User:Mim.cis/sandbox/Eulerian-Evolution-Lie-Bracket-Diffeomorphism-Group-CA

The orbit of shapes and forms in Computational Anatomy generated by the group action is made into a Riemannian orbit by introducing a metric assocaited to each point in the orbit defininng the norm

at each point of the tangent space of the group. Take as the metric in the group $$ \| \dot \phi \|_\phi \doteq \| \dot \phi \circ \phi^{-1} \|_V $$ implying integrated action from Computational anatomy given by the integrated running cost$$ J(\phi) \doteq \int_0^1 \| \dot \phi_t \circ \phi_t^{-1} \|_V^2 dt =\int_0^1 \| v_t \|_V^2 dt \. $$

Calculating the geodesic connections between elements in the orbit generated through the group action, is induced by generating shortest path connections between elements in the diffeormophism group. The shortest paths are defined by an action integral; first order variations of the solutions in the orbits of Computational Anatomy are based on computing critical points of an action integral computing metric length or energy of the path.

The Euler-Lagrange Equation in the Riemannian Orbit
The original derivation takes a first order perturbation of the flow at point $$ \phi \in Diff_V $$ according to $$ \phi_t^\epsilon \doteq (id + \epsilon \delta \phi) \circ \phi = \phi+\epsilon \delta \phi \circ \phi $$ fixing the boundary so that $$ \delta \phi_0= \delta \phi_1=0 \. $$ Deriving the Euler-Lagrange equation take the first order variation of the action integral: $$ \frac{d}{d \epsilon} J(\phi^\epsilon)|_{\epsilon=0} = \int_0^1 (Av_t |\delta v_t ) dt =0 $$. The Euler equation is a weak equation when$$Av \in V^*$$ is a distribution requiring the $$ad_v^*:V^* \rightarrow V^*$$ notation $$(ad_v^* (Av) | w) \doteq (Av|ad_v(w)), w \in V$$, with Lie bracket $$ad_v(\delta \phi) \doteq (Dv) \delta \phi -(D\delta \phi) v$$

\begin{cases} \text{distribution, generalized function case} & (\frac{d}{dt} Av_t + ad_{v_t}^* (Av_t)|w)= (\frac{d}{dt} Av_t|w) + (Av_t)|(dv_t)w-(dw)v_t) = 0 \, \ t \in [0,1] \ ; \\ \text{density case} &\frac{d}{dt} \mu_t +  (Dv_t)^T \mu_t +(D\mu_t)v_t + (\nabla \cdot v) \mu_t =0 \ . \end{cases} $$  The derivation calculates the perturbation $$ \delta v $$ on the vector fields $$ v^\epsilon = v + \epsilon \delta v \ $$   in terms of the derivative in time of the group perturbation $$ \frac{\partial}{\partial t} \delta \phi_t $$ adjusted by the Lie bracket of vector fields giving the correction
 * $$ ad_v:V \mapsto V $$ given by $$ ad_v(w)\doteq (Dv)w - (Dw)v,  v,w \in V ;$$

the Lie-bracket in this function setting involves the Jacobian matrix, unlike the matrix group. To calculate $$ \delta v $$ define the solution $$ \phi^\epsilon $$ to satisfy the flow
 * $$ \frac{d}{dt} \phi_t^\epsilon = v_t^\epsilon \circ \phi_t^\epsilon, \phi_0^\epsilon = id ,$$

giving

\frac{d}{dt} \phi_t^\epsilon = \frac{d}{dt} \phi_t + \epsilon \frac{d}{dt}(\delta \phi_t \circ \phi_t ) =v_t \circ \phi_t + (D \delta \phi_t)\circ \phi_t v_t \circ \phi_t \. $$ We also have
 * $$ \dot \phi_t^\epsilon = (v_t + \epsilon \delta v_t) \circ (\phi_t + \epsilon \delta \phi_t \circ \phi_t) \simeq v_t \circ \phi_t +\epsilon (Dv_t)\circ \phi_t \delta \phi_t \circ \phi_t + \delta v_t \circ \phi_t +o(\epsilon) \.

$$ Equating the above two equations gives the perturbation of the vector field in terms of the Lie brack adjustment
 * $$ \delta v_t = \frac{\partial}{\partial t} \delta \phi_t - ad_{v_t}(\delta \phi_t) \ . $$

This gives the first variation according to
 * $$ \frac{d}{d \epsilon} J(\phi^\epsilon)|_{\epsilon=0} = \int_0^1 (Av_t | \delta v_t ) dt =\int_0^1 (Av_t | \delta \phi_t - ad_{v_t}(\delta \phi_t) ) dt \, \text{for all} \delta \phi_t \in V \.

$$ Then the adjoint operator $$ ad_v^*: V^* \mapsto V^* $$ gives the Euler equation on general distributions $$ Av \in V^* $$; for all smooth $$ w \in V ,$$
 * $$ ( \frac{d}{dt} Av_t + ad_{v_t}^* (Av_t) | w ) = 0 \, $$

implying the Euler equation for all $$ t \in [0,1], $$

Equation ($$) is the weak version of Euler-equation for geodesics through the group. This equation has been called EPDiff, Euler-Poincare equation for diffeomorphisms and has been studied in the context of fluid mechanics

Matching or Registering Targets in CA
Adding a target term for matching given by the cost $$C(\phi) \doteq \int_0^1 (Av_t|v_t) dt + E(\phi_1)$$ gives the Euler equation with boundary term. Taking the variation gives

$$ \int_0^1 (Av_t |\frac{\partial}{\partial t} \delta \phi_t - (Dv \delta \phi-D\delta \phi v) ) dt +(\frac{\partial E(\phi)}{\partial \phi_1}| \delta \phi_1 ) =-\int_0^1 (\frac{\partial Av_t}{\partial t}+ad_{v_t}^*(Av_t) | \delta \phi_t ) dt +(Av_1 + \frac{\partial E(\phi)}{\partial \phi_1} | \delta \phi_1) $$.

Eulerian Momentum Density Evolution
For the smooth vector density case, with $$ Av_t = \mu_t dx $$ with
 * $$ (Av_t | w) =\int_X \mu_t \cdot w dx \ ,w \in V$$ then the Euler equation exists pointwise as first derived for the density case appear in

. Substituting the vector density $$ \mu_t $$ into the weak equation $$ \int_X \frac{d}{dt} \mu_t \cdot w dx + \int_X \mu_t \cdot (Dv_t \, w - Dw\, v_t ) dx= 0 $$, and integrating by parts gives the pointwise Euler equation for the density case:

Equation ($$) is the pointwise Euler-equation for geodesics.

For the smooth vector density case, with $$ Av_t = \mu_t dx $$ with $$ (Av_t | w) =\int_X \mu_t \cdot w dx \ ,w \in V$$ then the Euler equation exists pointwise for the density case as first derived in. Taking the weak equation and substituing with the vector density $$\mu_t $$ gives
 * $$ \int_X \frac{d}{dt} \mu_t \cdot w dx +

\int_X \mu_t(x) \cdot (Dv_t (x) w(x) -Dw(x) v_t(x)) dx =0, w \in V \. $$ and integrating by parts and using zero-boundary conditions gives
 * $$ \int_X ( \frac{d}{dt} \mu_t + (Dv_t)^T \mu_t +(D\mu_t)v_t + (\nabla \cdot v) \mu_t ) \cdot w dx = 0 \,  w \in V, $$

This gives the pointwise Euler equation for the density case:

Equation ($$) is the pointwise Euler-equation for geodesics.