Enforcing Constraints for Human Body Tracking

David DemirdjianArtificial Intelligence Laboratory, MIT

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Goal Real-time articulated body tracking from

stereo accounting for constraints on pose

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Approach

Differential tracking: assuming the articulated body pose t-1 is known, estimate the pose t (or

equivalently the set of limb rigid motions i=(tii) between

poses t-1 and t) that minimizes the distance between the articulated model and the observed 3D data

tracking as a constrained optimization problem

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Approach

Differential tracking: assuming the articulated body pose t-1 is known, estimate the pose t (or

equivalently the set of limb rigid motions i=(tii) between

poses t-1 and t) that minimizes the distance between the articulated model and the observed 3D data

tracking as a constrained optimization problem– Solve unconstrained optimization problem– Project solution on constraint surface

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Projection-based approach

unconstrained optimum)

human motion manifold

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Approach

Estimate limb motions i=(tii) independently using standard multi-object tracking algorithm

Projection: find the closest body motion =(i’) to =(i) that satisfies human body constraints: – articulated constraints – other constraints: joint limit, …

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Previous work

Particle sampling: Sidenbladh & al. ECCV’00

Sminchisescu & Triggs CVPR’01

Differential tracking: Plankers & Fua ICCV’99

Jojic & al. ICCV’99

Delamarre & Faugeras ICCV’99

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Plan

Unconstrained problem Articulated constraints enforcing Other constraints Tracking results Application (Multimodal interface) Conclusion

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Multi-object tracking

Assuming the articulated body pose t-1 is known, estimate the set of limb rigid motions i=(tii) minimizes the distance between the (moved) limb and the observed 3D data

Consists in estimating limb motions i=(tii) independently:

– Estimate visible 3D mesh of each limb– Current implementation uses the ICP algorithm to

register each limb w.r.t 3D data

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Iterative Closest Point

3D registration– find the rigid transformation that maps shape St (limb model) to

shape Sr (3D data)

SrSt

),(minarg1

2*

n

i

rt SSGd

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Iterative Closest Point

Matching points• For all points in St, we search for the closest point in Sr by

computing the distance and keep the closest

SrSt

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Iterative Closest Point

Energy function minimization• Estimate the rigid transformation that minimizes the sum of

squared distances between corresponding matched points

SrSt

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Iterative Closest Point

Energy function minimization• Estimate the rigid transformation that minimizes the sum of

squared distances between corresponding matched points

SrSt

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Iterative Closest Point

Optimal rigid transformation (and uncertainty ) found by combining all the elementary displacements

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Plan

Unconstrained problem Articulated constraints enforcing Other constraints Tracking results Application (Multimodal interface) Conclusion

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ProjectionThe unconstrained optimal body motion is given by

=(1, 2 … N)With uncertainty

=(1, 2 … N)

)()( 12 TE

with =(1’, 2’ … N’) satisfying articulated constraints

Articulated constraints enforcement: find that minimizes the Mahalanobis distance:

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Articulated motion estimation

If Mij is a joint between objects i and j:

 Mij joint

(Ri,ti)

(Rj,tj)obj. i

obj. j

Motion of point Mij

on limb i

Motion of point Mij

on limb j=

)(')(' ijjiji MM

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Articulated motion estimation

If Mij is a joint between objects i and j:

 Mij joint

(Ri,ti)

(Rj,tj)obj. i

obj. j

Motion of point Mij

on limb i

Motion of point Mij

on limb j=

'''' jijjiiji tMRtMR

)(')(' ijjiji MM

WOMOT 2003

Articulated motion estimation

If Mij is a joint between objects i and j:

 Mij joint

(Ri,ti)

(Rj,tj)obj. i

obj. j

Motion of point Mij

on limb i

Motion of point Mij

on limb j=

0'')''(][

0'']''[

')]'[(')]'[(

''''

jijiij

jiijji

jijjiiji

jijjiiji

ttM

ttM

tMItMI

tMRtMR

[.]x denotes skew-symmetric matrix

)(')(' ijjiji MM

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Articulated motion estimation

If Mij is a joint between objects i and j:

 Mij joint

(Ri,ti)

(Rj,tj)obj. i

obj. j

Motion of point Mij

on limb i

Motion of point Mij

on limb j=

0'')''(][ jijiij ttM

[.]x denotes skew-symmetric matrix

)(')(' ijjiji MM

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Articulated motion estimation

If Mij is a joint between objects i and j:

 Mij joint

(Ri,ti)

(Rj,tj)obj. i

obj. j

Motion of point Mij

on limb i

Motion of point Mij

on limb j=

0 ijS

)(')(' ijjiji MM

0'')''(][ jijiij ttM

=(1’, 2’ … N’)

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Articulated motion estimation

If Mij is a joint between objects i and j:

 Mij joint

(Ri,ti)

(Rj,tj)obj. i

obj. j

Motion of point Mij

on limb i

Motion of point Mij

on limb j=

(Stack for all joints)0

)(')(' ijjiji MM

0 ijS

0'')''(][ jijiij ttM

=(1’, 2’ … N’)

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Articulated motion estimation

All the joint constraints can be written as a linear constraint:

0

is a linear combination of vectors in the nullspace of Therefore there exists a matrix V such that:

V

V can be estimated by SVD of

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Articulated motion estimation

)()( 12 VVE T

 

111 )( TT VVVVP

)()( 12 TE

unconstrainedmotion

articulatedmotion

Find minimum of E2 in nullspace of

P

…

(linear projection)

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Plan

Unconstrained problem Articulated constraints enforcing Other constraints Tracking results Application (Multimodal interface) Conclusion

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Other constraints

Constraints:– Static: Joint angle bounds, gravity law, …– Dynamic: Maximum velocity, …

Motivation:– Using more constraints to reduce local minima

and therefore increase tracking robustness– Application context can reduce tremendously

the dimension of the pose space

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Other constraints

)()( *1*2 TE

Pose constraints modeled by a (user-defined) function f, such that valid poses correspond to f()>0

ex: f()=min(g1(), g2(), … gN()) with g1() = angle(l-arm, l-forearm) – min_angle

g2() = max_angle - angle(l-arm, l-forearm)….

Constraints enforcement: find * that minimizes the Mahalanobis distance:

with * satisfying Ft-1(*)=f( *(t-1))>0

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Other constraints

)()( *1*2 TE

articulated motionarticulated constrained

motion

** V

)()(

)()(*1*2

*1*2

VVE

VVVVETT

T

with (local parameterization)

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Constrained optimization algorithm

Alternate between binary and stochastic searches

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Constrained optimization algorithm

Alternate between binary and stochastic searches

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Constrained optimization algorithm

Alternate between binary and stochastic searches

E2 = E0

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Constrained optimization algorithm

Alternate between binary and stochastic searches

E2 = E1

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Constrained optimization algorithm

Alternate between binary and stochastic searches

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TRACKING SEQUENCE

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Future work

Quantitative measurement(comparing results with tethered motion capture system)

Appearance/Shape information(learning color distribution + shape of limbs)

Motion/gesture(including dynamic constraints)

Learning human motion constraints (instead of giving them explicitly.. [ICCV’03])

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Applications Multimodal Human-Computer Interaction

(gesture + speech)

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Conclusion

Projection-based approach for articulated body tracking– articulated constraints enforced by (linearly)

projecting unconstrained limb motion on articulated motion manifold

– other constraints enforced using a stochastic constrained optimization algorithm