Tracking Particles with Large Displacements using Energy ...

Tracking Particles with Large Displacements using Energy Minimization

Rostislav Boltyanskiy,1 Jason W. Merrill,2 and Eric R. Dufresne3

1Department of Physics, Yale University, New Haven, Connecticut 06520, USA2Desmos, Inc., San Francisco, California 94103, USA

3Department of Materials, ETH Zurich, 8092 Zurich, CH.(Dated: August 1, 2018)

We describe a method to track particles undergoing large displacements. Starting with a list ofparticle positions sampled at different time points, we assign particle identities by minimizing thesum across all particles of the trace of the square of the strain tensor. This method of trackingcorresponds to minimizing the stored energy in an elastic solid or the dissipated energy in a viscousfluid. Our energy-minimizing approach extends the advantages of particle tracking to situationswhere particle imaging velocimetry and digital imaging correlation are typically required. Thisapproach is much more reliable than the standard squared-displacement minimizing approach forspatially-correlated displacements that are larger than the typical interparticle spacing. Thus, it issuitable for particles embedded in a material undergoing large deformations. On the other hand,squared-displacement minimization is more effective for particles undergoing uncorrelated randommotion. In the Supplement, we include a flexible MATLAB particle tracker that implements eitherapproach. This implementation returns an estimation of the strain tensor for each particle, inaddition to its identification.

INTRODUCTION

In a wide range of pure and applied sciences, the mo-tion of objects needs to be tracked over time. The centraldifficulty is that while an object’s trajectory is continuousthrough space and time, its position can only be sampledat discrete moments. When the objects of interest arefar away from each other or otherwise distinguishable,tracking is simple. However, when many identical ob-jects (which we refer to as particles) are near each other,tracking can be difficult.

Particle tracking is employed across physics, biology,and many other fields. In soft matter physics, it hasshed light on mechanical properties like rheology [1], andrevealed microscopic processes underlying phase transi-tions [2], among many other applications. In biology,particle tracking is used across many length scales fromthe movement of single molecules, to the transport oforganelles within cells, to movement of whole organismssuch as flies, birds, fish, and humans [3–6].

In the absence of Brownian motion, particles embed-ded in a material reveal its deformation field. In fluidmechanics, particle tracking reveals flow fields in theLagrangian frame of reference ([7]). Traction force mi-croscopy (TFM) quantifies the forces applied to a solidsurface by tracking and analyzing motion of particles em-bedded within the solid. TFM was originally developedto quantify the forces exerted by adherent cells ([8]), buthas recently been applied to a wide range of problems inbiology and physics [9].

The standard method of particle tracking assigns iden-tities by minimizing the sum of squared displacementsacross time points [10]. While this method is rigorouslycorrect for objects undergoing Brownian motion, it worksvery well across a wide range of applications, providedthat the particles move a distance that is small com-pared to the typical inter-particle separation. The im-

provement of this basic tracking approach has been anactive area of research in recent years, driven by appli-cations in the biomedical community. For a useful com-parison of these particle tracking approaches, see [11].When the displacements are large, it is helpful to employa tracking algorithm that exploits knowledge of the sys-tem’s kinematics. For example, particle tracking in denseturbulent fluid flows has been greatly improved by em-ploying a “predictive tracker,” which exploits the inertialcharacter of high Reynolds number fluid flow [12].

Particle Imaging Velocimetry (PIV) and Digital Imag-ing Correlation (DIC) are widely used to assess the flowof fluids and deformation of solids [13–15]. These meth-ods assign velocities or displacements to locations in amaterial by cross-correlating patches of an image acrosstime points. This is a robust and successful approach,even for systems with large displacements. However, it isinappropriate in cases where individual particle identitiesneed to be followed over time or where the displacementsneed to be known at the resolution of individual particles.

Here, we extend particle tracking to the large-displacement regime where PIV and DIC are usually em-ployed. Our algorithm is optimized for tracking the mo-tion of particles undergoing large spatially-correlated dis-placements, typical for tracers embedded in a deformedelastic solid or flowing viscous fluid. We start by posingthe tracking problem generally, and review the maximumlikelihood method employed to track Brownian particles.Then, we describe our energy minimization approach anddirectly compare the results of the two methods for sim-ulated and real data.

