Computational Movement Analysis Lecture 1: Similarity Joachim Gudmundsson

Computational Movement Analysis

Lecture 1: SimilarityJoachim Gudmundsson

Goal: Extract and make sense of information from trajectories (GPS, mobile, video, rfid…).

Provide fundamental software tools to simplify and support qualitative analysis.

Our model

Input: Locations sampled over time

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Our model

Input: Locations sampled over timeAssumptions:

a) Straight-line movement between consecutive samplesb) Constant velocity

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Brownian bridge movement model

Input: Locations sampled over timeOther models? Brownian bridge

Brownian motion conditioned under starting and ending position

Probability that object visits region A

Our model

Input: a sequence of (x,y,t) - coordinates and a time stamp for example GPS data, mobile phone, radar, …

Additional information

Input: a sequence of (x, y, t, z, temperature, speed, direction…)

Trajectory data 2005

Animal Country Number Frequency

Wolves Spain 15 DailyWolves Finland 16 4 hours

Elks Sweden 25 30 minReindeer Finland 19 8 hoursRed deer Germany 7 4 hoursLeopards South

Africa32 Daily

Mountain Goats

USA 32 3 hours

Objects Number

Frequency

Football 23 25HzSupermarke

tlarge 100Hz

Mobile phones

huge 5 min

Trajectory data 2011

Applications: Behavioural ecology

› Migration patterns› “Popular places”› Home region› …

› Migration patterns› “Popular places”› Home range› Flock movements› …

[Joint work: Georgina Wilcox, Kevin buchin, Maike Buchin,

Ran Nathan and Anael Engel]

• Interaction between jackdaws? • Effect of external factors ?

Behavioural ecology

Short term:- Annual migration behaviour- Automatically detect activity- Home range

Long term:

- Foraging search strategies - Factors that may influence animal movement - How changes in the environment affects wildlife

Applications: Sport

Data Accuracy: 100 mm Frequency: 25Hz

[Joint work: Thomas Wolle and many others]

Applications: Sport

Tools to support sports analysts and coaches• Evaluate a players ability to pass (receive, execute…)• Evaluate a players ability to move into open areas• Evaluate a players ability to follow instructions

• Detect frequent movement patterns• Detect correlations between players’ movement• Detect frequent passing patterns

• Analyse turn-overs • Which events change the intensity of the game?• …

Applications: Hurricanes

Goal: Early detection of hurricanes that may hit land

Applications: Transport

› Collision prevention› Maximize unloading/loading

Applications: Shopping behaviour

› Detect movement patterns› Change structure of shop to

increase sales?

Examples of general problems

General problems • Repetitive patterns • Popular places• Trajectory segmentation• Trajectory simplification• Trajectory clustering• Subtrajectory clustering• Movement patterns

• Single file• Formations• Leadership• Meetings

• …

Seminar schedule

Lecture 1: Similarity

Lecture 2: Curve clustering

Lecture 3: Simplification

Lecture 4: Movement patterns

Lecture 5: Trajectory segmentation

Fundamental tools: clustering

When are two trajectories similar?

In most cases the spatial similarity is much more important than the temporal aspect.

Similarity of curves…measurements

Agrawal, Lin, Sawhney & Shim, 1995 Alt & Godau, 1995Yi, Jagadish & Faloutsos, 1998Gaffney & Smyth, 1999Kalpakis, Gada & Puttagunta, 2001Vlachos, Gunopulos & Kollios, 2002Gunopoulos, 2002Mamoulis, Cao, Kollios, Hadjieleftheriou, Tao & Cheung, 2004Chen, Ozsu & Oria, 2005Nanni & Pedreschi, 2006Van Kreveld & Luo, 2007Gaffney, Robertson, Smyth, Camargo & Ghil, 2007Lee, Han & Whang, 2007Buchin, Buchin, van Kreveld & Luo, 2009Agarwal, Aranov, van Kreveld, Löffler & Silveira, 2010Sankaraman, Agarwal, Molhave, Pan and Boedihardjo, 2013...

Similarity measures

Input: Two polygonal curves P and Q of size m in Rd

Approach: Map curve into a point in dm-dimensional space

Similarity measures

Similar?

