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Image Analysis & Retrieval
CS/EE 5590 Special Topics (Class Ids: 44873, 44874)
Fall 2016, M/W 4-5:15pm@Bloch 0012
Lec 15
Graph Laplacian Embedding
Zhu Li
Dept of CSEE, UMKC
Office: FH560E, Email: [email protected], Ph: x 2346.
http://l.web.umkc.edu/lizhu
p.1Z. Li, Image Analysis & Understanding, 2016
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Outline
Recap:
Eigenface
Fisherface
Graph Embedding
Laplacianface
Graph Fourier Transform
Summary
p.2Z. Li, Image Analysis & Understanding, 2016
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Subspace Learning for Face Recognition
Project face images to a subspace with basis A
Matlab: x=faces*A(:,1:kd)
eigf1
eigf2
eigf3
= 10.9* + 0.4* + 4.7*
p.3Z. Li, Image Analysis & Understanding, 2016
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PCA & Fisher’s Linear Discriminant
• Between-class scatter
• Within-class scatter
• Where
– c is the number of classes
– i is the mean of class i
– | i | is number of samples of i..
T
ii
c
i
iBS ))((1
c
i x
T
ikikW
ik
xS1
))((
1
2
12
p.4Z. Li, Image Analysis & Understanding, 2016
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Eigen vs Fisher Projection
p.5
• PCA (Eigenfaces)
Maximizes projected total scatter
• Fisher’s Linear Discriminant
Maximizes ratio of projected between-class to projected within-class scatter, solved by the generalized Eigen problem:
WSWW T
T
WPCA maxarg
WSW
WSWW
W
T
B
T
Wfld maxarg
12
PCA
Fisher𝑆𝐵𝑊 = 𝜆𝑆𝑊𝑊
Z. Li, Image Analysis & Understanding, 2016
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LDA implementation
myLDA.m
p.6
% compute class mean
mx = mean(x);
ids = unique(y); m = length(ids);
Sb = zeros(kd, kd);
for k=1:m
indx = find(y==ids(k)); nk(k) = length(indx);
% class mean
m_cx(k,:) = mean(x(indx, :));
% between class scatter
Sb = Sb + nk(k)*(m_cx(k,:) - mx)'*(m_cx(k,:) - mx);
end
% compute intra-class scatter
Sw = zeros(kd, kd);
for k=1:m
indx = find(y==ids(k)); nk(k) = length(indx);
% remove mean
xk = x(indx, :) - repmat(m_cx(k,:), [nk(k), 1]);
% adding up
Sw = Sw + (xk'*xk);
end
Z. Li, Image Analysis & Understanding, 2016
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LDA Implementation
Find projection by generalized Eigen problem solution
p.7
% generalized eigen problem
[A, v]=eigs(Sb, Sw);
if dbg
figure(31);
subplot(2,2,1); imagesc(Sb);
colormap('gray'); title('S_b');
subplot(2,2,2); imagesc(Sw);
colormap('gray'); title('S_w');
z = x*A; dist = pdist2(z, z);
subplot(2,2,3); imagesc(dist);
title('dist(j,k)');
subplot(2,2,4); stem(diag(v), '.'); grid on;
hold on; title('eig v');
end
𝑆𝐵𝑊 = 𝜆𝑆𝑊𝑊
Z. Li, Image Analysis & Understanding, 2016
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Fisherface Basis
It is interesting to compare Fisherface with Eigenfacebasis
p.8
FisherfaceEigenface
Z. Li, Image Analysis & Understanding, 2016
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Fisherface Performance
Fisher vs Eigenface performance: (fisherface.m)
1200 face images, 144 subjects
Eigen kd=32
p.9
144 subjectsROC
Z. Li, Image Analysis & Understanding, 2016
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Outline
Recap:
Eigenface
Fisherface
Graph Embedding
Locality Preserving Projection
Laplacian Face
Graph Fourier Transform
Summary
p.10Z. Li, Image Analysis & Understanding, 2016
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Locality Preserving Projection
Recall the dimension reduction formulations: find w, s.t y=wx:
PCA:
LDA:
p.11
max𝑤
wTSw
S = SB + SW
Z. Li, Image Analysis & Understanding, 2016
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LPP Formulation –Affinity
To preserve local affinity relationship
Affinity map
Selection of heat kernel size and threshold are important
Hint: affinity matrix should be sparse
p.12
affinity histogram
Z. Li, Image Analysis & Understanding, 2016
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LPP Formulation –Affinity Supervised
How to utilize the label info ?