OPTIMAL ASSIGNMENT

Suppose a set of particles, {p(k)(ti)}, where k is theparticle index, are found at locations {~x(k)(ti)} at time

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energy requires correct particle identifications. Even afew missed or incorrect trajectories can make large dif-ferences in the calculated energy. Figure 3b shows strainenergy heat maps as calculated from trajectories usingthe di↵usion tracker (top) and the strain tracker (bot-tom).

Many applications in soft matter physics involve thestudy of material deformation. Figure 3c shows an exam-ple of a silicone gel coated with fluorescent particles beingstretched equi-biaxially. While the di↵usion tracker canhandle regions were the displacements are small, parti-cle identification is inaccurate for larger displacements.The strain tracker identifies particles e↵ectively acrossthe entire field. (Fig. 3c).

Particle tracking is a also common technique in ad-dressing drift and studying fluid flow. In application withintentional or unintentional drift, particle tracking can beused to characterize substrate displacements. Figure 3dshows an example of a substrate, coated with fluores-cent microspheres, that drifted between two time points.While the displacements are too large for the di↵usion

tracker everywhere, they are accurately captured by thestrain tracker.

It is possible to combine the strengths of the di↵u-sion tracker and the strain tracker in one combinationtracker. In such an algorithm, the cost matrix can bebuilt with appropriate weights assigned to minimizationof net squared strain and net squared displacements.This could be very useful in applications where both dif-fusive and non-di↵usive displacements are present.

ACKNOWLEDGEMENTS

We thank Madhusudhan Venkadesan and KatharineJensen for helpful discussions.

cl,m = Tr(✏(l,m))2

cl,n = 1

[1] Reza Ardekani, Anurag Biyani, Justin E. Dalton, Ju-lia B. Saltz, Michelle N. Arbeitman, John Tower, SergeyNuzhdin, and Simon Tavare. Three-dimensional trackingand behaviour monitoring of multiple fruit flies. Journalof the Royal Society Interface, 10(78):13pp.–13pp., 2013.0.

[2] J. C. Crocker and D. G. Grier. Methods of digi-tal video microscopy for colloidal studies. Journal ofColloid and Interface Science, 179(1):298–310, 1996.Times Cited: 1337 Grier, David/C-5761-2008; Crocker,John/D-4991-2012 Grier, David/0000-0002-4382-5139;Crocker, John/0000-0001-6239-6010 1347.

[3] M. L. Falk and J. S. Langer. Dynamics of viscoplas-tic deformation in amorphous solids. Physical Re-view E, 57(6):7192–7205, 1998. Times Cited: 711Falk, Michael/A-8478-2008 Falk, Michael/0000-0002-8383-4259 721.

[4] Elizabeth R. Jerison, Ye Xu, Larry A. Wilen, and Eric R.Dufresne. Deformation of an elastic substrate by a three-phase contact line. Physical Review Letters, 106(18),2011. Times Cited: 60 Xu, Ye/E-7963-2011; Dufresne,Eric/A-7760-2009 Xu, Ye/0000-0003-4322-244X; 61.

[5] Yael Katz, Kolbjorn Tunstrom, Christos C. Ioannou,Cristian Huepe, and Iain D. Couzin. Inferring the struc-ture and dynamics of interactions in schooling fish. Pro-ceedings of the National Academy of Sciences of theUnited States of America, 108(46):18720–18725, 2011.Times Cited: 120 Huepe, Cristian/E-3653-2014; Ioan-nou, Christos/ Huepe, Cristian/0000-0002-6495-8387;Ioannou, Christos/0000-0002-9739-889X 125.

[6] H. W. Kuhn. The hungarian method for the assignmentproblem. Naval Research Logistics, 52(1):7–21, 2005.Times Cited: 57 65.

