Globally Optimal Consensus Set Maximization
Through Rotation Search
Jean-Charles Bazin(1) , Yongduek Seo(2) and Marc Pollefeys(1)
(1): Institute for Visual Computing, ETHZ, Switzerland
(2): Department of Media Technology, Sogang University, South Korea
Abstract. A popular approach to detect outliers in a data set is to
find the largest consensus set, that is to say maximizing the number
of inliers and estimating the underlying model. RANSAC is the most
widely used method for this aim but is non-deterministic and does not
guarantee to return the optimal solution. In this paper, we consider a
rotation model and we present a new approach that performs consensus
set maximization in a mathematically guaranteed globally optimal way.
We solve the problem by a branch-and-bound framework associated with
a rotation space search. Our mathematical formulation can be applied
for various computer vision tasks such as panoramic image stitching, 3D
registration with a rotating range sensor and line clustering and vanishing
point estimation. Experimental results with synthetic and real data sets
have successfully confirmed the validity of our approach.
1
Introduction
Outlier removal is a key issue in computer vision [1]. The main idea to detect the
outliers is to find the largest consensus set, that is to say maximizing the number
of inliers and estimating the underlying model [1–4]. In this paper, we consider
rotation model, which is important in several computer vision applications such
as panoramic image stitching [5], line clustering and vanishing point estimation
[6] and 3D registration with a rotating range sensor. RANSAC [2] is the most
widely used and well-known method for the task of outlier removal. It starts by
randomly selecting a minimal number of data to hypothesize a model and then
counts the number of inliers verifying the hypothesized model up to a residual
threshold [1]. After a certain number of iterations, it returns the model maximizing the number of inliers and the inlier set. Several variants of RANSAC have
been proposed such as MLESAC, LO-RANSAC, preemptive-RANSAC, among
many others [7–10]. While great results can be obtained, RANSAC and its variants are not guaranteed to fulfill the exact goal of consensus set maximization:
they might not return the optimal solution in terms of number of inliers. Furthermore, their randomized nature make them somewhat unpredictable: different
runs might return different results.
Some works tried to overcome these issues by using algorithms based on
mathematical optimization. For example, Li [3] proposed a reformulation method
based on convex and concave envelops [11] with a branch-and-bound technique.
2
Jean-Charles Bazin, Yongduek Seo and Marc Pollefeys
However this technique can deal with only linear algebraic cost and linear constraints, i.e. it cannot handle rotation model and angular distance. Olsson et
al. proposed a post-validation step to check whether a candidate solution is the
optimal one [12]. However this approach does not give an upper bound on the
number of trials needed (or the number of candidate solutions to compute and
test) to reach the optimal solution.
The remaining of this paper is organized as follows. In Section 2, we introduce
our mathematical formulation of consensus set maximization with a rotation
model. In Section 3, we present our optimization procedure based on branchand-bound. Finally, experimental results are presented in Section 4.
2
Mathematical formulation
We note ui and vi the ith corresponding measurements (also called a match)
of two sets related by a rotation R. Let also note ∠(u, v) the angle formed by
the vectors u and v, and in the range [0, π]. In noise-free and outlier-free cases,
Rui = vi ∀i and ∠(Rui , vi ) = 0∀i. When noise and outliers exist, these relations
are not verified anymore. Similarly to the popular “residual tolerance method”
[2], we define a match (ui , vi ) an inlier if the distance is lower than a residual
tolerance δ, i.e. ∠(Rui , vi ) ≤ δ. Otherwise the match is considered an outlier.
