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Efficient Globally Op5mal Consensus Maximisa5on with Tree Search
Tat-‐Jun Chin School of Computer Science The University of Adelaide
by T.-‐J. Chin
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Maximum consensus
by T.-‐J. Chin
max
✓, I✓X|I|
subject to ri(✓) ✏ 8xi 2 I,
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Example 1: Line fiMng
by T.-‐J. Chin
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Example 2: Triangula5on
by T.-‐J. Chin
Fig. 5. The triangulation problem: Assuming that the maximum reprojection erroris less than some value �, the sought point X must lie in the intersection of a set ofcones. If � is set too small, then the cones do not have a common intersection (left). If� is set too large, then the cones intersect in a convex region in space, and the desiredsolution X must lie in this region (right). The optimal value of � lies between thesetwo extremes, and can be found by a binary search (bisection) testing successive valuesof �. For more details, refer to the text.
2. If we define a cost function
Cost1(X) = maxi
\(X�Oi,vi) ,
then the sublevel set S�(Cost1) is simply the intersection of the conesC�(Oi,vi), which is convex for all �. This by definition says that Cost1(X)is a quasi-convex function of X.
3. Finding the optimum
minX
Cost1(X) = minX
maxi
\(X�Oi,vi)
is accomplished by a binary search over possible values of �, where for eachvalue of � we solve an SOCP feasibility problem, (determine whether a set ofcones have a common intersection). Such a problem is known as a minimaxor L1 optimization problem.
Generally speaking, this procedure generalizes to arbitrary quasi-convex op-timization problems; they may be solved by binary search involving a convexfeasibility problem at each step. If we have a set of individual cost functionsfi(X), perhaps each associated with a single measurement, and each of themquasi-convex, then the maximum of these cost functions maxi fi(X) is also quasi-convex, as illustrated in Fig 4. In this case, the minimax problem of findingminX maxi fi(X) is solvable by binary search.
Reconstruction with known rotations. Another problem that may besolved by very similar means to the triangulation problem is that of Structureand Motion with known rotations, which is illustrated in Fig 6.
Figure 3. Two images from the dinosaur sequence, and the resulting reconstruction.
Figure 4. Two images from the house sequence, and the resulting reconstruction.
Figure 5. Two images from the church sequence, and the resulting reconstruction.
[15] Y. Seo, H. Lee, and S. W. Lee. Outlier removal by convexoptimization for l-infinity approaches. In PSIVT ’09: PacificRim Symposium on Advances in Image and Video Technol-ogy, 2009.
[16] K. Sim and R. Hartley. Removing outliers using the L1-norm. In Conf. Computer Vision and Pattern Recognition,pages 485–492, New York City, USA, 2006.
[17] H. Stewenius, C. Engels, and D. Nister. Recent develop-
ments on direct relative orientation. ISPRS Journal of Pho-togrammetry and Remote Sensing, 60:284–294, 2006.
[18] J. F. Sturm. Using SeDuMi 1.02, a Matlab toolbox for opti-mization over symmetric cones. Optimization Methods andSoftware, 11-12:625–653, 1999.
[19] J. Tropp. Just relax: convex programming methods for iden-tifying sparse signals in noise. Information Theory, IEEETransactions on, 52(3):1030–1051, March 2006.
Figure 3. Two images from the dinosaur sequence, and the resulting reconstruction.
Figure 4. Two images from the house sequence, and the resulting reconstruction.
Figure 5. Two images from the church sequence, and the resulting reconstruction.
[15] Y. Seo, H. Lee, and S. W. Lee. Outlier removal by convexoptimization for l-infinity approaches. In PSIVT ’09: PacificRim Symposium on Advances in Image and Video Technol-ogy, 2009.
[16] K. Sim and R. Hartley. Removing outliers using the L1-norm. In Conf. Computer Vision and Pattern Recognition,pages 485–492, New York City, USA, 2006.
