cell family history
tracking is not useful for dictyostelium cells which, like the neural cells it is a model
for, rarely undergo mitosis, in stark contrast to cancerous cells. Due to major accuracy
concerns, this method is not robust enough for large-scale cell tracking.
Figure 2.5: CancerCellTracker shows many ID switches even in the first few frames.

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2.2.2 Macrophage tracker (Hungarian algorithm)
This macrophage cell tracker [9] is based on one of the simplest bipartite graph
matching algorithm, the Hungarian algorithm, and is hence a good candidate for our
large-scale cell tracking problem. The proposed method is an automated single
macrophage tracking method for a dataset of a mouse brain (11.7 T time-lapse MR
images). The method detects macrophages using background subtraction algorithm,
and tracks macrophages using the Hungarian algorithm, backed up by a Kalman filter
for association. The results showed that the proposed method detected and tracked
macrophages in MR images successfully. The below table (Table 2.1) from the paper
shows an example matching obtained by the Hungarian algorithm between cells in
frame ‘𝑡’ and frame ‘𝑡 + 1’.
Table 2.1: An example matching using the Hungarian algorithm. The Red numbers
represent the highest probability match for the cells in ‘𝑡 + 1’, and the light red boxes
represent the final match chosen, after resolving conflicts [9].
However, despite the application of the Kalman filter in this method, the issues with
this method remain the same as for the optimized Hungarian algorithm (Hungarian-on-
KNN) discussed in the Introduction section. This algorithm is still 𝑂(𝑛 ) in time
complexity after limiting the scope of matching to the nearest neighbours, and hence,
despite the accuracy of association, is rendered unsuitable.
2.2.3 Candidate cell region detection-based association
In this paper [24], the proposed method tracks cells under highly confluent conditions
by using the candidate cell region detection-based association approach. While the
association phase is also a linear programming based approach like in our case, the
detection phase is it’s unique point. Unlike conventional segmentation-based association
tracking methods, the proposed method uses the tracking results from the previous
frame to segment the cell regions at the current frame. First, candidate cell regions are

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detected, and while there may be many false positives, there are very few false
negatives. Next, optimized detection results are selected from the candidate regions and
associated with the tracking results of the previous frame by resolving a linear
programming problem.
This non-deep learning approach is very for our current research, as it has the
potential to be fast enough for the requisite number of cells. Many ideas discussed in
the proposed method section are similar to those discussed in this paper.
2.2.4 Probabilistic event detection of stem cells
In this non-deep learning approach [7], a probabilistic model for event detection is
proposed which can simultaneously identify spatio-temporal patch sequences that
contain a mitotic event and also specify the time and location of cell birth events. The
approach significantly outperforms previous approaches in terms of both detection
accuracy and computational efficiency, when applied to multipotent C3H10T1/2
mesenchymal and C2C12 myoblastic stem cell populations.
However, this is a method specialized for cells which undergo mitosis very
frequently, (as in the case of cancer cells shown in Figure 2.5), as the semi-Markov
model used adds a lot of unnecessary computation if the method is used on cells with
infrequent mitosis events.
2.2.5 Tracking using Particle filters and Time-to-Event
analysis
This paper [25] proposes a method for spatiotemporal tracking and analysis of FUCCI
(fluorescent ubiquitination-based cell cycle indicator)-labelled cells from time-lapse
videos. To address the colour transition of the FUCCI-labelled cell with the cell cycle
progression, the proposed method simultaneously estimates the location and the cell
cycle phase of the target cell. Furthermore, to analyse the cell phase transition, this
paper proposes to apply multistate time-to-event analysis to the information obtained
through our tracking method. This paper demonstrates the usefulness of the method
with application to FUCCI-labelled HuH7 cells (human hepatocellular carcinoma cell
line).
The method was deemed to be increasingly complex for the problem at hand, since
cell-cycle analysis is unnecessary for our research. The particle filters are a useful idea