Heat map mapping is good for intra-class affinity modelling, but how about intra-class affinity ?
One direct solution is to set affinity to zero for intra class pairs
p.13
% LPP - compute affinity
f_dist1 = pdist2(x1, x1);
% heat kernel size
mdist = mean(f_dist1(:)); h = -
log(0.15)/mdist;
S1 = exp(-h*f_dist1);
id_dist = pdist2(ids, ids);
subplot(2,2,3); imagesc(id_dist);
title('label distance');
S2=S1; S2(find(id_dist~=0)) = 0;
subplot(2,2,4); imagesc(S1);
colormap('gray'); title('affinity-
supervised');
Z. Li, Image Analysis & Understanding, 2016
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LPP- Affinity Preserving Projection
Find a projection that best preserves the affinity matrix
p.14
min𝑊
𝑖
𝑗
𝑆𝑖,𝑗 𝑦𝑖 − 𝑦𝑗2, 𝑠. 𝑡. 𝑦 = 𝑊𝑥
min𝑊
𝑖
𝑗
𝑆𝑖,𝑗 𝑊𝑥𝑖 −𝑊𝑥𝑗2
𝑠. 𝑡, 𝑆𝑖,𝑗 = −exp 𝑥𝑗 − 𝑥𝑖
2
ℎ, 𝑖𝑓 𝑥𝑗 − 𝑥𝑖 ≤ 𝜃
0, 𝑒𝑙𝑠𝑒
Z. Li, Image Analysis & Understanding, 2016
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LPP Formulation
L = D-S:
nxn matrix, called graph Laplacian
Normalizing factor: nxn D
Diagonal matrix, entry Dii = sum of col/row affinity
The larger the value, the more important data point is
Constraint on D:
p.15Z. Li, Image Analysis & Understanding, 2016
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Generalized Eigen Problem
Now the formulation is,
Lagranian
by KKT (Karush-Khun-Tucker) Condition, it is solved by a generalized Eigen problem
p.16
𝐿 𝑤, 𝜆 = 𝑤𝑇𝑋𝐿𝑋𝑇𝑤 − 𝜆(𝑤𝑇𝑋𝐷𝑋𝑤 − 1)
Z. Li, Image Analysis & Understanding, 2016
X He, S Yan, Y Hu, P Niyogi, HJ Zhang, “Face Recognition Using Laplacianface”, IEEE Trans PAMI, vol. 27 (3), 328-340, 2005.
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Matlab Implementation
laplacianface.m
p.17
%LPP
n_face = 1200; n_subj = length(unique(ids(1:n_face)));
% eigenface
kd=32; x1 = faces(1:n_face,:)*A1(:,1:kd); ids=ids(1:n_face);
% LPP - compute affinity
f_dist1 = pdist2(x1, x1);
% heat kernel size
mdist = mean(f_dist1(:)); h = -log(0.15)/mdist;
S1 = exp(-h*f_dist1);
figure(32); subplot(2,2,1); imagesc(f_dist1); colormap('gray'); title('d(x_i, d_j)');
subplot(2,2,2); imagesc(S1); colormap('gray'); title('affinity');
%subplot(2,2,3); grid on; hold on; [h_aff, v_aff]=hist(S(:), 40); plot(v_aff, h_aff,
'.-');
% utilize supervised info
id_dist = pdist2(ids, ids);
subplot(2,2,3); imagesc(id_dist); title('label distance');
S2=S1; S2(find(id_dist~=0)) = 0;
subplot(2,2,4); imagesc(S1); colormap('gray'); title('affinity-supervised');
% laplacian face
lpp_opt.PCARatio = 1;
[A2, eigv2]=LPP(S2, lpp_opt, x1);
Z. Li, Image Analysis & Understanding, 2016
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Laplacian Face
Now, we can model face as a LPP projection:
p.18
Eigenface Laplacian Face
Z. Li, Image Analysis & Understanding, 2016
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Laplacian vs Eigenface
1200 faces, 144 subjects
p.19Z. Li, Image Analysis & Understanding, 2016
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LPP and PCA
Graph Embedding is an unifying theory on dimension reduction
PCA becomes special case of LPP, if we do not enforce local affinity
p.20Z. Li, Image Analysis & Understanding, 2016
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LPP and LDA
How about LDA ?
