Efficient Algorithms for Shared
Camera Control
Vladlen Koltun, UC Berkeley
Joint work with
Sariel Har-Peled, Dezhen Song and Ken Goldberg
Robotic Webcameras
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Positioned at sites of
wide interest
Pan, tilt, zoom can be
controlled on-line
Existing implementations:
users queue
ShareCam
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Shared on-line
camera control
Geometric
algorithms for
finding the
optimal camera
frame
Related Work
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Shared on-line control over a single
mechanism
– Industrial robot arm [Goldberg, Chen, et al.]
– Waste cleanup system [Cannon, McDonald,
et al.]
– Tele-Actor [Goldberg, Song, et al.]
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Algorithms we describe are relevant in
these scenarios as well
Contribution
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Song, van der Stappen and Goldberg
(2002), O(n2)
Our work
– Exact - O(n3/2 log3n)
– ε-approximation - O(n logn)
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Discretizing zoom values
– Fixes the size of the camera rectangle
– Two degrees of freedom (center point)
Finding the Optimal Frame
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Idea: Maximize user satisfaction
Find a way to evaluate the satisfaction of
a given user with a given camera
rectangle
Compute rectangle that maximizes
cumulative user satisfaction
Individual User Satisfaction
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Intersection over maximum metric:
Intersection Area
Maximum Area
user
camera
Individual Satisfaction Function
Intersection (useri,cam(x,y))
SATi(x,y) =
Maximum (useri,cam(x,y))
useri
(x,y)
Individual Satisfaction Function
Intersection (useri,cam(x,y))
SATi(x,y) =
Maximum (useri,cam(x,y))
useri
(x,y)
Global Satisfaction Function
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SAT(x,y) = ∑i SATi(x,y)
We are searching for (x,y) that
maximizes SAT(x,y)
Approximation Algorithm
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For a given ε>0
and every
individual
satisfaction
function,
consider its level
sets at O(1/ε)
heights
Approximation Algorithm
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For a given ε>0
and every
individual
satisfaction
function,
consider its level
sets at O(1/ε)
heights
Approximation Algorithm
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Construct
rectilinear paths
between the
level sets
Decompose into
rectangles
Approximation Algorithm
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The weighted
rectangles
“almost”
approximate the
satisfaction
function
Approximation Algorithm
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We have a collection of weighted rectangles
We locate the most heavily covered point
The global satisfaction value of this point is
at least (1-ε) of the optimum
Approximation Algorithm
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To locate the heaviest point
– Sweep the plane with a vertical line
– Rectangles define weighted intervals on the line
– Maintain the heaviest point on the line using a
segment tree
– During the sweep, intervals are inserted and
removed
Analysis
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The number of slices is O( log(1/ε)/ε )
The number of rectangles (per function)
is O( [log(1/ε)/ε]2 )
The overall number of rectangles is
N = O( n [log(1/ε)/ε]2 )
Every update of the segment tree
during the sweep is O(log N)
Overall running time is O(N log N)
Exact Algorithm
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Sweep the
satisfaction
functions with a
vertical line
Maintain the global
satisfaction
function along the
line with a KDS
Exact Algorithm
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To avoid handling O(n2) events,
partition points into n1/2 batches
Each batch maintains an additional
value: the sum of long functions
When a long function changes we just
update this value
When a short function changes we
rebuild the kinetic data structure
Exact Algorithm
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The number of events processed by
each kinetic data structure is ≈O(n1/2)
The kinetic data structures are rebuilt
O(n) times
The overall running time is O(n3/2 log3n)
Implementation
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The approximation algorithm becomes
faster than the algorithm of Song et al.
only when n is in the hundreds of
thousands.
The bottleneck is updating the segment
tree during the sweep. Each insertion or
deletion takes O(log N), but there are
2N of them, and N is at least 50n
Implementation
Work in progress
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Hardware-assisted algorithm
Continuous zoom range
More than one camera frame
Dynamic maintenance, obstacles,
“weighted” users ...
Thank you