Analyze Random Walks from Video

Track randomly moving particles and classify their trajectories

In this example we are going to track a set of randomly moving particles. We will analyze and classify the particles according to whether their motion is purely Brownian.

Import the video:

In[1]:=

video=

;

Extract the particles displaying motion:

In[2]:=

findCentroid[frame_]:=Point/@Values[ComponentMeasurements[Binarize[frame,.5],"Centroid",CornerNeighborsFalse]];particles=VideoMapList[findCentroid[#Image]&,video];Short[particles]

Out[2]//Short=

{{Point[{28.5,834.5}],Point[{88.5,832.5}],399,Point[{392.5,24.5}],Point[{517.5,21.5}]},{1},97,{1},{}}

Track the moving particles:

In[3]:=

tracks=VideoObjectTracking[particles,Method{"RunningBuffer","OcclusionThreshold"4}]

Out[3]=

ObjectTrackingData

Type: Points

Length: 101

Times: None

Data not saved. Save now



Create and plot trajectories for the particles:

In[4]:=

allTrajectories=DeleteMissing/@Values[tracks["Position"]];

In[5]:=

ListLinePlotallTrajectories,



Out[5]=

Classify trajectories according to the mean-squared displacement (MSD):

In[6]:=

MSD[pos_,t_:30]:=Module{transposedlist,length,listreshape,lr},transposedlist=Table[Thread[{pos,RotateLeft[pos,i]}]〚1;;-(i+1)〛,{i,t}];length=Dimensions/@transposedlist;Mean/@Tablelistreshape=ArrayReshape[transposedlist〚i〛,length〚i〛];lr=Function[x,

Apply[EuclideanDistance][Reverse[x]]

]/@listreshape1;;All,{i,t};

The time lag (second argument of MSD) is chosen as 1/3 of the trajectory length. Moreover, only longer trajectories with a length > 35 are selected for computing the mean-squared-displacement:

In[7]:=

trajlength=35;longertrajectories=Select[allTrajectories,Length[#]≥trajlength&];msd=MSD[#,Floor[1/3*Length[#]]]&/@longertrajectories;

Plot the MSDs of the particles:

In[8]:=

ListLinePlot[msd,AxesLabel{"TimeStep","Mean Square Displacement"}]

Out[8]=

Most trajectories for mean-squared displacement exhibit a linear trend, implying a Brownian motion, while a few trajectories deviate away from the straight line and possibly imply a super-diffusive regime. Moreover, for some trajectories the horizontal slope at larger timestep may also point to particles following a sub-diffusive motion.

Plot a few of the longer trajectories:

In[9]:=

paths=MaximalBy[longertrajectories,Length,100];

In[10]:=

Multicolumn[ListLinePlot[#,AxesFalse,PlotStyleRandomColor[]]&/@paths]

Out[10]=

Perform a linear fit to the MSDs of the particles to acquire gradients:

In[11]:=

gradients=a/.FindFit[#,ax+b,{a,b},x]&/@msd;

Cluster the gradients to define the different regimes of the particle motion, most likely linked to super diffusive, Brownian and sub-diffusive regimes:

In[12]:=

FeatureSpacePlot[Thread[ClusteringComponents[gradients]gradients],PlotStylePointSize[0.01]]

Out[12]=

Compute the Hurst exponent to analyze any memory effects in the motion process:

In[13]:=

eproc=Quiet@EstimatedProcess[#,FractionalBrownianMotionProcess[h]]&/@allTrajectories;

In[14]:=

hurstinds=Chop[Cases[DeleteCases[eproc,_EstimatedProcess],FractionalBrownianMotionProcess[__,hurst_]hurst],10^-8];

In[15]:=

Histogram[hurstinds]

Out[15]=

Classify trajectories according to the Hurst exponent criterion.

Get the number of particles following the process in which the future increments are negatively correlated. The Hurst exponent between 0-0.5 suggest that particles having larger spatial shift at time 't' will likely have smaller step at 't+1' and subsequently larger spatial shift at 't+2':

In[16]:=

Count[hurstinds,x_/;0≤x<0.5]

Out[16]=

782

Get the number of particles exhibiting a Wiener Process or following Brownian motion. The autocorrelation decays exponentially towards 0. There is no memory effect in such cases:

In[17]:=

Count[hurstinds,x_/;x0.5]

Out[17]=

Get the number of particles following a process wherein the future increments are positively correlated. This implies that particles that have larger spatial steps now will tend to retain larger spatial steps in the future:

In[18]:=

Count[hurstinds,x_/;x>0.5]

Out[18]=

1265

Publisher Information

Contributed by: Ali Hashmi, PhD.
Technical Consultant, Wolfram Research

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

Wolfram Language Example Repository

Analyze Random Walks from Video

See Also

Publisher Information

License Information

Analyze Random Walks from Video

See Also

Related Symbols

Publisher Information

License Information