Jacobian matrix H

[a-github-learner]

Nice work！ II have a question about how to obtain the Jacobian matrix $\boldsymbol{H}_{k,j}^n$？Looking forward to your response.

[Ji1Xingyu]

Please see attached for the derivation of the Jacobian matrix. Also I have updated the sup_material.pdf to include this derivation.

[a-github-learner]

Thank you for your timely reply and detailed derivation! I have two additional questions:

1. Dose $^{G}\hat{\boldsymbol{p}}_j$ in your sup_material.pdf has the same meaning as $^G\mathbf{\hat{\mu}}_j^n$, which means the centroid of the voxel? Or does it simply denote the position of points in the global frame?
2. In $\left(\Lambda_j^n\right)^{-1}=\left(\mathbf{Q}_j^n\right)^T \mathbf{V}_k^{-1} \mathbf{Q}_j^n$, since $\mathbf{V}_k$ is a diagonal matrix, how is matrix decomposition performed on it? Is $\mathbf{Q}_j^n$ an identity matrix？

[Ji1Xingyu]

1. ${}^G\hat{\boldsymbol{p}}_j$ denotes the point ${}^{L_k}\hat{\boldsymbol{p}}_j$ after being transformed to the global frame.
2. Actually, the matrix decomposition is performed on $\left( {\Lambda}_j^n \right)^{-1}$. Moreover, since it is a diagonal matrix, this decomposition can be simplified as solving the following equation on each diagonal element: $q_k = \sqrt{\frac{v_k}{\lambda_k}}$.