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PLANE RECTIFICATION THROUGH ROBUST VANISHING POINT TRACKING USING THE
EXPECTATION-MAXIMIZATION ALGORITHM
Marcos Nieto and Luis Salgado
Grupo de Tratamiento de Im´ genes - E. T. S. Ing. Telecomunicaci´ n
a o
Universidad Polit´ cnica de Madrid - Madrid (Spain)
e
{mnd, lsa}@gti.ssr.upm.es
ABSTRACT outliers. Specific methods that work on the image plane, using accu-
mulators [5], or transforms to the polar space [6] lacks the required
This paper introduces a new strategy for plane rectification in robustness for this scenario since they do not separate inliers from
sequences of images, based on the Expectation-Maximization (EM) outliers [7], which may cause very inaccurate estimations. General
algorithm. Our approach is able to compute simultaneously the pa- methods, which work on the projective plane [8][9], do not specifi-
rameters of the dominant vanishing point in the image plane and the cally treat this scenario, and typically work very well for vanishing
most significant lines passing through it. It is based on a novel defini- points far away from the image boundaries. Besides, these strategies
tion of the likelihood distribution of the gradient image considering typically require the knowledge of the calibration matrix to be able
both the position and the orientation of the gradient pixels. Besides, to treat the points at the infinity.
the mixture model in which the EM algorithm operates is extended, In this paper we propose a robust tracking system, based on the
compared to other works, to consider an additional component to Expectation-Maximization algorithm [10] that is able to determine
control the presence of outliers. the position of the dominant vanishing point of the image plane as
Some synthetic data tests are described to show the robustness well as the required set of converging lines to perform the plane rec-
and efficiency of the proposed method. The plane rectification re- tification at each time instant. The main advantages of the proposal
sults show that the method is able to remove the perspective and come from the usage of the EM algorithm. On the one hand, it is
affine distortion of real traffic sequences without the need to com- an iterative procedure that is guaranteed to converge, and that can be
pute two vanishing points. used as a tracking system, as each estimate can be used in the next
Index Terms— Plane rectification, vanishing point, Expectation- time instant as initialization. On the other hand, a mixture model is
Maximization, outliers rejection defined that considers an implicit classification between inliers and
outliers, which makes the method very robust. Besides, the likeli-
hood function uses the information of position and orientation of the
1. INTRODUCTION
gradient pixels, which improves the models of previous approaches
only based on position [7].
In the field of traffic applications the rectification of the road plane
has been extensively used as a great help to perform metric measure-
ments in static surveillance [1], and onboard systems [2]. 2. VANISHING POINT TRACKING
Plane rectification can be done using the information contained
in vanishing points to compute the road-plane to image-plane ho- This section introduces the likelihood model for gradient pixels,
mography. Typically, two orthogonal vanishing points are detected which can be obtained with any method that approximates the gra-
and used to determine the line at the infinity and thus, the projective dient, such as the Sobel operator. The EM algorithm leads to the
distortion can be removed [3]. However, in road scenarios, there is equations that describe the optimization problem, considering also a
typically information about just one dominant vanishing point, hence distribution to handle the presence of outliers.
there are no reliable methods to compute such rectification. Fortu-
nately, as shown by Schaffalitzky [4], three or more lines that are 2.1. Likelihood model
parallel (thus converging into a single vanishing point) and equally
spaced, can determine in closed form the line at the infinity, from Let xi = {ri , gi } be a data sample composed by ri = (xi , yi ) ,
which, knowing the camera calibration matrix, an orthogonal van- a point of the image plane with significant gradient magnitude, and
ishing point in the plane can be obtained. This way only one vanish- the gradient vector gi = (gxi , gyi ) .
ing point determined by such three lines must be detected to perform Lines in the image plane are parameterized in the homogeneous
plane rectification. In the road scenario, three lane markings corre- form as lj = (aj , bj , cj ) , subject to a2 + b2 = 1. The perpendic-
j j
sponding to two adjacent lanes satisfy this condition. ular distance from a point, ri , to a line, lj , is defined as fρ (ri , lj ) =
The estimation of the dominant vanishing point inside the lim- aj xi + bj yi + cj . In the case the line passes through a vanishing
its of the image can be of great complexity, considering the reduced point v = (vx , vy ) , then cj = −aj vx − bj vy and the function fρ
amount of information of the scenario and the large proportion of depends on the parameters lj and v:
fρ (ri , lj , v) = aj (xi − vx ) + bj (yi − vy ) (1)
This work has been supported by the Ministerio de Ciencia e Innovaci´ n
o
of the Spanish Government under projects TEC2007-67764 (SmartVision) Therefore, a point lies in a line if fρ (ri , lj ) = 0 holds. Con-
and TEC2006-26845-E (Highway), and by the European Commission 6th sidering Gaussian measurement noise, the likelihood of a point, ri ,
FP under project IST-2004-027195 (I-WAY). belonging to a line j is defined as:
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![where Θ∗ and ωj are, respectively, the set of parameters and the a
∗
priori probabilities for component j at the previous iteration. From
1 1
p(ri |lj , v) = √ exp − 2 fρ (ri , lj , v)2 (2) here on, p(j|xi , Θ∗ ) will be denoted as γij .