[17] H. Stewenius, C. Engels, and D. Nister. Recent develop-
ments on direct relative orientation. ISPRS Journal of Pho-togrammetry and Remote Sensing, 60:284–294, 2006.
[18] J. F. Sturm. Using SeDuMi 1.02, a Matlab toolbox for opti-mization over symmetric cones. Optimization Methods andSoftware, 11-12:625–653, 1999.
[19] J. Tropp. Just relax: convex programming methods for iden-tifying sparse signals in noise. Information Theory, IEEETransactions on, 52(3):1030–1051, March 2006.
Olsson et al., Efficient op5miza5on for L-‐inQy problems using pseudoconvexity, ICCV 2007. Hartley and Kahl, Op5mal algorithms in mul5view geometry, ACCV 2007.
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Example 3: Homography fiMng
by T.-‐J. Chin Figure from hYp://sse.tongji.edu.cn/linzhang/CV14/Projects/panorama.htm.
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1
Running example: Linear regression
by T.-‐J. Chin
ri(✓) = |bi � ✓ai|
xi = [ai, bi]T
b(a) = ✓a
✏
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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1
RANSAC, minimal subset size = p
by T.-‐J. Chin
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Minmax problem • Minimise the maximum residual:
• Same as L-‐infinity minimisa5on1,2:
• A.k.a. Chebyshev approxima5on/regression.
by T.-‐J. Chin
min
✓max
i|bi � ✓ai|
min✓
���������
2
6664
|b1 � ✓a1||b2 � ✓a2|
...|bN � ✓aN |
3
7775
���������1
1Hartley and Schaffalitzky, L-‐inQy minimiza5on in geometric reconstruc5on problems, CVPR 2004. 2Kahl and Hartley, Mul5ple-‐view geometry under the L-‐inQy-‐norm, PAMI 2008.
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1
Minmax problem
by T.-‐J. Chin
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−1 −0.5 0 0.5 1 1.5 2 2.50
0.5
1
1.5
Minmax problem
by T.-‐J. Chin
|bi � ✓ai|
✓
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−1 −0.5 0 0.5 1 1.5 2 2.50
0.5
1
1.5
Minmax problem
by T.-‐J. Chin
max
i|bi � ✓ai|
Global minimum
✓✓⇤
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−1 −0.5 0 0.5 1 1.5 2 2.50
0.5
1
1.5
Simplex algorithm
by T.-‐J. Chin
✓
min✓
�
s.t. |bi � ✓ai| �
✓init
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1
Combinatorial dimension = p+1
by T.-‐J. Chin
Ac5ve set or “basis”
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−1 −0.5 0 0.5 1 1.5 2 2.50
0.5
1
1.5
Combinatorial dimension = p+1
by T.-‐J. Chin
✓✓⇤
Ac5ve set or “basis”
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Maximum consensus
by T.-‐J. Chin
I⇤
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0.25 0.3 0.35 0.4 0.450.2
0.25
0.3
0.35
0.4
Maximum consensus
by T.-‐J. Chin
I⇤min
✓max
i2I⇤|bi � ✓ai|
� ✏
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An algorithm
• For all subsets of size (p+1); • Solve minmax problem on . • If maximum residual of is ; • If the coverage of is greater than the current largest; • Set as the coverage of .
by T.-‐J. Chin
✏
BB
B
B
B ⇢ X
I⇤
✓N
p+ 1
◆=
N !
(p+ 1)!(N � p� 1)!
