Optimisation in Multiple View Geometry: The L-infinity Way (Part...
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Optimisation in Multiple View Geometry:The L-infinity Way
(Part 2)
Tat-Jun Chin
The University of Adelaide
CVPR 2018 Tutorial
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Table of Contents
1 Recap: Quasiconvexity
2 LP-type problems
3 Approximation
4 Robust estimation
5 References
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Table of Contents
1 Recap: Quasiconvexity
2 LP-type problems
3 Approximation
4 Robust estimation
5 References
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Recap: Quasiconvexity
r(x) = π − θ(x)x ∈ upper halfplane
Animation of sublevel sets.
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Recap: QuasiconvexityLet {ri}Ni=1 be a set of quasiconvex functions, where each
ri : D 7→ R≥0
and D is a convex domain.Then, the point-wise maximum function
Q(x) = maxi
ri (x) =
∥∥∥∥∥∥∥ r1(x)
...rN(x)
∥∥∥∥∥∥∥∞
is also quasiconvex for x ∈ D.48 2. APPROXIMATE ALGORITHMS
-2 -1 0 1 2x
0
0.5
1
1.5
r(x)
s
(a)
-2 -1 0 1 2x
0
2
4
6
8
10
12
r(x)
s
(b)
Figure 2.16: (a) A quasiconvex function r(x) with the real line R as its domain. The s-sublevel
set Lrs is plotted as a red line segment on the x-axis. For any s � 0, Lr
s is a convex set, i.e., a
contiguous line segment. Note that this function is not convex. (b) A non-quasiconvex function
for comparison; in this function, not all Lrs for s � 0 are convex.
2.5 LP-TYPE PROBLEMS
This slightly lengthier section explores consensus maximisation under the framework of
LP-type problems. The motivation for including this relatively abstract topic is two-fold:
• LP-type problems are a generalisation of the Chebyshev approximation problem. As
we will see later in Section 2.5.3, the LP-type framework allows further theoretical
analysis of the outlier removal technique in Section 2.4.2.
• LP-type problems provide a unifying framework and convenient terminology to discuss
a class of “tractable” exact algorithms in Chapter 3.
We begin by defining LP-type problems and describing their common properties.
2.5.1 DEFINITION AND PROPERTIES
LP-type problems were first defined by Matousek et al. [1996] as a generalisation of linear
programming problems. In abstract terms, an LP-type problem (S, f) consists of a set of
constraints S, each involving d variables ✓, and an objective function f that maps subsets
of S to R+ [ {0}. Further, f satisfies the following properties:
Property 2.10 (Monotonicity) For every two sets P ✓ Q ✓ S, the inequalities f(P) f(Q) f(S) can be established.
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Quasiconvex programming
Minimising the point-wise maximum of a set of quasiconvex functions{ri}Ni=1 defined over a convex domain D, i.e.,
minx∈D
maxi
ri (x)
D. Eppstein. Quasiconvex programming. Combinatorial and Computational Geometry 25 (2005).
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Example: Mesh smoothing
Function ri (x) = π − θi (x)decreases with θi (x).
Solving
minx∈D
maxi
ri (x)
finds mesh vertex x that is asclose as possible to all sides.
Solution can only lie in D (theconvex region shaded in blue).
Amenta et al. Optimal point placement for mesh smoothing. Journal of Algorithms 30.2 (1999), pp. 302–322.
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Example: Multi-view triangulation (L-infinity)
Given camera matrix P and image point u,
r(x) =
∥∥∥∥∥P(1:2)x
P(3)x− u
∥∥∥∥∥2
=
∥∥[P(1:2) − uP(3)]x∥∥
2
P(3)x
gives the reprojection error of 3D point x ∈ R3. If P(3)x ≥ 0, then xlies in front of the camera.
Given N 2D points {ui}Ni=1 fromN views {Pi}Ni=1, solving
minx∈D
maxi
ri (x)
finds x that minimises thelargest reprojection error. Here
D ={x ∈ R3 | P(3)
i x ≥ 0}. Figure from Kahl and Hartley, ECCV’06 Tutorial.
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Example: Minimum enclosing circle (MEC)
Given a set of points {pi}Ni=1 on theplane, find the smallest circle thatencloses all the points:
minx∈R2
maxi‖x− pi‖2
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Example: Chebyshev approximation
Given a set of points {(ai , bi )}Ni=1 in (d + 1) dimensions, find thehyperplane x ∈ Rd that minimises the largest residual
minx∈Rd
maxi|aTi x− bi |
Special case: Minimum enclosing slab (MES)Setting ai = [pi 1]T and bi = qi ,where (pi , qi ) are points on theplane, performing Chebyshevapproximation amounts to findingthe thinnest slab that encloses allthe points.
