Timeline for Closed Form Solution for Optimization Problem over the Space of Rigid Transforms
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 21, 2018 at 22:12 | answer | added | DCM | timeline score: 1 | |
| Oct 21, 2018 at 0:20 | comment | added | user3749105 | All $\mathbf p_i$ are distinct 3D points not lying on a sphere. $M_i$ are of the form $M_i = \mathbf u_i {\mathbf u_i}^{\top} - I$, where $\mathbf u_i \in \mathbb R^3$ are unit-length vectrors and $I$ is the identity matrix. | |
| Oct 20, 2018 at 16:48 | comment | added | DCM | Something I should have said before: to avoid 'silly' cases, you probably want to insist that your $p_i$s are all distinct (which obviously requires $n>1$) and your $M_i$s are (at least) not all zero. Could you possibly provide some background? The answer to your question as currently stated is just `no' as currently stated if you're interested in a unique minimiser. One other case to consider; what happens if all your $M_i$ are isometries and your $p_i$ all lie on a sphere? | |
| Oct 20, 2018 at 0:26 | comment | added | user3749105 | Recall that I am interested in a closed form solution :) But in any case, let us know about your results. | |
| Oct 19, 2018 at 22:56 | comment | added | DCM | Re. constraints, we do have the coordinate functions $x_{ij}: \mathbb{R}^{4\times 4} \to \mathbb{R}, A\mapsto A_{ij}$ to help out with these, so I'd be surprised if it's really that bad (the equations you get out of the Lagrange multiplier method might be horrible of course). Alternatively, there is a (I think surjective) exponential map $\mathbb{R}^3 \times \mathbb{R}^3 \to SE(3)$ available; it's use would make this an unconstrained problem on $\mathbb{R}^3 \times \mathbb{R}^3$. I'll stop being so lazy and try some of this tomorrow :) | |
| Oct 19, 2018 at 17:21 | comment | added | user3749105 | Thanks for the comment, but I do not see how this can help... The constraints for keeping A on the manifold of rigid motions seem to be the hard part. | |
| Oct 18, 2018 at 21:05 | comment | added | DCM | To clarify what I mean above: I'd be inclined (perhaps unhelpfully) to think of this as minimising $f:\mathbb{R}^{4 \times 4}\to \mathbb{R}, A\mapsto \sum_i\left|\left( \begin{array}{cc} M_i & 0 \\ 0 & 1 \end{array}\right) A \left( \begin{array}{c} p_i \\ 1 \end{array}\right)\right|^2$ subject to some constraints which keep $A$ on the manifold of rigid motions. I don't claim to have any expertise in this area, but I like the question so I thought I'd post in case it helps (sorry if you've already tried something like this). | |
| Oct 18, 2018 at 20:49 | comment | added | DCM | This looks a bit like a Lagrange multiplier type problem no? | |
| Oct 18, 2018 at 18:30 | history | asked | user3749105 | CC BY-SA 4.0 |