2πσρ,j 2σρ,j
Regarding the M-step, the estimation of the a priori probabilities
where σρ,j is the standard deviation of the normal distribution. is given by ωj = N N γij , which is, as expected, independent
ˆ 1
i=1
For each data sample, the gradient vector gi depicts the normal of the rest of parameters of the model. For the maximization related
to the direction of the edge at ri . The difference between the orien- to the line parameters, Lagrange multipliers are required to add the
tation of a line j, which is given by the vector mj = (−bj , aj ) , restriction a2 + b2 = 1 which leads to a linear system of equations:
g m
and the one of the data sample is defined as fφ (gi , lj ) = gii |mjj .
Cnj = λnj (8)
Considering, as in the case of the position, a normal distribu-
tion on the orientation, the likelihood of a vector gi to represent the ˆ −2 σ −2
where nj = (aj , bj ) , and C = σρ A+ˆφ B. Matrices A and B
orientation of a line is given by: contain, respectively, the information given by the weighted samples
of position and orientation. For the sake of simplicity in the notation,
1 1 we define the weighted sample mean of a generic data set {zi }N , as
i=1
p(gi |lj ) = √ exp − 2 fφ (gi , lj )2 (3) ξ(z) = N zi γij ( N γij )−1 , yielding the following matrices:
2πσφ,j 2σφ,j i=1 i=1
Assuming that the position of data samples and their orientation
are independent, the joint likelihood of xi to represent a line lj is: ξ(˜2 ) − ξ 2 (˜)
x x ξ(˜y ) − ξ(˜)ξ(˜)
x˜ x y
A= (9)
ξ(˜y ) − ξ(˜)ξ(˜)
x˜ x y ξ(˜2 ) − ξ 2 (˜)
y y
p(xi |lj , v) = p(ri |lj , v)p(gi |lj ) (4)
where random variables, x and y include the vanishing point com-
˜ ˜
which is the expression that links each data sample with a hypothe- ponents: xi = xi − vx and yi = yi − vy , and:
˜ ˜
sized line oriented in the direction of a vanishing point.
2
ξ(gy ) −ξ(gx gy )
B= 2 (10)
2.2. Mixture model likelihood −ξ(gx gy ) ξ(gx )
If we consider now M lines, the likelihood of a data sample is given The maximization of the vanishing point is obtained as:
by the mixture model:
M N −1 M N
M v=
ˆ Nj γij ri Nj γij (11)
p(xi ; Θ, Ω) = ωj p(xi |lj , v) (5) j=1 i=1 j=1 i=1
j=1
where Nj = nj nj is a 2 × 2 matrix. As expected, the maximum
where Θ = {v, {lj }M } and Ω = {ωj }M , are the parameters
j=1 j=1 likelihood estimation of aj and bj depend on v as in (9), and vicev-
of the likelihood distribution. Note that the weighting factors must ersa, as shown in (11). Therefore, the maximization process must be
M
satisfy j=1 ωj = 1, as they can be seen as the a priori probability done in a two-steps process, which is known as gradual M-step [7].
values for each component of the mixture. That way it is necessary to proceed fixing one of this parameters and
The joint probability density of all the data samples, assumed updating the other, so that in two steps, both parameters are updated.
N
to be i.i.d., is given by p(X |Θ) = i=1 p(xi ; Θ, Ω). The log-
likelihood function transform the product into a summation, expand-
ing the expression of the mixture defined in (5): 2.4. Outliers control
N M In our EM scheme proposal, the presence of outliers is handled with
log p(X |Θ) = log ωj pj (xi |lj , v) (6) the inclusion of an additional component, pM (xi |v), for the mix-
i=1 j=1
ture model. That way, M components are assumed by the EM to
represent the data samples drawn from the M most significant lines
The MLE problem to be solved is to find the parameters Θ that
in the scene. The rest of the data samples are absorbed by the outlier
maximize log p((X |Θ)). However, partial derivatives of the log-
distribution. Hence, in our approach, the complete mixture model
likelihood with respect each parameter yield expressions that depend
has M = M + 1 components.
on the derivative of a natural logarithm of a sum, which is not ana-
The mixture model (5) is rewritten as:
lytically tractable.
