3D Geometry for Computer Graphics

3D Geometry forComputer Graphics Class 3

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

Finding cluster representatives

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!

Managing high-dimensional data • Data base of face scans (3D geometry + texture) • 10,000 points in each scan • x, y, z, R, G, B  6 numbers for each point • Thus, each scan is a 10,000*6 = 60,000-dimensional vector!

Managing high-dimensional data • How to find interesting axes is this 60000-dimensional space? • axes that measures age, gender, etc… • There is hope: the faces are likely to be governed by a small set of parameters (much less than 60,000…) age axis gender axis FaceGen demo

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 where Y is dn matrix with yk as columns (k = 1, 2, …, n) Y YT

Variance of projected points • In a way, S measures variance (= scatterness) of the data in different directions. • Let’s look at a line L through the center of mass m, and project our points xi onto it. The variance of the projected points x’iis: L L L L Original set Small variance Large variance

Variance of projected points • Given a direction v, ||v|| = 1, the projection of xi onto L = m + vt is: xi x’i v L m

Variance of projected points • So,

Directions of maximal variance • So, we have: var(L) = <Sv, v> • Theorem: Letf : {v Rd | ||v|| = 1} R, f (v) = <Sv, v>(and S is a symmetric matrix). Then, the extrema of f are attained at the eigenvectors of S. • So, eigenvectors of S are directions of maximal/minimal variance!

Summary so far • We take the centered data points y1, y2, …, yn Rd • Construct the scatter matrix • S measures the variance of the data points • Eigenvectors of S are directions of maximal variance.

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

Principal components • Eigenvectors that correspond to big eigenvalues are the directions in which the data has strong components (= large variance). • If the eigenvalues are more or less the same – there is no 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

See you next time

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

3D Geometry for Computer Graphics