Lecture’2:’’ A’Case’Study’of’Computer’Vision’– Face...
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Lecture 2 - Fei-Fei Li Lecture 2: A Case Study of Computer Vision – Face Recogni<on Professor FeiFei Li Stanford Vision Lab 24Sep12 1
Transcript of Lecture’2:’’ A’Case’Study’of’Computer’Vision’– Face...
Lecture 2 - !!!
Fei-Fei Li!
Lecture 2: A Case Study of Computer Vision –
Face Recogni<on
Professor Fei-‐Fei Li Stanford Vision Lab
24-‐Sep-‐12 1
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What we will learn today
• Recogni<on problems • Face discrimina<on
– Principal Component Analysis (PCA) and Eigenfaces (Problem Set 1 (Q1))
– Linear Discriminant Analysis (LDA) and Fisherfaces (supplementary materials)
• Face detec<on – SVM (Problem Set 0 (Q2)) – Boos<ng
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Courtesy of Johannes M. Zanker
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“Faces” in the brain
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“Faces” in the brain
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Kanwisher, et al. 1997
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• Digital photography
Face Recogni<on
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Face Recogni<on • Digital photography • Surveillance
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• Digital photography • Surveillance • Album organiza<on
Face Recogni<on
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Face Recogni<on • Digital photography • Surveillance • Album organiza<on • Person tracking/id.
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• Digital photography • Surveillance • Album organiza<on • Person tracking/id. • Emo<ons and expressions
Face Recogni<on
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• Digital photography • Surveillance • Album organiza<on • Person tracking/id. • Emo<ons and expressions
• Security/warfare • Tele-‐conferencing • Etc.
Face Recogni<on
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vs.
Iden%fica%on
What’s ‘recogni<on’?
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What’s ‘recogni<on’?
vs.
Categoriza%on Iden<fica<on
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Categoriza<on Iden<fica<on
No
localiza%
on
Yes, there are faces
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Categoriza<on Iden<fica<on
No
localiza%
on Yes, there is John Lennon
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Categoriza<on Iden<fica<on
No
localiza<
on
Detec%on
or
Localizatoin
John Lennon
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Categoriza<on Iden<fica<on
No
localiza<
on
Detec%on
or
Localizatoin
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Pose recogni<on
Categoriza<on Iden<fica<on
No
localiza<
on
Detec%on
or
Localizatoin
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Categoriza<on Iden<fica<on
No
localiza<
on
1. PCA & Eigenfaces (Turk & Pentland, 1991) 2. LDA & Fisherfaces (Bellumeur et al. 1997)
3. AdaBoost (Viola & Jones, 2001)
Detec<on
or
Localizatoin
Milestone Face Recogni<on methods
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Eigenfaces and Fishfaces – Principle Component Analysis (PCA) – Linear Discriminant Analysis (LDA)
1. Turk and Pentland, Eigenfaces for Recogni<on, Journal of Cogni6ve Neuroscience 3 (1): 71–86. 2. P. Belhumeur, J. Hespanha, and D. Kriegman. "Eigenfaces vs. Fisherfaces: Recogni<on Using Class Specific Linear Projec<on". IEEE Transac6ons on pa>ern analysis and machine intelligence 19 (7): 711. 1997.
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The Space of Faces
• An image is a point in a high dimensional space – If represented in grayscale intensity, an N x M image is a point in RNM [T
hanks to Ch
uck Dy
er, Steve Seitz, N
ishino]
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• Images in the possible set are highly correlated.
• So, compress them to a low-‐dimensional subspace that captures key appearance characteris<cs of the visual DOFs.
Key Idea
}x̂{=χ
• USE PCA for es<ma<ng the sub-‐space (dimensionality reduc<on)
• Compare two faces by projec<ng the images into the subspace and measuring the EUCLIDEAN distance between them.