M
2.3. EM algorithm p(xi ; Θ, Ω) = ωM pM (ri |v) + ωj p(xi |lj , v) (12)
j=1
The EM strategy offers a solution to this MLE problem [10]. The
EM algorithm proceeds iteratively, applying two steps: E-step, that where the components j = 1, 2, . . . , M are the ones already de-
obtains the conditional probability of each data sample to belong to scribed, and pM (xi |v), is the outlier distribution. We propose to
each of the components of the mixture model; and M-step, which define this distribution as the sum of a uniform distribution of value
searches for the values of the target parameters that maximize the ρ = (πW H)−1 that integrates to one within the data range, and a
likelihood function. normal distribution on the distance of the data samples position, ri ,
The conditional probability of the samples to belong to a specific to the vanishing point, d2 (ri , v) = (xi − vx )2 + (yi − vy )2 :
component j is given by:
∗
ωj pj (xi |lj , v) 1 1
p(j|xi , Θ∗ ) = M ∗
(7) pM (ri |v) = ρ + √ exp − 2 d2 (ri , v) (13)
k=1 ωk pk (xi |lk , v) 2πσ 2σ](https://image.slidesharecdn.com/planerectificationthroughrobustvanishingpointtrackingusingtheexpectationmaximizationalgorithm-111115083617-phpapp02/85/Plane-rectification-through-robust-vanishing-point-tracking-using-the-expectation-maximization-algorithm-2-320.jpg){kind=tracking}
![0 0
Table 1. Mean number of iterations for pEM and gpEM (100 tests).
50 50
Parameters noise # it. pEM # it. gpEM
{σa,b = 0.10, σvx,y = 10} 8.4 4.7 100
y
100
y
{σa,b = 0.15, σvx,y = 15} 9.2 5.1 150 150
{σa,b = 0.20, σvx,y = 20} 12.7 6.5
{σa,b = 0.25, σvx,y = 25} 16.4 7.8 200
0 50 100 150 200
200
0 50 100 150 200
x x
(a) (b)
Bad initialization 3 converging lines
2.5 0
The uniform component models the general uncertainty about pEM
vpEM
3
outliers, while the second term models that data samples near the 50
Mean log likelihood
vanishing point are more likely outliers (as they are images of very 3.5
100
y
remote elements of the scene). The equations of the EM algorithm 4
150
are still valid by just updating the mixture model with the outlier 4.5
distribution. 5
0 5 10 15 20
Iteration
25 30 35
200
0 50 100
x
150 200
(c) (d)
3. HOMOGRAPHY COMPUTATION
Fig. 1. Synthetic data test: (a) the ground truth model with N =
The road plane rectification can be done by means of a projective 100 data samples drawn with Gaussian noise; (b) initialization, in
transform, known as homography, that links, up to scale, points in dotted lines, and final estimation in solid lines; (c)likelihood value at
the plane coordinate system and their image in the image plane. Typ- each iteration in solid lines for the proposed method, in dotted lines
ically the homography is computed using the DLT algorithm [11], for the method without using the orientation information; (d) adding
using at least four correspondences between points of the image 300 outliers, the estimation is still correct.
plane and road plane, or in an stratified manner, knowing two van-
ishing points and other information about the scene, such as known
length or angle ratios[3]. samples are drawn, corrupted by Gaussian noise. For these experi-
In this paper we propose to use the DLT method, for which we ments we define σρ = 5 (pixels) and σφ = 0.1 (rads), which is in
need to find four image points that correspond to points in the road line to the expected noise present in a real image.
that form a rectangle. This way we can remove both the projective An example initialization and final estimation are shown in
and the affine distortion, which is enough for the purposes of the Fig. 1 (b). As previously mentioned, the algorithm iterates until
application (the only remaining distortion is an anisotropic scaling convergence is achieved. We define the convergence related to the
and rotation). In order to do so, we use the information of the lines mean log-likelihood value, which is given by:
obtained with the EM vanishing point tracker, that we assume to
be equally spaced in the plane. To compute the line at the infinity, N M
¯ 1
without need to compute two orthogonal vanishing points, we follow L(Θ|X ) = log ωj p(xi |lj , v) (15)
the method of Schaffalitzky [4]: N i=1 j=1
l∞ = det[l1 , l2 , l]l3 − det[l, l2 , l3 ]l1 (14) At each iteration, indexed as k, the improvement of mean log-
¯
L −L¯
¯
likelihood is checked as ΔLk = kL k−1 . When ΔLk is below ¯
where l is any 3-vector for which the determinants are nonzero. ¯k−1
Knowing the camera calibration matrix, K, we can translate a certain threshold, , the process is considered to have converged.
both the line at the infinity and the known vanishing point, v1 , to Typically it is accurately enough to use = 10−5 . In Fig. 1 (c), a
the camera coordinate system, as l∞ = K l∞ and v1 = K −1 v1 comparison of the convergence of the gpEM and the pEM is shown.
and compute the position of the orthogonal vanishing point as It is clear that our method achieves faster convergence for this exam-
v2 = l∞ × v1 . This way we can trace two lines converging in ple, showing just a small oscillation at the firsts iterations due to the
v2 that intersect with two of the lines obtained with the EM. The gradual M-step.
intersection of these lines are the four required points. This experiment was repeated several times, using different ini-
tializations of the parameter set, with similar results. For conve-
4. RESULTS nience, the parameters were initialized randomly, using zero mean
Gaussian noise and a defined standard deviation for each parameter.
