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PCA: cleanup the doc

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Pierre Alliez 2008-03-26 14:30:26 +00:00
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Principal_component_analysis/doc_tex/Principal_component_analysis/intro.tex
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This \cgal\ package provides functions to compute global information
on the shape of a set of 2D or 3D objects such as points. It provides
the computation of axis-aligned bounding boxes for point sets,
barycenters of weighted point sets and centroids as well as linear
least squares fitting for sets of points, circles, rectangles,
segments and triangles in 2D and in addition to these, cuboids,
spheres and tetrahedrons in 3D. It also provides the computation of
centroids and linear least squares fitting for all $n-1$ manifolds of
these $n$ dimensional geometries ($n = 2$ or $3$). It assumes the set
of kernel primitive elements to be stored into an iterator range of a
container.\\
This \cgal\ package provides functions to compute global information on the shape of a set of 2D or 3D objects such as points. It provides the computation of axis-aligned bounding boxes for point sets, and barycenters of weighted point sets. The set of kernel primitive elements are assumed to be stored into an iterator range of a container.\\
In addition, it computes centroids as well as linear least squares fitting for sets of points, circles, rectangles, segments and triangles in 2D and in points, segments, triangles, cuboids, spheres and tetrahedra in 3D. For each object type in 2D and 3D, it allows the user to specify the dimension of the object fitted. For example, it is possible to fit a set of balls by providing an iterator range of spheres and specifying dimension 3. It is also possible to fit only the edges of a set of tetrahedra by specifying dimension 1, etc.
\section{Definitions}
A \emph{bounding box} for a set of objects is a cuboid that completely
contains the set. An \emph{axis-aligned bounding box} is a bounding
box aligned with the axes of the coordinate system.\\
A \emph{bounding box} for a set of objects is a cuboid that completely contains the set. An \emph{axis-aligned bounding box} is a bounding box aligned with the axes of the coordinate system.\\
A \emph{centroid} is defined as average of position. A
\emph{barycenter} of weighted point sets is defined as weighted
average of position. When all weights are equal the barycenter
coincides with the centroid.\\
A \emph{centroid} is defined as average of position. A \emph{barycenter} of weighted point sets is defined as weighted
average of position. When all weights are equal the barycenter coincides with the centroid.\\
Given a point set, \emph{linear least squares fitting} amounts to find
the linear sub-space which minimizes the sum of squared distances from
the points to their projection onto this linear sub-space. This
problem is equivalent to search for the linear sub-space which
maximizes the variance of projected points, the latter being obtained
by eigen decomposition of the covariance matrix of the point
set. Eigenvectors corresponding to large eigenvalues are the
directions in which the data has strong component, or equivalently
large variance. If eigenvalues are the same there is no preferable
sub-space.\\
Given a point set, \emph{linear least squares fitting} amounts to find the linear sub-space which minimizes the sum of squared distances from the points to their projection onto this linear sub-space. This problem is equivalent to search for the linear sub-space which maximizes the variance of projected points, the latter being obtained by eigen decomposition of the covariance matrix of the point set. Eigenvectors corresponding to large eigenvalues are the
directions in which the data has strong component, or equivalently large variance. If eigenvalues are the same there is no preferable sub-space.\\
% Given a triangle set, \emph{linear least squares fitting} amounts to find the linear sub-space which minimizes the sum of squared
% distances from all points in the set to their projection onto this linear
% sub-space. This problem is equivalent to the one of fitting a linear sub-space to a point set, except that the covariance matrix is now derived from a continuous integral over the triangles instead of a discrete sum over the points.
Given an object set, \emph{linear least squares fitting} amounts to find the linear sub-space which minimizes the sum of squared
distances from all points in the set to their projection onto this linear
sub-space. This problem is equivalent to the one of fitting a linear sub-space to a point set, except that the covariance matrix is now derived from a continuous integral over the objects instead of a discrete sum over the points.
Given an object set, \emph{linear least squares fitting} amounts to find the linear sub-space which minimizes the sum of squared distances from all points in the set to their projection onto this linear sub-space. This problem is equivalent to the one of fitting a linear sub-space to a point set, except that the covariance matrix is now derived (closed form formula) from a continuous integral over the objects instead of a discrete sum over the points.
\begin{center}