Handling affine transformation : Polar Decomposition

Key-pose can be given as matrix \(\mathrm{M}\) containing rotation, but also scaling and shearing.

\(\Rightarrow\) Cannot convert \(M\) to quaternion (not a rotation).

\(\Rightarrow\) Separate rotation component from shearing and scaling component using polar decomposition.

[K. Shoemake and T. Duff, Matrix Animation and Polar Decomposition. Graphics Interface 92.]

- Polar decomposition: \(\mathrm{M} = \mathrm{R}\;\mathrm{D}\)
- - \(\mathrm{R}\): Rotation matrix
- - \(\mathrm{D}\): Positive semi-definite matrix

- Polar decomposition is obtained from SVD
- \(\mathrm{SVD}(\mathrm{M}) = W \Sigma \, V^{T}\) with
  \(\mathrm{R}=\mathrm{W}\mathrm{V}^T\)
  , \(\;\;\mathrm{D}=\mathrm{V} \Sigma \mathrm{V}^T\)

- Numerically, \(\mathrm{R}\) can be computed using the following iterative scheme
- \(R_0 = \mathrm{M}\,,\; R _ {i+1}=0.5 \, (R _ i+(R_i^{-1}) ^ {T})\)

pictures/linear.png

pictures/polar.png