Distributed Consensus Algorithms on Manifolds. Roberto Tron, Bijan Afsari, René Vidal Johns Hopkins University

Transcription

1 Distributed Consensus Algorithms on Manifolds Roberto Tron, Bijan Afsari, René Vidal Johns Hopkins University

2 Distributed algorithms in controls Mo#on coordina#on Forma#on control Collabora#ve mapping Network op#miza#on

3 Consensus algorithms Global agreement from local interacfons

4

5 New SeJngs

6 Cameras Today 24 hrs/min 30 million cameras 4 bn hrs/week Tasks Detection, Segmentation, Motion Estimation, Stabilization, Registration, Categorization Application Areas End Users National Security Governments Recreation Consumer

7 Camera Networks Classical approach One camera wired to one computer MulFple cameras wired to a central processing unit Processing than by a human operator or by a central computer Problems Flexibility: Wiring makes it hard to deploy new cameras Robustness: central node failure = enfre system failure Scaling: processing and bandwidth requirements do not scale well

8 Camera Sensor Networks Motes Small, wireless devices basery powered limited memory and compufng power ApplicaFons Surveillance Environmental monitoring Smart homes Ques#on Can we deploy vision algorithms on camera sensor networks?

9 Challenges to Computer Vision Algorithms Tradi#onal computer vision algorithms ExisFng algorithms are centralized: all images are sent to one node for processing Computer Vision Sensor networks have limited resources Limited processing power and memory Slow wireless channel Nodes can have limited communicafon range Challenge TradiFonal computer vision algorithms require resources not available in a camera sensor network

10 Challenges to Sensor Network Algorithms Tradi#onal distributed algorithms for sensor networks Existing algorithms have been designed for processing simple scalar measurements Sensor Networks In computer vision applicafons Measurements (images) are high- dimensional Measurements are corrupted by noise, outliers EsFmates are non- Euclidean (e.g. rotafons) Challenge TradiFonal sensor network algorithms cannot be directly used for computer vision applicafons

11 Toward Distributed Computer Vision Algorithms Centralized algorithms Considerable complexity Computer Vision Sensor Networks Distributed algorithms Only simple problems Our goal Develop distributed computer vision algorithms that Are efficient: local processing + short communications Converge to the centralized solution Can handle outliers, packet losses, data on manifolds

12 Consensus on manifolds example y z x z z x y Data on y non- linear x manifolds!

13 Previous work Specific Manifolds Sphere [OlfaF- Saber 2006] N- Torus [Kuramoto model, 1975], [Vicksek model, 1995], [Scardovi, Sepulchre 2007] Extrinsic approach Embedding + ProjecFons [SarleSe, Sepulchre 2009], [Hatanaka, Bullo 2010], [Igarashi, Fujita 2010] CoordinaFon Lie groups [SarleSe, Sepulchre 2010] Rigid mofons [SarleSe, Sepulchre 2009], [Thunberg, Hu 2011], [Bai, Wen ], [Igarashi, Spong, 2012], [Smith, Leonard 2001] LocalizaFon Planar case [Piovan, Bullo 2008] Centralized [Shirmohammadi, Taylor 2010]

14 Our work 1. Consensus on any Riemannian manifold with bounded curvature 2. Distributed image- based localizafon from images Distributed opfmizafon

15 Outline Introduc#on Consensus in Euclidean spaces Riemannian geometry

16 Riemannian consensus Choice of step size Local convergence results Almost- global convergence on SO(3)

17 Image- based camera network localiza#on y x z z y x Setup cost funcfon Link with Riemannian Consensus Results

18 Review of Euclidean consensus

19 NotaFon Graph Measurements States

20 OpFmizaFon cost funcfon

21 OpFmizaFon cost funcfon

22 Algorithm = Gradient Descent

23 Algorithm = Gradient Descent

24 How to choose the step size = bound on max eval of Hessian of = max # of neighbors 2 µ max (ϕ)

25 Gradient Descent

26 Gradient Descent

27 Gradient Descent

28 Gradient Descent

29 Gradient Descent

30 Gradient Descent mean{x i (t)} = mean{x i (t + 1)} \ {

31 Convergence to the mean Iterations x \

32 Review of Riemannian geometry

33 NotaFon

34 \ma ^2(

35 Riemannian Consensus

36 NotaFon Graph Measurements States On manifold

37 OpFmizaFon cost funcfon

38 OpFmizaFon cost funcfon Not convex! Consensus configurations are global minima

39 OpFmizaFon cost funcfon

40 Gradient descent

41 Gradient descent

42 How to choose the step size Theorem = max # of neighbors = bound on max eval of Hessian of Tron, Afsari, Vidal - Riemannian Consensus for Manifolds with Bounded Curvature to appear in Transactions on Automatic Control, µ max (ϕ)