=1
(p+ 1)!N(N � 1) . . . (N � p) ⌘ O(Np+1)
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−1 −0.5 0 0.5 1 1.5 2 2.50
0.5
1
1.5
Minmax problem
by T.-‐J. Chin
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0.5 1 1.5 20
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Minmax problem
by T.-‐J. Chin
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0.5 1 1.5 20
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Recursive minmax
by T.-‐J. Chin
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0.5 1 1.5 20
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Recursive minmax
by T.-‐J. Chin
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0.5 1 1.5 20
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0.5
Recursive minmax
by T.-‐J. Chin
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0.5 1 1.5 20
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Recursive minmax
by T.-‐J. Chin
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It’s a tree!
by T.-‐J. Chin
Level 0
Level 1
Level 2
Level 3Level 4
Adjacent bases
f(B)
θ(B)
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Tree depth or level
by T.-‐J. Chin
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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1
Level 0
Level 1
Level 2
Level 3Level 4
Level-0 basisf(B)
θ(B)
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Tree depth or level
by T.-‐J. Chin
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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Level 0
Level 1
Level 2
Level 3Level 4
Level-3 basis
f(B)
θ(B)
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Maximum consensus II
by T.-‐J. Chin
minB
l(B) s.t. f(B) ✏.
Level 0
Level 1
Level 2
Level 3Level 4
f(B)
θ(B)
ε
Feasible
28 of 45
Breadth-‐first search (BFS)
by T.-‐J. Chin
Level 0
Level 1
Level 2
Level 3Level 4
f(B)
θ(B)
ε
Feasible
29 of 45
A* algorithm
• Basis expansion is priori5sed by
by T.-‐J. Chin
: Level of basis . : An es5mate of the number of steps remaining to feasibility.
e(B) = l(B) + h(B)
l(B)
h(B)
B
Hart et al., A formal basis for the heuris5c determina5on of minimum cost paths, IEEE Trans. on Systems Science and Cyberne5cs, 4(2):100–107, 1968.
30 of 45
A* algorithm
by T.-‐J. Chin
A basis Bl(B)
h(B)
Level 0
Level 1
Level 2
Level 3Level 4
f(B)
θ(B)
ε
Feasible
31 of 45
A* algorithm
by T.-‐J. Chin
Level 0
Level 1
Level 2
Level 3Level 4
f(B)
θ(B)
ε
Feasible
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Heuris5c func5on
by T.-‐J. Chin
O = {B1}
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Heuris5c func5on
by T.-‐J. Chin
O = {B1,B2}
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Heuris5c func5on
by T.-‐J. Chin
O = {B1,B2,B3}
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Heuris5c func5on
by T.-‐J. Chin
O = {B1,B2,B3,B4}
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Heuris5c func5on
by T.-‐J. Chin
O = {B1,B2,B3,B4,B5}
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Heuris5c func5on
by T.-‐J. Chin
h(B) = |{B1,B2,B3,B4,B5}| = 5
38 of 45 by T.-‐J. Chin
Defini&on (Admissibility): A heuris5c is admissible if it sa5sfies where is the true remaining cost from to feasibility.
h(B) � 0 and h(B) h⇤(B)
h⇤(B) B
Theorem: A* algorithm is op5mal if is admissible. h(B)
39 of 45
Results
by T.-‐J. Chin
0 0.2 0.4 0.6 0.8 1 1.2−0.6
−0.4
−0.2
0
0.2
0.4
A* result (global) vs RANSAC result
2D PointsRANSAC fitA* fitRANSAC inliersCommon inliersA* inliers
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Results
by T.-‐J. Chin
0 5 10 15 20 250
5
10
15
20
Number of outliers
Run
time
(s)
RANSACMaxFSMatousekBFSA*
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1
Limita5on: Outlier ra5o
by T.-‐J. Chin
h(B) = |{B1,B2,B3,B4,B5}| = 5
1
p+ 1
42 of 45
Figure 3. Two images from the dinosaur sequence, and the resulting reconstruction.
Figure 4. Two images from the house sequence, and the resulting reconstruction.
Figure 5. Two images from the church sequence, and the resulting reconstruction.
[15] Y. Seo, H. Lee, and S. W. Lee. Outlier removal by convexoptimization for l-infinity approaches. In PSIVT ’09: PacificRim Symposium on Advances in Image and Video Technol-ogy, 2009.