[7] S. Munevar, Y. L. Wang, and M. Dembo. Traction forcemicroscopy of migrating normal and h-ras transformed3t3 fibroblasts. Biophysical Journal, 80(4):1744–1757,

2001. Times Cited: 240 Dembo, Micah/C-2755-2013 247.[8] N. T. Ouellette, H. T. Xu, and E. Bodenschatz. A quan-

titative study of three-dimensional lagrangian particletracking algorithms. Experiments in Fluids, 40(2):301–313, 2006. Times Cited: 117 Xu, Haitao/C-9857-2010; Ouellette, Nicholas/D-8541-2011; Bodenschatz,Eberhard/C-6603-2009 Ouellette, Nicholas/0000-0002-5172-0361; Bodenschatz, Eberhard/0000-0002-2901-0144117.

[9] Y. Peng, Z. R. Wang, A. M. Alsayed, A. G. Yodh, andY. Han. Melting of multilayer colloidal crystals confinedbetween two walls. Physical Review E, 83(1), 2011. TimesCited: 12 1 12.

[10] Mikel Rodriguez, Ivan Laptev, Josef Sivic, Jean-Yves Au-dibert, and Ieee. Density-aware person detection andtracking in crowds. 2011 Ieee International Confer-ence on Computer Vision (Iccv), pages 2423–2430, 2011.Times Cited: 19 IEEE International Conference on Com-puter Vision (ICCV) Nov 06-13, 2011 Barcelona, SPAINIEEE; Toyota; Google; Microsoft Res; Siemens; Techni-color; Adobe; Alcatel Lucent; Gentex Corp; Kooaba Im-age Recognit; Mitsubishi Elect; Mobileye; Object Video(OV); Toshiba; Xerox; Zeiss; 2d3; SATURNUS 19 978-1-4577-1102-2.

[11] Todd M. Squires and Thomas G. Mason. Fluid mechanicsof microrheology. Annual Review of Fluid Mechanics,42:413–438, 2010. Times Cited: 161 162 978-0-8243-0742-4.

[12] Andrew D. Straw, Kristin Branson, Titus R. Neumann,and Michael H. Dickinson. Multi-camera real-time three-dimensional tracking of multiple flying animals. Jour-nal of the Royal Society Interface, 8(56):395–409, 2011.Times Cited: 50 Straw, Andrew/A-1244-2007 Straw,Andrew/0000-0001-8381-0858 50.

[13] Robert W. Style, Rostislav Boltyanskiy, Guy K. Ger-man, Callen Hyland, Christopher W. MacMinn, Aaron F.

4



Particle tracking is a also common technique in ad-dressing drift and studying fluid flow. In application withintentional or unintentional drift, particle tracking can beused to characterize substrate displacements. Figure 3dshows an example of a substrate, coated with fluores-cent microspheres, that drifted between two time points.While the displacements are too large for the di↵usion

tracker everywhere, they are accurately captured by thestrain tracker.

It is possible to combine the strengths of the di↵u-sion tracker and the strain tracker in one combinationtracker. In such an algorithm, the cost matrix can bebuilt with appropriate weights assigned to minimizationof net squared strain and net squared displacements.This could be very useful in applications where both dif-fusive and non-di↵usive displacements are present.

ACKNOWLEDGEMENTS


cl,m = Tr(✏(l,m))2

cl,n = 1







[7] S. Munevar, Y. L. Wang, and M. Dembo. Traction forcemicroscopy of migrating normal and h-ras transformed3t3 fibroblasts. Biophysical Journal, 80(4):1744–1757,

2001. Times Cited: 240 Dembo, Micah/C-2755-2013 247.[8] N. T. Ouellette, H. T. Xu, and E. Bodenschatz. A quan-

titative study of three-dimensional lagrangian particletracking algorithms. Experiments in Fluids, 40(2):301–313, 2006. Times Cited: 117 Xu, Haitao/C-9857-2010; Ouellette, Nicholas/D-8541-2011; Bodenschatz,Eberhard/C-6603-2009 Ouellette, Nicholas/0000-0002-5172-0361; Bodenschatz, Eberhard/0000-0002-2901-0144117.




[12] Andrew D. Straw, Kristin Branson, Titus R. Neumann,and Michael H. Dickinson. Multi-camera real-time three-dimensional tracking of multiple flying animals. Jour-nal of the Royal Society Interface, 8(56):395–409, 2011.Times Cited: 50 Straw, Andrew/A-1244-2007 Straw,Andrew/0000-0001-8381-0858 50.