Width of slab is measured vertically(i.e., along the q-axis).
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Table of Contents
1 Recap: Quasiconvexity
2 LP-type problems
3 Approximation
4 Robust estimation
5 References
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Linear programming (LP)-type problems
An LP-type problem consists ofI A set of “constraints” S;I An objective function f that measures the cost of any subset of S.
The goal is to compute f (S).
Two main properties:
Monotonicity
For any two subsets A and B of S, if A ⊂ B then f (A) ≤ f (B).
Locality
For any A ⊆ S, p ∈ S and q ∈ S, if f (A) = f (A ∪ {p}) = f (A ∪ {q}),then f (A) = f (A ∪ {p, q}).
D. Eppstein. Quasiconvex programming. Combinatorial and Computational Geometry 25 (2005).
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Examples: MEC and MES
Minimum enclosing circle (MEC):
S = {pi}Ni=1 and f (S) gives the radius of MEC(S).
Monotonicity
If A ⊂ B ⊆ S, then MEC(A) cannot be larger than MEC(B).
Locality
If f (A) = f (A ∪ {pi}), then point pi lies in MEC(A).If f (A) = f (A ∪ {qi}), then point qi also lies in MEC(A).Clearly {pi ,qi} also lie in MEC(A), hence f (A) = f (A ∪ {pi ,qi}).
Minimum enclosing slab (MES):
S = {(ai , bi )}Ni=1 and f (S) gives the width of MES(S).
Replace MEC with MES in the arguments above.
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Basis
A basis of an LP-type problem (S, f ) is a subset B ⊆ S such thatf (A) < f (B) for every A ⊂ B.
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Examples: MEC and MES
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Support set
The support set K of an LP-type problem (S, f ) is a basis of the problemsuch that f (S) = f (K).
Further properties:
Solving an LP-type problem amounts to finding its support set.
The cost/residual of all items in K are equal w.r.t. the estimate.
Items not in the support set (the subset S \ K) can be ignored.
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Example: MEC
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Example: MES
Points in the support set K have equal (maximum) distance to theChebyshev fit.
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Combinatorial dimension
The upper bound on the size of a basis in an LP-type problem (S, f ).
Any quasiconvex program forms an LP-type problem of combinatorialdimension is 2d + 1.
If each component ri is continuously shrinking, then the combinatorialdimension is d + 1.
Sim and Harley showed that functions of the form
r(x) =‖Ax + b‖2
pTx + q
are continuously shrinking for pTx + q > 0. Call problems of thisform quasiconvex geometric problems.
Amenta et al. Optimal point placement for mesh smoothing. Journal of Algorithms 30.2 (1999), pp. 302–322.
K. Sim and R. Hartley. Removing outliers using the l-infinity norm. CVPR 2006.
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Example: L-infinity triangulationS = {(Pi ,ui )}Ni=1 is a set of N camera matrices and image points, and
f (S) = minx∈D
maxi∈S
∥∥∥[P(1:2)i − uiP
(3)i
]x∥∥∥
2
P(3)i x
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Solving LP-type problems
function solve lptype(S,f ,C):
1: if S = C then2: return C.3: end if4: c ← A random item from S \ C.5: B ← solve lptype(S \ {c}, f , C).6: if f (B) 6= f (B ∪ {c}) then7: B ← basis(B ∪ {c}).8: B ← solve lptype(S,f ,B).9: end if
10: return B
// Core 1: violation test// Core 2: basis updating
Matousek et al. A subexponential bound for linear programming. Algorithmica 16 (1996), pp. 498–516.
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Solving LP-type problems
function solve lptype(S,f ,C):
1: if S = C then2: return C.3: end if4: c ← A random item from S \ C.5: B ← solve lptype(S \ {c}, f , C).6: if f (B) 6= f (B ∪ {c}) then7: B ← basis(B ∪ {c}).8: B ← solve lptype(S,f ,B).9: end if
10: return B
H. Li. Efficient reduction of L-infinity geometry problems. CVPR 2009.
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Table of Contents
1 Recap: Quasiconvexity
2 LP-type problems
3 Approximation
4 Robust estimation
5 References
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Another algorithm
Keep expanding C and updating xC using the point with the largestresidual until all points are covered.
function solve lptype mv(S,f ):
1: C ← Randomly select one item from S.2: while true do3: xC ← arg minx∈D maxi∈C ri (x). // Solve on current coreset C.4: i∗ ← arg maxi∈S ri (xC). // Find most violating point.5: if ri∗(xC) ≤ f (C) then6: Break. // Most violating point already covered; done.7: end if8: C ← C ∪ {i∗}. // Insert one point.9: end while
10: return xC
Clarkson. Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm. SODA 2008.