In this section we first compare the proposed EM procedure with the Table 1 shows the results, in terms of the mean number of required
one of Minagawa [7]. For fair comparisons, we add the outlier distri- iterations to converge, for 100 executions of the proposed method,
bution to this approach and test the benefits of using the orientation and the method we compare with. The contents of the table reveal
information to the models. We will denote this method as point-EM that the required number of iterations of the pEM increases dramati-
(or pEM), and ours as gradient-point-EM (gpEM). In second place, cally as the initialization get worse, while for the gpEM, the number
we present some results of the plane rectification achieved through of iterations is approximately delimited and in all cases much more
the computation of the vanishing point and the line at the infinity as lower than for pEM. The conclusion is that the utilization of gradient
described in previous sections. information makes the system much faster and less dependent on the
deviation of the initialization.
4.1. Synthetic data tests Another test is done to demonstrate the need for the outlier dis-
tribution. Its addition to the mixture model aims to absorb the mis-
For the experiments, let’s consider three lines converging into a com- matching between the data set and the model. The performance of
mon vanishing point, as shown in Fig. 1 (a), for which a set of N data our method against outliers randomly distributed among the image is](https://image.slidesharecdn.com/planerectificationthroughrobustvanishingpointtrackingusingtheexpectationmaximizationalgorithm-111115083617-phpapp02/85/Plane-rectification-through-robust-vanishing-point-tracking-using-the-expectation-maximization-algorithm-3-320.jpg){kind=tracking}
![shown in Fig. 1 (d). As shown, the data set of the example in (a) con-
tains a large number of outliers. The outlier distribution makes that
most part of outliers are discarded for the estimation process of the
hypothesized lines, so that the result is correct. This method shows
excellent results in all cases and only begins to suffer significant er-
rors when the proportion of outliers reaches 75%. The color code
is the following: blue, red and green lines are fitted using the cor-
responding colored data samples, which are automatically selected.
The black data samples are those marked as outliers.
4.2. Plane rectification results
l
8
We have applied our method to different road sequences, with a for- l0
l1 l2
ward looking camera installed inside a vehicle. Fig. 2 shows some
example images of the computed sequences.
The EM works as a tracker, where each instantaneous estimation
is used as initialization for the next time instant, since the motion be-
tween two consecutive frames is low. For this reason, initializations
are quite close to the optimal solution and the EM algorithm finds it
with just very few iterations, typically between two and three.
As shown in the examples of Fig. 2 (a), where outliers are de- (a) (b) (c)
picted as black dots, the method is very robust, leading to the correct
estimation of the vanishing point, since outliers are absorbed by the Fig. 2. Examples of the application of the proposed EM algorithm
outlier distribution and thus do not corrupt neither the estimation of for plane rectification in onboard and static traffic sequences: (a)
the vanishing point nor the parameters of the lines. This is particu- EM-based classification of the data; (b) computation of l∞ and v2
larly significant in the example of the last row of Fig. 2. This way, and the four control points; and (c) the virtual fronto-parallel view of
the estimation of the target parameters is very steady along time, and the road plane, with the projected control points and detected lines.
thus the proposed method allows to compute a road plane rectifica-
tion adapted to the motion of the vehicle. Therefore, this proposal
overcomes problems of other approaches that also compute road rec- in IEEE Int. Conf. on Image Processing, ICIP 2008, 2008, pp.
tification but without updating the information of the homography 777–780.
[2]. [2] M. Bertozzi and A. Broggi, “Gold: A parallel real-time stereo
This strategy can also be applied to static surveillance scenarios, vision system for generic obstacle and lane detection,” IEEE
to avoid the computation of the second vanishing point, for which Trans. on Image Processing, vol. 7, No. 1, pp. 62–81, 1998.
typically there is not enough information, or its hard to obtain [1]. [3] D. Liebowitz and A. Zisserman, “Metric rectification for per-
Fig. 2 (c) shows examples of the obtained plane rectification. spective images of planes,” IEEE Conf. on Computer Vision
The two first rows depict a typical road, with well painted lane mark- and Pattern Recognition, vol. 0, pp. 482, 1998.
ings and straight lanes. The third one is more complex as there is [4] F. Schaffalitzky and A. Zisserman, “Planar grouping for au-
less information about lane markings and there is also curvature in tomatic detection of vanishing lines and points,” Image and
the road, which is much more simple to estimate using the rectified Vision Computing, vol. 18, pp. 647–658, 2000.
images.
