Laplacian deformation least square solution

Need to minimize \(\|\mathrm{M}\,q - b\|^2\)
- Similar to solve the linear system \((\mathrm{M}^T\,\mathrm{M})\, q = \mathrm{M}^T\,b\) (Normal equations)

\(\mathrm{M}^T\mathrm{M}\) is sparse (diagonally dominant)

- Can use standard sparse solver (CG, Jacobi, Gauss Seidel, etc.)
- Reminder: Never try to compute the inverse matrix ! Only solve the linear system.
- Special solvers (QR factorization) doesn't even need to compute the normal equations

System can be solved independently on \(x,y,z\) coordinates. Same matrix \(\mathrm{M}\)
- \(\mathrm{M}\,q_x = b_x\), \(\;\;\mathrm{M}\,q_y = b_y\), \(\;\;\mathrm{M}\,q_z = b_z\)

Changing constraints position \(c_i\) only impact rhs \(b\)
- \(\rightarrow\) \(\mathrm{M}\) can be factorized once, and reused during interactive deformation

Changing constrained vertex impact \(\mathrm{M}\)
- \(\rightarrow\) need to be re-factorized