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for large-scale cell tracking, allowing global association in relatively quick time, and this
is hence one of the main proposed improvements to our cell tracker for the future.
2.3 General multi-object tracking methods
2.3.1 DeepMOT
DeepMOT [10] specializes in removing false positives and false negatives in general
multi-object tracking, while also preventing unnecessary ID changes during the cell
track. It achieves this by using a trained Bidirectional RNN (Bi-RNN) to perform the
assignment on a distance (cost) matrix). The detailed pipeline for this method is shown
in Figure 2.6.
Figure 2.6: The pipeline of DeepMOT. The distance matrix D is flattened and fed to a
Bi-RNN at two steps of the process for learning, and the recovered matrix is taken as
the assignment matrix A [10].
There are four steps to achieve this extremely accurate matching between the
objects. First, we randomly sample a pair of consecutive frames from the training video
sequences. These two images together with their ground-truth bounding boxes
constitute one training instance. Then, for each such instance, we first initialize the
tracks with ground-truth bounding boxes (at time 𝑡) and run the forward pass to
obtain the track's bounding-box predictions in the following video frame (time 𝑡 + 1).
To mimic the effect of imperfect detections, we add random perturbations to the
ground-truth bounding. Thirdly, we compute D (distance matrix in Figure 2.7) and use
the proposed Deep Hungarian Network (a Bi-RNN) to compute A (the assignment
matrix). Finally, we compute our proxy loss based on D and A. This provides us with a
gradient that accounts for the assignment, and that is used to update the weights of
the tracker.
Thus, the weights of the tracker are updated as the tracking results are generated,
simplifying the process of training and testing. Figure 2.7 depicts the flow of data and

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the back-propagation of gradients with the Deep Hungarian Network (the Bi-RNN) in
the centre. Figure 2.8 is a frame of the result obtained.
Figure 2.7: The pipeline of DeepMOT including detection, association and training. The
pink box represents the training stage, where the DHN and the DeepMOT Loss
alternate till the training is complete [10].
As is evident in the small example frames in Figure 2.7 and the training data
proposed in [10], this method which uses a complex Bi-RNN architecture is suitable
only for a few dozen objects in-frame. Because of the use of a complex network, it is
very well suited for large-sized objects with many features to learn from. Neither of
these match with the conditions of large-scale cell tracking however, where the goal is a
100,000 cells with a very low resolution of imaging, few features and constantly
changing shapes.
2.3.2 Multi-object tracking using dynamical graph matching
This association part of this method is based on a tracking algorithm to address the
interactions among objects, and to track them individually for a static video camera
[11]. It is achieved by constructing an invariant bipartite graph to model the dynamics
of the tracking process, of which the nodes are classified into objects and profiles as
shown in Figure 2.9(a),(b) and (c).

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Figure 2.9: (a) End of frame ‘𝑡 − 1’. The matching information is used to update the
profiles used in frame ‘𝑡’. (b) Beginning of frame ‘𝑡’. The bipartite graph has only 2
profile nodes. (c) Now 3 objects are detected so the graph now has 3 new object nodes.
(d) A bipartite matching is carried out. Since there is one unmatched object node, a
profile node is created to match it as in (e) and (f) [11].
The matching occurs in as shown in Figure 2.9(d),(e) and (f), and includes ad-hoc
approaches to solve object interaction issues. First, the matching cost between a profile
node and an object node is measured by the Kullback-Leibler distance on the image
data distribution of the nodes. Now, if there exist unmatched object nodes, then new
profiles are is created for each newly detected object. When an object leaves the scene,
its corresponding profile will become unmatched. After finding the optimal bipartite
matching, we check all profiles to see if there are some profiles getting too close to
others. Those profiles will be marked as ”TBM” (to-be-merged). In the next image
frame, the TBM profiles will be processed separately to handle possible interactions.
This simple assignment method is designed for ever-changing numbers of cells and is
quite appropriate for the given dataset. However, as the number of objects for the video
data used in this research is small, the computational complexity has not been
considered. A simpler form of the TBM profile processing feature was implemented in
the form of a ‘merge dictionary’ in the tracklet association stage (Chapter 3.4).

