User:Mim.cis/sandbox/Hamiltonian-Control-CA

In Computational Anatomy (CA) the diffeomorphic flows $$\phi_t, t \in [0,1]$$ are used to flow the coordinate systems, and the vector fields $$v_t, t \in [0,1]$$ are viewed as the control within the anatomical space. The flows and vector fields are related via the Lagrangian and Eulerian ordinary differential equations $$\dot \phi_t = v_t \circ \phi_t$$ determining the motions of coordinates. For purposes of calculating geodesic flows between coordinate systems, CA has a least action principle on the kinetic energy which the flows satisfy, with Hamilton's principle of least action giving the geodesics satisfying an Euler-Lagrange equation for flows of minimal energy. The Lagrangian kinetic energy in CA generalizes the kinetic energy of free bodies from classical mechanics which derives Newton's laws of motion for free bodies. In CA, since the velocities of the flows of particles are in a function space, the Lagrangian kinetic energy is more akin to an $$L^2$$ Hilbert space norm as relevant for the Euler equations de scribing incompressible fluids mechanics. However, the Hilbert space norm central to CA is a Sobolev norm, for flows in $$\mathbb{R}^3$$ minimallly supporting $$>2$$ derivatives in integral square for each component of the vector field; this ensures the flows are diffeomorphisms which strongly separates it from the inviscid fluid case.

The Eulerian diffeomorphic shape momentum in CA which is conserved and satisfies the Euler-Lagrange equation for least-action on the Lagrangian kinetic energy departs from the Euler equations (fluid dynamics) for inviscid fluids for which momentum is point supported in space. At any point in space diffeomorphic shape momentum is a differential combination of co-localized spatial velocities implying the vector fields of the geodesics are smooth superpositions of Green's kernels. In the early work of CA all of the equations of motion were derived from the Lagrangian on the kinetic energy. Interestingly much of the recent work has been on the geodesic flows of submanifolds of landmarks, surfaces and subvolumes of $${\mathbb R}^3$$ including subcortical surfaces of the brain. The manifolds and submanifolds being controlled are of lower dimension then the diffeomorphisms themselves. This is exploited in the Hamiltonian view in which the generalized momentum distribution is reparameterized in terms of a co-state variable, often termed the conjugate momentum, canonical momentum or conjugate state.

=Langrangian and Hamiltonian in Computational Anatomy= The model used in Hamiltonian control in Computational anatomy is that anatomical shapes and forms are,a linear state space embedding shapes $$\mathcal {Q} \subset \mathcal{M}$$ The model is that of a dynamical system, with elements in the space of shape reached by flowing the state under diffeomorphic flows $$ t \mapsto q_t \doteq \varphi_t \cdot q_0 \in \mathcal{Q} $$, and the control the vector field $$ t \mapsto v_t \in V$$ with Lagrangian and Eulerian flow dynamics $$\dot \phi_t = v_t \circ \phi_t$$. The control and state are related via the infinitesimal action of the flow $$ \dot q_t = v_t \cdot q_t, q_{t=0}=q_0. $$

To ensure smooth flows of diffeomorphisms, the vector fields are at least 1-time continuously differentiable in space , with the space of vector fields $$(V, \| \cdot \|_V )$$ modelled as a reproducing kernel Hilbert space (RKHS), with norm  $$ \| v\|_V^2 \doteq (Av|v) $$ for $$ A: V \rightarrow V^*  $$ a 1-1, differential operator. .

The Lagrangian for the control of the system is the kinetic energy:

the Hamiltonian has added Lagrange multiplier $$ p_t \in \mathcal{Q}^* $$. constraining $$ \dot q_t = v_t\cdot q_t$$ :

H(q_t,p_t,v_t)=(p_t|v_t \cdot q_t)-\frac{1}{2}(Av_t|v_t) .$$

For each state the $$(p | q), p \in \mathcal {Q}^*, q\in \mathcal {Q}$$ takes a different form; a sum of landmark points or an integral over surface parameterization or subvolume integral. The Pontryagin maximum principle gives the optimizing vector field $$ \hat v_t, t \in [0,1] $$, and the reduced Hamiltonian $$ H(q_t,p_t) \doteq \max_v H( q_t, p_t,v) \. $$

The velocity associated to the geodesics take the form

$$v_{\max} = \arg \max_{v \in V} H(q,p,v)$$ with

Dense Image Matching
Examine $$ q \doteq \varphi \in \mathbb{Q} \doteq Diff_V $$. Then $$ q_t \doteq \varphi_t \cdot I, q_0 = I $$ with $$ \dot q_t = \dot \varphi_t = v_t \circ \varphi_t, \varphi_0=id $$. Let the domain of integration be $${\mathbb R}^3$$, then

H(\phi_t,p_t,v_t) =\int_{{\mathbb R}^3} p_t \cdot (v_t \circ \phi_t) dx - \frac{1}{2} (Av_t | v_t) $$. The Hamiltonian takes the form $$ H(\phi_t,p_t)= \frac{1}{2} \int_X p_t(x) \cdot \int_X K(\phi_t(x),\phi_t(y)) p_t(y) dy dx   $$

The Eulerian momentum has a density with the geodesic velocity given as follows:

\begin{cases} & v_t (\cdot) = \int_{\mathbb{R}^3} K(\cdot, \phi_t(y)) p_t(y) dy \\ & Av_t = \mu_t dx \ \text{with} \ \mu_t = p_t \circ \phi_t^{-1} | D \phi_t^{-1}|\\ & \dot p_t = -(Dv_t)_{|\phi_t}^T \ p_t \end{cases} $$

Image Matching with Reduced State via Advective Action
Examine $$ q \doteq I \circ \varphi^{-1} \in \mathbb{Q} \doteq \mathcal{I} $$. Then $$ q_t \doteq I \circ \phi_t^{-1}, q_0 = I $$ with $$ \dot q_t = - \nabla q_t \cdot v_t $$. Let the domain of integration be $${\mathbb R}^3$$, then

H(\phi_t,p_t,v_t) =\int_{{\mathbb R}^3} p_t (- \nabla q_t \cdot v_t) dx - \frac{1}{2} (Av_t | v_t) $$. The Hamiltonian takes the form $$ H(\phi_t,p_t)= \frac{1}{2} \int_X p_t(x)\nabla q_t(x) \cdot \int_X K(x,y) p_t(y)\nabla q_t(y) dy dx   $$

The Eulerian momentum has a density with the geodesic velocity given as follows:

\begin{cases} & v_t (\cdot) = -\int_{\mathbb{R}^3} K(\cdot, y) p_t(y)\nabla q_t(y) dy \\ & Av_t = \mu_t dx \ \text{with} \ \mu_t = -p_t \nabla q_t \\ & \dot p_t = - \nabla \cdot (p_t v_t) \end{cases} $$