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

Dense Image Matching in CA via LDDMM
The original dense image matching algorithm took variations with respect to the vector fields;

Image $$ I(x), x \in X $$ with action $$\phi \cdot I \doteq I \circ \phi^{-1}$$, endpoint $$

E(\phi) \doteq \| I\circ \phi^{-1} -I^\prime\|^2

$$, momentum density $$ \mu \doteq (\mu_1,\mu_2,\mu_3) $$, $$ (Av|w) = \int_X \mu \cdot w dx $$:

\begin{cases} & \text{Endpoint Condition:} \ \ \ \ \ \ \ \ \ \ \ \  \  Av_1 =\mu_1 dx, \mu_1 =(I\circ \phi_1^{-1}-I^\prime ) \nabla (I \circ \phi_1^{-1})  \ ,\\ &\text{Conservation equation:} \ \ \ \ \ \ \ Av_t =\mu_t \, dx, \ \mu_t = (D \phi_t^{-1})^T \mu_0 \circ \phi_t^{-1}|D \phi_t^{-1}|, \ \mu_0=(I - I^\prime \circ \phi_1) \nabla I |D \phi_1| \\ \end{cases} $$