Let M represent the set of matches: M = {(ui , vi ), i = 1 . . . N } where N is
the number of matches. The set M is partitioned into an inlier-set MI ⊆ M
containing the inlier matches and an outlier-set MO ⊆ S with MO = M −
MI . The cardinality of MI corresponds to the number of inlier matches. The
consensus set maximization with a rotation model is formulated as:
max
MI ,R
card(MI )
s.t. ∠(Rui , vi ) ≤ δ, ∀i ∈ MI ⊆ M
R ∈ SO(3)
(1a)
(1b)
(1c)
To solve this consensus set maximization problem, RANSAC is definitively the
most popular method, but as explained in Section 1, it is non-deterministic and
might not reach the optimal solution. In a numerical optimization point of view,
it is not simple to process the above formulation. Following the approach of Li
[3], let us introduce a binary variable yi for each match such that yi = 1 if the
ith match is an inlier and yi = 0 otherwise (i.e. outlier). We can now reformulate
system (1) in the following equivalent way:
N
∑
max
yi
(2a)
y,R
s.t.
i
yi ∠(Rui , vi ) ≤ yi δ, ∀i = 1 . . . N
(2b)
yi ∈ {0, 1}, ∀i = 1 . . . N
R ∈ SO(3)
(2c)
(2d)
When yi = 1, the inlier/outlier constraint at Eq (2b) corresponds to Eq (1b).
When yi = 0, the constraint is simply eliminated (i.e. always true since δ > 0).
Globally Optimal Consensus Set Maximization Through Rotation Search
3
System (2) still constitutes a challenging optimization problem. First, it contains two families of unknowns: binary variables (y) and continuous geometric
model parameters (R). Second, it is non-linear because of the distance and the
unknowns (y and R) are multiplied to each other at Eq (2b). Third, it is nonconvex. Finally, fulfilling the rotation constraint at Eq (2d) requires specific care.
3
Rotation Search Algorithm
In this section, we explain our algorithm to solve system (2), and thus system (1),
in a globally optimal way. It is based on branch-and-bound framework and a rotation space search. We start by a brief introduction to branch-and-bound, then
explain how to compute the bounds and finally present our search procedure.
3.1
Branch-and-bound
Branch-and-bound (noted B&B) is a general framework for global optimization
[13]. The main idea is the following: the definition space of the model to be
estimated is partitioned into smaller sub-spaces that are iteratively discarded or
refined. When it is certain, via a feasibility test, that a sub-space does not contain
the optimal solution, then this sub-space is discarded. On the contrary, if it might
contain the solution, then it is partitioned into smaller sub-spaces. Iteratively,
the size of the sub-spaces decreases and the estimated solution converges to the
optimal solution. The process stops when the desired accuracy level is reached.
B&B has gained popularity quite recently in computer vision [14–20].
3.2
Bounds
The feasibility test is generally conducted by the computation of bounds, that is
the best (and optionally the worst) solution that can be obtained in a sub-space.
In the following, we explain how to compute the bounds for our application.
Computation By adopting the angle-axis representation, any rotations can be
represented by a point in a ball Bπ of radius π. For example, a point β in this ball
represents the rotation whose axis is β/∥β∥ and angle ∥β∥ [20]. Let decompose
Bπ into a set of cubes. The question becomes: how to compute the bounds of
the inlier cardinality that can be obtained for any rotations in a cube D. The
answer is: the lower and upper bounds are the solutions of the systems (3) and
(4) respectively, where R̄ is the (known) rotation corresponding to the center of
the cube D and σ is the half length side of D (cf proof below).
max
y
N
∑
yi
(3a)
i
s.t. yi ∠(R̄ui , vi ) ≤ yi δ
(3b)
yi ∈ {0, 1}, ∀i = 1 . . . N (3c)
max
y
N
∑
yi
(4a)
i
(
√ )
s.t. yi ∠(R̄ui , vi ) ≤ yi δ + 3σ (4b)
yi ∈ {0, 1}, ∀i = 1 . . . N
(4c)
4
Jean-Charles Bazin, Yongduek Seo and Marc Pollefeys
The unknowns of the systems (3) and (4) are the binary y. It is interesting to
note that an Integer Programming solver, which is computationally expensive,
does not have to be run. Indeed, we can simply find wether yi is 1 or 0 by
checking whether a matching pair verifies the inlier distance constraint or not:
– ∠(R̄ui , vi ) ≤ δ for√the lower bound at system (3)
– ∠(R̄ui , vi ) ≤ δ + 3σ for the upper bound at system (4)
Proof Let us now prove that systems (3) and (4) provide valid lower and upper
bounds in the current cube D, that is to say that the optimal inlier cardinality
c that can be obtained in D is between these lower l and upper u bounds, i.e.
l ≤ c ≤ u. For writing simplification, let note Ropt the rotation of D providing c.
