Sensor Scheduling and Energy Allocation For Lifetime Maximization in User-Centric Visual Sensor Networks

Transcription

1 1 Sensor Scheduling and Energy Allocation For Lifetime Maximization in User-Centric Visual Sensor Networks Chao Yu IP Lab, University of Rochester December 4, 2008

2 2 1 Introduction User-Centric VSN Camera Scheduling and Energy Allocation 2 Modeling Network Lifetime Formulation Abstraction 3 Lifetime-Maximizing Camera Scheduling 4 Lifetime-Maximizing Energy Allocation 5 System Setup and Simulations 6 Concluding Remarks

3 3 Outline 1 Introduction User-Centric VSN Camera Scheduling and Energy Allocation 2 Modeling Network Lifetime Formulation Abstraction 3 Lifetime-Maximizing Camera Scheduling 4 Lifetime-Maximizing Energy Allocation 5 System Setup and Simulations 6 Concluding Remarks

4 4 User-Centric VSN R monitored by cameras {C} N i=1. C i covers a sub-region V i cameras only communicate with CP. CP receives sequence of user request for U, select camera to provide data, synthesize Ũ Lifetime: the duration a certain percentage (e.g. 90%) of R is covered by at least one camera

5 5 Camera Scheduling and Energy Allocation Questions to ask

6 5 Camera Scheduling and Energy Allocation Questions to ask A formulation of the network lifetime

7 5 Camera Scheduling and Energy Allocation Questions to ask A formulation of the network lifetime Given a desired view, multiple cameras contain the data how to select a camera to provide the data

8 5 Camera Scheduling and Energy Allocation Questions to ask A formulation of the network lifetime Given a desired view, multiple cameras contain the data how to select a camera to provide the data Given total available energy w t how to allocation w t among cameras

9 6 Outline 1 Introduction User-Centric VSN Camera Scheduling and Energy Allocation 2 Modeling Network Lifetime Formulation Abstraction 3 Lifetime-Maximizing Camera Scheduling 4 Lifetime-Maximizing Energy Allocation 5 System Setup and Simulations 6 Concluding Remarks

10 7 Notations divide R into M r blocks: {R j,j [M r ]} coverage matrix: B r R N Mr, where B r i,j def = I(R j V i ), if region R j covered by camera C i, B r i,j = 1, otherwise 0. U: M u blocks {U j,j [M u ]} B u R N Mu, where B u i,j def = I(U j V i ) (1) Figure: The discretization of the target plane R and desired view U.

11 7 Notations p j : probability that R j is requested by the user, M r j=1 (p j) = 1. p = [p 1 p 2... p Mr ] T w t j : energy of camera C j at time t. w t = [w t 1 w t 2... w t M r ] T L(p,w t,b r ): the network lifetime E[L(p,w t,b r )]: the expected network lifetime Figure: The discretization of the target plane R and desired view U.

12 8 Camera Scheduling Strategy At time t, if C s is selected, w t updated as w t+1 s. optimal camera selection strategy: maximize E[L(p,w t+1 s,b r )].

13 8 Camera Scheduling Strategy coverage energy:the sum of the energies of all the cameras that cover R j. m t+1 s = B r w t+1 s, the j th entry m t+1 j,s represents the coverage energy of R j approximate: L(m t+1 s,p) the optimal camera scheduling strategy at time t: s t = argmax i λ u (j) E[L(m t+1 i,p)]

14 9 Abstraction box B i contains m i balls, i [M] at each request, a ball is taken from B i with probability p i. After L requests, one of these boxes first become empty characterization of E[L(m,p)]

15 10 Exact Solution for M = 2 Proposition 1 For M = 2, the p.m.f of L can be written as: Pr(L = l) = I(m 1 l (m 1 + m 2 1))α(l m 1 ;m 1,p) +I(m 2 l (m 1 + m 2 1))α(l m 2 ;m 2,1 p) where α(k;τ,b) represents the p.m.f of a negative binomial distribution [M.Hilbe, 2007], ( ) k + τ 1 α(k;τ,b) = b τ (1 b) k k

16 11 Exact Solution for M = 2 Proposition 1 (cont d) the expectation of L can be obtained as E[L] = β(l m 1,m 1,p) + β(l m 2,m 2,1 p) where β(k;j,b) = jf α(k;j,b) (k + 1)α(k + 1;j,b) b (1) and F α (k;j,b) def = k i=0 α(k;j,b)

17 12 Recursive Calculate of L for M > 2 consider a sequence of M experiments in the k th experiment k(2 k M) only the first k boxes {B i } k i=1 are utilized a ball requested from the i th box with normalized probability p i = p i P k i=1 p i. let L k denote the number of requests after which one of the boxes {B i } k i=1 first becomes empty, then immediately we see L = L M recursively calculate the p.m.f of L k (2 k M) from the p.m.f of L k 1.

