8/16/99 computer vision and modeling. 8/16/99 principal components with svd
TRANSCRIPT
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Computer Vision and ModelingComputer Vision and Modeling
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Principal Components with SVDPrincipal Components with SVD
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Linear Dimension Reduction:Linear Dimension Reduction:
High-dimensionalInput Space
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Linear Subspace:Linear Subspace:
+=
+ 1.7=
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Linear Subspace:Linear Subspace:
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Principal Components Analysis:Principal Components Analysis:
xWy ~
N
nT mnys
1
22 )][(
TN
nT xxS )~()~(
1
TTT WWSs 2
m
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Examples:Examples:
Data:
Kirby, Weisser, Dangelmayer 1993
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Examples:Examples:
Data:
PCA
New Basis Vectors
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Examples:Examples:
Data:
PCA
EigenLips
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Examples:Examples:
Face Recognition with Eigenfaces (Turk+Pentland, ):
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Examples:Examples:
Face Recognition System (Moghaddam+Pentland):
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Examples: Visual CortexExamples: Visual Cortex
Hubel
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Examples: Visual CortexExamples: Visual Cortex
Hubel
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Examples: Receptive FieldsExamples: Receptive Fields
Hubel
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Examples: Receptive FieldsExamples: Receptive Fields
Hancock et al: The principal components of natural images
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Examples: Receptive FieldsExamples: Receptive Fields
Hancock et al: The principal components of natural images
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Examples:Examples:
Active Appearance Models (AAM): (Cootes et al)
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Examples:Examples:
Active Appearance Models (AAM): (Cootes et al)
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Examples:Examples:
Active Appearance Models (AAM): (Cootes et al)
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Examples:Examples:
3D Morphable Models (Blanz+Vetter)
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Examples:Examples:
3D Morphable Models (Blanz+Vetter)
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ReviewReview
E(V)V V
Constrain-
Analytically derived:Affine, Twist/Exponential Map
Learned:Linear/non-linear
Sub-Spaces
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S = (p ,…,p )
E(S) Constrain
1 n
Non-Rigid Constrained Spaces
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Non-Rigid Constrained Spaces
Nonlinear Manifolds:
Linear Subspaces:
• Small Basis Set
• Principal Components Analysis
Mixture Models
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Examples:Examples:
Eigen Tracking (Black and Jepson)
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Examples:Examples:
Shape Models for tracking:
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More generic Feature/Shape Models:
Visual Motion Contours:Blake, Isard, Reynard
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More generic Feature/Shape Models:
Visual Motion Contours:Blake, Isard, Reynard
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Linear Discriminant Analysis:Linear Discriminant Analysis:
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Fisher’s linear discriminant:Fisher’s linear discriminant:
21
))(())(( 1111Cn
Tnn
Cn
TnnW xxxxS T
BS ))(( 1212
wSw
WSwJ
WT
BT
KCn
nK
k xN
1
)( 121
WSw
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Example: Eigenfaces vs FisherfacesExample: Eigenfaces vs Fisherfaces
Glasses or not Glasses ?
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Example: Eigenfaces vs FisherfacesExample: Eigenfaces vs Fisherfaces
Input New Axis
Belhumeur, Hespanha, Kriegman 1997
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Nonlinear Manifolds
Nonlinear Manifolds:
Linear Subspaces:
• Small Basis Set
• Principal Components Analysis
Mixture Models