Proving that the lower bound is valid is trivial: since c is the optimal cardinality with respect to a residual threshold δ among all R ∈ D and since R̄ is a
particular element of D, then it is obvious that the cardinality l obtained by R̄
with the same residual threshold is such that l ≤ c.
For the proof of the upper
√ bound, it has been shown by Hartley and Kahl [20]
that maxR (∠(R̄u, Ru)) ≤ 3σ where R is any rotation in the cube D of half
length side σ and u any vectors. Let (ui′ , vi′ ) be an inlier match by Ropt , i.e.
∠(Ropt ui′ , vi′ ) ≤ δ. Therefore we can write:
∠(R̄ui′ , vi′ )
≤ ∠(Ropt ui′ , vi′ ) + ∠(R̄ui′ , Ropt ui′ )
√
≤ δ + 3σ
(5)
(6)
It means that if a match is an inlier by Ropt with respect to the residual threshold
√
δ, then it is also an inlier by R̄ with respect to the residual threshold δ + 3σ.
Therefore system (4) provides at least c inliers, i.e. u ≥ c.
Finally, B&B is guaranteed to converge to the globally optimal solution when
the bounds are valid [13]. Since we proved the bounds are valid, it proves, in
turn, that our proposed B&B-based algorithm is globally optimal.
3.3
Search procedure
Our B&B procedure is formalized in Alg1 using a breadth-first search. The cube
list L is simply initialized with the rotation ball Bπ . At each B&B iteration,
each cube of L is subdivided into 8 sub-cubes of half size and the associated
bounds are computed. Then we remove from the list all the cubes whose upper
bound is lower than the maximum lower bound l∗ obtained so far because it
means that, even in the best case, it is sure that they cannot obtain an inlier
cardinality higher than l∗ . The procedure stops when at least one cube whose
lower bound equals the maximum upper bound is obtained. Finally the rotation
corresponding to the center of that cube is returned, i.e. this is a rotation that
globally maximizes the number of inliers.
4
Experiments
In Section 3.2, we proved that our algorithm is mathematically guaranteed to
obtain the globally optimal solution. In addition, we present, in the following,
Globally Optimal Consensus Set Maximization Through Rotation Search
5
Algorithm 1 B&B for inlier maximization through rotation search
Initialize the cube list L with the rotation ball Bπ
repeat
Subdivision (σ ← σ/2) of each cube Di of L
for each cube Di of L do
Get the rotation RDi of the cube center
Compute the lower li and upper ui bounds (cf Section 3.2)
end for
l∗ = maxi li , i∗ = arg maxi ui , u∗ = ui∗ , R∗ = RDi∗
Remove all the cubes from L such that ui < l∗
until ∃i such that li = u∗ (i.e. at least one cube whose lower-bound is u∗ )
Return: R∗ (i.e. the rotation maximizing the nb of inliers)
experimental results for both synthesized and real data to illustrate the feasibility and convergence of our approach. The algorithm has been implemented in
Matlab and run on a laptop equipped with an Intel i7-2820QM 2.3GHz CPU (a
single core was used) and 8GB RAM. We applied the algorithm on the whole
rotation space Bπ to show the worst case scenario (i.e. no a priori or initial
solution). We compared our approach to RANSAC and the quality of the results
is measured by the number of detected inliers. As RANSAC is not deterministic, we repeated the experiments 1000 times with the same input to obtain the
inlier cardinality distribution and where the number of RANSAC iterations is
automatically computed with the true outlier ratio (if not available, e.g. for real
data, set to 60%), a guaranteed accuracy of 99% and the minimal sampling (2
points for rotation) [1]. Our algorithm runs in a fully automatic manner and
converges in a couple of seconds.
