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Direct Edge Align
Robust Camera Motion Estimation usingDirect Edge Alignment and Sub-gradient
Method
Manohar Kuse & Shaojie Shen
Robotics Institute (RI)Hong Kong University of Science & Technology
October 26, 2015
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Direct Edge Align
OverviewIntroduction
Proposed FormulationRelative Pose EstimationEnergy FormulationLie ManifoldsDT Relaxation
Numerical SolutionSub-Gradient MethodComputation of a Sub-gradientFast Convergence
Results
Concluding Remarks
References
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Introduction
Visual Odometry : Problem Definition
I RGB Images : I0, I1, · · · In.I Depth Images : Z0,Z1, · · ·Zn.I Determine camera pose (k
k−1R, kk−1t) between adjacent
frames.I k
k−1R ∈ SO(3) is a rotation matrix.I k
k−1t ∈ R3 is a translation matrix.
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Introduction
Literature Review for Odometry from RGB-D Camera
I Feature-based methods.I Direct methods (or feature-less methods).
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Introduction
Feature-based methods
I Extract features eg. SIFT, SURF, Harris etc.I Find correspondences between images.I Solve Perspective-N-Point problem.I Huang et al. [4], Dryanovski et al. [2].I Disadvantages :
Inaccurate, since it discards most of image information.Sensitive to incorrect correspondences.Comparitively unstable
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Introduction
Direct Methods
I Do not involve feature extraction.I Use of entire image information for pose determination.I Based on minimizing the photometric error to get best
pose.I Relatively new approach Kerl et al. [5].I Disadvantages :
Sensitive to illumination variation and noise.Small basin of attraction.Non-differentiable cost function.
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Introduction
Analysis on Non-differentiability of Cost functionsI Previous work by Kerl et al. [5] suggest to minimize the
linearized photometric error cost.
r(ξ ,r Pi) = In(Π[exp(ξ ) r Pi ])− Ir (r ui)
ξ∗ = argmin
ξ
∑i
rlin(ξ ,r Pi)2.
I They proposed to use the Gauss-Newton method tominimize this function.
I Gauss-Newton method is not guaranteed to converge.I We argue about the non-differentiability of the cost function
proposed by Kerl et al. [5].I Thus, we propose to use the sub-gradient method
(numerical method for optimization of non-differentiablefunctions).
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Introduction
(a)
50 100 150 200 250 3000
50
100
150
200
250
Locations along Green Line
Image Inte
nsity
(b)
Figure : Highlighting the non-differentiability of the function In atobject transition points
r(ξ ,r Pi) = In Π τ(ξ ,r Pi)
Theorem (Continuity and Composition)if g is continuous at a and f is continuous at g(a), only then f gis continuous at a.
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Proposed Formulation
OverviewIntroduction
Proposed FormulationRelative Pose EstimationEnergy FormulationLie ManifoldsDT Relaxation
Numerical SolutionSub-Gradient MethodComputation of a Sub-gradientFast Convergence
Results
Concluding Remarks
References
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Proposed Formulation
Notations
I RGB Image Ik : Ω⊂ R2 7→ R.I Depth Image Zk : Ω⊂ R2 7→ R.I 3D Scene Point kP ∈ R3 at k th timestamp.I Projection Function ku = Π(kP). Π : R3 7→ R2.I Inverse Projection kP = Π(ku,Zk (ku)). Π : (R2,R) 7→ R3.I Edge image point r ei ∈ R2 and its inverse projection
r Ei ∈ R3.I Distance transform of edge-map V (k) : R2 7→ R.
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Proposed Formulation
Relative Pose Estimation
Relative Pose Estimation
Figure : Notations and Conventions
I Estimation of rigid transformation between a Referenceframe (r) and Current Frame (n) as an OptimizationProblem.
I Reference frame periodically refreshed.I Relative poses chained to obtained robot state estimate.
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Proposed Formulation
Energy Formulation
Energy FormulationI Proposed Geometric Energy function :
Sum of the squared distances between,a) re-projections (of edge points from reference image)ANDb) nearest edge points in current image:
f (R,T) = ∑i
minj
D2( Π[RT (r Pi −T)], nuj).
minimizeR,T
f (R,T)
subject to R ∈ SO(3)
I Non-linear and Non-convex
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Proposed Formulation
Lie Manifolds
Optimization with Unitary Constraints
I Theory of Optimization under unitary constraints [6]proposes to use an appropriate manifold.
