Efficient Globally Op5mal Consensus Maximisa5on with Tree Search

Transcription

1 Efficient Globally Op5mal Consensus Maximisa5on with Tree Search Tat- Jun Chin School of Computer Science The University of Adelaide 1 of 45

2 Maximum consensus max, I X I subject to r i ( ) apple 8x i 2 I, 2 of 45

3 Example 1: Line fimng 3 of 45

4 Example 2: Triangula5on Figure 3. Two images from the dinosaur sequence, and the resulting reconstruction. e dinosaur sequence, and the resulting reconstruction. Figure 4. Two images from the house sequence, and the resulting reconstruction. Fig. 5. The triangulation problem: Assum is less than some value, the sought point X cones. If is set too small, then the cones do Olsson et al., Efficient op5miza5on for L- inqy problems using pseudoconvexity, ICCV 27. by T.- J. Chin 4 of 45 ismul5view set too large, then the cones intersect in the house sequence, and the resulting Hartley areconstruction. nd Kahl, Op5mal algorithms in geometry, ACCV 27.

5 Example 3: Homography fimng Figure from hyp://sse.tongji.edu.cn/linzhang/cv14/projects/panorama.htm. 5 of 45

6 Running example: Linear regression 1.9 b(a) = a r i ( ) = b i a i x i =[a i,b i ] T of 45

7 RANSAC, minimal subset size = p of 45

8 Minmax problem Minimise the maximum residual: min max i b i a i Same as L- infinity minimisa5on 1,2 : min b 1 a 1 b 2 a 2. b N a N A.k.a. Chebyshev approxima5on/regression Hartley and Schaffalitzky, L- inqy minimiza5on in geometric reconstruc5on problems, CVPR Kahl and Hartley, Mul5ple- view geometry under the L- inqy- norm, PAMI of 45

9 Minmax problem of 45

10 Minmax problem b i a i of 45

11 Minmax problem 1.5 max i b i a i 1 Global minimum of 45

12 Simplex algorithm 1.5 min s.t. b i a i apple init 12 of 45

13 Combinatorial dimension = p Ac5ve set or basis of 45

14 Combinatorial dimension = p Ac5ve set or basis of 45

15 Maximum consensus I of 45

16 Maximum consensus.4 apple min max b i2i i a i apple I of 45

17 An algorithm For all subsets B X of size (p+1); Solve minmax problem on B. If maximum residual of B is apple ; If the coverage of current largest; is greater than the Set I as the coverage of B. N p +1 = = B N! (p + 1)!(N p 1)! 1 (p + 1)! N(N 1)...(N p) O(N p+1 ) 17 of 45

18 Minmax problem of 45

19 Minmax problem of 45

20 Recursive minmax of 45

21 Recursive minmax of 45

22 Recursive minmax of 45

23 Recursive minmax of 45

24 It s a tree! f(b) Adjacent bases Level Level 1 Level 2 Level 3 Level 4 θ(b) 24 of 45

25 Tree depth or level f(b) Level- basis Level 1.9 Level Level 2.6 Level 3 Level θ(b) of 45

26 Tree depth or level f(b) Level-3 basis Level Level Level 2.6 Level 3 Level θ(b) of 45

27 Maximum consensus II min B l(b) s.t. f(b) apple. f(b) Level Level 1 Level 2 Level 3 Level 4 ε Feasible θ(b) 27 of 45

28 Breadth- first search (BFS) f(b) Level Level 1 Level 2 Level 3 Level 4 ε Feasible θ(b) 28 of 45

29 A* algorithm Basis expansion is priori5sed by e(b) =l(b)+h(b) l(b) : Level of basis B. h(b) : An es5mate of the number of steps remaining to feasibility. Hart et al., A formal basis for the heuris5c determina5on of minimum cost paths, IEEE Trans. on Systems Science and Cyberne5cs, 4(2):1 17, of 45