[5] T. Tuytelaars, M. Proesmans, and L. Van Gool, “The cascaded
hough transform,” in in IEEE Proc. International Conference
5. CONCLUSIONS on Image Processing 1998, 1998, pp. 736–739.
[6] V. Cantoni, L. Lombardi, M. Porta, and N. Sicard, “Vanishing
In this paper we have introduced a new approach towards plane rec-
point detection: representation analysis and new approaches,”
tification, using the EM algorithm for the task of vanishing point
in in Proc. International Conference on Image Analysis and
tracking in sequences of images. Our strategy allows to compute si-
Processing (ICIAP 2001), 2001, pp. 26–28.
multaneously the dominant vanishing point of the scene as well as
the most significant lines in that orientation. The proposed method [7] A. Minagawa, N. Tagawa, T. Moriya, and T. Gotoh, “Vanish-
is very robust against large amounts of outliers, thanks to the inte- ing point and vanishing line estimation with line clustering,”
gration of an outlier distribution into the EM framework. IEICE Trans. Inf. Syst., vol. E83-D, no. 7, July 2000.
The plane rectification is achieved by means of the computation [8] J. Kosecka and W. Zhang, “Video compass,” in Proceedings of
of the line at the infinity, avoiding to compute a second vanishing the European Conference on Computer Vision (ECCV), 2002.
point, which is not typically feasible in traffic scenarios. The results [9] R. Pflugfelder, Self-calibrating cameras in video surveillance,
show that the proposed method is able to remove the projective and Graz University of Technology, May 2008.
the affine distortion of the images, which results on a stable genera-
[10] A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum
tion of fronto-parallel views of the road plane.
likelihood from incomplete data via the em algorithm,” Journal
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[11] R. Hartley and A. Zisserman, Multiple View Geometry in com-
[1] C. Maduro, K. Batista, P. Peixoto, and J. Batista, “Estimation
puter vision, Cambridge University Press, 2001.
of vehicle velocity and traffic intensity using rectified images.,”](https://image.slidesharecdn.com/planerectificationthroughrobustvanishingpointtrackingusingtheexpectationmaximizationalgorithm-111115083617-phpapp02/85/Plane-rectification-through-robust-vanishing-point-tracking-using-the-expectation-maximization-algorithm-4-320.jpg){kind=tracking}
Plane rectification through robust vanishing point tracking using the expectation maximization algorithm
- 1. 3URFHHGLQJV RI ,((( WK ,QWHUQDWLRQDO RQIHUHQFH RQ ,PDJH 3URFHVVLQJ 6HSWHPEHU +RQJ .RQJ PLANE RECTIFICATION THROUGH ROBUST VANISHING POINT TRACKING USING THE EXPECTATION-MAXIMIZATION ALGORITHM Marcos Nieto and Luis Salgado Grupo de Tratamiento de Im´ genes - E. T. S. Ing. Telecomunicaci´ n a o Universidad Polit´ cnica de Madrid - Madrid (Spain) e {mnd, lsa}@gti.ssr.upm.es ABSTRACT outliers. Specific methods that work on the image plane, using accu- mulators [5], or transforms to the polar space [6] lacks the required This paper introduces a new strategy for plane rectification in robustness for this scenario since they do not separate inliers from sequences of images, based on the Expectation-Maximization (EM) outliers [7], which may cause very inaccurate estimations. General algorithm. Our approach is able to compute simultaneously the pa- methods, which work on the projective plane [8][9], do not specifi- rameters of the dominant vanishing point in the image plane and the cally treat this scenario, and typically work very well for vanishing most significant lines passing through it. It is based on a novel defini- points far away from the image boundaries. Besides, these strategies tion of the likelihood distribution of the gradient image considering typically require the knowledge of the calibration matrix to be able both the position and the orientation of the gradient pixels. Besides, to treat the points at the infinity. the mixture model in which the EM algorithm operates is extended, In this paper we propose a robust tracking system, based on the compared to other works, to consider an additional component to Expectation-Maximization algorithm [10] that is able to determine control the presence of outliers. the position of the dominant vanishing point of the image plane as Some synthetic data tests are described to show the robustness well as the required set of converging lines to perform the plane rec- and efficiency of the proposed method. The plane rectification re- tification at each time instant. The main advantages of the proposal sults show that the method is able to remove the perspective and come from the usage of the EM algorithm. On the one hand, it is affine distortion of real traffic sequences without the need to com- an iterative procedure that is guaranteed to converge, and that can be pute two vanishing points. used as a tracking system, as each estimate can be used in the next Index Terms— Plane rectification, vanishing point, Expectation- time instant as initialization. On the other hand, a mixture model is Maximization, outliers rejection defined that considers an implicit classification between inliers and outliers, which makes the method very robust. Besides, the likeli- hood function uses the information of position and orientation of the 1. INTRODUCTION gradient pixels, which improves the models of previous approaches only based on position [7]. In the field of traffic applications the rectification of the road plane has been extensively used as a great help to perform metric measure- ments in static surveillance [1], and onboard systems [2]. 