Computer Graphics

Computer Graphics Recitation 6

Last week - eigendecomposition • We want to learn how the transformation A works: A

Last week - eigendecomposition • If we look at arbitrary vectors, it doesn’t tell us much. A

Spectra and diagonalization • Moreover, if A is symmetric, the eigenvectors are orthogonal (and there’s always an eigenbasis). A A = UUT = Aui = iui

In real life… • Matrices that have eigenbasis are rare – general transformations involve also rotations, not only scalings. • We want to understand how general transformations behave. • Need a generalization of eigendecomposition  SVD: A = UVT • Before we learn SVD, we’ll see Principal Component Analysis – usage of spectral analysis to analyze the shape and dimensionality of scattered data.

The plan for today • First we’ll see some applications of PCA • Then look at the theory.

PCA – the general idea y’ x’ • PCA finds an orthogonal basis that best represents given data set. • The sum of distances2 from the x’ axis is minimized. y x

PCA – the general idea • PCA finds an orthogonal basis that best represents given data set. • PCA finds a best approximating plane (again, in terms of distances2) z 3D point set in standard basis y x

PCA – the general idea • PCA finds an orthogonal basis that best represents given data set. • PCA finds a best approximating plane (again, in terms of distances2) 3D point set in standard basis

Application: finding tight bounding box • An axis-aligned bounding box: agrees with the axes y maxY minX maxX x minY

Usage of bounding boxes (bounding volumes) • Serve as very simple “approximation” of the object • Fast collision detection, visibility queries • Whenever we need to know the dimensions (size) of the object • The models consist of thousands of polygons • To quickly test that they don’t intersect, the bounding boxes are tested • Sometimes a hierarchy of BB’s is used • The tighter the BB – the less “false alarms” we have

Application: finding tight bounding box • Oriented bounding box: we find better axes! x’ y’

Application: finding tight bounding box • This is not the optimal bounding box z y x

Application: finding tight bounding box • Oriented bounding box: we find better axes!

Notations • Denote our data points by x1, x2, …, xn Rd

The origin of the new axes • The origin is zero-order approximation of our data set (a point) • It will be the center of mass: • It can be shown that:

Scatter matrix • Denote yi = xi – m, i = 1, 2, …, n = d  d

Scatter matrix - eigendecomposition • S is symmetric S has eigendecomposition: S = VVT S v1 1 2 v2 = v1 v2 vn n vn The eigenvectors form orthogonal basis

Principal components • S measures the “scatterness” of the data. • Eigenvectors that correspond to big eigenvalues are the directions in which the data has strong components. • If the eigenvalues are more or less the same – there is not preferable direction.

Principal components • There’s no preferable direction • S looks like this: • Any vector is an eigenvector • There is a clear preferable direction • S looks like this: •  is close to zero, much smaller than .

How to use what we got • For finding oriented bounding box – we simply compute the bounding box with respect to the axes defined by the eigenvectors. The origin is at the mean point m. v3 v2 v1

For approximation y v2 v1 x y y x x The projected data set approximates the original data set This line segment approximates the original data set

For approximation • In general dimension d, the eigenvalues are sorted in descending order: 1  2  …  d • The eigenvectors are sorted accordingly. • To get an approximation of dimension d’ < d, we take the d’ first eigenvectors and look at the subspace they span (d’ = 1 is a line, d’ = 2 is a plane…)

For approximation • To get an approximating set, we project the original data points onto the chosen subspace: xi = m + 1v1 + 2v2 +…+ d’vd’ +…+dvd Projection: xi’ = m + 1v1 + 2v2 +…+ d’vd’ +0vd’+1+…+ 0vd

Optimality of approximation • The approximation is optimal in least-squares sense. It gives the minimal of: • The projected points have maximal variance. Original set projection on arbitrary line projection on v1 axis

Technical remarks: • i  0, i = 1,…,d (such matrices are called positive semi-definite). So we can indeed sort by the magnitude of i • Theorem: i  0  <Sv, v>  0 v Proof: Therefore, i 0  <Sv, v>  0 v

Technical remarks: • In our case, indeed <Sv, v>  0 v • This is because S can be represented as S = XXT, X is dn matrix • So:

Technical remarks: • S = XXT, X is dn matrix = = d  d = XXT =

See you next time

Computer Graphics