Riemannian geometry · Definition

Minimum energy

Riemannian geometryoptimizationLDDMM

Before you start you need to know

The minimum energy summarizes the cost of the best admissible deformation, that is, the one requiring the least effort among those satisfying \(\mathcal{A}\).

Definition

The minimum energy \(E_{\min}\) is the lowest cost among admissible deformations; in LDDMM it measures how much effort the field generating the transformation requires.

\[E_{\min}=\min_{\varphi\in\mathcal{A}}E(\varphi)\]

By definition, \(E_{\min}\le E(\varphi)\) for every \(\varphi\in\mathcal{A}\). The minimum is attained when there is an admissible deformation achieving that lowest cost.

Suppose a template must reach a target. If \(E(\varphi_A)=2\) and \(E(\varphi_B)=5\), both admissible, then \(E_{\min}\le 2\) and \(\varphi_A\) is the better candidate between these two.

A common mistake is comparing energies of deformations that do not reach the target: a very low energy is useless if it is not in \(\mathcal{A}\). That is why the minimum is taken only over the admissible set.

Think about it

If the admissible set grows, can \(E_{\min}\) increase?

No: with more candidates the minimum can only stay the same or decrease.

Exercises 11

  1. If \(\mathcal{A}\) is the set of admissible deformations, what does \(E_{\min}=\min_{\varphi\in\mathcal{A}}E(\varphi)\) express?
  2. Two transformations reach the same target: \(E(\varphi_A)=2\) and \(E(\varphi_B)=5\). If both are admissible, what can be stated?
  3. If the norm used to measure \(v_t\) is changed, what can change even if the endpoints are the same?
  4. A very low-energy deformation does not satisfy the condition of reaching the target. Why can it not be the LDDMM solution?
  5. What directly determines the value of the energy of a flow in LDDMM?
  6. What does it mean for a deformation to be admissible in LDDMM?
  7. What is the name of the smallest value of \(E\) over the admissible set?
  8. What does the field \(v_t\) represent in LDDMM?
  9. What does a lower energy between two admissible deformations indicate?
  10. Two admissible deformations \(\varphi_1\) and \(\varphi_2\) each have \(E=4\). What can be stated?
  11. Let \(E(\varphi_k)=(k-3)^2+2\) for \(k=1,\dots,5\), with \(\mathcal{A}=\{\varphi_1,\varphi_4,\varphi_5\}\). What is \(E_{\min}\)?

What comes next

References

  • M. Faisal Beg, Michael I. Miller, Alain Trouvé, Laurent Younes, Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms, 2005.