$$ p_{transformed} = \underbrace{\sum_i w_iM_i(t) M_i^{-1}(0)}_\text{not a rigid matrix} p $$

This results in collapse artifacts:

Bending:
elbow collapse |
Twisting:
candy wrapper effect |

Moreover, it is hard, if not impossible, to design joints with plausible deformations in a large range. The following image shows armpit bulging occuring while the arm is lift far above the reference position.

Armpit bulging |

Even Dual Quaternion Skinning creates artefacts in bending:

Bulging artefact in Dual
Quaternion Skinning (courtesy of L.Kavan) |

$$ \begin{equation} \label{eq:blend shapes}

S = \sum_i w_i S_i

\end{equation}

$$

where $i$ spans the number of example shapes. The limitations are:

- There are as many dimensions (sliders to tune) as example shapes.
- Example shapes interfere with each other:

Blendshape artefact: smirk counteracts raise. (courtesy of
J.P.Lewis talk slides) |

Example of two-dimensional pose
space in facial animation. |

This requires mathematical tools to interpolate shapes based on an arbitrary, uneven distribution of samples: scattered interpolation.

For $p$ poses and $n$ vertices in 3d, RBF interpolation requires the factoring of a $p\times p$ matrix, and then the associated solution of $3n$ equation systems at initialization time. During run-time, each interpolation requires the evaluation of a formula similar to (\ref{eq:blend shapes}), where $i$ spans the dimensions of the pose space.

PSD layered on skinning. |

PSD has been used in facial animation to superimpose fine wrinkles on top of a coarse, physically-based face deformation.

Hybrid face animation pipeline. |

$$

\| f-f_j\|_v = \sqrt{ \sum_{i=1}^{F} \alpha_{v,i}( f_i - f_{i,j} )^2 }

$$

where $v$ is the vertex index, $F$ the number of examples, and $f_{i,j}$ is the strain of the $i$'th feature edge in the $j$'th pose. The weights $\alpha$ are assigned based on proximity to the feature edges as illustrated in the following figure.

Weight kernel used to compute
per-vertex distance. |

Francois Faure, University of Grenoble. Main page