30 A* algorithm f(b) A basis B l(b) Level Level 1 h(b) Level 2 Level 3 Level 4 ε Feasible θ(b) 3 of 45

31 A* algorithm f(b) Level Level 1 Level 2 Level 3 Level 4 ε Feasible θ(b) 31 of 45

32 Heuris5c func5on O = {B 1 } of 45

33 Heuris5c func5on O = {B 1, B 2 } of 45

34 Heuris5c func5on O = {B 1, B 2, B 3 } of 45

35 Heuris5c func5on O = {B 1, B 2, B 3, B 4 } of 45

36 Heuris5c func5on O = {B 1, B 2, B 3, B 4, B 5 } of 45

37 Heuris5c func5on h(b) = {B 1, B 2, B 3, B 4, B 5 } = of 45

38 Defini&on (Admissibility): A heuris5c is admissible if it sa5sfies h(b) and h(b) apple h (B) where h (B) is the true remaining cost from to feasibility. B Theorem: A* algorithm is op5mal if h(b) is admissible. 38 of 45

39 Results A* result (global) vs RANSAC result D Points RANSAC fit A* fit RANSAC inliers Common inliers A* inliers of 45

40 Results 2 Runtime (s) RANSAC MaxFS Matousek BFS A* Number of outliers 4 of 45

41 Limita5on: Outlier ra5o apple 1 p h(b) = {B 1, B 2, B 3, B 4, B 5 } = of 45

42 Other residual func5ons? dinosaur sequence, and the resulting reconstruction. Triangula&on Reprojec5on error: e house sequence, and the resulting reconstruction. k(pi,1:2 xi Pi,3 ) k ri ( ) = Pi,3 by T.- J. Chin Homography fi<ng Transfer error: ri ( ) = k( 1:2 ui 3 )u i k 3 u i 42 of 45

43 Pseudoconvex residual α init 43 of 45

44 Combinatorial dimension = p+1 44 of 45

45 Thank you! A lvaro Parra Bustos received a BSc Eng. (26), a Computer Science Engineer degree (28) and a M.Sc. degree in Computer Science (211) from Universidad de Chile (SantiEX CLASS FILES, VOL. 6, NO. 1, JANUARY 27 ago, Chile). He is currently a PhD student within the Australian Centre for Visual Technologies (ACVT) in the University of Adelaide, Australia. ching and similarity, ACM TOG, vol. 25, pp , His main research interests include point cloud registration, 3D computer vision and optimisad B. Vemuri, A robust algorithm for point set registration methods in computer vision. mixture of Gaussians, in ICCV, 25. nko and X. Song, Point set registration: coherent point E TPAMI, vol. 32, no. 1, pp , 21. ia, A. Patterson, and K. Daniilidis, Fully automatic n of 3D point clouds, in CVPR, 26. d, N. Mitra, L. Guibas, and H. Pottmann, Robust global n, in Eurographics, 25. O. Enqvist, and F. Kahl, A polynomial-time bound ng and registration with outliers, in CVPR, 28. Enqvist, and F. Kahl, Optimal geometric under Tat- Jun Chin Pulak fitting Purkait Anders Eriksson ted l2-norm, Adelaide in CVPR, 213.Tat-Jun Chin Adelaide QUT received a B.Eng. in Mechatronics R. Hartley, The 3D 3D registration problem revisited, Engineering from Universiti Teknologi Malaysia 7. in 23 and subsequently in 27 a PhD in Com, K. Josephson, and F. Kahl, Optimal correspondences puter Systems Engineering from Monash Uniwise constraints, in ICCV, 29. versity, Victoria, Australia. He was a Research anu and M. Hebert, A spectral technique for correfellow at the Institute for Infocomm Research problems using pairwise constraints, in ICCV, 27. in Singapore Since 28 he is a by Tand.- J. Chin F. Kahl, M. Oskarsson, Branch-and-bound methlecturer at The University of Adelaide, Australia. Anders Eriksson receivd electrical engineering aa mathematics from LundF 2 and 28, respecg senior research associao Adelaide, Australia. Hiss clude optimization theorl ods applied to the fieldsa machine learning. ( U h Science, The Universit David SScience. uter Computer He (ARC)Adelaide College of Exper of International Journal on the editorial boards International Journal of or the Asian Conferenc is currently co-chair of Processing (ICIP213). 45 of 45