How to use PCA to reconstruct 3D Mesh？ Given a set of human body mesh {S_i}, for each S_i, the points are N*3, we can do PCA on that. We can use PCA to reduce the dimension of {S_i}, for example to k dimension. Let the main direction of PCA be [D1, D2, ..., Dk], then the body geometry can be represented by S = W1 * D1 + W2 * D2 + ... + Wk * Dk, where W = [W1, W2, ..., W3] is the weights.

But how can we get the weight W of one specific mesh S_i in {S_i}?
Given a matrix $D \in \mathbb{R}^{3N\times k}$ where each column is normalized $D_i$, You can get $W = D^TS_i$, and later it can be used to reconstruct the shape with $S_i = DW$ (With some error if you truncated some basis). The acquisition of weights is equivalent to calculating dot products between each PCA basis and $S_i$. What is done here is you project $S_i$ onto each orthogonal basis $D_i$, and calculate the length of the projected vector, so you are basically calculating the required contribution of each basis to construct the given $S_i$.

If you are using unnormalized PCA basis, you can decompose the unnormalized PCA basis matrix into a diagonal matrix an an orthonormal matrix (Say, $\Lambda D$), then use a similar approach to obtain the contribution of each basis. In this case, you'll have to use $\Lambda^{-1}D^T$ instead of $D^T$.