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Chapter 3
Proposed method
3.1 Pipeline
The proposed method in order to solve the time and space complexity issue with large-
scale cell tracking is the Sparse Sinkhorn-Knopp algorithm. The pipeline is as shown in
Figure 3.1, and has the same broad structure of detection followed by tracklet
formation and ultimately global association of these tracklets.
Figure 3.1: The pipeline of the proposed method. It follows the same pattern as the
generic tracking pipeline in Figure 1.2.
3.2 Detection phase
We begin with a detection phase that is independent of the association phase, based on
a blob (maxima) detection algorithm on Laplacian of Gaussian (LoG) filtered frames
with non-maxima suppression. This method was chosen for being among the quickest
available methods. The LoG filter helps in boosting the brightness of the peaks
compared to the surroundings. The method used computes the LoG images with
successively increasing standard deviations and stacks them up in a cube data
structure. The blobs are the local maxima in this cube after non-maxima suppression..
The equation for the LoG filter is as follows directly from spatial image processing
theory, with an added Gaussian filter of standard deviation 𝜎. Various 𝜎 will be used in
the method. 𝑥, 𝑦 are the two dimensional variables used to specify the filter’s equation.

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The 2D function looks as shown in Figure 3.2, and has the effect of highlighting
edges and peaks, the latter being our target of detection.
Figure 3.2: The LoG filter calculates the second derivatve of the image in the 2D space
domain. This leads to highlighted edges, and therefore, peaks.
The usual disadvantage of the LoG approach is that detecting larger blobs is
especially slower because of larger kernel sizes during convolution. However, since D.
discoideum cells very have similar sizes, the kernel size can be limited to a small range,
and the detection can occur smoothly in the expected 𝑂(𝑛) time. Alternative methods
are by applying the Difference of Gaussian (DoG) or the Determinant of Hessian (DoH)
filters instead, to speed up the detection even further. However, a simple comparison of
these three filters (Figure 3.3) shows the massive loss in accuracy for the DoG and DoH
filters.

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Figure 3.3: A comparison of the LoG, the DoG and the DoH filters used in peak
extraction. The latter two while faster and widely used as approximations, are both
largely inaccurate for our low resolution target data.
These detected blobs are then sorted into an adjustable number of bins as shown in
Figure 3.4, based on their position on the frame, with bin 1 starting on the top left.
This is done so as to allow for optimized Sinkhorn-Knopp matches later on, by
considering matches for a cell in frame ‘𝑡’ only from frame ‘𝑡 − 1’ cells in the same bin
or the surrounding bins. Hence, each detected blob consist of four parameters, the x
and y coordinates, the radius of the circular blob, and the bin in which the blob is
present.
Figure 3.4: The division of all the cells in each frame into bins of equal sizes, allowing
for the removal of the vast majority of the candidate matches faster than a KNN
approach. Bin numbering starts from the top left.

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3.3 Tracklets generation
3.3.1 Features and cost matrix
As this is a bipartite matching or assignment problem, the costs or dissimilarities
between cells in frame ‘𝑡’ and frame ‘𝑡 − 1’ must be arranged in a cost matrix. The
features used for calculating dissimilarity are the distance 𝑑 between the two cells in
the adjacent frames, the radii of the blobs, the brightness at the peaks (averaged with
the surrounding pixels), and the net brightness of all pixels that make up the cell blobs.
The last feature is useful because it is directly proportional to the amount of fluorescent
protein present inside the cell. The blob radii, peak brightness values and brightness
sum values are all averaged for each detection of each cell, to remove variance and get
more accurate data. A simple cost function that provides decent results is as shown
below.
Features: Euclidean distance moved 𝑑, radii 𝑟 and 𝑟 , peak brightness values 𝑏 and
𝑏 , brightness sum values 𝑠 and 𝑠 (distances in pixels, brightness from 0 to 255, 𝑤 are
the weights).
𝐶𝑜𝑠𝑡 = 𝑑 + 𝑤 × (𝑟 − 𝑟 ) + 𝑤 × (𝑏 − 𝑏 ) + 𝑤 × (𝑠 − 𝑠 )
Figure 3.5 is an example cost matrix when the cells are divided into 100 bins.
Figure 3.5: An example cost matrix when the cells are sorted into 100 bins. The
columns represent the cell IDs in frame ‘𝑡 − 1’, and the rows represent the temporary
IDs before matching in frame ‘𝑡’.