I Projection of the iterate onto the manifold gives a naturalframework to model the non-convex unitary (hereorthogonal) constraint.
I Lie manifolds widely used to model a rotation matrix.
ξ = (t;w)T ∈ R6.
τ(r Pi ,ξ ) =[exp(ξ )
]−1r Pi = RT (r Pi −T).
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Proposed Formulation
Lie Manifolds
Unconstrained Reformulation
I Using the Lie manifolds the problem can be reformulatedas an unconstrained optimization problem.
minimizeξ
∑i
minj
D2( Π[τ(r Pi ,ξ )], nuj).
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Proposed Formulation
DT Relaxation
Relaxation using Distance TransformI Observation : If edge points pre-selected (from reference
frame), minj D(ui ,uj) is exactly the definition of theDistance Transform [3].
I Using the Lie Manifold and the Distance Transformrelaxation.
υei (ξ ) = V (n)( Π[τ(Π(r ei ,Zr (r ei)),ξ )])
f (ξ ) = ∑∀ei(υei (ξ ))2.
I Optimal estimates of ξ obtained as
ξ∗ = argmin
ξ
∑∀ei(υei (ξ ))2
I ξ ∗ back-projected using logarithm-map from the LieManifold to obtain the optimal rigid transform between theframes.
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Direct Edge Align
Numerical Solution
OverviewIntroduction
Proposed FormulationRelative Pose EstimationEnergy FormulationLie ManifoldsDT Relaxation
Numerical SolutionSub-Gradient MethodComputation of a Sub-gradientFast Convergence
Results
Concluding Remarks
References
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Numerical Solution
Sub-Gradient Method
Sub-Gradient Method
I Numerical method for minimizing non-differentiablefunctions (Boyd et al. [1] )
I First order methodI Concept of sub-gradients (a set). Generalizing the concept
of gradients.
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Numerical Solution
Sub-Gradient Method
Sub-Gradient MethodI Start with an initial estimate (ξ (0)).I Update along negative sub-gradient direction
ξ(k+1) = ξ
(k) + (−αk ) (k).
ξa + ξb := log(exp(ξa) exp(ξb)).
I Step length fixed ahead of time.
αp :=η
(p + 1); p = 0, · · · ,k
I Keep track of best estimates.
f (k)best = min(f (ξ(0)), f (ξ
(1)), . . . , f (ξ(k))).
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Numerical Solution
Computation of a Sub-gradient
Computation of a Sub-gradientI All c which satisfy following inequality are the
sub-gradients of υei (ξ ) wrt ξ at ξ = ξ (k).
limξ→ξ (k)
υei (ξ )−υei (ξ(k))≥ lim
ξ→ξ (k)c · (ξ −ξ
(k)).
I Since, υei (ξ ) = V (n) Π τ(ξ ,r Pi)
Jei :=∂V (n)
∂ei· ∂ei
∂Ei· ∂Ei
∂ξ
∣∣∣∣ξ=ξ (k)
.
I 3rd term computed using approximation of rotation matrix
Ei(ξ(k) + δξ ) u [R(I3 + bδwcx )]T
[r Ei − T−δ t].
∂Ei
∂ξ
∣∣∣∣ξ=ξ (k)
= RT[−I3 br Ei − Tcx
]
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Numerical Solution
Computation of a Sub-gradient
Computation of a Sub-gradient
f (ξ ) = ∑∀ei(υei (ξ ))2.
(k) = ∑∀ei2 υei (ξ
(k)) Jei .
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Numerical Solution
Fast Convergence
Heavy Ball Method
I Technique to speedup convergence of a gradient methodI Use the update direction s(k) instead of (k).
s(k) = (1−β )(k) + βs(k−1).
