User:Mim.cis/sandbox/Landmark Matching in CA via LDDMM

Landmark or Point Matching With Correspondence in CA via LDDMM:
Landmarked shape $$ m\doteq \{ m_1,m_2,\dots \} $$ with action $$ \phi \cdot m \doteq \phi(m) $$, with endpoint $$ E(\phi) \doteq  \textstyle \sum_u \displaystyle \|\phi(m_u)-m_u^\prime \|^2 $$, momentum a singular distribution $$ \displaystyle Av \in V^*\textstyle $$.

\begin{cases} & \text{Endpoint Condition:} \ \ \ \ \ \ \ \ \ \ \ \  \ Av_1= \sum_{u=1}^n p_1(u)\delta_{\phi_1(m_u)}, p_1(i)=(m_u^\prime-\phi_1(m_u))  ,\\ &\text{Conservation equation:} \ \ \ \ \ \ \  Av_t = \sum_{u=1}^n p_t(u)\delta_{\phi_t(m_u)} ,p_t(u) = (D \phi_t^{-1})_{|\phi_t(m_u)}^T p_0(u),p_0(u) = D\phi_1^T (m_u^\prime-\phi_1(m_u)), \end{cases} $$ with the weight evolution also given by $$ p_t(u) = (D\phi_{t1})_{|\phi_t(m_u)}^T p_1(u)\, \phi_{t1} \doteq \phi_1 \circ \phi_t^{-1} $$