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3.3.2 Nearest neighbour matching
The simplest method for matching the bipartite graph of cells is the K nearest
neighbour algorithm. In this approach, each cell in frame ‘𝑡’ is first matched with its
closest cell in frame ‘𝑡 − 1’ terms of the cost function. If two or more cells match to the
same cell in frame ‘𝑡 − 1’, an ad-hoc algorithm sorts out these cells to match their next
closest cells in frame ‘𝑡’, considering up to K nearest neighbours. If in the previous case,
however, the cost function for each of the K nearest neighbours is above a certain
threshold, then the cell is regarded as different from any corresponding cell in frame
‘𝑡 − 1’, and is given a new ID number.
These new IDs largely occur in three cases, in the case of a genuinely new cell (cell
mitosis or new cell appearance from the edges) or in case of a misdetection (a false
negative, causing track discontinuity) or a mismatch of a neighbouring cell, which may
cause no valid matches to remain for the current cell.
3.3.3 Hungarian matching
We use the cost matrix discussed in section 3.3.1 as the bipartite graph’s weights in the
assignment problem. As the algorithm needs square matrix, any extra rows (if frame ‘𝑡’
has more detected blobs) will have corresponding columns added with zero cost. The
Hungarian algorithm consisting of four steps is then applied to find the optimal match.
Step 1: Subtract row minima
For each row, find the lowest element and subtract it from each element in that row.
Step 2: Subtract column minima
Similarly, for each column, find the lowest element and subtract it from each element
in that column.
Step 3: Cover all zeros with a minimum number of lines
Cover all zeros in the resulting matrix using a minimum number of horizontal and
vertical lines. If n lines are required, an optimal assignment exists among the zeros. The
algorithm stops. If less than n lines are required, continue with Step 4.
Step 4: Create additional zeros
Find the smallest element (call it k) that is not covered by a line in Step 3.
Subtract k from all uncovered elements, and add k to elements that are covered twice.

21
The match then obtained is processed in the same manner as in the Nearest Neighbour
matching algorithm, to remove cases of multiple cells in ‘𝑡’ matching to the same cell in
‘𝑡 − 1’ and to generate new IDs, wherever necessary.
3.3.4 Sinkhorn-Knopp matching
The Sinkhorn-Knopp algorithm is an optimal transport algorithm based on a linear
programming approach to maximize the sum of entropies. This algorithm was chosen
because it solves the entropic regularization optimal transport problem at a speed that
is several orders of magnitude faster than other transportation solvers[12, 13]. Unlike
the Hungarian algorithm, a square matrix is not necessary, however in the steps shown
below a square matrix of size 𝑛 × 𝑛 is assumed as input, along with two probability
vectors 𝑝 and 𝑞. These probability vectors correspond to the total brightness of the cell.
When two cells visually merge, the Sinkhorn algorithm detects that their brightness has
been added, and matches these merger cells to the merged cell. The algorithm has 8
steps shown below in pseudocode format.
Algorithm 1: Sinkhorn-Knopp algorithm
Inputs: Probability vectors 𝑝 and 𝑞, cost matrix 𝑐, hyperparameter 𝜆
For each row, find the lowest element and subtract it from each element in that row.
1: Calculate entropic regularized matrix 𝐾 where 𝐾 ← 𝑒𝑥𝑝(−𝜆. 𝑐 )
2: Initialize 𝑛-dimensional vector 𝑢 as [1,1, … .1]
3: While not convergence do
4: 𝑣 ← 𝑞./(𝐾 𝑢)
5: 𝑢 ← 𝑝./(𝐾𝑣)
6: end while
7: 𝑈 ← 𝑑𝑖𝑎𝑔(𝑢), 𝑉 ← 𝑑𝑖𝑎𝑔(𝑣)
8: 𝑚∗
← 𝑈𝐾𝑉
Output: Probability matrix 𝑚∗
, with values from 0 to 1 for each combination of a
current frame cell and a previous frame cell, signifying the probability that this
combination is a match.
The hyperparameter 𝜆 is the regularizer, which allows for matching over a more
convex function as it’s value increases, speeding up the convergence (Figure 3.6).

