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ShareCam Part II: Approximate and Distributed Algorithms for a Collaboratively Controlled

Robotic Webcam

Supported in part by the National Science Foundation

Dezhen Song, Ken GoldbergUC Berkeley, United States

Anatoly PashkevichState University of Informatics and

Radioelectronics, Belarus

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Robot System Taxonomy (Tanie, Matsuhira, Chong 00)

Single Operator, Single Robot (SOSR):

Single Operator, Multiple Robot (SOMR):

Multiple Operator, Multiple Robot (MOMR):

Multiple Operator, Single Robot (MOSR):

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Contents

• Related work

• Problem definition

• Algorithm– Approximation bound– Distributed algorithm

• Results

• Future work

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Related Work• Facilities Location Problems

– Megiddo and Supowit [84]

– Eppstein [97]

– Halperin et al. [02]

• Rectangle Fitting – Grossi and Italiano [99,00]

– Agarwal and Erickson [99]

– Mount et al [96]

– Kapelio et al [95]

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Related Work

• Similarity Measures – Kavraki [98]

– Broder et al [98, 00]

– Veltkamp and Hagedoorn [00]

• Distributed robot algorithms – Sagawa et al [01], Safaric[01]

– Parker[02], Bulter et al. [01]

– Mumolo et al [00], Hayes et al [01]

– Agassounon et al [01], Chen [99]

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Related Work• Existing algorithms for ShareCam

– Song, van der Stappen, Goldberg [02] O(n2)

– Har-Peled, Koltun, Song, Goldberg [03] O(n log n)

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One Optimal Frame

ShareCam Problem: Given n requests, find optimal frame

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Problem Definition• Assumptions

– Camera has fixed aspect ratio: 4 x 3– Candidate frame c = [x, y, z] t

– (x, y) R2 (continuous set)– z Z (continuous set)

(x, y)3z

4z

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Problem Definition

Requested frames: ri=[xi, yi, zi], i=1,…,n

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Problem Definition• “Satisfaction” for user i: 0 Si 1

Si = 0 Si = 1

= c ri c = ri

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• Measure user i’s satisfaction:

)1),/min(()/(

1,)(

)(min

)(

)(),(

zzap

csize

rsize

rArea

rcAreacrs

iii

i

i

ii

Satisfaction Metrics

Requested frame ri Area= ai

Candidate frame c

Area = api

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Optimization Problem

n

iii

crcs

1

),(max

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Algorithm Overview

• Grid based approach• Derive approximation bound

– Price to pay for enlarging a candidate frame– Optimal frame must be enclosed by a large

frame on the sampling lattice. The size difference depends on lattice resolution

– Bound depends on inputs and lattice resolution

• Distributed algorithm

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Approximation Algorithm

n)dd

whgO(

spacing resolution :

spacing lattice :

z2

zd

d x

y

d

Compute S(x,y) at lattice of sample points:

w, h : width and height,g: Resolution range

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Approximation Bound

Requested frames

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Approximation Bound

c

Requested frames

Candidate frame

z

ncrscs

z

zzap

crs

n

iii

iii

ii

1

),()(

/1

)/)(/(

),(

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Approximation Bound

ca

cb

Requested frames

Candidate frames

bb

aa

zncs

zncs

/)(

/)(

b

a

a

b

z

z

cs

cs

)(

)(

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Approximation Bound

b

a

a

b

z

z

cs

cs

)(

)(

ca

cb

Requested regions

Candidate frames

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Approximation Algorithm

ca

cb

za

a

b

a

z

dz

z

z

z

dd

2

3set

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Approximation Algorithm

– Run Time: – O(n / 3)

c* : Optimal frame

: Optimal at lattice (Algorithm output)

c~

: Smallest frame at lattice that encloses c*

c

)ˆ()~()( * cscscs

)(

)ˆ(

)(

)~(1

** cs

cs

cs

cs

zdz

z

2...

min

min

1

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Distributed Algorithms

•Server O(n+1/3)•Client O(1/3)•Robustness to dropouts…

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Distributed Lattice• Define Final Lattice (Define d)

dd

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Distributed Lattice• Divide Lattice point based on n (Assume n=4)

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Distributed Lattice• Sub lattice for each user

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Robustness to client failures

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Results

• A demo with 6 inputs

t

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Current & Future Work - Satellite Application

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Current & future work - Functional Box Sums

• Efficient reporting of

n

i i

ii

n

ii φArea

φAreaωsS

11 )()Φ(

)Φ()Φ(

iφ

Φ

jφ

kφ[Zhang et al 2002]

)log)/1(( 23 nnO

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