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 Goldberg UC Berkeley, United States Anatoly Pashkevich State University of Informatics and Radioelectronics, Belarus

2 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]

6 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]

7 Related Work Existing algorithms for ShareCam –Song, van der Stappen, Goldberg [02] O(n 2 ) –Har-Peled, Koltun, Song, Goldberg [03] O(n log n)

8 One Optimal Frame ShareCam Problem: Given n requests, find optimal frame

9 Problem Definition Assumptions –Camera has fixed aspect ratio: 4 x 3 –Candidate frame c = [x, y, z] t –(x, y)  R 2 (continuous set) – z  Z (continuous set) (x, y) 3z 4z

10 Problem Definition Requested frames : r i =[x i, y i, z i ], i=1,…,n

11 Problem Definition “Satisfaction” for user i: 0  S i  1 S i = 0 S i = 1  = c  r i c = r i

12 Measure user i’s satisfaction: Satisfaction Metrics Requested frame r i Area= a i Candidate frame c Area = a pipi

13 Optimization Problem

14 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

15 Approximation Algorithm x y d Compute S(x,y) at lattice of sample points: w, h : width and height, g: Resolution range

16 Approximation Bound Requested frames

17 Approximation Bound c Requested frames Candidate frame

18 Approximation Bound caca cbcb Requested frames Candidate frames

19 Approximation Bound caca cbcb Requested regions Candidate frames

20 Approximation Algorithm caca cbcb

21 Approximation Algorithm –Run Time: –O(n /  3 ) c * : Optimal frame : Optimal at lattice (Algorithm output) : Smallest frame at lattice that encloses c *

22 Distributed Algorithms Server O(n+1/  3 ) Client O(1/  3 ) Robustness to dropouts…

23 Distributed Lattice Define Final Lattice (Define d) d d

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

25 Distributed Lattice Sub lattice for each user

26 Robustness to client failures

27 Results A demo with 6 inputs t

28 Current & Future Work - Satellite Application

29 Current & future work - Functional Box Sums Efficient reporting of [Zhang et al 2002]

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