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Lecture 13-14Face Recognition – Subspace/Manifold Learning

Tae-Kyun Kim

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EE4-62 MLCVFace Image Tagging and Retrieval

• Face tagging at commercial weblogs

• Key issues– User interaction for face tags– Representation of a long- time

accumulated data– Online and efficient learning

• Active research area in Face Recognition Test and MPEG-7 for face image retrieval and automatic passport control• Our proposal promoted to

MPEG7 ISO/IEC standard

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Principal Component Analysis (PCA)- Maximum Variance Formulation of PCA- Minimum-error formulation of PCA

- Probabilistic PCA

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Maximum Variance Formulation of PCA

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Minimum-error formulation of PCA

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Applications of PCA to Face Recognition

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(Recap) Geometrical interpretation of PCA• Principal components are the vectors in the direction of the

maximum variance of the projection samples.

• Each two-dimensional data point is transformed to a single variable z1 representing the projection of the data point onto the eigenvector u1.

• The data points projected onto u1 has the max variance.• Infer the inherent structure of high dimensional data.• The intrinsic dimensionality of data is much smaller.

• For given 2D data points, u1 and u2 are found as PCs

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Eigenfaces• Collect a set of face images• Normalize for scale, orientation (using eye locations)

• Construct the covariance matrix and obtain eigenvectors

w

h

D=wh

NDRX

,...,1

1 xxXXXN

S T

MDRUUSU ,

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Eigenfaces• Project data onto the

subspace

• Reconstruction is obtained as

• Use the distance to the subspace for face recognition

DMRZXUZ NMT ,,

UZXUzuzxM

iii

~,~

1

x~||~|| xx

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x

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Matlab Demos – Face Recognition by PCA

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• Face Images• Eigen-vectors and Eigen-value plot• Face image reconstruction• Projection coefficients (visualisation of high-

dimensional data)• Face recognition

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Probabilistic PCA

• A subspace is spanned by the orthonormal basis (eigenvectors computed from covariance matrix)

• Can interpret each observation with a generative model

• Estimate (approximately) the probability of generating each observation with Gaussian distribution,

PCA: uniform prior on the subspace PPCA: Gaussian dist.

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Continuous Latent Variables

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Probabilistic PCAEE4-62 MLCV

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Maximum likelihood PCA

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Limitations of PCA

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Unsupervised learning

PCA vs LDA (Linear Discriminant Analysis)

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Linear model

Linear Manifold = Subspace Nonlinear

Manifold

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PCA vs Kernel PCA

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Gaussian Distribution Assumption

IC1

IC2

PC1

PC2

PCA vs ICA (Independent Component Analysis)

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(also by ICA)