Seo and Hartley. A Fast Method to Minimize L Error Norm for Geometric Vision Problems. ICCV 2007.
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Another algorithm
Keep expanding C and updating xC using the point with the largestresidual until all points are covered.
function solve lptype mv(S,f ):
1: C ← Randomly select one item from S.2: while true do3: xC ← arg minx∈D maxi∈C ri (x). // Warm start using prev. xC .4: i∗ ← arg maxi∈S ri (xC). // Find most violating point.5: if ri∗(xC) ≤ f (C) then6: Break. // Most violating point already covered; done.7: end if8: C ← C ∪ {i∗}. // Insert one point.9: end while
10: return xC
Clarkson. Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm. SODA 2008.
Seo and Hartley. A Fast Method to Minimize L Error Norm for Geometric Vision Problems. ICCV 2007.
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Example: MEC
Input data S: Initial C:
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Example: MEC
Found most violating point: Insert into C and update MEC:
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Example: MEC
Found most violating point: Insert into C and update MEC:
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Example: MEC
Found most violating point: Insert into C and update MEC:
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Example: MEC
Found most violating point: Insert into C and update MEC:
No more violating points—done.
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Coresets
Red circle = MEC on current C.Blue circle = MEC on S.
errorC := maxi∈S ri (xC).
errorS := f (S) = maxi∈S ri (xS).
By construction errorC ≥ errorS , but we can also guarantee that
errorCerrorS
≤ (1 + ε), where ε =2
|S|
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Coresets
Blue = Theoretical boundRed = Actual error ratios
Q. Zhang and T.-J. Chin. Coresets for triangulation. IEEE TPAMI (2017). doi: 10.1109/TPAMI.2017.2750672.
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Coresets
Q. Zhang and T.-J. Chin. Coresets for triangulation. IEEE TPAMI (2017). doi: 10.1109/TPAMI.2017.2750672.
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Coresets
Q. Zhang and T.-J. Chin. Coresets for triangulation. IEEE TPAMI (2017). doi: 10.1109/TPAMI.2017.2750672.
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Table of Contents
1 Recap: Quasiconvexity
2 LP-type problems
3 Approximation
4 Robust estimation
5 References
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Sensitivity to outliers
Minimising the largest residual amounts to fitting on the outliers.Hence, L-infinity estimation is not robust.
Nonetheless, the L-infinity approach provides a good framework toanalyse and develop robust estimation algorithms.
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Robust estimation
Let (S, f ) be an LP-type problem with quasiconvex geometricresiduals {ri}Ni=1, where some of the residuals correspond to outliers.
Solve for x by ignoring residuals that are greater than ε:
maxx∈D, I⊆S
|I|
s.t. ri (x) ≤ ε, i ∈ I.(Consensus maximisation)
Let I∗ be the maximum consensus set. Observe that
minx∈D
maxi∈I∗
ri (x)
is the LP-type problem (I∗, f ), and by construction
f (I∗) ≤ ε.
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Simple algorithm (enumeration)
Let K be the support set of (I∗, f ). Since I∗ ⊆ S, K is also a basisof (S, f ). =⇒ Find K by enumerating all bases of (S, f ).
Require: LP-type problem (S, f ), inlier threshold ε.1: ψ ← 0, K ← ∅.2: for all (d + 1)-subsets B of S do3: if f (B) ≤ ε then4: if coverage(B)> ψ then5: ψ ← coverage(B). // Needs a bit of explanation.6: K ← B.7: end if8: end if9: end for
10: return K.
Number of iterations =( Nd+1
). Including checking coverage, runtime
is O(Nd+2T (d)), where T (d) is time to solve f (B).
Olsson et al. A polynomial-time bound for matching and registration with outliers. CVPR 2008.
Enqvist et al. Robust fitting for multiple view geometry. ECCV 2012.
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Computational hardness
NP-hardness
Consensus maximisation is NP-hard.
Proof: Turing reduction from Least Median Squares.=⇒ There are no polynomial time algorithms.