4.1
Synthesized data
We randomly generated two sets of N 3D points related by any random 3D
rotation in the whole SO(3). We applied a Gaussian noise to the coordinates of
the points and corrupted a certain percentage p of the points to create outliers.
Representative generated data is shown in Fig 1.
Figure 2 compares the number of inliers obtained by the proposed approach
and RANSAC. Our method obtains 30 inliers, which corresponds to the number
of synthesized inliers. On the contrary, RANSAC spans between 2 and 30 inliers,
and manages to detect the 30 inliers in only 1% of the experiments. This limited
performance of RANSAC might be surprising, especially given the high value of
the guaranteed accuracy. This can be explained by two reasons. First, the success
of RANSAC is related to the data noise: we performed experiments without any
noise and in this case, the RANSAC success rate corresponded to the guaranteed
accuracy. Second, RANSAC can hypothesize only models directly supported by
the randomly selected minimal points. Therefore if the optimal rotation cannot
be hypothesized by the minimal points then RANSAC cannot return the optimal number of inliers. We performed extensive experiments with various data
amounts (N = 20 ∼ 500 3D points) and proportion of outliers (p = 0% ∼ 80%).
6
Jean-Charles Bazin, Yongduek Seo and Marc Pollefeys
Z
1.5
1
Z
Y
Y
0.5
X
0
X
−0.5
1
0.5
0
−0.5
−1
0
1
2
3
4
5
6
Fig. 1. Representative result of registration of two sets of 3D points related by a rotation and containing outliers. The green (reciprocally red) lines correspond to the
inlier (reciprocally outlier) matches automatically found by our approach. For a better
visualization, only 30 matches are displayed and a set of points is translated.
In all the experiments, our algorithm always converged to (at least1 ) the number
of synthesized inliers and our estimated number of inliers was always higher or
equal than the one obtained by RANSAC.
In Figure 2-a, one might also note that the lower and upper bounds converge which illustrates the convergence of the proposed algorithm and proves
the bounds are valid. For completeness, Figure 2-(c,d) illustrate the evolution of
the volume and the number of cubes. It shows the volume drops continuously
and the number of cubes remains limited. Note that this number of cubes does
not have to be strictly decreasing but must remain relatively low for memory
efficiency, which is our case.
4.2
Real data: registration
We tested our algorithm on two parts of the “Bunny” 3D model from the Stanford 3D Scanning Repository2 in the context of rotational registration. These
two parts have a limited overlap and are related by a rotation. We applied the
method of Darom and Keller [21] on the two 3D sets to extract 3D feature points
and their associated descriptors, which provides candidate matches between the
two 3D sets. Figure 3-a represents the inlier and outlier matches automatically
detected by the proposed algorithm with an accuracy δ of 0.5◦ . Figure 3-(b,c)
illustrate the registration of the two sets by the estimated rotation (without any
additional refinement, e.g. [22], in order to show the quality of our B&B solution). Figure 4 compares the number of inliers obtained by the proposed method
1
2
in rare cases, it might happen that the optimal number of inliers (returned by our
algorithm) is higher than the number of synthesized inliers because some outliers
might create a new dominant model, especially for high outlier ratio.
http://graphics.stanford.edu/data/3Dscanrep
Globally Optimal Consensus Set Maximization Through Rotation Search
distribution of the nb of inliers
30
20
10
upper
lower
0
1
2
3
4
Iterations
5
percentage of experiments
Upper and lower bounds
Convergence of bounds
20
15
10
5
0
6
5
10
15
20
25
Nb of detected inliers
(a)
30
(b)
Convergence of volume
2
7
evolution of the nb of cubes
Number of cubes
remaining volume (in%)
10
0
10
−2
10
3
10
2
10
−4
10
1
1
2
3
4
Iterations
5
6
(c)
10
1
2
3
4
Iterations
5
6
(d)
Fig. 2. Experimental results on synthesized data. (a): convergence of the bounds of
the proposed B&B approach. (b): distribution of the number of inliers detected by
RANSAC on the same problem instance. (c): convergence of the volume of the search
space. (d): evolution of the number of cubes.
and RANSAC. Our method converges to 8 inliers, whereas RANSAC spans between 2 and 8 inliers. RANSAC performs poorly on this dataset because of the
large proportion of outliers (8 inliers out of 1191 points) and the data noise. In
contrast our approach can handle this challenging case without any difficulties.