EIGENFACES: [Turk and Pentland 91]
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x1
x2
1 2
3
4
5
6
x1
x2
1 2
3
4
5
6
X1’
PCA projec<on
USE PCA for es<ma<ng the sub-‐space
• Computes n-‐dim subspace such that the projec<on of the data points onto the subspace has the largest variance among all n-‐dim subspaces.
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x1
x2
1st principal component
2rd principal component
USE PCA for es<ma<ng the sub-‐space
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Orthonormal
PCA Mathema<cal Formula<on
Define a transforma<on, W,
m-‐dimensional n-‐dimensional
= Data Scarer matrix Tj
N
1jjT )xx)(xx(S −−∑=
=
N...2,1jxWy jT
j ==
PCA = eigenvalue decomposi<on of a data covariance matrix
= Transf. data scarer matrix WSW)yy)(yy(S~ TTT
j
N
1jjT =−−∑=
=
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𝑊↓𝑜𝑝𝑡 =𝑎𝑟𝑔𝑚𝑎𝑥↓𝑊 |𝑊↑𝑇 𝑆↓𝑇 𝑊| =[𝑣↓1 𝑣↓2 ⋅⋅⋅ 𝑣↓𝑚 ] Eigenvectors of ST
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Image space
Face space
• Maximize the scarer of the training images in face space
• Computes n-‐dim subspace such that the projec<on of the data points onto the subspace has the largest variance among all n-‐dim subspaces.
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Eigenfaces
• PCA extracts the eigenvectors of ST – Gives a set of vectors v1, v2, v3, ...
– Each one of these vectors is a direc<on in face space:
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Projec<ng onto the Eigenfaces • The eigenfaces v1, ..., vK span the space of faces
– A face is converted to eigenface coordinates by
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Algorithm
1. Align training images x1, x2, …, xN
2. Compute average face u = 1/N Σ xi 3. Compute the difference image
φi = xi – u
Training
Note that each image is formulated into a long vector!
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Tes%ng 1. Take query image X 2. Project X into Eigenface space (W = {eigenfaces}) and compute projec<on ωi = W (X – u), 3. Compare projec<on ωi with all training N projec<ons ai
ST = (1/N) Σ φi φiT = BBT, B=[φ1, φ2 … φN]
4. Compute the covariance matrix (total scarer matrix)
5. Compute the eigenvectors of the covariance matrix ST 6. Compute training projec<ons a1, a2... aN
Algorithm
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Illustra<on of Eigenfaces
These are the first 4 eigenvectors from a training set of 400 images (ORL Face Database).
• The visualiza<on of eigenvectors:
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Eigenfaces look somewhat like generic faces.
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• Only selec<ng the top P eigenfaces à reduces the dimensionality.
• Fewer eigenfaces result in more informa<on loss, and hence less discrimina<on between faces.
Reconstruc<on and Errors
P = 4
P = 200
P = 400
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Summary for Eigenface
Pros • Non-‐itera<ve, globally op<mal solu<on
Limita<ons
• PCA projec<on is op%mal for reconstruc%on from a low dimensional basis, but may NOT be op%mal for discrimina%on…
• See supplementary materials for “Linear Discrimina<ve Analysis”, aka “Fisherfaces”
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• Global appearance method: not robust to misalignment, or background varia<on
Limita<ons
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Limita<ons • PCA assumes that the data has a Gaussian distribu<on (mean
µ, covariance matrix Σ)
The shape of this dataset is not well described by its principal components Cred
it slide
: S. Lazeb
nik
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PCA-‐SIFT: A More Dis<nc<ve Representa<on for Local Image Descriptors -‐ Y Ke, R Sukthankar -‐ IEEE CVPR 04
Extensions
• PCA-‐SIFT
• Generalized PCA: R. Vidal, Y. Ma, and S. Sastry. Generalized Principal Component Analysis (GPCA). IEEE Transac<ons on Parern Analysis and Machine Intelligence, volume 27, number 12, pages 1 -‐ 15, 2005.