22
Figure 3.6: The matching occurs faster over a more convex function as the value of the
hyperparameter rises from left to right. The blue and red curves represent the
probability vector input [22].
The results upon application of this algorithm were different from the KNN and the
Hungarian algorithm, with more continuous tracks, as described in section 4.2 where
specific examples for the different methods are compared.
3.3.5 Sparse Sinkhorn-Knopp matching
The algorithm in 3.3.4 is optimized by modifying the cost matrix in an advantageous
way. Only the cells in the current or neighboring bins (set to 20 in total at maximum)
will be considered for calculation of the regularized costs (entropies), and the rest of the
combinations will have exactly zero entropy (infinite cost). This allows us to form a
sparse entropic regularized cost matrix as shown below in Figure 3.6, which is much
simpler for calculations than the entropic regularized original cost matrix.
This sparse regularized cost matrix is then stored in a Column Sparse Row format
and the same procedure as in the original Sinkhorn-Knopp algorithm is followed.

23
Figure 3.6: The modified entropic regularized cost matrix is extremely sparse with only
three thin lines, representing the cells in the current bin and the 8 neighbouring bins
taken as candidates. Columns are previous frame IDs, rows are current frame IDs.
3.4 Tracklets association
For the proposed method, the global association of the tracklets formed in the previous
subsections is based on ad-hoc methods. These ad-hoc methods include simple case-by-
case analyses such as disallowing matching to cells beyond practically feasible
distances, preventing non-maximum detections from overtaking existing cells (causing
ID switches) and storing a list of merged blobs (Merge Dictionary) and dispersing the
correct IDs to the blobs at the time of their demerge.
The prevention of non-maxima breaking cell tracks is done by the following simple
method, leading to fixing many previously broken tracks.
Algorithm 2: Prevention of non-maxima overtaking of tracks
1: if new ID added 𝑘 frames ago
2: and old (correct) ID nearby is not updated from 𝑘 − 1 or more frames ago
3: then remove new (incorrect) ID and reassign old (correct) ID

24
An example of a non-maxima detection (ID 736) overtaking an existing cell track
(ID 539) is shown in Figure 3.7, with the resolved track on the right. Cell 539 (deep
blue track) is now correctly the only cell following the specific track.
Figure 3.7: An example where the blue track is overtaken by a false positive non-
maximum detection, leading to a yellow track with a different ID.
The Merge Dictionary is a Heuristic method for detecting (visual) merging of cells.
First, a merge is detected when matching the cells in the new frame as follows:
Algorithm 3: Registering merges
1: if (cell not detected this frame):
2: and if it touched another detected cell in the last frame
3: then cells merged, IDs added to dictionary
(touch = distance between centres < sum of radii+1-2 pixels)
The cell in front is determined by which of the multiple cells is the closest KNN
match to the blob in the same position last frame. This front cell’s ID now represents a
merged cell, and the other ID(s) are stored in a data structure to be distributed upon
detection of a demerge as follows:
Algorithm 4: Detecting demerges
1: if (new cell detected this frame):
2: and touches a Merged cell in this frame
3: and brightness/radius similar to merged cell
4: then cell separation detected,
5: IDs in Merged Cell dictionary distributed appropriately
An example of the working of the merge dictionary is shown in Figure 3.8. The
demerge is detected correctly, and the track for cell 442 is recovered.

25
Figure 3.8: An example where two cells merge to form a single blob and then demerge
to continue their tracks. The track for ID 442 is recovered, using the merge dictionary.