4.3
Real data: panorama stitching
We now present results in the context of panoramic image stitching [5]. When
the scene is far away, two images are related by the infinite homography H∞ .
When the intrinsic calibration matrix K is known [1], H∞ reduces to a rotation
that we aim to estimate. To obtain correspondences between the two images, we
extracted and matched SIFT features [23] and applied K −1 to compute the associated ray matches. Figure 5-a illustrates the inlier and outlier matches found by
the proposed approach. Once the inliers are found, any reprojection error minimization method can be applied [5, 22]. The final stitching result is shown in Figure 5-b. Figure 5-(c,d) compare the number of inliers obtained by the proposed
8
Jean-Charles Bazin, Yongduek Seo and Marc Pollefeys
(a)
(b)
(c)
Fig. 3. Registration result between two 3D point sets of the “Bunny” model. (a): inlier
(green lines) and outlier (red lines) matches detected by the proposed approach. (b,c):
3D views after 3D registration of the two 3D point sets by the estimated rotation,
without any additional refinement.
distribution of the nb of inliers
upper
lower
120
100
80
60
40
20
0
2
4
6
Iterations
8
percentage of experiments
Upper and lower bounds
Convergence of bounds
80
60
40
20
0
1
2 3 4 5 6 7
Nb of detected inliers
8
Fig. 4. Experimental results for the “Bunny” model. (a): convergence of the bounds of
the proposed B&B approach. (b): distribution of the number of inliers by RANSAC.
approach and RANSAC. Our approach finds 654 inliers while RANSAC spans
between 651 and 654 inliers and manages to detect 654 inliers in less than 1%
of the RANSAC runs.
Globally Optimal Consensus Set Maximization Through Rotation Search
9
(a)
(b)
Convergence of bounds
distribution of the nb of inliers
800
600
400
200
0
upper
lower
2
4
6
Iterations
(c)
8
percentage of experiments
Upper and lower bounds
100
80
60
40
20
0
651
652
653
654
Nb of detected inliers
(d)
Fig. 5. Result of panoramic image stitching. (a): inlier (green lines) and outlier (red
lines) matches detected by our approach between the two input images. (b): stitching
of the two images to build a panoramic view. (c): convergence of the bounds of our
B&B approach. (d): distribution of the number of inliers detected by RANSAC.
10
4.4
Jean-Charles Bazin, Yongduek Seo and Marc Pollefeys
Real data: line clustering
Given a set of input lines, line clustering aims to find which line belongs to which
vanishing point (VP) and estimate the VPs [24]. Very recently, Bazin et al. [25]
proposed a method to maximize the number of clustered lines in this context. It
provides interesting results but is based on interval analysis [26], which provides
loose bounds and, similarly to the big-M method [11], is subject to numerical
inaccuracies. The formulation of system (2) can be modified so that our framework can be applied for line clustering. In the following, we consider intrinsically
calibrated images and orthogonal vanishing points (Manhattan world [6, 27]).
Mathematical formulation: let ni be the normal vector of the ith input
line with i = 1 . . . N and vj be the j th VP with j = 1 . . . M where
1 ≤ M ≤ 3. A
line-VP pair composed of (ni , vj ) is considered an inlier if ∠(ni , vj ) − π2 ≤ δ.