• Tensor Faces: "Mul<linear Analysis of Image Ensembles: TensorFaces," M.A.O. Vasilescu, D. Terzopoulos, Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002
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Slide not used
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Categoriza<on Iden<fica<on
No
localiza<
on
1. PCA & Eigenfaces (Turk & Pentland, 1991) 2. LDA & Fisherfaces (Bellumeur et al. 1997)
3. AdaBoost (Viola & Jones, 2001)
Detec<on
or
Localizatoin
Milestone Face Recogni<on methods
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Features
Decision boundary
Computer screen
Background
In some feature space
Detec<ng foreground objects: A binary classifica<on formula<on
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Linear classifiers • Find linear func<on (hyperplane) to separate posi<ve and nega<ve examples
0:negative0:positive
<+⋅
≥+⋅
bb
ii
ii
wxxwxx
Which hyperplane is best?
w, b
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Support vector machines • Find hyperplane that maximizes the margin between the
posi<ve and nega<ve examples
Margin Support vectors
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• Datasets that are linearly separable work out great:
•
•
• But what if the dataset is just too hard?
• We can map it to a higher-‐dimensional space:
0 x
0 x
0 x
x2
Nonlinear SVMs
Slide credit: Andrew Moore
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Φ: x → φ(x)
Nonlinear SVMs • General idea: the original input space can always be mapped
to some higher-‐dimensional feature space where the training set is separable:
li~ing transforma<on
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Slide credit: Andrew Moore
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Each data point has
a class label:
wt =1 and a weight:
+1 ( )
-‐1 ( ) yt =
Boos<ng
• It is a sequen<al procedure:
xt=1
xt=2
xt
Y. Freund and R. Schapire, A short introduction to boosting, Journal of Japanese Society for Artificial Intelligence, 14(5):771-780, September, 1999.
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Toy example Weak learners from the family of lines
h => p(error) = 0.5 it is at chance
Each data point has
a class label:
wt =1 and a weight:
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+1 ( )
-‐1 ( ) yt =
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Toy example
This one seems to be the best
This is a ‘weak classifier’: It performs slightly berer than chance.
Each data point has
a class label:
wt =1 and a weight:
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+1 ( )
-‐1 ( ) yt =
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Toy example
Each data point has
a class label:
wt wt exp{-‐yt Ht}
We update the weights:
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+1 ( )
-‐1 ( ) yt =
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Toy example
Each data point has
a class label:
wt wt exp{-‐yt Ht}
We update the weights:
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+1 ( )
-‐1 ( ) yt =
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Toy example
Each data point has
a class label:
wt wt exp{-‐yt Ht}
We update the weights:
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+1 ( )
-‐1 ( ) yt =
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Toy example
Each data point has
a class label:
wt wt exp{-‐yt Ht}
We update the weights:
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+1 ( )
-‐1 ( ) yt =
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Toy example
The strong (non-‐ linear) classifier is built as the combina<on of all the weak (linear) classifiers.
f1 f2
f3
f4
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• Defines a classifier using an addi<ve model:
Boos<ng
Strong classifier
Weak classifier
Weight Features vector
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ℎ(𝑥)= 𝛼↓1 ℎ↓1 (𝑥)+ 𝛼↓2 ℎ↓2 (𝑥)+ 𝛼↓3 ℎ↓3 (𝑥)+…
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• Defines a classifier using an addi<ve model:
• We need to define a family of weak classifiers
Boos<ng
Strong classifier
Weak classifier
Weight Features vector
form a family of weak classifiers
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ℎ(𝑥)= 𝛼↓1 ℎ↓1 (𝑥)+ 𝛼↓2 ℎ↓2 (𝑥)+ 𝛼↓3 ℎ↓3 (𝑥)+…
ℎ↓𝑘 (𝑥)
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• A simple algorithm for learning robust classifiers – Freund & Shapire, 1995 – Friedman, Has<e, Tibshhirani, 1998
• Provides efficient algorithm for sparse visual feature selec<on – Tieu & Viola, 2000 – Viola & Jones, 2003
• Easy to implement, not requires external op<miza<on tools.
Why boos<ng?
