Gradient Calculations for Nonrigid Surface Registration

The goal of rigid registration is to align a source surface $X$ to a target surface $Y$. The alignment process involves iteratively transforming $X$ closer and closer to $Y$, such that $X=Z^0\to Z^1\to Z^2\to \mathrm{.}\mathrm{.}\mathrm{.}\to Z^T=Y$. By applying two simplifications, we can trace the minimization problem back to solving a system of linear equations.