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ShareCam Part II: Approximate and Distributed Algorithms for a Collaboratively Controlled Robotic Webcam. Dezhen Song, Ken Goldberg UC Berkeley, United States Anatoly Pashkevich State University of Informatics and Radioelectronics, Belarus. - PowerPoint PPT Presentation
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1
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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4
Contents
• Related work
• Problem definition
• Algorithm– Approximation bound– Distributed algorithm
• Results
• Future work
5
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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