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• Weak learners
value of rectangle feature
threshold
1 if ( )( )
0 otherwisej j
j
f xh x
θ>⎧= ⎨⎩
Boos<ng -‐ mathema<cs
1 1
11 ( )( ) 2
0 otherwise
T Tt t tt th x
h xα α
= =
⎧ ≥⎪= ⎨⎪⎩
∑ ∑
• Final strong classifier
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Weak classifier
• 4 kind of Rectangle filters • Value = ∑ (pixels in white area) – ∑ (pixels in black area)
Credit slide: S. Lazebnik
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Source
Result
Credit slide: S. Lazebnik
Weak classifier
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….. 1( ,1)x 2( ,1)x 3( ,0)x 4( ,0)x
( , )n nx y1α
5( ,0)x 6( ,0)x
Weak classifier threshold
1 if ( )( )
0 otherwisej j
j
f xh x
θ>⎧= ⎨⎩
1. Evaluate each rectangle filter on each example
Viola & Jones algorithm
0.8 0.7 0.2 0.3 0.8 0.1
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P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. CVPR 2001.
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• For a 24x24 detec<on region,
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Viola & Jones algorithm
P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. CVPR 2001.
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b. For each feature, j ( )j i j i iiw h x yε = −∑
1 if ( )( )
0 otherwisej j
j
f xh x
θ>⎧= ⎨⎩
c. Choose the classifier, ht with the lowest error tε
1 ( )1, ,
t i ih x yt i t i tw w β − −+ =
1t
tt
εβ
ε=
−
2. Select best filter/threshold combina<on ,
,,1
t it i n
t jj
ww
w=
←∑a. Normalize the weights
3. Reweight examples
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Viola & Jones algorithm
P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. CVPR 2001.
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4. The final strong classifier is
1 1
11 ( )( ) 2
0 otherwise
T Tt t tt th x
h xα α
= =
⎧ ≥⎪= ⎨⎪⎩
∑ ∑1logtt
αβ
=
The final hypothesis is a weighted linear combina<on of the T hypotheses where the weights are inversely propor<onal to the training errors
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Viola & Jones algorithm
P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. CVPR 2001.
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Feature selec<on
Discrimina<ve features Non discrimina<ve features
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• A “paradigma<c” method for real-‐<me object detec<on
• Training is slow, but detec<on is very fast • Key ideas
– Integral images for fast feature evalua<on – Boos6ng for feature selec<on – A>en6onal cascade for fast rejec<on of non-‐face windows
Cred
it slide
: S. Lazeb
nik
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Viola & Jones algorithm
P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. CVPR 2001.
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• For each round of boos<ng: 1. Evaluate each rectangle filter on each example 2. Select best filter/threshold combina<on 3. Reweight examples
Boos<ng for face detec<on
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The implemented system • Training Data
– 5000 faces • All frontal, rescaled to 24x24 pixels
– 300 million non-‐faces • 9500 non-‐face images
– Faces are normalized • Scale, transla<on
• Many varia<ons – Across individuals – Illumina<on – Pose
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P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. CVPR 2001.
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System performance • Training <me: “weeks” on 466 MHz Sun worksta<on • 38 layers, total of 6061 features • Average of 10 features evaluated per window on test set • “On a 700 Mhz Pen<um III processor, the face detector can
process a 384 by 288 pixel image in about .067 seconds” – 15 Hz – 15 <mes faster than previous detector of comparable accuracy
(Rowley et al., 1998)
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P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. CVPR 2001.
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Output of Face Detector on Test Images
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P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. CVPR 2001.
