The Legendre Transform for Hamiltonians?

[lupine]

I'm trying to implement and extend the work of Emmanuel Prados (http://www-sop.inria.fr/odyssee/research/prados-faugeras:04b/thesis.htm). I'm trying to follow how Appendix A, "How to transform a convex Hamiltonian into a HJB Hamiltonian; Legendre Transform", works for the provided example. I've contacted the author, but thought I might try here and see if anybody else could help me out in the meantime/

Given a Hamiltonian of the form

$$H_{R/T}^{orth}(x,p) = I(x) \sqrt{1+|p|^{2}} + p \cdot l - \gamma$$

we can apparently use the Legendre transform to obtain

$$H_{R/T}^{orth*} = -\sqrt{I(x)^2 - |a-l|^2}+\gamma \qquad x\in\overline{B}(1,I(x))$$

The process given for this transform is:

1. Find the gradient $$\nabla_p H$$
2. Solve the equation $$\nabla_p H(p_0)=0$$ ($$p_0$$ depends on x)
3. Calculate $$H(x,p_0)$$

Here, p is a two-element vector (corresponding to the gradient of the surface $$p = \nabla u(x)$$, I(x) is the image intensity at pixel x, l is a two-element vector corresponding to the light source direction (constant), and gamma is a further light constant.

Finding the gradient seems pretty simple -- plug in the equation into any CAS package gives:

$$\frac{dH}{dp}=\dot{H}(p) = \left[ \left( \frac{I(x) p_x}{\sqrt{1+p_{x}^{2}+p_{y}^{2}}} + l_x \right), \left( \frac{I(x) p_y}{\sqrt{1+p_{x}^{2}+p_{y}^{2}}} + l_y \right)\right]$$

The next step, solving $$\nabla_p H(p_0)=0$$, is where I start coming undone. Since we now effectively have two unknowns, if we then substitute $$p_0=\left(p_{0_x},p_{0_y}\right)$$ for p, when solving for $$p_0$$, the variables will not be independent between the two components.

This is really confusing me, and I hope I've made it straight-forward enough for others to read and understand.

Thanks

Lupine

 

[jvicens]

Dear Lupine,

I can understand your confusion with the process given for the Legendre transform. It is indeed a complex mathematical concept, but it is essential for understanding and extending the work of Emmanuel Prados.

To solve the equation \nabla_p H(p_0)=0, we need to use the first-order optimality condition, which states that the gradient of a function at a stationary point is equal to zero. In this case, the function is the Hamiltonian H_{R/T}^{orth}(x,p), and the stationary point is p_0. Therefore, we need to set the gradient of the Hamiltonian at p_0 equal to zero and solve for p_0. This will give us the desired p_0 for use in the Legendre transform.

Substituting p=p_0 in the gradient of the Hamiltonian, we get:

\dot{H}(p_0) = \left[ \left( \frac{I(x) p_{0_x}}{\sqrt{1+p_{0_x}^{2}+p_{0_y}^{2}}} + l_x \right), \left( \frac{I(x) p_{0_y}}{\sqrt{1+p_{0_x}^{2}+p_{0_y}^{2}}} + l_y \right)\right] = 0

Now, since p_0 is a two-element vector, we have two equations and two unknowns. We can solve for these unknowns by equating the coefficients of p_{0_x} and p_{0_y} to zero:

\frac{I(x)}{\sqrt{1+p_{0_x}^{2}+p_{0_y}^{2}}} + l_x = 0

\frac{I(x)}{\sqrt{1+p_{0_x}^{2}+p_{0_y}^{2}}} + l_y = 0

From these equations, we can see that p_{0_x} and p_{0_y} are not independent, and we can solve for them by substituting one equation into the other:

\frac{I(x)}{\sqrt{1+p_{0_x}^{2}+p_{0_y}^{2}}} = -l_x = -l_y

Squaring both sides and rearranging, we get:

p_{0_x}^{2} + p_{0_y}^{2} =

 

[tacman]

The Legendre transform is a mathematical operation that converts one function into another function. In the case of Hamiltonians, it is used to transform a convex Hamiltonian into a Hamilton-Jacobi-Bellman (HJB) Hamiltonian. This is useful in solving certain problems in optimal control and dynamic programming.

In the case of the example provided, the Legendre transform is used to transform the Hamiltonian H_{R/T}^{orth}(x,p) into H_{R/T}^{orth*}(x,p). This is done by finding the gradient of H_{R/T}^{orth}(x,p) and solving the equation \nabla_p H(p_0)=0 for p_0, which depends on x. This results in the transformed Hamiltonian H_{R/T}^{orth*}(x,p), which can then be used in solving the problem at hand.

The process of solving \nabla_p H(p_0)=0 can be challenging, as it involves solving a system of equations with two unknowns. However, there are various techniques and methods that can be used to solve such systems, such as substitution or elimination. It may also be helpful to consult with a mathematician or expert in this field for further assistance.

In conclusion, the Legendre transform is an important tool in solving problems involving Hamiltonians, and it is worth investing time and effort to understand and implement it correctly. I hope this helps and I wish you success in your work!