[13] Robert W. Style, Rostislav Boltyanskiy, Guy K. Ger-man, Callen Hyland, Christopher W. MacMinn, Aaron F.

4



Particle tracking is a also common technique in ad-dressing drift and studying fluid flow. In application withintentional or unintentional drift, particle tracking can beused to characterize substrate displacements. Figure 3dshows an example of a substrate, coated with fluores-cent microspheres, that drifted between two time points.While the displacements are too large for the di↵usiontracker everywhere, they are accurately captured by thestrain tracker.

It is possible to combine the strengths of the di↵u-

sion tracker and the strain tracker in one combinationtracker. In such an algorithm, the cost matrix can bebuilt with appropriate weights assigned to minimizationof net squared strain and net squared displacements.This could be very useful in applications where both dif-fusive and non-di↵usive displacements are present.

ACKNOWLEDGEMENTS


cl,m = Tr(✏(l,m))2

cl,n = 1

pl(ti)

pm(tf )

pn(tf )







[7] S. Munevar, Y. L. Wang, and M. Dembo. Traction forcemicroscopy of migrating normal and h-ras transformed3t3 fibroblasts. Biophysical Journal, 80(4):1744–1757,2001. Times Cited: 240 Dembo, Micah/C-2755-2013 247.

[8] N. T. Ouellette, H. T. Xu, and E. Bodenschatz. A quan-titative study of three-dimensional lagrangian particletracking algorithms. Experiments in Fluids, 40(2):301–313, 2006. Times Cited: 117 Xu, Haitao/C-9857-2010; Ouellette, Nicholas/D-8541-2011; Bodenschatz,Eberhard/C-6603-2009 Ouellette, Nicholas/0000-0002-5172-0361; Bodenschatz, Eberhard/0000-0002-2901-0144117.




[12] Andrew D. Straw, Kristin Branson, Titus R. Neumann,and Michael H. Dickinson. Multi-camera real-time three-

4






ACKNOWLEDGEMENTS


cl,m = Tr(✏(l,m))2

cl,n = 1

pl(ti)

pm(tf )

pn(tf )













4






ACKNOWLEDGEMENTS


cl,m = Tr(✏(l,m))2

cl,n = 1

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pm(tf )

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FIG. 1: Schematic of strain-based particle tracking. (a)Particles at time ti. Dotted circle represents a region

around the particle of interest, p(l)(ti), with radius rmax

within which the particle neighbors are considered. (b)Particles at time tf with candidate particles circled as

in (a). (c) Particle p(l)(ti) and its neighbors overlappedwith p(m)(tf ) and its neighbors. Arrows show tracked

displacements between neighbors of the candidateparticles. (d) Particle p(l)(ti) and its neighbors

overlapped with p(n)(tf ) and its neighbors. Arrowsshow tracked displacements between neighbors of the

candidate particles.

ti and a set of particles {p(k)(tf )} are found at {~x(k)(tf )}at time tf . We wish to connect particle positions acrosstime to make trajectories. The particle identities are rep-resented by a list, T , where Tl = m if particle p(l)(ti) attime ti becomes particle p(m)(tf ) at time tf .

We find an optimal assignment of identities by mini-mizing a cost function. We associate a cost, cl,m, to eachpossible particle pairing. The total cost, C, is the sum ofthe costs of individual pairings across all of the particles,C =

∑l cl,T (l). We assign identities that minimize the

cost using the Hungarian algorithm. [16, 17].

Since assigning particles is a combinatoric process, itrapidly becomes computationally overwhelming as thesystem size increases. To avoid this difficulty, it is es-sential to limit the number of combinations by ruling outunphysical assignments, as described below.