26
Chapter 4
Experiments
4.1 Details on the dataset
In order to scientifically understand the mechanisms of these rare cell-driven systems,
simultaneous observation of huge populations of cells is essential. Hence, the dataset
consists of large pixel frames, with sub-million cells. Each frame is taken at an interval
of 30 seconds. The fluorescent imaging system named AMATERAS (A Multi-
scale/modal Analytical Tool for Every Rare Activity in Singularity) is used, which is a
novel type of imager, trans-scale-scope, which allows us to observe cell dynamics with
sub-cell resolution in a field of view greater than a centimetre [15].
The provided data[15, 16] has over a 100,000 dictyostelium cells laced with two
different fluorescent protiens (Flamindo and RFP), allowing for dual color imaging of
the amoeba, as shown in Figure 4.1.
Figure 4.1: The two proteins used in the fluorescent imaging. The method used is dual
color imaging, with a mixture of both, Flamindo2 and RFP proteins.
Flamindo2 shows strong reactivity to the cAMP released by the cells changing from
bright green to dark as cAMP concentration increases, and RFP shows weak reactivity

27
to it, maintining a reddish colour. In combination, these two show a change from
yellow to red based on cAMP concentration.
A comparison of the number of bright spots in the Flamindo2 and RFP frames below
confirms the high reactivity of Flamindo2 and the low reactivity of RFP.
Figure 4.2: The difference in the reactivity of Flamindo2 and RFP to cAMP is clear
from the raw monochrome data, as the former has many more bright spots.
4.2 Examples
We compare the results of the Nearest Neighbour matching algorithm and the proposed
Sparse Sinkhorn-Knopp matching algorithm in Figure 4.3 by looking at corresponding
frames, and at a substantial number of cells (over 600), for enough visual test cases. A
more detailed look at some of the examples in Figure 4.3 shows us the differences in the
results for these two methods.

28
Figure 4.3: A direct comparison of the output frame 13 of Flamindo2 data for the NN
and the sparse sinkhorn methods. The green highlights are examples of the
improvement observed, while the red area shows the pitfall of the proposed method.

29
Looking at the example results, a comparison of the Nearest Neighbour matching
and the Sparse Sinkhorn matching can be made. Below are the ‘plus’ and ‘minus’
points for the proposed algorithm in comparison to the ad-hoc nearest neighbour
implementation.
+ Sinkhorn gave better accuracy especially for longer tracks (Green in Figure 4.3).
+ Greater track continuity.
+ 2-3 times the computational speed. (the Nearest Neighbour algorithm can also be
implemented in 𝑂(𝑛) time and space using bin sorting of cells).
Worse performance in terms of new IDs generated (Red in Figure 4.3).
Fewer incorrect matches but reliance on new IDs in crowded areas.
4.3 Protocol
The experiments conducted to determine the tracking accuracy and the processing time
follow a simple protocol. The metrics used for tracking accuracy are the percentage of
new IDs generated every frame, the IDS (ID switch) metric, the precision and the
recall.
IDS: Number of times that a tracked trajectory changes its matched ground-truth
identity, in the sample size chosen [17].
Recall: Ratio of correctly matched detections to ground-truth detections [17].
Precision: Ratio of correctly matched detections to total result detections [17].
For calculating IDS, 50 random cells are chosen and the ground truth results are
compared to the results of the tracker over the interval of 15 frames. For precision and
recall, the algorithm was ran for random areas of the overall frame containing about 50
cells, and the mismatches till frame 15 were counted. The data shown in the graphs for
tracking accuracy and processing time is averaged over 30 runs, and 10-40 runs
(depending on the number of cells) respectively.

30
4.4 Tracking accuracy
The overall accuracy metric used during development was the number of new IDs
generated per frame, since this is an upper bound for all misdetections and mismatches.
More common and useful metrics were also calculated, as shown in the second half of
this subsection. Since new ID cells are those which had no decent match in the previous
frame, they either arise from being genuinely new (mitosis or entry at edges), from
being false positives, or from mismatches in the neighbouring area.
As false positives are rare in the LoG filter peak detection method used (false negatives
are more common), this percentage of new IDs is a simple and useful metric to compare
the accuracy of the tracking part for different algorithms.
Figure 4.4 shows the percentage of new IDs in each frame for the KNN, the Sinkhorn-
Knopp and the Sparse Sinkhorn-Knopp algorithms, averaged over a 50 runs.
Figure 4.4: The percentage of new IDs generated for the first 17 frames, averaged over
30 runs. This is a loose upper bound to the number of mismatches.
The percentage of new IDs per frame is slightly higher with the faster method or
with smaller sizes of bins, signifying a trade-off of matching accuracy and
computational cost. Since cell mitosis can be assumed to be at a constant rate in terms
of percentage of cells, it is clearly below 1%, as is discernible from the KNN new ID
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
%
New
cell
IDs
Frame number
Sparse Sinkhorn Sinkhorn-Knopp KNN

