LOCAL RIGIDITY OF HYPERBOLIC MANIFOLDS WITH GEODESIC BOUNDARY
Graph Rigidity for Near-Coplanar Structure from …Graph Rigidity for Near-Coplanar Structure from...
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Graph Rigidity for Near-Coplanar Structure from Motion 2 Queensland University of Technology, Australia 1 CSIRO ICT Centre, Australia Jack Valmadre 1,2 Simon Lucey 1,2 Sridha Sridharan 2 10 -3 10 -2 10 -1 10 -1 10 0 10 1 10 2 10 3 10 4 σ (Magnitude of Gaussian noise) Coplanar Tetrahedron Average length error (%) Factorisation Quartic Graph rigidity 10 -3 10 -2 10 -1 10 -1 10 0 10 1 10 2 10 3 10 4 σ (Magnitude of Gaussian noise) Tetrahedron Average length error (%) Factorisation Quartic Graph rigidity Input image Factorisation Graph rigidity Input image Pose likelihood ij 1 s t q t ij z t i z t j image plane A four-point rigid torso can be used to estimate human pose [1]. However, since it is almost coplanar, factorisation using the SVD will not give an affine reconstruction. Weak perspective projection and Pythagoras’ theorem provide a system of equations. Non-convex objective. Introducing a relaxation and using the nuclear norm heursitic to minimise rank [2], a convex objective can be obtained. Results in situations with a small number of coplanar points are more accurate. Rigid pose estimate critically affects bone length estimates through scale. Pose likelihood learned from motion capture to reduce remaining ambiguity. [1] X. Wei and J. Chai, Modelling 3D human poses from uncalibrated monocular images. ICCV, 2009. [2] M. Fazel, Matrix rank minimization with applications. PhD thesis, Stanford University, 2002.
Transcript of Graph Rigidity for Near-Coplanar Structure from …Graph Rigidity for Near-Coplanar Structure from...
Graph Rigidity for Near-Coplanar Structure from Motion2Queensland University of Technology, Australia1CSIRO ICT Centre, Australia
Jack Valmadre1,2 Simon Lucey1,2Sridha Sridharan2
10−3
10−2
10−1
10−1
100
101
102
103
104
σ (Magnitude of Gaussian noise)
Coplanar Tetrahedron
Ave
rag
e le
ng
th e
rro
r (%
)
Factorisation
Quartic
Graph rigidity
10−3
10−2
10−1
10−1
100
101
102
103
104
σ (Magnitude of Gaussian noise)
Tetrahedron
Ave
rag
e le
ng
th e
rro
r (%
)
Factorisation
Quartic
Graph rigidity
Inpu
t im
age
Fact
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atio
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raph
rigi
dity
Inpu
t im
age
Pose
like
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A four-point rigid torso can be used to estimate human pose [1]. However, since it is almost coplanar, factorisation using the SVD will not give an a�ne reconstruction.
Weak perspective projection and Pythagoras’ theorem provide a system of equations.
Non-convex objective.
Introducing a relaxation and using the nuclear norm heursitic to minimise rank [2], a convex objective can be obtained. Results in situations with a small number of coplanar points are more accurate.
Rigid pose estimate critically a�ects bone length estimates through scale.
Pose likelihood learned from motion capture to reduce remaining ambiguity.
[1] X. Wei and J. Chai, Modelling 3D human poses from uncalibrated monocular images. ICCV, 2009.
[2] M. Fazel, Matrix rank minimization with applications. PhD thesis, Stanford University, 2002.
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