[16] K. Sim and R. Hartley. Removing outliers using the L1-norm. In Conf. Computer Vision and Pattern Recognition,pages 485–492, New York City, USA, 2006.
[17] H. Stewenius, C. Engels, and D. Nister. Recent develop-
ments on direct relative orientation. ISPRS Journal of Pho-togrammetry and Remote Sensing, 60:284–294, 2006.
[18] J. F. Sturm. Using SeDuMi 1.02, a Matlab toolbox for opti-mization over symmetric cones. Optimization Methods andSoftware, 11-12:625–653, 1999.
[19] J. Tropp. Just relax: convex programming methods for iden-tifying sparse signals in noise. Information Theory, IEEETransactions on, 52(3):1030–1051, March 2006.
Other residual func5ons?
Triangula&on
Reprojec5on error:
Homography fi<ng
Transfer error:
by T.-‐J. Chin
ri(✓) =k(Pi,1:2 � xiPi,3)✓k
Pi,3✓ri(✓) =
k(✓1:2 � u0i✓3)uik
✓3ui
43 of 45
Pseudoconvex residual
by T.-‐J. Chin
α
✓init ✓⇤
44 of 45
Combinatorial dimension = p+1
by T.-‐J. Chin
45 of 45
Thank you!
by T.-‐J. Chin
JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 14
shape matching and similarity,” ACM TOG, vol. 25, pp. 130–150,2006.
[21] B. Jian and B. Vemuri, “A robust algorithm for point set registra-tion using mixture of Gaussians,” in ICCV, 2005.
[22] A. Mynorenko and X. Song, “Point set registration: coherent pointdrift,” IEEE TPAMI, vol. 32, no. 1, pp. 2262–2275, 2010.
[23] A. Makadia, A. Patterson, and K. Daniilidis, “Fully automaticregistration of 3D point clouds,” in CVPR, 2006.
[24] N. Gelfand, N. Mitra, L. Guibas, and H. Pottmann, “Robust globalregistration,” in Eurographics, 2005.
[25] C. Olsson, O. Enqvist, and F. Kahl, “A polynomial-time boundfor natching and registration with outliers,” in CVPR, 2008.
[26] E. Ask, O. Enqvist, and F. Kahl, “Optimal geometric fitting underthe truncated l2-norm,” in CVPR, 2013.
[27] H. Li and R. Hartley, “The 3D–3D registration problem revisited,”in ICCV, 2007.
[28] O. Enqvist, K. Josephson, and F. Kahl, “Optimal correspondencesfrom pairwise constraints,” in ICCV, 2009.
[29] M. Leordeanu and M. Hebert, “A spectral technique for corre-spondence problems using pairwise constraints,” in ICCV, 2007.
[30] C. Olsson, F. Kahl, and M. Oskarsson, “Branch-and-bound meth-ods for Euclidean registration problems,” IEEE TPAMI, vol. 31,no. 5, pp. 783–794, 2009.
[31] J.-C. Bazin, H. Li, I. S. Kweon, C. Demonceaux, P. Vasseur, andK. Ikeuchi, “A branch and bound approach to correspondenceand grouping problem,” IEEE TPAMI, 2013.
[32] J. O’Rourke, Computational geometry in C. Cambridge UniversityPress, 1998.
[33] S. Rusinkiewicz and M. Levoy, “Efficient variants of the ICPalgorithm,” in Int’l Conf. on 3-D Digital Imaging and Modeling(3DIM), 2001.
[34] B. Curless and M. Levoy, “A volumetric method for buildingcomplex models from range images,” SIGGRAPH, 1996.
Alvaro Parra Bustos received a BSc Eng.(2006), a Computer Science Engineer degree(2008) and a M.Sc. degree in Computer Sci-ence (2011) from Universidad de Chile (Santi-ago, Chile). He is currently a PhD student withinthe Australian Centre for Visual Technologies(ACVT) in the University of Adelaide, Australia.His main research interests include point cloudregistration, 3D computer vision and optimisa-tion methods in computer vision.
