Handling affine transformation : Polar Decomposition

Key-pose transforms are often given as \(4\times 4\) matrices.
How can we interpolate b/w affine transforms?

- - Splitting linear \(\mathrm{M}\)/translation part is easy.
- - Problem: interpolating the linear part \(\mathrm{M}\)
- - mixes rotation, scaling, shearing

\(\Rightarrow\) Split \(\mathrm{M}\) into: rotation part, and scaling/shearing.

- - Interpolate rotation with quaternion (ex. SLERP/LERP)
- - Scaling/shearing using componentwise interpolation (ex. linear).

- Polar decomposition: \(\mathrm{M} = \mathrm{R}\;\mathrm{D}\)
- - \(\mathrm{R}\): Rotation matrix
- - \(\mathrm{D}\): Positive semi-definite matrix
K. Shoemake and T. Duff, Matrix Animation and Polar Decomposition. Graphics Interface 92.

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

- Or 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