Following this post
Jacobian matrix of the Rodrigues' formula (exponential map)
What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$?
Basically, I want to optimize
$argmin_{\mathbf T} \sum_i \mathbf{e}_i(\mathbf{T}), i = 0..N $ with
- $\mathbf{T} \in SE(3)$
- $\mathbf{e}_i(\mathbf{T}) = exp(a_i log(\mathbf{T}))\cdot P_i – P^*_i $
- $a_i \in [0,1]$
- $P_i, P_i^* \in \mathbb{R}^3$
This can be tought as a variation of the classical ICP (http://en.wikipedia.org/wiki/Iterative_closest_point) problem, where each point $P_i$ is captured from a different pose linearly interpolated (with known factor $a_i$) between the origin and the pose $\mathbf{T}$ that is subject of the estimation.
I want to use a Gauss-Newton like approach, thus I can reformulate the problem in terms of the algebra $\mathfrak{se3}$, i.e.,
$argmin_{\mathbf \omega} \sum_i \mathbf{e}_i(\mathbf{\omega}), i = 0..N $ with
- $\mathbf{\omega} \in \mathfrak{se3}$ (i.e., $\in \mathbb{R}^6$)
- $\mathbf{e}_i(\mathbf{\omega}) = exp(a_i log(exp(\omega)\cdot\mathbf{T}))\cdot P_i – P^*_i $
and I need to evaluate $\left.\frac{\partial \mathbf{e}_i(\mathbf{\omega})}{\partial \mathbf{\omega}}\right|_{\mathbf{\omega} = 0}$
the problem w.r.t. the original question is that the most external $exp (\cdot)$ function is evaluated in a generic point and not in $0$.
I tried succesfully to use the Pade approximation theorem to compute the generic $\frac{\partial \mathbf{e}_i(\mathbf{\omega})}{\partial \mathbf{\omega}}$, but I want something in closed form, if possible!
This comment (Jacobian matrix of the Rodrigues' formula (exponential map)) in the original question states that $\frac{\partial \mathbf{e}_i(\mathbf{\omega})}{\partial \mathbf{\omega}_k} = exp(\omega) \cdot \mathbf{G}_k$. Emplyoing this formula the convergence is not always reached (while the approximated Jacobians works fine), thus I think there is something wrong with this Jacobian.
Notice that to simplify the problem, everything may be formulated in $SO(3)$ at the moment.
Best Answer
I suggest reading this. It uses a variation of Lucas-Kanade to calculate the pose between two cameras. There is also the formula for the Jacobian of the exponential map not on $0$ in the paper (I don't know how it's derived though).