CVPR2008 tutorial generalized pca

Generalized Principal Component Analysis
Tutorial @ CVPR 2008
Yi Ma René Vidal
ECE Department Center for Imaging Science
University of Illinois Institute for Computational Medicine
Urbana Champaign Johns Hopkins University

Data segmentation and clustering
• Given a set of points, separate them into multiple groups

• Discriminative methods: learn boundary
• Generative methods: learn mixture model, using, e.g.
Expectation Maximization

Dimensionality reduction and clustering
• In many problems data is high-dimensional: can reduce
dimensionality using, e.g. Principal Component Analysis

• Image compression
• Recognition
– Faces (Eigenfaces)
• Image segmentation
– Intensity (black-white)
– Texture

Segmentation problems in dynamic vision
• Segmentation of video and dynamic textures

• Segmentation of rigid-body motions

Segmentation problems in dynamic vision
• Segmentation of rigid-body motions from dynamic textures

Clustering data on non Euclidean spaces
• Clustering data on non Euclidean spaces
– Mixtures of linear spaces
– Mixtures of algebraic varieties
– Mixtures of Lie groups

• “Chicken-and-egg” problems
– Given segmentation, estimate models
– Given models, segment the data
– Initialization?

• Need to combine
– Algebra/geometry, dynamics and statistics

Outline of the tutorial
• Introduction (8.00-8.15)
• Part I: Theory (8.15-9.45)
– Basic GPCA theory and algorithms (8.15-9.00)
– Advanced statistical methods for GPCA (9.00-9.45)

• Questions (9.45-10.00)
• Break (10.00-10.30)
• Part II: Applications (10.30-12.00)
– Applications to motion and video segmentation (10.30-11.15)
– Applications to image representation & segmentation (11.15-12.00)

• Questions (12.00-12.15)

Part I: Theory
• Introduction to GPCA (8.00-8.15)

• Basic GPCA theory and algorithms (8.15-9.00)
– Review of PCA and extensions
– Introductory cases: line, plane and hyperplane segmentation
– Segmentation of a known number of subspaces
– Segmentation of an unknown number of subspaces

• Advanced statistical and methods for GPCA (9.00-9.45)
– Lossy coding of samples from a subspace
– Minimum coding length principle for data segmentation
– Agglomerative lossy coding for subspace clustering

Part II: Applications in computer vision
• Applications to motion & video segmentation (10.30-11.15)
– 2-D and 3-D motion segmentation
– Temporal video segmentation
– Dynamic texture segmentation

• Applications to image representation and segmentation
(11.15-12.00)
– Multi-scale hybrid linear models for sparse
image representation
– Hybrid linear models for image segmentation

References: Springer-Verlag 2008

Slides, MATLAB code, papers
Slides: http://www.vision.jhu.edu/gpca/cvpr08-tutorial-gpca.htm
Code: http://perception.csl.uiuc.edu/gpca

Part I
René Vidal
Center for Imaging Science
Institute for Computational Medicine
Johns Hopkins University

Principal Component Analysis (PCA)
• Given a set of points x1, x2, …, xN
– Geometric PCA: find a subspace S passing through them
– Statistical PCA: find projection directions that maximize the variance

• Solution (Beltrami’1873, Jordan’1874, Hotelling’33, Eckart-Householder-Young’36)

Basis for S

• Applications: data compression, regression, computer
vision (eigenfaces), pattern recognition, genomics

Extensions of PCA
• Higher order SVD (Tucker’66, Davis’02)

• Independent Component Analysis (Common ‘94)

• Probabilistic PCA (Tipping-Bishop ’99)
– Identify subspace from noisy data
– Gaussian noise: standard PCA
– Noise in exponential family (Collins et al.’01)

• Nonlinear dimensionality reduction
– Multidimensional scaling (Torgerson’58)
– Locally linear embedding (Roweis-Saul ’00)
– Isomap (Tenenbaum ’00)

• Nonlinear PCA (Scholkopf-Smola-Muller ’98)
– Identify nonlinear manifold by applying PCA to
data embedded in high-dimensional space

• Principal Curves and Principal Geodesic Analysis
(Hastie-Stuetzle’89, Tishbirany ‘92, Fletcher ‘04)

• Given a set of points lying in multiple subspaces, identify
– The number of subspaces and their dimensions
– A basis for each subspace
– The segmentation of the data points

• “Chicken-and-egg” problem
– Given segmentation, estimate subspaces
– Given subspaces, segment the data

Prior work on subspace clustering
• Iterative algorithms:
– K-subspace (Ho et al. ’03),
– RANSAC, subspace selection and growing (Leonardis et al. ’02)