43 \v Spaces of non- negafve constant curvature All other spaces might depend on max distance between neighboring states

44 Gradient Descent

45 Gradient Descent

46 Gradient Descent

47 Gradient Descent

48 Convergence Theorem Measurements not too dispersed Step size small enough Convergence to consensus configurafons Tron, Afsari, Vidal - Riemannian Consensus for Manifolds with Bounded Curvature to appear in Transactions on Automatic Control, 2012

49 1. Given a set containing all global minimizers S = (x 1,...,x N ) M N : x 0 M s.t. max d M (x i,x 0 ) <r i r = 1 2 min inj M, π S M N 2. Guarantee iterates do not leave this set Tron, Afsari, Vidal - Average Consensus on Riemannian Manifolds with Bounded Curvature IEEE Conference on Decision and Control 2011

50 Convergence Theorem Spaces of constant, non- negafve curvature Converge to a point inside convex hull Point need not be the global Frechet mean Tron, Afsari, Vidal - Riemannian Consensus for Manifolds with Bounded Curvature to appear in Transactions on Automatic Control, 2012

51 Convergence General result If inifal measurements are inside Sconv And step size is small enough Then, Riemannian Consensus converges to a set in Sconv Constant non- negafve curvature Convergence set is enlarged to Sr The convergence result is to a single consensus state instead of to a set The consensus state is shown to lie in the convex hull of the inifal measurements

52 Experiments Sphere(7) N=15 nodes, 4- regular graph

53 Experiments SO(7) N=15 nodes, 4- regular graph

54 Experiments SO(3) Closed geodesic N=15 nodes, 4- regular graph, closed geodesic

55 x 1 x 4 x 2 Local minimum x 3

56 d^

57 Tron, Afsari, Vidal Intrinsic consensus on SO(3) with almost-global convergence to appear in Conference on Decision and Control 2012 d^

58 Tron, Afsari, Vidal Intrinsic consensus on SO(3) with almost-global convergence to appear in Conference on Decision and Control 2012 f_{

59 x 1 x 4 x 2 Saddle point x 3

60

61 Almost- global convergence Theorem On SO(3), the only set of stable equilibria is the set of global minimizers Tron, Afsari, Vidal Intrinsic consensus on SO(3) with almost-global convergence to appear in Conference on Decision and Control 2012

62 Experiments Squared distance N=15 nodes, 4- regular graph, closed geodesic

63 Experiments Reshaped distance

64 Average number of iterafons for a K- regular network with N nodes N K Avg. It

65 Image- based camera network localizafon

66 y x z z y x

67 Vision graph A B C

68 Input (\til

69 Output y Camera j x z Camera i z y g i =(R i,t i ) x

70 \m (3 min {g ij } (i,j) E f d SE(3) (g ij, g ij ) d^

71 Challenge 1: Inconsistent Transf. g 12 g23 g 31

72 OpFmizaFon problem min {g ij } s.t. (i,j) E f d SE(3) (g ij, g ij ) are consistent \m (3 d^

73 Pose reparametrizafon Theorem are consistent g ij z y g j z y g i x x Tron, Vidal Distributed image-based 3-D localization in camera sensor networks Conference on Decision and Control 2009

74 \m (3 OpFmizaFon problem min {g ij } (i,j) E f d SE(3) (g ij, g ij ) d^

75 \m (3 f d_{ d OpFmizaFon problem min {g ij } (i,j) E f d SE(3) (g 1 g i j, g ij ) f\b

76 OpFmizaFon problem

77 OpFmizaFon problem

78 Challenge 2: Scale Ambiguity y y x z x z z y z y x x

79 Fix shortest edge length y x z z y 1 unit x

80 OpFmizaFon problem How to get global minimizer? Tron, Vidal - Distributed Image-Based 3-D Localization of Camera Sensor Networks in Conference on Decision and Control 2009

81 Ini#aliza#on of rota#ons \m (R

82 Ini#aliza#on of transla#ons \m R \t

83 Complete op#miza#on min ϕ({r i,t i, λ ij }) {R i,t i,λ ij } s.t. λ ij 1 \min {R_i, \lam

84 Gradient descent y x z z y x

85 How to choose the step size Theorem = max # of neighbors = bound on max eval of Hessian of

86 Noiseless case Theorem In noiseless case, equivalent to Riemannian consensus

87 Noiseless case Theorem In noiseless case, non- convex, but no local minima!

88 Experiments on synthefc data Experiment: 8 cameras, 30 scene points, 4- regular graph 1. Camera poses and points 2. Images + Noise 3. RelaFve poses 4. LocalizaFon