MINIMIZATION OFSQUARED-DISPLACEMENTS

The standard particle-tracking algorithm minimizesthe sum of the squared displacements across all of theparticles [10]. This yields the most likely assignment ofidentities when particles undergo Brownian motion. Inthat case, the probability that the kth particle will bedisplaced by ∆~x(k) in a time interval, ∆t, is:

P (∆~x(k),∆t) ∼ exp

(−∣∣∆~x(k)∣∣24dD∆t

)(1)

Here, D is the diffusion coefficient and d is the numberof spatial dimensions [18]. Therefore, the probability ofobserving a specific set of displacements of N identicalparticles moving independently is:

P =

N∏k=1

P (∆~x(k),∆t)

∼ exp

(−

N∑k=1

∣∣∆~x(k)∣∣24dD∆t

)(2)

The most likely assignment of particle identities max-imizes the total probability, P , which is equivalent to

minimizing the total squared-displacementN∑

k=1

∣∣∆~x(k)∣∣2.

Furthermore, we can assign costs for potential pairingsas cl,m = |~x(l)(ti) − ~x(m)(tf )|2, and minimize the costfunction C, as defined in the previous section.

To accelerate the data analysis, one needs to rule outunphysical assignments. Typically, one specifies a max-imum displacement a particle could have between timepoints, and assigns an infinite cost to all of the elementsof c that correspond to displacements that would exceedthis maximum. These elements are ignored in the com-binatorial optimization.

In practice, minimizing the squared-distance worksvery well whenever particle displacements are small com-pared to the distance of a particle to its nearest neigh-bors.

MINIMIZATION OF ENERGY

In the case of large displacements, square-distanceminimizing trackers may not work effectively. In manycases, however, these displacements may have strong spa-tial correlations. For example, particles embedded in arigid material undergoing translation and rotation arefixed relative to their neighbors. Similarly, displacementsof particles embedded in a liquid or solid undergoinglarge deformations are typically strongly correlated totheir neighbors. In both these cases, we do not wish topenalize displacements in the laboratory frame, but dis-placements relative to neighboring particles.

In continuum mechanics, the variation of displace-ments over space is characterized by the strain. In thecase of a linear, isotropic, elastic solid of Young’s modu-lus, E, the stored energy is

Uel =E

2

∫vol

Tr ε2dV (3)

3

where ε and ~u are the strain and displacement fields,respectively. In component form, these two are relatedas εij = 1

2 (∂iuj + ∂jui) [19]. Analogously, for a fluid ofdynamic viscosity µ sheared with a strain rate ε, the rateof energy dissipation is [20]:

Ufl = 2µ

∫vol

Tr ε2dV (4)

If one is considering only two time points, then experi-mental estimates of ε and ε only differ by a pre-factor.

Motivated by these two physical examples, we pro-pose to assign particle identities by minimizing thestored/dissipated energy. The basic hypothesis is thattracking errors tend to exaggerate the strain/strain-rate,increasing the apparent stored/dissipated energy. There-fore, we implement a cost function cl,m = Tr(ε(l,m))2.The main challenge of the proposed algorithm is to iden-tify the strain associated with a given particle pairing.Calculating the strain requires positions of particles andtheir nearest neighbors at two times.

The nearest neighbors of particle p(k), can be found in2-D using the Delaunay triangulation. To ensure that thestrain is uniform over the region where it is calculated, weonly consider neighbors within a distance rmax of p(k). InFig. 1(a,b), neighbors included in the strain calculationare within the dotted circles.

We use the method of Falk and Langer [21] to calculatethe strain associated with candidate particle pairs andtheir neighbors. The strain about a particle at position~x from the position of neighbors ~x(n) can be estimatedas follows:

Xij =∑n

(x(n)i (tf )− xi(tf )) · (x(n)j (ti)− xj(ti)), (5)

Yij =∑n

(x(n)i (ti)− xi(ti)) · (x(n)j (ti)− xj(ti)), (6)

Λij = XikY−1jk , (7)

εij =1

2(Λij + Λji)− δij . (8)

Note that in order to calculate strain accurately, at leastd neighbors must be included for particle tracking in d-dimensions.