Tat-Jun Chin received a B.Eng. in MechatronicsEngineering from Universiti Teknologi Malaysia
in 2003 and subsequently in 2007 a PhD in Com-puter Systems Engineering from Monash Uni-versity, Victoria, Australia. He was a ResearchFellow at the Institute for Infocomm Researchin Singapore 2007- 2008. Since 2008 he is aLecturer at The University of Adelaide, Australia.His research interests include robust estimation,computational geometry, and statistical learningmethods in Computer Vision.
Anders Eriksson received the M.Sc. degree inelectrical engineering and the PhD degree inmathematics from Lund University, Sweden, in2000 and 2008, respectively. Currently, he is asenior research associate at the University ofAdelaide, Australia. His research interests in-clude optimization theory and numerical meth-ods applied to the fields of computer vision andmachine learning.
Hongdong Li received the M.Sc. and PhD de-grees in information and electronics engineer-ing in 1996 and 2000, respectively, both fromZhejiang University, China. Since 2004 he hasjoined the Australian National University (ANU)as a research fellow and is also seconded to Na-tional ICT Australia (NICTA) Canberra Labs. Heis currently a faculty member with the ResearchSchool of Engineering, ANU. He is the recipientof the CVPR’12 Best Paper Award. His currentresearch interests include geometric computer
vision, pattern recognition, computer graphics, and combinatorial opti-mization.
David Suter received a B.Sc. degree inapplied mathematics and physics (TheFlinders University of South Australia 1977), aGrad. Dip. Comp. (Royal Melbourne Instituteof Technology 1984)), and a Ph.D. in computerscience (La Trobe University, 1991). He was aLecturer at La Trobe from 1988 to 1991; anda Senior Lecturer (1992), Associate Professor(2001), and Professor (2006-2008) at MonashUniversity, Melbourne, Australia. Since 2008 hehas been a professor in the school of Computer
Science, The University of Adelaide. He is head of the School ofComputer Science. He served on the Australian Research Council(ARC) College of Experts from 2008-2010. He is on the editorial boardof International Journal of Computer Vision. He has previously servedon the editorial boards of Machine Vision and Applications and theInternational Journal of Image and Graphics. He was General co-Chairor the Asian Conference on Computer Vision (Melbourne 2002) andis currently co-Chair of the IEEE International Conference on ImageProcessing (ICIP2013).
JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 14
shape matching and similarity,” ACM TOG, vol. 25, pp. 130–150,2006.
[21] B. Jian and B. Vemuri, “A robust algorithm for point set registra-tion using mixture of Gaussians,” in ICCV, 2005.
[22] A. Mynorenko and X. Song, “Point set registration: coherent pointdrift,” IEEE TPAMI, vol. 32, no. 1, pp. 2262–2275, 2010.
[23] A. Makadia, A. Patterson, and K. Daniilidis, “Fully automaticregistration of 3D point clouds,” in CVPR, 2006.
[24] N. Gelfand, N. Mitra, L. Guibas, and H. Pottmann, “Robust globalregistration,” in Eurographics, 2005.
[25] C. Olsson, O. Enqvist, and F. Kahl, “A polynomial-time boundfor natching and registration with outliers,” in CVPR, 2008.
[26] E. Ask, O. Enqvist, and F. Kahl, “Optimal geometric fitting underthe truncated l2-norm,” in CVPR, 2013.
[27] H. Li and R. Hartley, “The 3D–3D registration problem revisited,”in ICCV, 2007.
[28] O. Enqvist, K. Josephson, and F. Kahl, “Optimal correspondencesfrom pairwise constraints,” in ICCV, 2009.