89 Experiments on synthefc data RotaFon errors [degrees] Noise 0 pixels 1 pixels 3 pixels IniFal Final 0.00 ± ± ± ± ± ±0.03 Geometric variance of scale rafos Noise 0 pixels 1 pixels 3 pixels Final TranslaFon direcfon errors [degrees] Noise 0 pixels 1 pixels 3 pixels IniFal Final 0.00 ± ± ± ± ± ±0.03

90 Experiments on real data TRON et al.: DISTRIBUTED IMAGE-BASED 3-D LOCALIZATION OF CAMERA SENSOR NETWORKS Fig Images with features used by Bundler to compute the localization diagonal and with positive entries (because M M is symmetric and positive definite, hence unitarily diagonalizable). 1 Define the change of coordinates y = Σ 2 U y. Then we can rewrite the cost as q ϕ(l) = y y = (y )2i. (35) j

91 TRON et al.: DISTRIBUTED IMAGE-BASED 3-D LOCALIZATION OF CAMERA SENSOR NETWORKS Fig Images with features used by Bundler to compute the localization diagonal and with positive entries (because M M is symmetric and positive definite, hence unitarily diagonalizable). 1 Define the change of coordinates y = Σ 2 U y. Then we can rewrite the cost as q ϕ(l) = y y = (y )2i. (35) j 1 i=1 At the same time, the update equation can be written as y(l + 1) = y(l) εm M y(l) = (I εm M )y(l). By substituting M M with its decomposition (and also remembering that diagonal matrices commute), we can make the same change of coordinates as before. After some manipulations we obtain that q ϕ(l + 1) = y (I εσ)2 y = (1 εσi )2 (y )2i. (36) 1 j The denotes the center of the circles. Fig. 4. An illustration for the second part of the proof for Theorem 2 showing an example of the Gers gorin discs for (I εm M ). The region where the eigenvalues could be located is highlighted in solid gray. i=1 Each term of (36) is positive and is smaller than the corresponding term in (35) if and only if 1 εσi < 1 i = 1,..., p (37) 2 which is equivalent to 0 < ε < σmax. If ε satisfies these conditions, then ϕ(l+1) ϕ(l), with equality only if ϕ(l) = 0 (i.e., when the algorithm is at the global optimum). This completes the first part of the proof. For the second part, one can notice that requiring condition (37) is the same as requiring that all the eigenvalues of the matrix (I εm M ) are inside the unit circle. Since M M is a symmetric, positive semidefinite matrix, we know that its eigeinvalues σ are real and non-negative. This means that the eigenvalues of (I εm M ) are less or equal to one for any positive ε. We therefore need to make sure that these eigenvalues are greater than 1. One way to achieve this is by choosing ε such that all Gers gorin discs do not extend further than 1. This can be mathematically written as p (1 ε(m M )i,i ) ε (M M )i,j > 1 j=1,j =i disc center disc radius (38) for all i = 1,..., p. By expliciting ε, the main result of the Theorem can be obtained. As a remark, this Theorem generalize the proof of convergence for standard consensus (see, for instance, [6]). In fact, for standard consensus we have M = C, M M = L, where L is the graph Laplacian. Using our Theorem we get ε (0, 1G ), in accordance with previous results in the literature. C. Proof of Theorem 3 Define M and y as in (26) and (27). As explained in Section III-B, our algorithm is essentially a projected gradient descent procedure to minimize the cost ϕt (l) = 12 M y(l) 2. The proof is essentially based on the application of an extended version of Theorem 2. We need to introduce now some additional notation. Let v be a vector of norm one and d > 0 a scalar. Define the halfspace S = {x Rp : x v d}. Given a vector y Rp, the projection of y on S can be expressed as: y if y v d projs (y) = (39) (I vv )y + dv otherwise Snavely, Seitz, Szeliski. Photo Tourism: Exploring image collections in 3D in ACM Transactions on Graphics

92 Experiments on real data

93 Experiments on real data RotaFon errors per edge TranslaFon direcfon errors per edge

94 Conclusion Riemannian Consensus Theory Sufficient convergence on any Riemannian manifold Almost- global convergence on SO(3) Applica#ons Image- based camera network localizafon

95 Future work Convergence rate Incorporate uncertainfes Dynamic case Time- varying measurements Pose + velocity Improved localizafon

96 Thanks!