Recall within class scatter:
p.21
This is i-th classData covariance
Li has diagonal entry of 1/ni,Equal affinity among data points
Z. Li, Image Analysis & Understanding, 2016
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LPP and LDA
Now consider the between class scatter
C is the data covariance, regardless of label
L is graph Laplacian computed from the affinity rule that,
p.22
𝑆 𝑖, 𝑗 =
1
𝑛𝑘, 𝑖𝑓 𝑥𝑖 , 𝑥𝑗𝑎𝑟𝑒 𝑓𝑟𝑜𝑚 𝑐𝑙𝑎𝑠𝑠 𝑘
0, 𝑒𝑙𝑠𝑒
Z. Li, Image Analysis & Understanding, 2016
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LDA as a special case of LPP
The same generalized Eigen problem
p.23Z. Li, Image Analysis & Understanding, 2016
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Graph Embedding Interpretation of PCA/LDA/LPP
Affinity graph S, determines the embedding subspace W, via
PCA and LDA are special cases of Graph Embedding
PCA:
LDA
LPP
p.24
𝑆𝑖,𝑗 = −exp 𝑥𝑗 − 𝑥𝑖
2
ℎ, 𝑖𝑓 𝑥𝑗 − 𝑥𝑖 ≤ 𝜃
0, 𝑒𝑙𝑠𝑒
𝑆𝑖,𝑗 =
1
𝑛𝑘, 𝑖𝑓 𝑥𝑖 , 𝑥𝑗 ∈ 𝐶𝑘
0, 𝑒𝑙𝑠𝑒
𝑆𝑖,𝑗 = 1/𝑛
Z. Li, Adv. Multimedia Communciation, 2016
Fall
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Applications: facial expression embedding
Facial expressions embedded in a 2-d space via LPP
p.25
frown
sad
happy
neutral
Z. Li, Image Analysis & Understanding, 2016
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Application: Compression of SIFT
Compression of SIFT, preserve matching relationship, rather than reconstruction:
Z. Li, Image Analysis & Understanding, 2016 p.26
𝐴 = argmin𝑊
𝑘
𝑗
𝑠𝑗,𝑘 𝑊𝑥𝑗 −𝑊𝑥𝑘2
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Homework-3: Subspace Methods
Objective:
Understand the graph embedding connections among popular subspace methods like PCA, LDA and LPP
Practical experiences with serious size data set
Data Set:
https://umkc.app.box.com/s/0qu7tc3jb88at2h53l1dpcuqkt9pn7ww
417 subjects, 6650 image face data set, pre-processed to 20x20 pel images, intensity normalized to [0, 1]
Add your own face images, 10~15, frontal
Tasks:
Compute Eigenface, Fisherface and Laplacianface models
ROC plot on verification performance
mAP for retrieval/identification performance
Z. Li, Image Analysis & Understanding, 2016 p.27
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HW-3 test run
Laplacian face is powerful.
Z. Li, Image Analysis & Understanding, 2016 p.28
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Graph Fourier Transform
David I. Shuman, Sunil K. Narang, Pascal Frossard, Antonio Ortega, Pierre Vandergheynst:The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains. IEEE Signal Process. Mag. 30(3): 83-98 (2013)
Z. Li, Image Analysis & Understanding, 2016 p.29
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Signal on Graph
non-uniformly sampled
p.30Z. Li, Adv. Multimedia Communciation, 2016
Fall
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Graph Fourier Transform
GFT is different from Laplacian Embedding:
p.31Z. Li, Adv. Multimedia Communciation, 2016
Fall
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GFT Example
Graph Laplacian
p.32Z. Li, Adv. Multimedia Communciation, 2016
Fall
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Normalized Graph Laplacian
Normalize by edge pair degree
p.33Z. Li, Adv. Multimedia Communciation, 2016
Fall
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Graph Frourier Transform
Analogous to FT
p.34Z. Li, Adv. Multimedia Communciation, 2016
Fall
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Graph Spectrum
p.35Z. Li, Adv. Multimedia Communciation, 2016
Fall
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Graph Signal Smoothness
Quadratic form on L:
p.36Z. Li, Adv. Multimedia Communciation, 2016
Fall
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Summary
Graph Laplacian Embedding is an unifying theory for feature space dimension reduction PCA is a special case of graph embedding
o Fully connected affinity map, equal importance
LDA is a special case of graph embeddingo Fully connected intra class
o Zero affinity inter class
LPP: preserves pair wise affinity.
GFT: eigen vectors of graph Laplacian, has Fourier Transform like characteristics.
Many applications in Face recognition
Pose estimation
Facial expression modeling
Compression of Graph signals.
p.37Z. Li, Image Analysis & Understanding, 2016