• Probabilistic approaches: learn the parameters of a mixture
model using e.g. EM
– Mixtures of PPCA: (Tipping-Bishop ‘99):
– Multi-Stage Learning (Kanatani’04)

• Initialization
– Geometric approaches: 2 planes in R3 (Shizawa-Maze ’91)
– Factorization approaches: independent subspaces of equal
dimension (Boult-Brown ‘91, Costeira-Kanade ‘98, Kanatani ’01)
– Spectral clustering based approaches: (Yan-Pollefeys’06)

Basic ideas behind GPCA
• Towards an analytic solution to subspace clustering
– Can we estimate ALL models simultaneously using ALL data?
– When can we do so analytically? In closed form?
– Is there a formula for the number of models?

• Will consider the most general case
– Subspaces of unknown and possibly different dimensions
– Subspaces may intersect arbitrarily (not only at the origin)

• GPCA is an algebraic geometric approach to data segmentation
– Number of subspaces = degree of a polynomial
– Subspace basis = derivatives of a polynomial
– Subspace clustering is algebraically equivalent to
• Polynomial fitting
• Polynomial differentiation

Applications of GPCA in computer vision
• Geometry
– Vanishing points
• Image compression
• Segmentation
– Intensity (black-white)
– Texture
– Motion (2-D, 3-D)
– Video (host-guest)
• Recognition
– Faces (Eigenfaces)
• Man - Woman
– Human Gaits
– Dynamic Textures
• Water-bird

• Biomedical imaging
• Hybrid systems identification

Introductory example: algebraic clustering in 1D

• Number of groups?

• How to compute n, c, b’s?
– Number of clusters

– Cluster centers

– Solution is unique if

– Solution is closed form if

• What about dimension 2?

• What about higher dimensions?
– Complex numbers in higher dimensions?
– How to find roots of a polynomial of quaternions?

• Instead
– Project data onto one or two dimensional space
– Apply same algorithm to projected data

Representing one subspace
• One plane

• One line

• One subspace can be represented with
– Set of linear equations
– Set of polynomials of degree 1

Representing n subspaces
• Two planes

• One plane and one line
– Plane:
– Line:

De Morgan’s rule

• A union of n subspaces can be represented with a set of
homogeneous polynomials of degree n

Fitting polynomials to data points
• Polynomials can be written linearly in terms of the vector of coefficients
by using polynomial embedding

Veronese map

• Coefficients of the polynomials can be computed from nullspace of
embedded data
– Solve using least squares
– N = #data points

Finding a basis for each subspace
• Case of hyperplanes:
– Only one polynomial
– Number of subspaces
– Basis are normal vectors

Polynomial Factorization (GPCA-PFA) [CVPR 2003]
• Find roots of polynomial of degree in one variable
• Solve linear systems in variables
• Solution obtained in closed form for

• Problems
– Computing roots may be sensitive to noise
– The estimated polynomial may not perfectly factor with noisy
– Cannot be applied to subspaces of different dimensions
• Polynomials are estimated up to change of basis, hence they may not factor,
even with perfect data

Finding a basis for each subspace
Polynomial Differentiation (GPCA-PDA) [CVPR’04]

• To learn a mixture of subspaces we just need one positive
example per class

Choosing one point per subspace
• With noise and outliers
– Polynomials may not be a perfect union of subspaces

– Normals can estimated correctly by choosing points optimally
• Distance to closest subspace without knowing
segmentation?

GPCA for hyperplane segmentation
• Coefficients of the polynomial can be computed from null
space of embedded data matrix
– Solve using least squares
– N = #data points

• Number of subspaces can be computed from the rank of
embedded data matrix

• Normal to the subspaces can be computed
from the derivatives of the polynomial

GPCA for subspaces of different dimensions
• There are multiple polynomials
fitting the data

• The derivative of each
polynomial gives a different
normal vector

• Can obtain a basis for the
subspace by applying PCA to
normal vectors

GPCA for subspaces of different dimensions
• Apply polynomial embedding to projected data

• Obtain multiple subspace model by polynomial fitting

– Solve to obtain
– Need to know number of subspaces
• Obtain bases & dimensions by polynomial differentiation

• Optimally choose one point per subspace using distance

An example
• Given data lying in the union
of the two subspaces

• We can write the union as

• Therefore, the union can be represented with the two
polynomials

An example
• Can compute polynomials from

• Can compute normals from

Dealing with high-dimensional data
• Minimum number of points
– K = dimension of ambient space
– n = number of subspaces
Subspace 1
• In practice the dimension of
each subspace ki is much
smaller than K Subspace 2

– Number and dimension of the
subspaces is preserved by a
linear projection onto a
subspace of dimension

• Open problem: how to choose
– Can remove outliers by robustly projection?
fitting the subspace – PCA?

