TBD
video site
target: align 2 points into the same frame
PART1: Data Association & SVD
direct and optimal solution: \(\sum||y_n - \bar{x}_n||^2p_n \rightarrow min\)
properties: no initial guess and no better solution exist
weighted sums: \(x_0 = \frac{\sum x_n p_n}{\sum p_n}\) and \(y_0 = \frac{\sum y_n p_n}{\sum p_n}\)
cross-correlation matrix: \(H = \sum (y_n - y_0)(x_n - x_0)^T p_n\)
\(svd(H) = UDV^T\)
\(\Rightarrow R=VU^T, t = y_0 - Rx_0\)
use the local coordinates defined by the point set (set the origin as weighted mean \(y_0\))
\(\sum ||(y_n - y_0) - (R x_n + t - y_0)||^2 p_n \rightarrow min\)
start with \(\bar{x}_n - y_0 = Rx_n + t - y_0\)
rewrite the translation \(\bar{x}n - y_0 = R(x_n + R^Tt - R^Ty_0)\)
introduce a new variable \(x_0 = R^Ty_0 - R^Tt\)
\(\bar{x}_n - y_0 = R(x_n - x_0)\)
minimization problem: \(R^*, x^*_0 = \underset{R, x_0}{argmin}\sum || y_n - y_0 - R(x_n - x_0) ||^2 p_n\)
objective function: \(\Phi(x_0, R) = \sum [(y_n - y_0) - R(x_n - x_0)]^T[(y_n - y_0) - R(x_n - x_0)] p_n\)
let \(\frac{\delta \Phi}{\delta x_0} = 0 \Rightarrow x_0 = \frac{\sum x_np_n}{p_n}\)
then let \(\frac{\delta \Phi}{\delta x_0} = 0 \Rightarrow R^* = \underset{R}{argmax}\sum b_n^TRa_np_n\) \((b_n = y_n - y_0, a_n = x_n - x_0)\)
using the trace?: \(R^* = \underset{R}{argmax}\ tr(RH)\)
\(H\) is the covariance matrix: \(H = \sum (a_nb_n^T)p_n\)