XP (slice-wise polynomial)
Consensus maximisation is XP in the dimension d .
Proof: Use enumeration method to get runtime of O(Nd+2T (d)).=⇒ For a fixed d , can get polynomial time in N.
W[1]-hardness
Consensus maximisation is W[1]-hard in the dimension d .
Proof: FPT reduction from K-clique.=⇒ There are no algorithms that are faster than O(N f (d)).
Chin et al. Robust Fitting in Computer Vision: Easy or Hard? CoRR abs/1802.06464 (2018).
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The model is fitted to the outliers...
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Recursive support set removal
For any instance (S, f ), if f (S) > ε, then there are outliers.
Since f (K) = f (S) > ε, where K is the support set, it is reasonable tosuspect that K contains outliers.
Idea: Recursively remove support sets of S until f (S) ≤ ε.Require: LP-type problem (S, f ), inlier threshold ε.
1: while true do2: K ← Support set of S.3: if f (K) > ε then4: S ← S \ K.5: else6: Break.7: end if8: end while9: return K.
Sim and Hartley. Removing Outliers Using The L-infty Norm. CVPR 2006.
Olsson et al. Outlier removal using duality. CVPR 2010.
Yu et al. An adversarial optimization approach to efficient outlier removal. ICCV 2011.
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Recursive support set removal
Chin and Suter. The maximum consensus problem: recent algorithmic advances. Morgan & Claypool Publishers, Feb. 2017.
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Recursive support set removal
Chin and Suter. The maximum consensus problem: recent algorithmic advances. Morgan & Claypool Publishers, Feb. 2017.
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Recursive support set removal
Chin and Suter. The maximum consensus problem: recent algorithmic advances. Morgan & Claypool Publishers, Feb. 2017.
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Recursive support set removal (bad case)
Chin and Suter. The maximum consensus problem: recent algorithmic advances. Morgan & Claypool Publishers, Feb. 2017.
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“True” inliers and outliers
Call items in I∗ the “true” inliers, and items in O∗ = S \ I∗ the“true” outliers.
Existence of true outliers
Let C be a subset of S. If
f (C) > ε,
then C contains at least one item from O∗.The LP-type properties allow us to prove the above easily: iff (C) > ε, then by monotonicity
f (I∗ ∪ C) ≥ f (C) > ε. (1)
If C is a subset of I∗, then by locality
f (I∗ ∪ C) = f (I∗) ≤ ε (2)
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“True” inliers and outliers
Recursively removing support sets is sort of justified, but it alsoremoves true inliers in each iteration.
For combinatorial dimension d + 1, in the worst case, d true inliersare removed in each iteration.
To avoid losing too many inliers, the inlier proportion should begreater than
d
d + 1.
=⇒ Method is not advisable for high-dimensional or high-outlierrate problems.
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Running example: Robust subspace fitting (1D)
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In parameter space
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In parameter space
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Bases for LP-type problem
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Bases for LP-type problem (closeup)
Every basis can be reached from the support set via a directed path.
Matousek. On geometric optimization with few violated constraints. Discrete comput. geom. 14.1 (1995), pp. 365–384.
H. Li. A practical algorithm for l-infinity triangulation with outliers. CVPR 2007.
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Tree structure
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Level of basis
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Consensus maximisation
Chin et al. Efficient globally optimal consensus maximisation with tree search. CVPR 2015.
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Tree search
Breadth first search (BFS) and A* search.
Chin et al. Efficient globally optimal consensus maximisation with tree search. CVPR 2015.
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A* search
A* search prioritises the bases to expand by
eval(B) = level(B) + h(B),
where h is a heuristic function.
h estimates (but cannot overestimate) number of levels remaining.
Performing recursive support set removal with B as the initial supportset, and counting the number of removals, gives a valid heuristic.
Chin et al. Efficient globally optimal consensus maximisation with tree search. CVPR 2015.
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Fixed parameter tractability
Fixed parameter tractability
Consensus maximisation is FPT in the dimension d and number of outliers|O∗|.
Proof: Use BFS to search the basis tree to get runtime ofO((d + 1)|O
∗|L(N, d)), where L(N, d) is time to solve LP-type problem.
Chin et al. Robust Fitting in Computer Vision: Easy or Hard? CoRR abs/1802.06464 (2018).
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Monte Carlo Tree Search
Use randomisation to explore the tree quickly.
Leads to asymmetric treegrowth (tends to exploremore promising branches).