2. VANISHING POINT TRACKING Plane rectification can be done using the information contained in vanishing points to compute the road-plane to image-plane ho- This section introduces the likelihood model for gradient pixels, mography. Typically, two orthogonal vanishing points are detected which can be obtained with any method that approximates the gra- and used to determine the line at the infinity and thus, the projective dient, such as the Sobel operator. The EM algorithm leads to the distortion can be removed [3]. However, in road scenarios, there is equations that describe the optimization problem, considering also a typically information about just one dominant vanishing point, hence distribution to handle the presence of outliers. there are no reliable methods to compute such rectification. Fortu- nately, as shown by Schaffalitzky [4], three or more lines that are 2.1. Likelihood model parallel (thus converging into a single vanishing point) and equally spaced, can determine in closed form the line at the infinity, from Let xi = {ri , gi } be a data sample composed by ri = (xi , yi ) , which, knowing the camera calibration matrix, an orthogonal van- a point of the image plane with significant gradient magnitude, and ishing point in the plane can be obtained. This way only one vanish- the gradient vector gi = (gxi , gyi ) . ing point determined by such three lines must be detected to perform Lines in the image plane are parameterized in the homogeneous plane rectification. In the road scenario, three lane markings corre- form as lj = (aj , bj , cj ) , subject to a2 + b2 = 1. The perpendic- j j sponding to two adjacent lanes satisfy this condition. ular distance from a point, ri , to a line, lj , is defined as fρ (ri , lj ) = The estimation of the dominant vanishing point inside the lim- aj xi + bj yi + cj . In the case the line passes through a vanishing its of the image can be of great complexity, considering the reduced point v = (vx , vy ) , then cj = −aj vx − bj vy and the function fρ amount of information of the scenario and the large proportion of depends on the parameters lj and v: fρ (ri , lj , v) = aj (xi − vx ) + bj (yi − vy ) (1) This work has been supported by the Ministerio de Ciencia e Innovaci´ n o of the Spanish Government under projects TEC2007-67764 (SmartVision) Therefore, a point lies in a line if fρ (ri , lj ) = 0 holds. Con- and TEC2006-26845-E (Highway), and by the European Commission 6th sidering Gaussian measurement noise, the likelihood of a point, ri , FP under project IST-2004-027195 (I-WAY). belonging to a line j is defined as: ‹ ,((( ,,3
- 2. where Θ∗ and ωj are, respectively, the set of parameters and the a ∗ priori probabilities for component j at the previous iteration. From 1 1 p(ri |lj , v) = √ exp − 2 fρ (ri , lj , v)2 (2) here on, p(j|xi , Θ∗ ) will be denoted as γij . 2πσρ,j 2σρ,j Regarding the M-step, the estimation of the a priori probabilities where σρ,j is the standard deviation of the normal distribution. is given by ωj = N N γij , which is, as expected, independent ˆ 1 i=1 For each data sample, the gradient vector gi depicts the normal of the rest of parameters of the model. For the maximization related to the direction of the edge at ri . The difference between the orien- to the line parameters, Lagrange multipliers are required to add the tation of a line j, which is given by the vector mj = (−bj , aj ) , restriction a2 + b2 = 1 which leads to a linear system of equations: g m and the one of the data sample is defined as fφ (gi , lj ) = gii |mjj . Cnj = λnj (8) Considering, as in the case of the position, a normal distribu- tion on the orientation, the likelihood of a vector gi to represent the ˆ −2 σ −2 where nj = (aj , bj ) , and C = σρ A+ˆφ B. Matrices A and B orientation of a line is given by: contain, respectively, the information given by the weighted samples of position and orientation. For the sake of simplicity in the notation, 1 1 we define the weighted sample mean of a generic data set {zi }N , as i=1 p(gi |lj ) = √ exp − 2 fφ (gi , lj )2 (3) ξ(z) = N zi γij ( N γij )−1 , yielding the following matrices: 2πσφ,j 2σφ,j i=1 i=1 Assuming that the position of data samples and their orientation are independent, the joint likelihood of xi to represent a line lj is: ξ(˜2 ) − ξ 2 (˜) x x ξ(˜y ) − ξ(˜)ξ(˜) x˜ x y A= (9) ξ(˜y ) − ξ(˜)ξ(˜) x˜ x y ξ(˜2 ) − ξ 2 (˜) y y p(xi |lj , v) = p(ri |lj , v)p(gi |lj ) (4) where random variables, x and y include the vanishing point com- ˜ ˜ which is the expression that links each data sample with a hypothe- ponents: xi = xi − vx and yi = yi − vy , and: ˜ ˜ sized line oriented in the direction of a vanishing point. 