31
rates at the right end of the graph. Hence, the proposed Sparse Sinkhorn introduced
around as many new IDs as the regular Sinkhorn-Knopp algorithm and the simplest
KNN matching algorithm. This accuracy depends on the size of each bin (bin size used:
20px*20px). The larger the size of each bin, the lower the chance of not finding a
suitable match within the bound of the current and neighbouring bins. The IDS, Recall
and Precision calculated were as in Table 4.1.
Table 4.1: A comparison of the accuracy of the two methods tested, the proposed
method, and the simpler but slower KNN. The metrics were quite similar for both.
Metric
(50 cells, 15 frame interval)
Sparse Sinkhorn-
Knopp
KNN
IDS 44 40
Recall 65.5% 68.1%
Precision 76.3% 79.2%
A clearly visible pattern in the graph of Figure 4.4 is the decreasing nature of the
number of new IDs generated. This happens due to a combination of many reasons.
Firstly, the features (radius, peak brightness and brightness sum) of every cell are not
fully defined by the first few frames as they are stochastic, with a large variance, as
shown in the example of the changing peak brightness values for two cells in Figure 4.5
below. This causes many matches to go wrong but only in the first few frames.
Secondly, the various global association method discussed in 3.4 start to take effect
after the first few frames as well. The combined effect of clearer data and track
recovery methods leads to a drop in the number of cells without a suitable match (new
ID cells).
Figure 4.5: A graph of the raw peak brightness values of two random cells chosen from
the dataset. To reduce this variance, these features are averaged over each frame.

32
4.5 Computational cost compared to pure
Sinkhorn-Knopp algorithm
The processing in each frame consists of the detection part, the tracking part, and the
display or output part. The runtimes for each of these parts was measured separately.
As expected, the processing time of the detection and output parts is the same for each
matching algorithm from chapter 3, at about 10ms and 2ms respectively per cell for
numbers of cells above 200. Hence, a direct comparison of the computational cost of the
tracking part can be made as in the following graph.
Figure 4.6 shows the average runtime for the association phase (tracklet formation
and global association) per frame.
Figure 4.6: The time taken to complete the association of one frame averaged over 10-
40 runs. The reduction to linear time is clearly seen.
0
20
40
60
80
100
120
Time
per
frame
(s)
Number of cells
Sparse Sinkhorn Sinkhorn-Knopp

33
Chapter 5
Conclusions
In our thesis, the goal was to replace visual cell data analysis by a fast automatic
method to track pattern formation. For this purpose, the fastest approach to accurately
track cells was found out via elimination of all the different candidate methods, such as
the deep learning methods and the Hungarian algorithm. The proposed method was
based on a particularly fast optimal transport algorithm, the Sinkhorn-Knopp
algorithm, which allows for decently accurate matching of bipartite graphs in 𝑂(𝑛 )
time. This, however can be improved substantially by pruning off extremely unlikely
match candidate cells, to obtain a sparse entropic regularized cost matrix, which can
then speed up the convergence of the Sinkhorn-Knopp algorithm due to substantially
faster matrix computations.
However, as is clear in the results and the metrics that follow it, the algorithm may
be fast but it is in a bare-bones state, being prone to ID switching. To solve this issue,
future work such as the addition of a Kalman filter can be proposed. During the final
results, it was found that the proposed matching algorithm is much faster than the
detection method used, with the detection taking nearly 8 times as long for larger cell
numbers, and high memory usage. A modification of the applied detection method, and
further research into large-scale blob detection can also be proposed.

34
Acknowledgements
I want to thank professor Yasushi Yagi. His insights during our
meeting always push me to look further and try to be genuinely a researcher.
I want to show my deep appreciation for professor Yasushi Makihara for his insightful
guidance and continuous support.
I also want to thank Mr. Yang Yu for helping me with many technical issues, and for
suggesting new approaches to better implement the algorithms discussed.

35
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