[29] M. Leordeanu and M. Hebert, “A spectral technique for corre-spondence problems using pairwise constraints,” in ICCV, 2007.
[30] C. Olsson, F. Kahl, and M. Oskarsson, “Branch-and-bound meth-ods for Euclidean registration problems,” IEEE TPAMI, vol. 31,no. 5, pp. 783–794, 2009.
[31] J.-C. Bazin, H. Li, I. S. Kweon, C. Demonceaux, P. Vasseur, andK. Ikeuchi, “A branch and bound approach to correspondenceand grouping problem,” IEEE TPAMI, 2013.
[32] J. O’Rourke, Computational geometry in C. Cambridge UniversityPress, 1998.
[33] S. Rusinkiewicz and M. Levoy, “Efficient variants of the ICPalgorithm,” in Int’l Conf. on 3-D Digital Imaging and Modeling(3DIM), 2001.
[34] B. Curless and M. Levoy, “A volumetric method for buildingcomplex models from range images,” SIGGRAPH, 1996.
Alvaro Parra Bustos received a BSc Eng.(2006), a Computer Science Engineer degree(2008) and a M.Sc. degree in Computer Sci-ence (2011) from Universidad de Chile (Santi-ago, Chile). He is currently a PhD student withinthe Australian Centre for Visual Technologies(ACVT) in the University of Adelaide, Australia.His main research interests include point cloudregistration, 3D computer vision and optimisa-tion methods in computer vision.
Tat-Jun Chin received a B.Eng. in MechatronicsEngineering from Universiti Teknologi Malaysia
in 2003 and subsequently in 2007 a PhD in Com-puter Systems Engineering from Monash Uni-versity, Victoria, Australia. He was a ResearchFellow at the Institute for Infocomm Researchin Singapore 2007- 2008. Since 2008 he is aLecturer at The University of Adelaide, Australia.His research interests include robust estimation,computational geometry, and statistical learningmethods in Computer Vision.
Anders Eriksson received the M.Sc. degree inelectrical engineering and the PhD degree inmathematics from Lund University, Sweden, in2000 and 2008, respectively. Currently, he is asenior research associate at the University ofAdelaide, Australia. His research interests in-clude optimization theory and numerical meth-ods applied to the fields of computer vision andmachine learning.
Hongdong Li received the M.Sc. and PhD de-grees in information and electronics engineer-ing in 1996 and 2000, respectively, both fromZhejiang University, China. Since 2004 he hasjoined the Australian National University (ANU)as a research fellow and is also seconded to Na-tional ICT Australia (NICTA) Canberra Labs. Heis currently a faculty member with the ResearchSchool of Engineering, ANU. He is the recipientof the CVPR’12 Best Paper Award. His currentresearch interests include geometric computer
vision, pattern recognition, computer graphics, and combinatorial opti-mization.
David Suter received a B.Sc. degree inapplied mathematics and physics (TheFlinders University of South Australia 1977), aGrad. Dip. Comp. (Royal Melbourne Instituteof Technology 1984)), and a Ph.D. in computerscience (La Trobe University, 1991). He was aLecturer at La Trobe from 1988 to 1991; anda Senior Lecturer (1992), Associate Professor(2001), and Professor (2006-2008) at MonashUniversity, Melbourne, Australia. Since 2008 hehas been a professor in the school of Computer
Science, The University of Adelaide. He is head of the School ofComputer Science. He served on the Australian Research Council(ARC) College of Experts from 2008-2010. He is on the editorial boardof International Journal of Computer Vision. He has previously servedon the editorial boards of Machine Vision and Applications and theInternational Journal of Image and Graphics. He was General co-Chairor the Asian Conference on Computer Vision (Melbourne 2002) andis currently co-Chair of the IEEE International Conference on ImageProcessing (ICIP2013).
Pulak Purkait Adelaide
Anders Eriksson QUT
David Suter Adelaide
Tat-‐Jun Chin Adelaide
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