3
The geometry of linear transformations
A linear transform A always takes hyper-spheres to hyper-ellipses.
A
A
4
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
A
5
Spectral decomposition
If we are lucky: A = V VT, V orthogonal The eigenvectors of A are the axes of the ellipse
A
9
General linear transformations: SVD
In general A will also contain rotations, not just scales:
A
1
21 2 1 2
T
n n
n
A
u u u v v v
1 1 2
1
TA U V
13
SVD more formally
SVD exists for any matrix Formal definition:
For square matrices A Rnn, there exist orthogonal matrices U, V Rnn and a diagonal matrix , such that all the diagonal values i of are non-negative and
TA U V
=
A U TV
14
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
A U TV
=
Mn Mn nn nn
15
Full SVD
We can complete U to a full orthogonal matrix and pad by zeros accordingly
A U TV
=
Mn MM Mn nn
16
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…
17
SVD is the “working horse” of linear algebra
Must remember: SVD is expensive! For a nn matrix, SVD costs O(n3). For sparse matrices could sometimes be cheaper
18
Shape matching
We have two objects in correspondence Want to find the rigid transformation that aligns
them
20
Shape matching – formalization
Align two point sets
Find a translation vector t and rotation matrix R so that:
1 1 and, , ., ,n nQP p qp q
2
1
( ) is minimizedn
i ii
R
p tq
21
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
i ii i qp p qp q
1
nT
i ii
H
q p
TH U V
TR VU
R t p q
H is 22 or 33 matrix!
=
24
3D animations
Each frame is a 3D model (mesh) Connectivity – mesh faces Geometry – 3D coordinates of the vertices
25
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
26
Animation compression by dimensionality reduction
The geometry of each frame is a vector in R3N space (N = #vertices)
1
1
1
N
N
N
x
x
y
y
z
z
3N f
27
Animation compression by dimensionality reduction
Find a few vectors of R3N that will best represent our frame vectors!
1
1
1
N
N
N
x
x
y
y
z
z
=
1
2
f
1
2
f
VT
U 3Nf ff VT ff
1 2 f
28
Animation compression by dimensionality reduction
The first principal components are the important ones
1
1
1
N
N
N
x
x
y
y
z
z
=
u1 u2 u3
1
2
f
VT
1
2
3
0
0
29
1
1
1
N
N
N
x
x
y
y
z
z
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.
= u1 u2 u31 + 2 + 3
30
Animation compression by dimensionality reduction
Compressed representation: The chosen principal component vectors
Coefficients i for each frame
ui
Animation with only2 principal components
Animation with20 out of 400 principal
components
32
Eigenfaces
Each image is a vector in R250300
Want to find the principal axes – vectors that best represent the input database of images
33
Reconstruction with a few vectors
Represent each image by the first few (n) principal components
1 1 2 2 1 2, , ,n n n v u u u
34
Face recognition
Given a new image of a face, w R250300
Represent w using the first n PCA vectors:
Now find an image in the database whose representation in the PCA basis is the closest:
1 1 2 2 1 2, , ,n n n w u u u
1 2, , ,
, is the largest
n
w
w w
The angle between w and w’ is the smallestw w’
w’
w
35
Non-linear dimensionality reduction
More sophisticated methods can discover non-linear structures in the face datasets
Isomap,Science, Dec. 2000
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