Non-Rigid Shape and Motion Recovery: Degenerate Deformations

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Non-Rigid Shape and Motion Recovery Degenerate
Deformations

• Jing Xiao and Takeo Kanade
• CVPR 2004

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Problem Addressed

• Ambiguity in non-rigid SFM if only Rotation
  Constraints used (ECCV 2004)
• SFM recovery in Degenerate Deformations

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Basis Formulation for Non-Rigid Shape

• Shape at any time instant

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With Degenerate Deformation

• Of the K bases
• K1 are rank 1
• K2 are rank 2
• K3 are rank 3
• Kd K1 2xK2 3xK3
• WMB is not a unique decomposition
• Essentially the problem is to find G

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Degenerate contd

• First K3 triple columns of M correspond to
  non-degenerate basis and the rest to degenerate
  basis
• rj is a 3x1 eigen vector that corresponds to the
  degenerate basis shape
• Arises cause Bi (degenerate) can be factored as
  

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Rotation Constraints

• Denote by Qi, we have
• Qi has unknowns, so given enough frames
  we can find a solution?
• This is not true in general
• Qi can be written as where H satisfies

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Rotation cont

• is an arbitrary scalar and is a skew
  symmetric matrix
• The solution has degrees of
  freedom

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Basis Constraints

• Ambiguity cause any non-singular transformation
  on the bases gives another valid set of bases.
• To handle this choose the set of K3 frames that
  have smallest condition number (ECCV paper)
• Denote the chosen frames as the first K3 frames
• Coefficients will be

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Basis contd

• New constraints
• Finally combining Rotation
• and Basis constraints

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Constraints contd

• degrees of
  freedom and linear
  constraints.
• Therefore solution space has
  degrees of freedom.
• When ND is 0, there is a unique solution.

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Finding Qi

• If K2gt0 then
• If K20, a unique solution exists using the
  constraints.

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Results

• Synthetic data

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Results

• Real Data