18 13 Exact Solution for General Case M > 2 Proposition 2: Pr(L k = l) = 1(m 1 l π(k)) m k 1 X l 1 j «(p k )j (1 p k )l j Pr(L k 1 = l j) j=0 «l 1 +1(m k l π(k)) (p l m k )m k (1 p k )l m k Pr(L k 1 > (l m k )) k where π(k) def = ( kx m j k + 1). j=1 p.m.f of L 2,L 3,...,L M 1 are calculated in sequence to obtain L M direct evaluation: E[L k ] = π(k) l=m 1 lpr(l k = l) computationally prohibitive as k increases need for approximation

19 14 Approximate Evaluation of E[L] Proposition 3 The expectation of L can be obtained as E[L] = π(m) l=m 1 Ω M (m 1;l 1,p) + (m 1 1)Ω M (m 1;m 1 1,p) where Ω M ( ) represents the c.d.f of a multinomial distribution. efficient approximation for Ω M ( ) exists [Levin, 1981] allows approximation of E[L] without calculating Pr(L = l)

20 15 Experimental Evaluation Exact Approx Simulation m = m = m = m = Three boxes contain (m, m, 2m) balls, p = [0.25, 0.25, 0.5] Exact: the exact lifetime Approx: approximate lifetime using Proposition 3 Simulation: average lifetime from 200 Monte Carlo simulations. further approximation?

21 16 Asymptotic approximation of E(L) Proposition 4: When M = 2 and m 1,m 2 are sufficiently large, E[L] { min( m 1 p, m2 1 p ) m 1 p (m1+1)n(m1+1;m1,p) p (m2+1)n(m2+1;m2,1 p) 1 p m1 if p m2 1 p if m1 p = m2 1 p the approximation error reduces at an exponential rate

22 Experimental Evaluation p=0.25 p=0.4 p=0.45 p= p=0.5 Relative error in E[L] Relative error in E[L] Number of balls in each box Number of balls in each box (a) (b) B 1, B 2 contain balls m 1 = m 2, p = [p,1 p], Abscissa: the value of m 1,m 2, ordinate: relative error (a): simple approximation: min( m 1 p, m 2 1 p ) (b): refined approximation for the case m 1 p = m 2 1 p 17

23 18 Asymptotic Approximation for M > 2 E[L(m,p)] min( m 1 p 1, m 2 p 2,..., m M p M ) asymptotic approximation hot-spot exist: the difference between the two smallest values in { m i p i } N i=1 is not negligible very easy to calculate

24 19 Recap formulation of the expected network lifetime of a user-centric VSN exact solution for M = 2, exact solution for M > 2: recursive approach efficient approximation of E[L], M > 2 asymptotic analysis for E[L] Next, camera scheduling and energy allocation schemes

25 20 Outline 1 Introduction User-Centric VSN Camera Scheduling and Energy Allocation 2 Modeling Network Lifetime Formulation Abstraction 3 Lifetime-Maximizing Camera Scheduling 4 Lifetime-Maximizing Energy Allocation 5 System Setup and Simulations 6 Concluding Remarks

26 21 Camera Scheduling Strategy When block U j is requested: s = argmax{min( m i,1, m i,2,..., m i,m r )} (2) i λ u (j) p 1 p 2 p Mr interpretation: maximize the normalized energy of the hot-spot block in the monitored plane. if the hot-spot block doesn t belong to λ u (j), s = argmax{min( m i,k,k κ r (i))} i λ u (j) p k κ r (i) denotes the set of blocks in the monitored region R covered by camera C i

27 22 Camera Scheduling Strategy stochastically optimal for lifetime-maximization user interactions are explicitly modeled very intuitive interpretation computational efficient

28 23 Outline 1 Introduction User-Centric VSN Camera Scheduling and Energy Allocation 2 Modeling Network Lifetime Formulation Abstraction 3 Lifetime-Maximizing Camera Scheduling 4 Lifetime-Maximizing Energy Allocation 5 System Setup and Simulations 6 Concluding Remarks

29 Energy Allocation Strategy the expected lifetime can be given by E[L(m,p)] min( m 1 p 1, m 2 p 2,..., m N p N ), Optimized energy allocation strategy is a max-min optimization, max w s.t. min {f i } N i=1 (3) i N w i = w t i=1 f = PB r w w i 0 def where f i = m i p i for i = 1,2,...,N and f = [f 1,f 2,...,f N ], P is the diagonal matrix formed by the vector [ 1 p 1, 1 p 2,..., 1 p N ]. 24

30 25 Linear Programming Formulation introduce a new variable t, min w t s.t. N w i = w t i=1 f = PBw f i t,w i 0, for i = 1,2,...,N (4) efficient optimization in polynomial time

