Spring 2020: Venu: Haag 315, Time: M/W 4-5:15pm ECE 5582 Computer Vision Lec 15: Graph Embedding & Laplacianface Zhu Li Dept of CSEE, UMKC Office: FH560E, Email: [email protected], Ph: x 2346. http://l.web.umkc.edu/lizhu Z. Li: ECE 5582 Computer Vision, 2020 p.1 slides created with WPS Office Linux and EqualX LaTex equation editor
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Lec 15: Graph Embedding & Laplacianface · PCA is a special case of graph embedding oFully connected affinity map, equal importance LDA is a special case of graph embedding oFully
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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: ECE 5582 Computer Vision, 2020
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: ECE 5582 Computer Vision, 2020
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
���, �� = �1��
, �X X�, X��X� XX G �X��� �
0, �X��
Z. Li: ECE 5582 Computer Vision, 2020
LDA as a special case of LPP
The same generalized Eigen problem
p.23Z. Li: ECE 5582 Computer Vision, 2020
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
���� = �−exp��X� −X��
��ℎ , �X �X� −X�� ≤ �
0, �X��
���� = �1��
, �X X�, X� ∈ ��
0, �X��
���� = 1/�
Z. Li: ECE 5582 Computer Vision, 2020
Applications: facial expression embedding
Facial expressions embedded in a 2-d space via LPP
p.25
frown
sad
happy
neutral
Z. Li: ECE 5582 Computer Vision, 2020
Application: Compression of SIFT
Compression of SIFT, preserve matching relationship, rather than reconstruction:
Z. Li: ECE 5582 Computer Vision, 2020 p.26
� =argmin�
����������X� −�X��
�
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/0qu7tc3jb88at2h53l1dpcuqkt9pn
7ww 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: ECE 5582 Computer Vision, 2020 p.27
HW-3 test run
Laplacian face is powerful.
Z. Li: ECE 5582 Computer Vision, 2020 p.28
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: ECE 5582 Computer Vision, 2020 p.29
Signal on Graph
non-uniformly sampled
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Graph Fourier Transform
GFT is different from Laplacian Embedding:
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GFT Example
Graph Laplacian
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Normalized Graph Laplacian
Normalize by edge pair degree
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Graph Frourier Transform
Analogous to FT
p.34Z. Li: ECE 5582 Computer Vision, 2020
Graph Spectrum
Freq interpretation
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GetGFT.m
Implementation of GFT: affinity graph sparsity control is important
Z. Li: ECE 5582 Computer Vision, 2020 p.36
Graph Signal Smoothness
Quadratic form on L:
p.37Z. Li: ECE 5582 Computer Vision, 2020
most smooth: low freq dominating
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 embedding
o Fully connected intra classo 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.