Similarity measures

Drawbacks: P and Q must have same number points Only takes the vertices into consideration

Not similar

Similarity measures: Hausdorff

Input: Two polygonal curves P and Q in Rd

Distance: Hausdorff distance

H(PQ) = maxpP minqQ |p-q|

dH(P,Q) = max [H(PQ), H(QP)]

Drawbacks: Fail to capture the distance between curves.

Similarity measures: Fréchet

Other interesting measures are:• Dynamic Time Warping (DTW)• Longest Common Subsequence (LCSS)• Model-Driven matching (MDM)

Drawback: Non-metrics (triangle inequality does not hold) which makes clustering complicated.

Similarity measure: Fréchet Distance

Fréchet Distance measures the similarity of two curves.

Dog walking example- Person is walking his dog (person on one curve and the dog on other)- Allowed to control their speeds but not allowed to go backwards!- Fréchet distance of the curves: minimal leash length necessary for

both to walk the curves from beginning to end

Fréchet Distance

Fréchet Distance measures the similarity of two curves.

Dog walking example- Person is walking his dog (person on one curve and the dog on other)- Allowed to control their speeds but not allowed to go backwards!- Fréchet distance of the curves: minimal leash length necessary for

both to walk the curves from beginning to end

Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.

The Fréchet distance between P and Q is:

where and range over all continuous non-decreasing reparametrizations.

Note that (0)=p1, (1)=pn, (0)=q1 and (1)=qm.

Well-suited for the comparison of curves since it takes the continuity of the curves into account.

Fréchet Distance

(P,Q) =

The discrete Fréchet distance considers only positions of the leash where its endpoints are located at vertices of the two polygonal curves and never in the interior of an edge.

Discrete Fréchet Distance

May give large errors!

Compute the continuous Fréchet distance

Decision problem: F(P, Q) ≤ r ?

Compute the Fréchet distance

Freespace diagram of P and Qfree spac

How do we find a reparametrization between P and Q that realises the Fréchet distance r?

Recall that it has to be continuous, non-decreasing, q1 p1 and qm pn. Q

› q1 p1

› qm pn

› q1 p1

› qm pn

› must lie in the free space

Not valid!

› q1 p1

› qm pn

› continuous

› q1 p1

› qm pn

› continuous

› non-decreasing

› q1 p1

› qm pn

› continuous

› non-decreasing

If there exists an xy-non-decreasing path inside the freespace from the bottom left to the top right corner of the diagram then the Fréchet distance is at most r.

Free Space Diagram

Decision version Is the Fréchet distance between two paths at

most r?

Free Space Diagram

Decision version Is the Fréchet distance between two paths at

most r?

Free Space Diagram

If there exists a non-decreasing path in the free space from (q1,p1) to (qm,pn) which is non-decreasing in both coordinates, then curves P and Q have Fréchet distance at most r.

(q1,p1)

(qm, pn)

Free Space Diagram

(q1,p1)

(qm, pn)

Free Space Diagram

(q1,p1)

(qm, pn)

Free Space Diagram

(q1,p1)

(qm, pn)

Free Space Diagram

(q1,p1)

(qm, pn)

Free Space Diagram

(q1,p1)

(qm, pn)

Free Space Diagram

(q1,p1)

(qm, pn)

Free Space Diagram

(q1,p1)

(qm, pn)

Free Space Diagram

(q1,p1)

(qm, pn)

Decision algorithm: free space diagram

Algorithm:

1. Compute Free Space diagram O(mn) time

A square and an ellipsoid can intersect in at most 8 points. (critical points)

The free space in a cell is a convex region. One cell can be constructed in time O(1).

One cell can be constructed in time O(1). How?

Note that we do not need the exact shape. We only need the intersection points between the cell boundary and the ellipsoid.

Decision algorithm: free space algorithm

Algorithm: 1. Compute Free Space diagram

O(mn) time

A square and an ellipsoid can intersect in at most 8 points.The free space in a cell is a convex region. One cell can be constructed in time O(1).

Critical points

Decision algorithm: compute path

(q1,p1)

(qm,pn)

Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).

(q1,p1)

(qm,pn)

(q1,p1)

(qm,pn)

(q1,p1)

(qm,pn)

(q1,p1)

(qm,pn)

(q1,p1)

(qm,pn)

(q1,p1)

(qm,pn)

(q1,p1)

(qm,pn)

(q1,p1)

(qm,pn)

Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn). Build network O(mn) time.