To properly calculate strain around a particle, oneneeds to accurately track its neighbors across time. Con-sider a particle at time ti, p

(l)(ti) (in the center of thedotted circle in Fig. 1a) and a candidate correspondingparticle at time tf , p(m)(tf ) (in the center a dotted cir-

cle in Fig. 1b). To connect the neighbors of p(l)(ti) withthose of p(m)(tf ), we translate the candidate particles andtheir neighbors to the same location, as shown in Figure1(c,d). We then track the neighbors of the candidatepair by minimizing the square of the residual displace-ments. To accelerate the calculation, we rule out relativedisplacements that would exceed an expected maximumvalue of the strain. If the candidate pair produces at

least d tracked neighbors, we calculate the strain for thecandidate pairing according to Eq. (8) and assign thepairing a cost cl,m = Tr(ε(l,m))2. Otherwise, the pairingis ruled out by setting the cost to infinity. Once all pos-sible pairings have been assigned costs, the Hungarianalgorithm is used to minimize the total cost across all ofthe particles.

IMPLEMENTATION OF TRACKINGALGORITHMS

We implement the squared-distance-minimizing “dif-fusion tracker” and energy-minimizing “strain tracker”algorithms in MATLAB. In the supplement, we includethe main particle-tracking function, Tracker.m. We alsoinclude a script, Example.m, that allows the user to ex-plore the examples described below. Details regardinguse of the code are found in the comments as well asthe text of ReadMe.doc. A convenient by-product of ourtracker is that it returns the symmetrized strain matrixfor each tracked particle.

COMPARISON OF TRACKING STRATEGIESWITH SIMULATED DATA

We compare the performance of the diffusion andstrain trackers with simulated data. As shown in Figure2, we consider four types of particle motion: diffusion,translation, shear, and stretch. For each example, thefirst row of Figure 2 shows the positions of particles atthe first (black) and second (magenta) time points. Thesecond row shows the displacements for correctly trackedparticles in green. The third and fourth rows show thedisplacements determined by the diffusion tracker (red)and the strain tracker (blue).

Not surprisingly, the diffusion tracker out-performs thestrain tracker for the case of diffusion. For the exampleshown in Figure 2, the diffusion tracker returns resultsfor all of the particles. About 96% of these tracks arecorrect. On the other hand, the strain tracker providesidentifications for only 84% of the particles, and about15% of these are incorrect. These tracking errors im-pact quantitative measures of the particle motion. Eventhough the distributions of the particle displacements forthe two trackers look similar, Figure 3, tracking errorstend to introduce counts in the tails of the displacementdistribution that have a significant impact on the mean-squared particle displacement (MSD). In this example,the MSD calculated from the diffusion tracker is within2% of the correct value. However, the value calculatedusing the strain tracker is 30% greater than the correctvalue.

On the other hand, for the cases of large displacementsdue to translation, shear, and stretch, the strain tracker isa much more reliable choice. For the examples in Figure2, the diffusion tracker returns tracks for greater than

4

right: 73% wrong: 11%


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Diffusion Translation Stretch Shear R

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right: 72% wrong: <1%

FIG. 2: Comparison of the diffusion and strain trackers for simulated data of particles moving by diffusion (firstcolumn), translation (second column), shear (third column), and stretch (fourth column). The particle positions atthe first time point (black dots) and second time point (magenta dots) are shown in the first (top) row. The correctdisplacments (green arrows) are shown in the second row. The displacments calculated from identities returned bythe diffusion and strain trackers are shown in the third and fourth rows, respectively. In all cases, arrows are drawnto scale. In the third and fourth rows, the numbers in gray boxes correspond to percentages of particle trajectoriescalculated correctly and incorrectly by each tracker. The percentages do not add to 100% because the trackers were

not able to find partners for some particles.

84% of the particles, but less than 8% of them are correct.On the other hand, the strain tracker identifies greaterthan 73% of the particles and less than 3% of them areincorrect. Furthermore, for the case of the strain tracker,dropped particles and errors tend to be localized to theboundaries of the field of view, which are easily discardedwhen greater accuracy is required.

In these cases, the strain tracker accurately quanitfiesparticle displacements and strains. The expected dis-placements and strains for the cases of translation, shear,and stretch are shown as vertical green lines in Figure 3.

The red histograms display the values for each particlereturned by the diffusion tracker and the blue histogramsreport those from the strain tracker. The strain trackerconsistently returns the correct values, while the incor-rect tracks from the diffusion tracker return broadly dis-tributed incorrect values.