I We derive the convergence
fbest − f (ξ∗)≤
R2 + G2∑
kp=0 α2
p (1−β p)2
2∑kp=0 αpµ(p)
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Numerical Solution
Fast Convergence
Summary of Proposed Algorithm
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Direct Edge Align
Results
OverviewIntroduction
Proposed FormulationRelative Pose EstimationEnergy FormulationLie ManifoldsDT Relaxation
Numerical SolutionSub-Gradient MethodComputation of a Sub-gradientFast Convergence
Results
Concluding Remarks
References
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Results
Relative Pose Error
I GT Poses : Q1, · · · ,Qn ∈ SE(3)
I Estimated Poses : B1, · · · ,Bn ∈ SE(3)
I For an integer δ
Ei =(Q−1
i Qi+δ
)−1(B−1i Bi+δ
)I RMSE of the translation component of the sequence
E1, · · · ,En−δ
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Results
Sequence D-EA Kerl et al. [5]δ = 1 δ = 20 δ = 1 δ = 20
fr2/desk 0.0324 0.1529 0.0333 0.2217fr1/desk 0.0289 0.0948 0.0346 0.4286fr1/desk2 0.0335 0.1818 0.0343 0.3658fr1/floor 0.0355 0.1988 0.0330 0.3380fr1/room 0.0353 0.2514 0.0307 0.3399fr2/desk with person 0.0125 0.0594 0.0137 0.1516fr3/sitting halfsphere 0.0208 0.1462 0.0181 0.2599fr2/pioneer slam2 0.0593 0.4447 0.0847 0.4707
Table : RMSE values of the Relative Pose Errors for varioussequences.
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Results
100 200 300 400 500 6000
5
10
15
Frame#
||tr
an
s(E
i)|| 2
(in
mm
)
rel translation error for Kerl et al.rel translation error for D−EA
(a) Processing all frames
100 200 300 400 500 6000
5
10
15
Frame#
||tr
an
s(E
i)|| 2
(in
mm
)
rel translation error for Kerl et al.rel translation error for D−EA
(b) Processing only frames0,2,4,6 · · ·
100 200 300 400 500 6000
5
10
15
Frame#
||tr
an
s(E
i)|| 2
(in
mm
)
rel translation error for Kerl et al.rel translation error for D−EA
(c) Processing only frames0,3,6,9 · · ·
100 200 300 400 500 6000
5
10
15
Frame#
||tr
an
s(E
i)|| 2
(in
mm
)
rel translation error for Kerl et al.rel translation error for D−EA
(d) Processing only frames0,4,8,12 · · ·
Figure : Robustness for large motions. Relative pose estimation of‘fr1/desk’ by skipping frames. Best viewed in color.
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Results
Reference Frame
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Results
Current Frame
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Results
Energy at each Iteration
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Results
Iteration 0
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Results
Iteration 5
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Results
Iteration 10
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Results
Iteration 15
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Results
Iteration 20
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Results
Iteration 25
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Results
Iteration 30
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Results
Iteration 35
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Results
Iteration 40
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Results
Iteration 45
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Results
Iteration 50
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Concluding Remarks
OverviewIntroduction
Proposed FormulationRelative Pose EstimationEnergy FormulationLie ManifoldsDT Relaxation
Numerical SolutionSub-Gradient MethodComputation of a Sub-gradientFast Convergence
Results
Concluding Remarks
References
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Concluding Remarks
Concluding Remarks
I Feature-less approach for 6-DOF pose estimation.I Novel formulation based on Distance transform.I Robust to illumination changes, noise and has larger
convergence basin.I Address the problem of non-differentiability of cost
function.I Outperforms previous method for fast motion.
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Direct Edge Align
References
OverviewIntroduction
Proposed FormulationRelative Pose EstimationEnergy FormulationLie ManifoldsDT Relaxation
Numerical SolutionSub-Gradient MethodComputation of a Sub-gradientFast Convergence
Results
Concluding Remarks
References
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References
S. Boyd, L. Xiao, and A. Mutapcic.Subgradient methods.lecture notes of EE392o, Stanford University, AutumnQuarter, 2004:2004–2005, 2003.
I. Dryanovski, R. G. Valenti, and J. Xiao.Fast visual odometry and mapping from RGB-D data.In Robotics and Automation (ICRA), 2013 IEEEInternational Conference on, pages 2305–2310. IEEE,2013.
P. F. Felzenszwalb and D. P. Huttenlocher.Distance Transforms of Sampled Functions.Theory of computing, 8(1):415–428, 2012.
A. S. Huang, A. Bachrach, P. Henry, M. Krainin,D. Maturana, D. Fox, and N. Roy.
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References
Visual odometry and mapping for autonomous flight usingan RGB-D camera.In International Symposium on Robotics Research (ISRR),pages 1–16, 2011.
C. Kerl, J. Sturm, and D. Cremers.Robust odometry estimation for RGB-D cameras.In ICRA, pages 3748–3754. IEEE, 2013.
J. H. Manton.Optimization algorithms exploiting unitary constraints.IEEE Transactions on Signal Processing, 50(3):635–650,2002.