We say that ni is matched (clustered) to vj and we note (ni , vj ) ∈ MI . The
consensus set maximization can be written:
max
card(MI )
∠(ni , vj ) − π ≤ δ, ∀(i, j) ∈ MI ⊆ M
2
MI ,{v}
s.t.
and ∥vj ∥ = 1, ∀j and
vTj vk
= 0, ∀j, k with j ̸= k
(7a)
(7b)
(7c)
As observed by Bazin et al. [6], this system can be simplified by representing
the set of M orthogonal VPs by a single rotation R: vj = Rej where ej represents
the j th axis of the canonical basis, i.e. e1 = (1, 0, 0), e2 = (0, 1, 0) and e3 =
(0, 0, 1). Additionally, we introduce the clustering variables yi,j set to 1 if the
ith line is clustered to the j th VP and to 0 otherwise. This permits to define the
following system: N M
∑∑
max
yi,j
(8a)
y,R
i
j
π s.t. yi,j ∠(ni , Rej ) − ≤ yi,j δ, ∀i = 1 . . . N, j = 1 . . . M
2
yi,j ∈ {0, 1}, ∀i = 1 . . . N, j = 1 . . . M
0≤
M
∑
yi,j ≤ 1, ∀i = 1 . . . N
(8b)
(8c)
(8d)
j
R ∈ SO(3)
(8e)
The (optional) constraint at Eq (8d) enforces that a line can belong to at most
one VP. To solve system (8), we employ a strategy similar to Section 3.2 where
the rotation cube center is used. Concretely, the variables yi,j are set to 0 or 1
by simply testing the constraints:
– ∠(ni , R̄ej ) − π2 ≤ δ for√the lower bound
– ∠(ni , R̄ej ) − π2 ≤ δ + 3σ for the upper bound
The (optional) at-most-one constraint at Eq (8d) can be enforced afterwards
by simply setting only one yi,j to 1 given i, for example for the nearest VP:
Globally Optimal Consensus Set Maximization Through Rotation Search
11
Fig. 6. Representative line clustering results obtained by our algorithm on the York
urban database [28] for indoor and outdoor images, with 2 and 3 VPs. Top: input line
segments. Bottom: output line clustering obtained by the proposed algorithm.
Fig. 7. Distribution of the execution time of our algorithm for line clustering on all
the images of the York urban database [28].
j ∗ = arg minj ∠(ni , Rej ) − π2 . Finally we apply the B&B procedure presented
in Section 3.3. It returns the rotation maximizing the number of clustered lines,
the associated VPs, the clustering information (i.e. which line belongs to which
VP) and the outlier set (i.e. the lines belonging to none of the M VPs).
Results: we tested our algorithm on the York urban database [28] which
is composed of 102 images acquired in indoor and outdoor man-made environments. Each image contains the camera intrinsic calibration parameters, a set of
line segments manually extracted and the ground truth line clustering. For robustness, we removed the segments shorter than 30 pixels. Our algorithm takes
12
Jean-Charles Bazin, Yongduek Seo and Marc Pollefeys
as input the set of lines and returns the line clustering. We applied our algorithm
to each image and compared our results to the ground truth. This comparison
showed that, for the entire database, each line clustered by our algorithm corresponds to its ground truth clustering, which successfully validates our approach.
Some clustering results are shown in Fig 6. The distribution of the execution
time is illustrated in Fig 7: it shows our algorithm runs in just a few seconds.
5
Conclusion
This paper was dedicated to the problem of inlier/outlier detection under a
rotation model. RANSAC is a popular approach to solve this task but is nondeterministic (different runs might lead to different results) and might not return
the optimal solution. In contrast, we proposed a method that is mathematically
guaranteed to obtain the largest consensus set, i.e. the highest number of inliers,
in a globally optimal manner, which constitutes our main contribution. Extensive
experiments on synthesized data and challenging real images have successfully
demonstrated the validity of the proposed method.
Acknowledgments
This research, which has been partially carried out at BeingThere Centre, is
supported by the Singapore National Research Foundation under its International Research Centre @ Singapore Funding Initiative and administered by the
IDM Programme Office. It has also been funded in part by the ECs Seventh
Framework Programme (FP7/2007-2013) / ERC grant #210806 4D Video and
the Sogang University Research Grant of 2012 (201210029.1).
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