GPCA with spectral clustering
• Spectral clustering
– Build a similarity matrix between pairs of points
– Use eigenvectors to cluster data

• How to define a similarity for subspaces?
– Want points in the same subspace to be close
– Want points in different subspace to be far

• Use GPCA to get basis

• Distance: subspace angles

Comparison of PFA, PDA, K-sub, EM
18
PFA
K−sub
16
PDA
Error in the normals [degrees]

EM
14 PDA+K−sub
PDA+EM
12 PDA+K−sub+EM

10

8

6

4

2

0
0 1 2 3 4 5
Noise level [%]

Dealing with outliers
• GPCA with perfect data

• GPCA with outliers

• GPCA fails because PCA fails seek a robust estimate
of Null(Ln ) where Ln = [ n (x1 ), . . . , n
(xN )].

Three approaches to tackle outliers
• Probability-based: small-probability samples
– Probability plots: [Healy 1968, Cox 1968]
– PCs: [Rao 1964, Ganadesikan & Kettenring 1972]
– M-estimators: [Huber 1981, Camplbell 1980]
– Multivariate-trimming (MVT):
[Ganadesikan & Kettenring 1972]

• Influence-based: large influence on model parameters
– Parameter difference with and without a sample:
[Hampel et al. 1986, Critchley 1985]

• Consensus-based: not consistent with models of high consensus.
– Hough: [Ballard 1981, Lowe 1999]
– RANSAC: [Fischler & Bolles 1981, Torr 1997]
– Least Median Estimate (LME):
[Rousseeuw 1984, Steward 1999]

Robust GPCA
Simulation on Robust GPCA (parameters fixed at = 0.3rad and = 0.4
• RGPCA – Influence

(e) 12% (f) 32% (g) 48% (h) 12% (i) 32% (j) 48%

• RGPCA - MVT

(k) 12% (l) 32% (m) 48% (n) 12% (o) 32% (p) 48%

Robust GPCA
Comparison with RANSAC
• Accuracy

(q) (2,2,1) in 3 (r) (4,2,2,1) in 5 (s) (5,5,5) in 6

• Speed
Table: Average time of RANSAC and RGPCA with 24% outliers.

Arrangement (2,2,1) in 3 (4,2,2,1) in 5 (5,5,5) in 6

RANSAC 44s 5.1min 3.4min
MVT 46s 23min 8min
Influence 3min 58min 146min

Summary
• GPCA: algorithm for clustering subspaces
– Deals with unknown and possibly different dimensions
– Deals with arbitrary intersections among the subspaces

• Our approach is based on
– Projecting data onto a low-dimensional subspace
– Fitting polynomials to projected subspaces
– Differentiating polynomials to obtain a basis

• Applications in image processing and computer vision
– Image segmentation: intensity and texture
– Image compression
– Face recognition under varying illumination

For more information,

Vision, Dynamics and Learning Lab
@

Thank You!

via Lossy Coding and Compression
Yi Ma
Image Formation & Processing Group, Beckman
Decision & Control Group, Coordinated Science
Lab.
Electrical & Computer Engineering Department

University of Illinois at Urbana-Champaign

OUTLINE

MOTIVATION

PROBLEM FORMULATION AND EXISTING APPROACHES

SEGMENTATION VIA LOSSY DATA COMPRESSION

SIMULATIONS (AND EXPERIMENTS)

CONCLUSIONS AND FUTURE DIRECTIONS

MOTIVATION – Motion Segmentation in Computer Vision

Goal: Given a sequence of images of multiple moving objects, determine:
– 1. the number and types of motions (rigid-body, affine, linear, etc.)
2. the features that belong to the same motion.

QuickTime™ and a
Cinepak decompressor
are needed to see this picture.

The “chicken-and-egg” difficulty:
– Knowing the segmentation, estimating the motions is easy;
– Knowing the motions, segmenting the features is easy.

