I am currently working on the strain energy function for a particular graph.
The paper I am currently referencing is "Spatial Relations Preserving Character Motion Adaptation".
I am asking because I do not understand the formula of the thesis.
The formula is:
$E_L(V_i') = \sum_{j}\frac{1}{2}\left\|\delta_j-L(p'^i_j) \right\|^2 = \frac{1}{2}V_i '^TM_i^TM_iV_i -b_i^TM_iV'_i+\frac{1}{2}b_i^Tb_i $
In the formula above, $p$ represents the vertex $(x,y,z)$ at frame i. $M$ represents the Laplacian matrix and $b$ represents the Laplacian coordinates.
At this time, when there are 5 vertices, $V$ becomes a $5X3$ matrix, $M$ becomes a $5X5$ matrix, and $b$ becomes a $5X3$ matrix.
The matrix calculation shows the values of the final energy function in the form of a $3X3$ matrix. However, in order to minimize a function, I think it should come out in scalar form.
If you want matrix calculations to be displayed in scalar form, you can convert them to $1X15$ form ($V_i = (p_1^T...p_5^T)$) by swapping the elements of $V$. However, in this case, the Laplacian matrix $M$ must be derived in the form of $15X15$, and the Laplacian matrix is regarded as a square matrix according to the number of vertices.
I think $M$ is a specific matrix that has been modified, or I think that $x$, $y$, $z$ of $V$ should be separately processed when performing matrix calculation. I'd appreciate it if you could point out where I'm wrong about matrix calculations.
The norm could probably be the Forbenius Norm. You can find this paper, "Optimal Step Nonrigid ICP Algorithms for Surface Registration", for details of how the norm of a matrix is calculated to minimize the energy term. The Forbenius Norm is to calculate the norm of a matrix $\textbf{A}$. It is given by: $$||\textbf{A}_{m\times n}||^2 = \sum_m\sum_n|a_{ij}|^2$$