(q1,p1)

(qm,pn)

Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn). Build network O(mn) time. Find a path O(mn) time.

(q1,p1)

(qm,pn)

Theorem: Given two polygonal curves P, Q and r0 one can decide in O(mn) time, whether (P,Q) r.

How can we use the algorithm to compute the Fréchet distance?

Binary search on r?

But what do we do binary search on?

Theorem: Given two polygonal curves P, Q and r0 one can decide in O(mn) time, whether (P,Q) r.

How can we use the algorithm to compute the Fréchet distance?

In practice: Determine r bit by bit

O(mn log “accuracy”) time.

Assume r=0 and the let r grow. The free space will become larger and larger.

Aim: Determine the smallest r such that it contains a montone path from (q1,p1) to (qm,pn).

This can only occur when:1. r is minimal with (q1,p1) to (qm,pn) in the free space, 2. r is minimal when a new vertical or horizontal passage

opens up between two adjacent cells in the free space.3. r is minimal when a new vertical or horizontal passage

opens up between two non-adjacent cells in the free space.

1. r is minimal with (q1,p1) to (qm,pn) in the free space, 2. r is minimal when a new vertical or horizontal passage

Case 1 is easy to test in constant time.

Case 2:

Number of events?

Case 2:

Number of events: O(mn)

Case 1 is easy to test in constant time.Case 2: O(mn) eventsCase 3:

Case 1 is easy to test in constant time.Case 2: O(mn) eventsCase 3: Number of events?

Case 1 is easy to test in constant time.Case 2: O(mn) eventsCase 3: O(m2n+mn2)

Case 1: easy to test in constant time.

Case 2: O(mn) possible events

Case 3: O(m2n+mn2) possible events

Perform binary search on the events O((m2n+mn2) log mn) time.

Improvement: Use parametric search [Meggido’83, Cole’88]

O(mn log mn) time.

Parametric search

The decision version F(r) of the Fréchet distance depends on theparameter r, and it is monotone in r.

Aim: Find minimum r such that F(r) is true.

Our algorithm As can solve F(r) in Ts=O(mn) time.

Meggido’83 and Cole’88 (simplified): If there exists a parallel algorithm Ap that can solve F(r) in Tp time using P processors then there exists a sequential algorithm with running time O(PTp+TpTs+Ts log P).

For our problem we get: O(mn log mn) time using a parallel sorting network [n processors and O(log n) time]

Improvement: Use parametric search [Meggido’83, Cole’88]

O(mn log mn) time.

Summary: Lecture 1

Many different similarity measures.

We use the Fréchet distance!

Theorem: Given two polygonal curves P, Q and r 0 one can decide in O(mn) time, whether (P,Q) r.

Theorem: The Fréchet distance between two polygonal curves P and Q can be computed in O(mn log mn) time.

Open problems

1. Can the Fréchet distance be computed in subquadratic time or is there an (n2) lower bound?

2. Approximate the Fréchet distance in subquadratic time?3. Special cases in subquadratic time? If the curves are c-packed then a (1+ )-approximation of the Fréchet distance can be computed in O(cn/+cn log n). similar results for low density.

[Driemel, Har-Peled and Wenk’12]

Open problems

1. Can the Fréchet distance be computed in subquadratic time or is there an (n2) lower bound?

2. Approximate the Fréchet distance in subquadratic time?3. Special cases in subquadratic time? If the curves are c-packed then a (1+ )-approximation of the Fréchet distance can be computed in O(cn/+cn log n). similar results for low density.

[Driemel, Har-Peled and Wenk’12]

Part of P inside any ball B is at most c times the radius of B.

References

H. Alt and M. Godau.Computing the Fréchet distance between two polygonal curves. IJCGA, 1995.

K. Buchin, M. Buchin, W. Meulemans and W. Mulzer.Four Soviets Walk the Dog - with an Application to Alt's ConjectureSODA, 2014Time: O(n2 (log n)1/2 (loglog n)3/2)

R. van Oostrum and R. Veltkamp. Parametric search made practical. CGTA, 2004

Computational Movement Analysis Lecture 1: Similarity Joachim Gudmundsson

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