31 26 Outline 1 Introduction User-Centric VSN Camera Scheduling and Energy Allocation 2 Modeling Network Lifetime Formulation Abstraction 3 Lifetime-Maximizing Camera Scheduling 4 Lifetime-Maximizing Energy Allocation 5 System Setup and Simulations 6 Concluding Remarks

32 27 System Setup plane-based camera calibration image mosaicing

33 28 Image Mosaicing Figure: image coordinate of X h in the second (desired) view x 2 can be obtained from x 1 through the homography. x HX h H 1 1 x 1 = X h = H 1 2 x 2, x 1 = H 1 H 1 2 x 2

34 29 Simulation Setup target plane R: size 4m 3m N cameras placed randomly within a 4m 3m field, 3m from R Users viewpoints: a Markov random walk on a grid cameras and users views point toward R, random rotation within ±0.1 radian along each of the three axes. all images of (in pixel units) M u = 100, bilinear interpolation. for camera scheduling: focal length f = , N = 36 results averages over 100 simulations.

35 30 Simulation: Camera Scheduling three schemes compared OptCOV: proposed scheme MinANG: most similar viewing direction CovCOST: a heuristic defining a coverage cost [Yu et al., 2007] ξ i = 1 m j j λ r (i)

36 31 Simulation: Camera Scheduling OptCOV MinANG CovCOST coverage percentage Output Distortion (PSNR) OptCOV MinANG CovCOST time(t) time(t) (a) (b) Figure: Simulation results. (a): comparison of percentage coverage (b): comparison of distortion in output image. OptCOV significantly prolongs network lifetime

37 32 Snapshots of rendered images Figure: Snapshots of images rendered in the simulation. Top row: Mosaiced image using OptCOV at the 20 th, 27 th, and54 th user request. The desired views are fully covered. Middle: Using CovCOST, minor part of the desired view is not covered, denoted by black area in the mosaiced image. (d) Using MinANG, significant fraction of the desired view is not covered.

38 33 Simulation: Energy Allocation three energy allocation schemes compared: uniform allocation: UniForm using LP: LinOpt using max-min optimization: MaxMin

39 34 Simulation: Energy Allocation coverage percentage LP Optim. Max-Min Optim. Uniform time(t) Figure: Comparison of coverage from different energy allocation schemes. Optimized energy allocation significantly prolongs network lifetime LP optimization: 2.3 sec, Min-Max optimization: sec, Pentium IV 3.0G CPU, 1M memory, implemented in Matlab TM

40 35 Outline 1 Introduction User-Centric VSN Camera Scheduling and Energy Allocation 2 Modeling Network Lifetime Formulation Abstraction 3 Lifetime-Maximizing Camera Scheduling 4 Lifetime-Maximizing Energy Allocation 5 System Setup and Simulations 6 Concluding Remarks

41 36 Summary stochastic formulation of the expected network lifetime coverage/lifetime of a VSN differs from traditional WSN user interaction abstraction of expected network lifetime exact and approximate solutions: computational expensive asymptotic analysis lifetime-maximizing camera scheduling hot-spot of the network lifetime-maximizing energy allocation LP formulation

42 37 Remarks simplicity well known: binomial/multinomial known: negative binomial/multinomial proposed: bounded negative binomial/multinomial exact, approximate, asymptotic behaviors

43 37 Remarks simplicity well known: binomial/multinomial known: negative binomial/multinomial proposed: bounded negative binomial/multinomial exact, approximate, asymptotic behaviors generality other VSN applications: coverage information B r,b u by suitable discretization (e.g in 3D domain). e.g. task (storage, bandwidth) allocation among networked computing servers, sensor deployment problem: much larger parameter space, location, rotation

44 37 Remarks simplicity well known: binomial/multinomial known: negative binomial/multinomial proposed: bounded negative binomial/multinomial exact, approximate, asymptotic behaviors generality other VSN applications: coverage information B r,b u by suitable discretization (e.g in 3D domain). e.g. task (storage, bandwidth) allocation among networked computing servers, sensor deployment problem: much larger parameter space, location, rotation sub-optimality neglected the dependency of each block being requested mitigation: parameters updated afresh at each request

45 38 Ongoing work collaborative transmission between two sensors wyner-ziv coding techniques certain robustness to calibration error certain tolerance to non-planar scenes complexity: a fraction of JPEG compression

46 Figure: Wyner-Ziv image compression v.s. JPEG. 39 Ongoing work sigma=3.00 sigma=5.00 sigma=10.00 jpeg psnr data rate (1 for uncompressed)

47 40 Reference B. Levin. A representation for multinomial cumulative distribution functions. The Annals of Statistics, 9(5): , Joseph M.Hilbe. Negative Binomial Regression. Cambridge University Press, Chao Yu, Stanislava Soro, Gaurav Sharma, and Wendi Heinzelman. Lifetime-distortion trade-off in image sensor networks. In Proc. IEEE Intl. Conf. Image Proc., volume V, pages , Sept