Browne et al. A survey of Monte Carlo tree search methods. IEEE Trans. Computational Intelligence and AI in Games, 2012.
Le et al. RATSAC - Random Tree Sampling for Maximum Consensus Estimation. DICTA 2017.
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Approximability
APX-hardness
Consensus maximisation is APX-hard.
Proof: L-reduction from MAX-2SAT.=⇒ There are no polynomial time algorithms that can solve consensus
maximisation up to (1− δ)|I∗| for any pre-determined δ.
Chin et al. Robust Fitting in Computer Vision: Easy or Hard? CoRR abs/1802.06464 (2018).
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So what can be done?
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So what can be done?
RANSAC and variants (no guarantees, not necessarily efficient).
Use M-estimators and IRLS.
Deterministic refinement methods:I Penalty formulation + bilinear programming (Le et al.).I Iteratively reweighted L1 (Purkait et al.).
Le et al. An exact penalty method for locally convergent maximum consensus. CVPR 2017.
Purkait et al. Maximum consensus parameter estimation by reweighted L1 methods. EMMCVPR 2017.
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Table of Contents
1 Recap: Quasiconvexity
2 LP-type problems
3 Approximation
4 Robust estimation
5 References
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References I
N. Amenta, M. Bern, and D. Eppstein. “Optimal point placement for mesh smoothing”. In: Journal of Algorithms
30.2 (1999), pp. 302–322.
C. B. Browne et al. “A survey of Monte Carlo tree search methods”. In: IEEE Trans. Computational Intelligence and
AI in Games 4.1 (Mar. 2012), pp. 1–43.
T.-J. Chin, Z. Cai, and F. Neumann. “Robust Fitting in Computer Vision: Easy or Hard?” In: CoRR abs/1802.06464
(2018).
T.-J. Chin and D. Suter. The maximum consensus problem: recent algorithmic advances. Ed. by G. Medioni and
S. Dickinson. Synthesis Lectures on Computer Vision. Morgan & Claypool Publishers, San Rafael, CA, U.S.A., Feb. 2017.
T.-J. Chin et al. “Efficient globally optimal consensus maximisation with tree search”. In: CVPR. 2015.
K. L Clarkson. “Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm”. In: SODA. 2008.
K. L. Clarkson. “Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm”. In: ACM Trans. Algorithms
6.4 (2010), 63:1–63:30.
P.-A. Coquelin and R. Munos. “Bandit algorithms for tree search”. In: UAI. 2007.
O. Enqvist et al. “Robust fitting for multiple view geometry”. In: European Conference on Computer Vision (ECCV).
2012.
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References IID. Eppstein. “Quasiconvex programming”. In: Combinatorial and Computational Geometry 25 (2005).
H. Le, T.-J. Chin, and D. Suter. “An exact penalty method for locally convergent maximum consensus”. In: CVPR.
2017.
H. Le, T.-J. Chin, and D. Suter. “RATSAC - Random Tree Sampling for Maximum Consensus Estimation”. In:
DICTA. 2017.
H. Li. “A practical algorithm for l∞ triangulation with outliers”. In: CVPR. 2007.
H. Li. “Efficient reduction of L-infinity geometry problems”. In: CVPR. 2009.
J. Matousek. “On geometric optimization with few violated constraints”. In: Discrete comput. geom. 14.1 (1995),
pp. 365–384.
J. Matousek, M. Sharir, and E. Welzl. “A subexponential bound for linear programming”. In: Algorithmica 16 (1996),
pp. 498–516.
C. Olsson, O. Enqvist, and F. Kahl. “A polynomial-time bound for matching and registration with outliers”. In: CVPR.
2008.
C. Olsson, A. Eriksson, and R. Hartley. “Outlier removal using duality”. In: CVPR. 2010.
P. Purkait, C. Zach, and A. Eriksson. “Maximum consensus parameter estimation by reweighted L1 methods”. In:
EMMCVPR. 2017.
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References III
Y. Seo and R. I. Hartley. “A Fast Method to Minimize L Error Norm for Geometric Vision Problems”. In: ICCV. 2007.
K. Sim and R. Hartley. “Removing outliers using the l∞ norm”. In: CVPR. 2006.
J. Yu et al. “An adversarial optimization approach to efficient outlier removal”. In: ICCV. 2011.
Q. Zhang and T.-J. Chin. “Coresets for triangulation”. In: IEEE TPAMI (2017). doi: 10.1109/TPAMI.2017.2750672.
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