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report : PCA decomposition explanation

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Marion Lenfant 2015-03-11 22:49:29 +01:00
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report/subsections/junctions.tex
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@ -30,7 +30,26 @@ We begin by identifying the last circles of each spline of the junction (see fig
\end{center}
\end{figure}
To be able to do it the first step is to cut the circles in two parts to find the upside and downside of each of them. This can be done by computing the best fitting plane for the set of circles' center points (figure \ref{junction2}).
To be able to do it the first step is to cut the circles in two parts to find the upside and downside of each of them. This can be done by computing the best fitting plane for the set of circles' center points (figure \ref{junction2}).
The equation to find is easy when there is only 3 circles : first we compute the normal vector of the plane and then the constant element of the equation.
However when there is more than 3 circles to join, so more than three points to approximate, we need to use a PCA (Principal component analysis) decomposition to find the best fitting plane.
The first thing to do is to create X the matrix of the points coordinates.
\[
X =
\begin{pmatrix}
x_{1} & x_{2} & \cdots & x_{n} \\
y_{1} & y_{2} & \cdots & y_{n} \\
z_{1} & z_{2} & \cdots & z_{n}
\end{pmatrix}
\]
Then subtract it with the centroid matrix associated.
\[
X - X_{m}
\]
The normal vector of the best-fitting plane is the left singular vector corresponding to the least singular value of \[ XX^T \]
Then to compute the constant value of the equation we the centroid.
\begin{figure}[H]
\begin{center}