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FIG. 3: Histograms of individual particle displacements (a-d) and strain eigenvalues (e-h) from analysis of simulateddata in Figure 2. (a) and (b) are histograms of horizontal and vertical displacements, respectively, from simulated

diffusion data. (c) and (d) are histograms of horizontal and vertical displacements, respectively, from simulatedtranslation data. (e) and (f) are histograms of strain eignevalues from a simulated shear. (g) and (h) are histogramsof strain eigenvalues from a simulated stretch. In all panels, results from the diffusion tracker are in red, those of the

strain tracker are in blue, and the overlap is in purple. In green is an outline of the histogram of correctdisplacements in (a),(b) and a vertical line corresponding to the correct values of displacements and strains in

(c)-(h).

COMPARISON OF TRACKING STRATEGIESWITH EXPERIMENTAL DATA

In this section, we compare the diffusion and straintracker with some experimental data.

First, we assess both trackers’ ability to identify par-ticles during a relatively large and homogeneous strain.Here, we deform a silicone gel with embedded fluores-cent tracers using a macroscopic deformation. We imagethe tracers over a small field of view, where we expect areasonably uniform compression. We manually applied auniform affine transformation to the particle locations toestimate the strain, resulting in eigenvalues of -0.06 and -0.04. Then,we calculated particle displacements with thestrain and diffusion trackers, as shown in Figure 4(a,b).While both trackers return comparable results where thedisplacements are small (upper-left corner of the field ofview), they disagree where the displacements are large.In this region, the diffusion tracker returns an uncorre-lated displacement field, while the strain tracker returnsthe expected compression. Our strain tracker returns astrain tensor associated with the displacement of eachtracked particle. Histograms of the strain eigenvaluesof each particle are plotted in Figure 4 c-f. For bothtrackers, the peaks of the histogram agree well with theexpected values from the manual affine transformation.However, the diffusion tracker reports a much broaderdistribution, including some unphysical positive values.

Next, we consider an example from Traction Force Mi-croscopy (TFM) where the strains have a much strongerspatial heterogeneity. In TFM, forces exerted by smallobjects, such as cells, are quantified by measuring the de-formation of an elastic material they are adhered to. Toquantify the deformation, fluorescent particles are em-bedded in the elastic material. Displacements caused bya fibroblast cell adherent to a silicone gel are presentedin Figure 4 g,h. The cell is fluorescently tagged and isdisplayed in inverted contrast. Overlaid on top of thecell image are displacements of the surface underneaththe cell (scaled by a factor of 5), determined by the (g)the diffusion tracker and (h) strain tracker. While dis-placements measured by the strain tracker and diffusiontracker mostly agree, the diffusion tracker identifies someunexpected large strains near the center of the cell.

CONCLUSION

We have introduced a particle tracking algorithmbased on the minimization of energy. This approach out-performs the conventional squared-displacement mini-mizing particle tracker when the displacements are largerthan the interparticle spacing. On the other hand, ourstrain tracker may have difficulty when the strain changessignificantly over the typical interparticle spacing. Thiscan occur in the vicinity of strain singularities, such as

6

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40

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FIG. 4: Particle tracking examples with experimentaldata.

(a)-(b) Trajectories of particles embedded in a siliconegel undergoing uniform compression, calculated withthe diffusion tracker (a) and the strain tracker (b).

Arrows are scaled by a factor of 2. (c)-(f) Histograms ofstrain eigenvalues as calculated with the diffusion

tracker (c), (e) and the strain tracker (d),(f). (g)-(h) Acontractile fibroblast cell adherent on a silicone gel.

Arrows overlaid on top represent particle displacementsfrom cell traction forces as calculated with the diffusion

tracker (g), and the strain tracker (h). Arrows arescaled by a factor of 5. Scale bars are 20µm

cracks, and when particles are moving randomly, as inBrownian motion. Although, the code included in thesupplement is designed for two time points and two spa-tial dimensions, expanding to three dimensions and manytime points is a straightforward modification.

ACKNOWLEDGMENTS

We thank Madhusudhan Venkadesan, KatharineJensen, and Robert Style for helpful discussions and sup-port from the Army Research Office MultidisciplinaryUniversity Research Initiative (ARO MURI) W911NF-14-1-0403.

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