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Affine Approximation of Point Clouds

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A best Euclidean approximation of points

$\displaystyle p_i = (p_{i,1},\,\ldots,\,p_{i,n}),\quad i=1,\ldots,m, $

with a $ k$-dimensional affine subspace

$\displaystyle H:\ a + \mathrm{span}(v_1,\ldots,v_k) $

can be determined by minimizing the sum of the squared distances,

$\displaystyle e(P,H) = \sum_i \mathrm{dist}(p_i,H)^2. $

\includegraphics[width=0.7\linewidth]{Bild_approx_point_clouds}

The optimal affine subspace $ H$ is characterized as follows. The point $ a$ is the center of the points $ p_i$,

$\displaystyle a = \frac{1}{m} \sum_i p_i, $

and, if

$\displaystyle \left(\begin{array}{c} p_1 - a \\ \vdots \\ p_m - a \end{array}\right) = U\,\mathrm{diag}(s_1,\,s_2,\,\ldots)\,V^t $

is the singular value decomposition of the centered point set, then the basis vectors $ v_j$ are the first $ k$ columns of $ V$.

The error can be expressed in terms of the singular values:

$\displaystyle e(P,H) = \sum_{i>k} s_i^2. $

(Authors: Höllig/Pfeil/Walter)

Example:

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  automatically generated 7/ 2/2007