Subspace Representation for Face Recognition

Presenters:

Jian Li and Shaohua Zhou

Overview 4 different subspace

representations PCA, PPCA, LDA, and ICA

2 options Kernel v.s. Non-Kernel

2 databases with 3 different variations Pose, Facial expression, and

Illumination

Subspace representations

Training data X (d,n) X = [x1, x2, …, xn]

Subspace decomposition matrix W (d,m) W = [w1, w2, …, wm]

Representation Y (m,n) Y = W’ * X

PCA, PPCA, LDA and ICA

PCA, in an unsupervised manner, minimizes the representation error ||X – Y||.

LDA, in a supervised manner, minimizes the within-class distance while maximizing the between-class distance.

ICA, in an unsupervised manner, maximizes the independence between Y ’s.

Probabilistic PCA, coming late …

Kernel or Non-Kernel Often somewhere reduces to some

forms related to dot product Kernel trick

Replacing dot product by kernel function Mapping the original data space into

a high-dimensional feature space K(x,y) = <f(x) , f(y)> Gaussian kernel: exp(- 0.5 |x –

y|^2/sigma^2)

Gallery, Probe, Pre-processing Training dataset Testing dataset

Gallery: Reference images in testing Probe: Probe images in testing

Pre-processing Down-sampling Zero-mean-unit-variance x = { x - mean(x) } / var(x) Crop face region only

AT&T Database Pose variation 40 classes, 10 images/class, 28 by

23Set1

Set2

(Mirror of Set1)

FERET Database Facial expression and illumination

variation 200 classes, 3 images/class, 24 by

21Set1

Set2

Set3

Probabilistic PCA (PPCA) -- I PCA only extracts PCs thereby losing

probabilistic flavor PPCA add this by interpreting the

reconstruction error as confidence level y = u + W * x + e Different choices of e

Factor analysis, PPCA (Tipping and Bishop ’99) PCA

Probabilistic PCA (PPCA) -- II Assume e has covariance matrix,

pho*I R = U * D * U’ W = Um * (Dm – pho*I) ^(1/2) Pho = mean of the remaining eigenvalues

Implemented algorithm B. Moghaddam ’01

W = Um * (Dm) ^(1/2) - 2log P(y) = sum (Pci^2/Di) + e^2 / pho + const

Construct inter-person space

Probabilistic KPCA (PKPCA) Replace PCA by KPCA in the PPCA

algorithm Estimating e by computing sum of

all remaining PC’s.

ICA Independent face

PCA pre-whitening: X1 = U’ * X Y = W * X1

Independent facial expression Y = W * X’

Kernel ICA F. Bach and M. I. Jordan ‘01 ‘Kernel trick’ is played when

measuring independence Canonical correlation -- independence

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Experimental Setup Training Ranking the gallery based on the

distance or probability CMS curve

Distance Metric SAD, SQD, Correlation (mean

removed)

Tweaking Gaussian kernel width

Eigenfaces & Fisherfaces

Eigenfaces

Fisherfaces

Independent Basis Faces & Facial Features

Ind. Faces

Ind. Facial Features

Performance on pose variation

Performance on facial expression variation

Performance on illumination variation

Comparison of 4 methods

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PCA PPCA FDA ICA*

PoseExpressionIlluminationAverage

Comparison of Kernel/Non-kernel methods

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Non-Kernel Kernel

PoseExpressionIlluminationAverage

Computational load Training time:

PCA < LDA < PPCA < ICA KPCA < KLDA < PKPCA << KICA

Testing time: PCA = LDA = ICA < PPCA KPCA = KLDA = KICA < PKPCA