2 ξ(gy ) −ξ(gx gy ) B= 2 (10) 2.2. Mixture model likelihood −ξ(gx gy ) ξ(gx ) If we consider now M lines, the likelihood of a data sample is given The maximization of the vanishing point is obtained as: by the mixture model: M N −1 M N M v= ˆ Nj γij ri Nj γij (11) p(xi ; Θ, Ω) = ωj p(xi |lj , v) (5) j=1 i=1 j=1 i=1 j=1 where Nj = nj nj is a 2 × 2 matrix. As expected, the maximum where Θ = {v, {lj }M } and Ω = {ωj }M , are the parameters j=1 j=1 likelihood estimation of aj and bj depend on v as in (9), and vicev- of the likelihood distribution. Note that the weighting factors must ersa, as shown in (11). Therefore, the maximization process must be M satisfy j=1 ωj = 1, as they can be seen as the a priori probability done in a two-steps process, which is known as gradual M-step [7]. values for each component of the mixture. That way it is necessary to proceed fixing one of this parameters and The joint probability density of all the data samples, assumed updating the other, so that in two steps, both parameters are updated. N to be i.i.d., is given by p(X |Θ) = i=1 p(xi ; Θ, Ω). The log- likelihood function transform the product into a summation, expand- ing the expression of the mixture defined in (5): 2.4. Outliers control N M In our EM scheme proposal, the presence of outliers is handled with log p(X |Θ) = log ωj pj (xi |lj , v) (6) the inclusion of an additional component, pM (xi |v), for the mix- i=1 j=1 ture model. That way, M components are assumed by the EM to represent the data samples drawn from the M most significant lines The MLE problem to be solved is to find the parameters Θ that in the scene. The rest of the data samples are absorbed by the outlier maximize log p((X |Θ)). However, partial derivatives of the log- distribution. Hence, in our approach, the complete mixture model likelihood with respect each parameter yield expressions that depend has M = M + 1 components. on the derivative of a natural logarithm of a sum, which is not ana- The mixture model (5) is rewritten as: lytically tractable. M 2.3. EM algorithm p(xi ; Θ, Ω) = ωM pM (ri |v) + ωj p(xi |lj , v) (12) j=1 The EM strategy offers a solution to this MLE problem [10]. The EM algorithm proceeds iteratively, applying two steps: E-step, that where the components j = 1, 2, . . . , M are the ones already de- obtains the conditional probability of each data sample to belong to scribed, and pM (xi |v), is the outlier distribution. We propose to each of the components of the mixture model; and M-step, which define this distribution as the sum of a uniform distribution of value searches for the values of the target parameters that maximize the ρ = (πW H)−1 that integrates to one within the data range, and a likelihood function. normal distribution on the distance of the data samples position, ri , The conditional probability of the samples to belong to a specific to the vanishing point, d2 (ri , v) = (xi − vx )2 + (yi − vy )2 : component j is given by: ∗ ωj pj (xi |lj , v) 1 1 p(j|xi , Θ∗ ) = M ∗ (7) pM (ri |v) = ρ + √ exp − 2 d2 (ri , v) (13) k=1 ωk pk (xi |lk , v) 2πσ 2σ
- 3. 0 0 Table 1. Mean number of iterations for pEM and gpEM (100 tests). 50 50 Parameters noise # it. pEM # it. gpEM {σa,b = 0.10, σvx,y = 10} 8.4 4.7 100 y 100 y {σa,b = 0.15, σvx,y = 15} 9.2 5.1 150 150 {σa,b = 0.20, σvx,y = 20} 12.7 6.5 {σa,b = 0.25, σvx,y = 25} 16.4 7.8 200 0 50 100 150 200 200 0 50 100 150 200 x x (a) (b) Bad initialization 3 converging lines 2.5 0 The uniform component models the general uncertainty about pEM vpEM 3 outliers, while the second term models that data samples near the 50 Mean log likelihood vanishing point are more likely outliers (as they are images of very 3.5 100 y remote elements of the scene). The equations of the EM algorithm 4 150 are still valid by just updating the mixture model with the outlier 4.5 distribution. 5 0 5 10 15 20 Iteration 25 30 35 200 0 50 100 x 150 200 (c) (d) 3. HOMOGRAPHY COMPUTATION Fig. 1. Synthetic data test: (a) the ground truth model with N = The road plane rectification can be done by means of a projective 100 data samples drawn with Gaussian noise; (b) initialization, in transform, known as homography, that links, up to scale, points in dotted lines, and final estimation in solid lines; (c)likelihood value at the plane coordinate system and their image in the image plane. Typ- each iteration in solid lines for the proposed method, in dotted lines ically the homography is computed using the DLT algorithm [11], for the method without using the orientation information; (d) adding using at least four correspondences between points of the image 300 outliers, the estimation is still correct. plane and road plane, or in an stratified manner, knowing two van- ishing points and other information about the scene, such as known length or angle ratios[3]. samples are drawn, corrupted by Gaussian noise. For these experi- In this paper we propose to use the DLT method, for which we ments we define σρ = 5 (pixels) and σφ = 0.1 (rads), which is in need to find four image points that correspond to points in the road line to the expected noise present in a real image. that form a rectangle. This way we can remove both the projective An example initialization and final estimation are shown in and the affine distortion, which is enough for the purposes of the Fig. 1 (b). As previously mentioned, the algorithm iterates until application (the only remaining distortion is an anisotropic scaling convergence is achieved. We define the convergence related to the and rotation). In order to do so, we use the information of the lines mean log-likelihood value, which is given by: obtained with the EM vanishing point tracker, that we assume to be equally spaced in the plane. To compute the line at the infinity, N M ¯ 1 without need to compute two orthogonal vanishing points, we follow L(Θ|X ) = log ωj p(xi |lj , v) (15) the method of Schaffalitzky [4]: N i=1 j=1 l∞ = det[l1 , l2 , l]l3 − det[l, l2 , l3 ]l1 (14) At each iteration, indexed as k, the improvement of mean log- ¯ L −L¯ ¯ likelihood is checked as ΔLk = kL k−1 . When ΔLk is below ¯ where l