Globally Optimal Direction Fields I am trying to read the work by Knöppel et al and I have two questions:


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*The authors mention that the Dirichlet's energy is not a reliable measure of quality of direction fields, since under refinement the energy contributed by a singularity grows without bound. 
I see the point, however, right in the next sentence they re-formulate it as an eigenvalue problem without suppressing unbounded potential. Can someone please explain me how it became well-defined?

*Do I understand correctly that the optimality is guaranteed only for surfaces without boundaries/constraints? For example, I fail to see how they find the global minimum of the equation (10), any hints?
 A: *

*In equation 6 of the paper, they redefine the energy of a given n-direction field as the minimal dirichlet energy over any scaling of a unit n-direction field. i.e. the function $a: M \rightarrow \mathbb{R}$ in this equation is a function from the manifold $M$ to the real numbers $\mathbb{R}$.  The constraint $||a|| = 1$ is saying that the $L^2$ norm of the function $a$ is equal to 1. They then make the connection that minimizing this newly defined energy is equivalent to solving  the time-independent Schrödinger equation and appeal to results on this problem to conclude that the this newly defined energy for a directional field is indeed well defined.


In short, they don't reformulate the Dirichlet energy as an eigenvalue problem, but rather they reformulate the newly defined energy as an eigenvalue problem.
Edit: I should add that for the Dirichlet energy below
$\frac{1}{2}\int_M{|\nabla Z|^2}dA = \frac{1}{2}\int_M{|\nabla a|^2 + a^2|\omega|^2dA} = \frac{1}{2}\langle\langle(\Delta + |\omega|^2)a,a\rangle\rangle$
the scalar field $a$ can approach zero as the radius $r$ goes to zero, and so even though $|\omega|$ goes to infinity as $r$ goes to zero, the product $a^2|\omega|^2$ doesn't necessarily.


*The optimality is still guaranteed for surfaces with constraints, but the question is optimal in what sense? That is to say, in the case when they want to include alignment constraints, they are minimizing a different objective function, one that includes the alignment constraints as one of its terms. The parameter $t$ allows one to go through the spectrum from the alignment constraints being perfectly satisfied, to the smoothness energy being perfectly minimized.


Alternatively, if the set of alignment constraints were sparse (i.e. a boundary condition), one could conceivably apply hard constraints to the system while minimizing the smoothness energy only. 