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Other detec<on tasks
Facial Feature Localization
Male vs. female
Profile Detection
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Profile Detec<on
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Profile Features
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Face Image Databases
• Databases for face recogni<on can be best u<lized as training sets – Each image consists of an individual on a uniform and unclurered background
• Test Sets for face detec<on – MIT, CMU (frontal, profile), Kodak
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Experimental Results
• Test dataset – MIT+CMU frontal face test set – 130 images with 507 labeled frontal faces
MIT test set: 23 images with 149 faces Sung & poggio: detection rate 79.9% with 5 false positive AdaBoost: detection rate 77.8% with 5 false positives
False detection 10 31 50 65 78 95 110 167 422
AdaBoost 78.3 85.2 88.8 89.8 90.1 90.8 91.1 91.8 93.7
Neural-net 83.2 86.0 - - - 89.2 - 90.1 89.9
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Experimental Results
-> not significantly different accuracy
-> but the cascade class. almost 10 times faster
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Sharing features with Boos<ng
Matlab code
• Gentle boos<ng
• Object detector using a part based model
hrp://people.csail.mit.edu/torralba/iccv2005/
Sharing features: efficient boos<ng procedures for mul<class object detec<on A. Torralba, K. P. Murphy and W. T. Freeman Proceedings of the IEEE Computer Society Conference on Computer Vision and Parern Recogni<on (CVPR). pp 762-‐769, 2004.
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What we have learned today
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• Recogni<on problems • Face discrimina<on
– Principal Component Analysis (PCA) and Eigenfaces (Problem Set 1 (Q1))
– Linear Discriminant Analysis (LDA) and Fisherfaces (supplementary materials)
• Face detec<on – SVM (Problem Set 0 (Q2)) – Boos<ng
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Supplementary materials
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Fisher Faces: Linear Discriminant Analysis
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Linear Discriminant Analysis (LDA) Fisher’s Linear Discriminant (FLD)
• Eigenfaces exploit the max scarer of the training images in face space
• Fisherfaces arempt to maximise the between class scaPer, while minimising the within class scaPer.
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Illustra<on of the Projec<on
Poor Projec<on
x1
x2
x1
x2
w Using two classes as example:
Good
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Comparing with PCA
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Variables
• N Sample images: • c classes:
• Average of each class:
• Total average:
{ }Nxx ,,1
{ }cχχ ,,1
∑=∈ ikx
ki
i xN χ
µ1
∑==
N
kkxN 1
1µ
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Scarers
• Scarer of class i: ( )( )Tikx
iki xxSik
µµχ
−∑ −=∈
∑==
c
iiW SS
1
( )( )∑ −−==
c
i
TiiiBS
1µµµµχ
BWT SSS +=
• Within class scarer:
• Between class scarer:
• Total scarer:
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Illustra<on
2S
1S
BS
21 SSSW +=
x1
x2 Within class scarer
Between class scarer
( )( )Tikx
iki xxSik
µµχ
−∑ −=∈
∑==
c
iiW SS
1( )( )∑ −−=
=
c
i
TiiiBS
1µµµµχ
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Mathema<cal Formula<on (1)
• A~er projec<on:
• Between class scarer (of y’s): • Within class scarer (of y’s):
kT
k xWy =
WSWS BT
B =~
WSWS WT
W =~
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Illustra<on
2~S
1~S
BS~
21~~~ SSSW +=
x1
x2 kT
k xWy =
WSWS BT
B =~WSWS W
TW =~
∑==
c
iiW SS
1( )( )∑ −−=
=
c
i
TiiiBS
1µµµµχ
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Mathema<cal Formula<on • The desired projec<on:
WSW
WSW
SS
WW
TB
T
W
Bopt WW
max arg~~
max arg ==
miwSwS iWiiB ,,1 …== λ
• How is it found ? à Generalized Eigenvectors
• If Sw has full rank, the generalized eigenvectors are eigenvectors of SW-‐1 SB with largest eigen-‐values
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Lecture 2 - !!!
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Training/ Tes%ng
Projec<on in Eigenface Projec<on ωi = W opt (X – u), Wopt = {fisher-‐faces}
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Results: Eigenface vs. Fisherface (1)
• Variation in Facial Expression, Eyewear, and Lighting
• Input: 160 images of 16 people • Train: 159 images • Test: 1 image
With glasses
Without glasses
3 Lighting conditions 5 expressions
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Eigenface vs. Fisherface (2)
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