is any 3-vector for which the determinants are nonzero. ¯k−1 Knowing the camera calibration matrix, K, we can translate a certain threshold, , the process is considered to have converged. both the line at the infinity and the known vanishing point, v1 , to Typically it is accurately enough to use = 10−5 . In Fig. 1 (c), a the camera coordinate system, as l∞ = K l∞ and v1 = K −1 v1 comparison of the convergence of the gpEM and the pEM is shown. and compute the position of the orthogonal vanishing point as It is clear that our method achieves faster convergence for this exam- v2 = l∞ × v1 . This way we can trace two lines converging in ple, showing just a small oscillation at the firsts iterations due to the v2 that intersect with two of the lines obtained with the EM. The gradual M-step. intersection of these lines are the four required points. This experiment was repeated several times, using different ini- tializations of the parameter set, with similar results. For conve- 4. RESULTS nience, the parameters were initialized randomly, using zero mean Gaussian noise and a defined standard deviation for each parameter. In this section we first compare the proposed EM procedure with the Table 1 shows the results, in terms of the mean number of required one of Minagawa [7]. For fair comparisons, we add the outlier distri- iterations to converge, for 100 executions of the proposed method, bution to this approach and test the benefits of using the orientation and the method we compare with. The contents of the table reveal information to the models. We will denote this method as point-EM that the required number of iterations of the pEM increases dramati- (or pEM), and ours as gradient-point-EM (gpEM). In second place, cally as the initialization get worse, while for the gpEM, the number we present some results of the plane rectification achieved through of iterations is approximately delimited and in all cases much more the computation of the vanishing point and the line at the infinity as lower than for pEM. The conclusion is that the utilization of gradient described in previous sections. information makes the system much faster and less dependent on the deviation of the initialization. 4.1. Synthetic data tests Another test is done to demonstrate the need for the outlier dis- tribution. Its addition to the mixture model aims to absorb the mis- For the experiments, let’s consider three lines converging into a com- matching between the data set and the model. The performance of mon vanishing point, as shown in Fig. 1 (a), for which a set of N data our method against outliers randomly distributed among the image is
- 4. shown in Fig. 1 (d). As shown, the data set of the example in (a) con- tains a large number of outliers. The outlier distribution makes that most part of outliers are discarded for the estimation process of the hypothesized lines, so that the result is correct. This method shows excellent results in all cases and only begins to suffer significant er- rors when the proportion of outliers reaches 75%. The color code is the following: blue, red and green lines are fitted using the cor- responding colored data samples, which are automatically selected. The black data samples are those marked as outliers. 4.2. Plane rectification results l 8 We have applied our method to different road sequences, with a for- l0 l1 l2 ward looking camera installed inside a vehicle. Fig. 2 shows some example images of the computed sequences. The EM works as a tracker, where each instantaneous estimation is used as initialization for the next time instant, since the motion be- tween two consecutive frames is low. For this reason, initializations are quite close to the optimal solution and the EM algorithm finds it with just very few iterations, typically between two and three. As shown in the examples of Fig. 2 (a), where outliers are de- (a) (b) (c) picted as black dots, the method is very robust, leading to the correct estimation of the vanishing point, since outliers are absorbed by the Fig. 2. Examples of the application of the proposed EM algorithm outlier distribution and thus do not corrupt neither the estimation of for plane rectification in onboard and static traffic sequences: (a) the vanishing point nor the parameters of the lines. This is particu- EM-based classification of the data; (b) computation of l∞ and v2 larly significant in the example of the last row of Fig. 2. This way, and the four control points; and (c) the virtual fronto-parallel view of the estimation of the target parameters is very steady along time, and the road plane, with the projected control points and detected lines. thus the proposed method allows to compute a road plane rectifica- tion adapted to the motion of the vehicle. Therefore, this proposal overcomes problems of other approaches that also compute road rec- in IEEE Int. Conf. on Image Processing, ICIP 2008, 2008, pp. tification but without updating the information of the homography 777–780. [2]. [2] M. Bertozzi and A. Broggi, “Gold: A parallel real-time stereo This strategy can also be applied to static surveillance scenarios, vision system for generic obstacle and lane detection,” IEEE to avoid the computation of the second vanishing point, for which Trans. on Image Processing, vol. 7, No. 1, pp. 62–81, 1998. typically there is not enough information, or its hard to obtain [1]. [3] D. Liebowitz and A. 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