3D Geometry for Computer Graphics

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3D Geometry forComputer Graphics

• Class 7

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The plan today

• Review of SVD basics
• Connection PCA
• Applications
• Eigenfaces
• Animation compression

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The geometry of linear transformations

• A linear transform A always takes hyper-spheres
  to hyper-ellipses.

A
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The geometry of linear transformations

• Thus, one good way to understand what A does is
  to find which vectors are mapped to the main
  axes of the ellipsoid.

A
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Spectral decomposition

• If we are lucky A V ? VT, V orthogonal
• The eigenvectors of A are the axes of the ellipse

A
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Spectral decomposition

• The standard basis

y
x
Rotation VT
Au V ? VTu
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Spectral decomposition

• Scaling x ?1 x, y ?2 y

Scale ?
Au V ? VTu
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Spectral decomposition

• Rotate back

Rotate V
Au V ? VTu
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General linear transformations SVD

• In general A will also contain rotations, not
  just scales

?1
1
1
?2
A
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SVD

• The standard basis

y
x
Rotation VT
Au U ? VTu
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Spectral decomposition

• Scaling x ?1 x, y ?2 y

Scale ?
Au U ? VTu
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Spectral decomposition

• Rotate again

Rotate U
Au U ? VTu
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SVD more formally

• SVD exists for any matrix
• Formal definition
• For square matrices A ? Rn?n, there exist
  orthogonal matrices U, V ? Rn?n and a diagonal
  matrix ?, such that all the diagonal values ?i of
  ? are non-negative and

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Reduced SVD

• For rectangular matrices, we have two forms of
  SVD. The reduced SVD looks like this
• The columns of U are orthonormal
• Cheaper form for computation and storage

M?n
M?n
n?n
n?n

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Full SVD

• We can complete U to a full orthogonal matrix and
  pad ? by zeros accordingly

M?n
M?M
M?n
n?n

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SVD is the working horse of linear algebra

• There are numerical algorithms to compute SVD.
  Once you have it, you have many things
• Matrix inverse ? can solve square linear systems
• Numerical rank of a matrix
• Can solve least-squares systems
• PCA
• Many more

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SVD is the working horse of linear algebra

• Must remember SVD is expensive!
• For a n?n matrix, SVD costs O(n3).
• For sparse matrices could sometimes be cheaper

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Shape matching

• We have two objects in correspondence
• Want to find the rigid transformation that aligns
  them

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Shape matching

• When the objects are aligned, the lengths of the
  connecting lines are small.

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Shape matching formalization

• Align two point sets
• Find a translation vector t and rotation matrix R
  so that

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Summary of rigid alignment

• Translate the input points to the centroids
• Compute the covariance matrix
• Compute the SVD of H
• The optimal rotation is
• The translation vector is

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SVD for animation compression
Chicken animation
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3D animations

• Each frame is a 3D model (mesh)
• Connectivity mesh faces

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3D animations

• Each frame is a 3D model (mesh)
• Connectivity mesh faces
• Geometry 3D coordinates of the vertices

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3D animations

• Connectivity is usually constant (at least on
  large segments of the animation)
• The geometry changes in each frame ? vast amount
  of data, huge filesize!

13 seconds, 3000 vertices/frame, 26 MB
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Animation compression by dimensionality reduction

• The geometry of each frame is a vector in R3N
  space (N vertices)

3N ? f
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Animation compression by dimensionality reduction

• Find a few vectors of R3N that will best
  represent our frame vectors!

VT f?f
? f?f
U 3N?f
VT

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Animation compression by dimensionality reduction

• The first principal components are the important
  ones

u1
u2
u3

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Animation compression by dimensionality reduction

• Approximate each frame by linear combination of
  the first principal components
• The more components we use, the better the
  approximation
• Usually, the number of components needed is much
  smaller than f.

u3
u1
u2

?1
?2
?3
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Animation compression by dimensionality reduction
ui

• Compressed representation
• The chosen principal component vectors
• Coefficients ?i for each frame

Animation with only 2 principal components
Animation with 20 out of 400 principal components
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Eigenfaces

• Same principal components analysis can be applied
  to images

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Eigenfaces

• Each image is a vector in R250?300
• Want to find the principal axes vectors that
  best represent the input database of images

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Reconstruction with a few vectors

• Represent each image by the first few (n)
  principal components

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Face recognition

• Given a new image of a face, w ? R250?300
• Represent w using the first n PCA vectors
• Now find an image in the database whose
  representation in the PCA basis is the closest

The angle between w and w is the smallest
w
w
w
w
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Non-linear dimensionality reduction

• More sophisticated methods can discover
  non-linear structures in the face datasets

Isomap, Science, Dec. 2000
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See you next time