Vladlen Koltun, UC Berkeley

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Efficient Algorithms for Shared Camera Control

• Vladlen Koltun, UC Berkeley
• Joint work with
• Sariel Har-Peled, Dezhen Song and Ken Goldberg

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Robotic Webcameras

• Positioned at sites of wide interest
• Pan, tilt, zoom can be controlled on-line
• Existing implementations users queue

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ShareCam

• Shared on-line camera control
• Geometric algorithms for finding the optimal
  camera frame

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Related Work

• 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.
• Algorithms we describe are relevant in these
  scenarios as well

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Contribution

• Song, van der Stappen and Goldberg (2002), O(n2)
• Our work
• Exact - O(n3/2 log3n)
• e-approximation - O(n logn)
• Discretizing zoom values
• Fixes the size of the camera rectangle
• Two degrees of freedom (center point)

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Finding the Optimal Frame

• 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

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Individual User Satisfaction

• Intersection over maximum metric
• Intersection Area
• Maximum Area

user
camera
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Individual Satisfaction Function
Intersection (useri,cam(x,y)) Maximum
(useri,cam(x,y))
SATi(x,y)
useri
(x,y)
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Individual Satisfaction Function
Intersection (useri,cam(x,y)) Maximum
(useri,cam(x,y))
SATi(x,y)
useri
(x,y)
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Global Satisfaction Function

• SAT(x,y) ?i SATi(x,y)
• We are searching for (x,y) that maximizes SAT(x,y)

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Approximation Algorithm

• For a given egt0 and every individual satisfaction
  function, consider its level sets at O(1/e)
  heights

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Approximation Algorithm

• For a given egt0 and every individual satisfaction
  function, consider its level sets at O(1/e)
  heights

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Approximation Algorithm

• Construct rectilinear paths between the level
  sets
• Decompose into rectangles

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Approximation Algorithm

• The weighted rectangles almost approximate the
  satisfaction function

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Approximation Algorithm

• 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-e) of the optimum

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Approximation Algorithm

• 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

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Analysis

• The number of slices is O( log(1/e)/e )
• The number of rectangles (per function) is O(
  log(1/e)/e2 )
• The overall number of rectangles is N
  O( n log(1/e)/e2 )
• Every update of the segment tree during the sweep
  is O(log N)
• Overall running time is O(N log N)

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Exact Algorithm

• Sweep the satisfaction functions with a vertical
  line
• Maintain the global satisfaction function along
  the line with a KDS

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Exact Algorithm

• Treat these satisfaction values as points moving
  along vertical lines in 2-D
• Maintain the upper hull of these points using a
  kinetic data structure

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Exact Algorithm

• 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

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Exact Algorithm

• 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)

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Implementation

• 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

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Implementation
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Work in progress

• Hardware-assisted algorithm
• Continuous zoom range
• More than one camera frame
• Dynamic maintenance, obstacles, weighted users
  ...

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Thank you