A Unified Algebraic Approach to 2D and 3D Motion Segmentation, [Vidal-Ma, ECCV’

MOTIVATION – Image Segmentation
Goal: segment an image into multiple regions with homogeneous texture.

feature
s

Computer Human

Difficulty: A mixture of models of different dimensions or
complexities.
Multiscale Hybrid Linear Models for Lossy Image Representation, [Hong-Wright-Ma, TIP’

MOTIVATION – Video Segmentation
Goal: segmenting a video sequence into segments with “stationary” dynamics
Model: different
segments as outputs
from different (linear)
dynamical systems:
QuickTime™ and a
H.264 decompressor

Identification of Hybrid Linear Systems via Subspace Segmentation, [Huang-Wagner-Ma, C

MOTIVATION – Massive Multivariate Mixed Data

QuickTime™ and a
BMP decompressor

Face database Hyperspectral images Articulate motions

Hand written digits
Microarrays

SUBSPACE SEGMENTATION – Problem Formulation

Assumption: the data are noisy samples from an
arrangement of linear subspaces:

noise-free samples noisy samples samples with outliers

Difficulties:
– the dimensions of the subspaces can be different
– the data can be corrupted by noise or contaminated by outliers
– the number and dimensions of subspaces may be unknown

SUBSPACE SEGMENTATION – Statistical Approaches

Assume that the data
are i.i.d. samples from a mixture of
probabilistic distributions:

Solutions:
• Expectation Maximization (EM) for the maximum-likelihood estimate
[Dempster et. al.’77], e.g., Probabilistic PCA [Tipping-Bishop’99]:

• K-Means for a minimax-like estimate [Forgy’65, Jancey’66, MacQueen’67],
e.g., K-Subspaces [Ho and Kriegman’03]:

Essentially iterate between data segmentation and model estimation.

SUBSPACE SEGMENTATION – An Algebro-Geometric Approach

Idea: a union of linear subspaces is an
algebraic set -- the zero set of a set of
(homogeneous) polynomials:

Solution:
• Identify the set of polynomials of degree n that vanish on

• Gradients of the vanishing polynomials are normals to the
subspaces

Complexity exponential in the dimension and number of subspaces.

Generalized Principal Component Analysis, [Vidal-Ma-Sastry, IEEE Transactions PAMI’0

SUBSPACE SEGMENTATION – An Information-Theoretic Approach
Problem: If the number/dimension of subspaces not given and data
corrupted
by noise and outliers, how to determine the optimal subspaces that fit
Solutions: Model Selection Criteria?
the data?
– Minimum message length (MML) [Wallace-Boulton’68]
– Minimum description length (MDL) [Rissanen’78]
– Bayesian information criterion (BIC)
– Akaike information criterion (AIC) [Akaike’77]
– Geometric AIC [Kanatani’03], Robust AIC [Torr’98]

Key idea (MDL):
• a good balance between model complexity and data fidelity.
• minimize the length of codes that describe the model and the
data:

with a quantization error optimal for the model.

LOSSY DATA COMPRESSION
Questions:

– What is the “gain” or “loss” of segmenting or merging data?

– How does tolerance of error affect segmentation results?

Basic idea: whether the number of bits required to store “the
whole is more than the sum of its parts”?

LOSSY DATA COMPRESSION – Problem Formulation

– A coding scheme maps a set of vectors
to a sequence of bits, from which we can decode
The coding length is denoted as:

– Given a set of real-valued mixed data
the optimal segmentation
minimizes
the overall coding length:

where

LOSSY DATA COMPRESSION – Coding Length for Multivariate Data

Theorem.
Given with

is the number of bits needed to encode the data s.t.
.

A nearly optimal bound for even a small number
of vectors drawn from a subspace or a Gaussian
source.

Segmentation of Multivariate Mixed Data, [Ma-Derksen-Hong-Wright, PAMI’

LOSSY DATA COMPRESSION – Two Coding Schemes
Goal: code s.t. a mean squared error

Linear subspace Gaussian source

LOSSY DATA COMPRESSION – Properties of the Coding Length

1. Commutative Property:
For high-dimensional data, computing the coding length only needs
the kernel matrix:

2. Asymptotic Property:
At high SNR, this is the optimal rate distortion for a Gaussian source.

3. Invariant Property:
Harmonic Analysis is useful for data compression only when the data are
non-Gaussian or nonlinear ……… so is segmentation!

LOSSY DATA COMPRESSION – Why Segment?

partitioning:

sifting:

LOSSY DATA COMPRESSION – Probabilistic Segmentation?

Assign the ith point to the jth group with probability

Theorem. The expected coding length of the segmented data

is a concave function in Π over the domain of a convex polytope.

Minima are reached at the vertexes of the
polytope -- no probabilistic
segmentation!


LOSSY DATA COMPRESSION – Segmentation & Channel Capacity

A MIMO additive white Gaussian noise (AWGN) channel

has the capacity:

If allowing probabilistic grouping of transmitters, the expected
capacity

is a concave function in Π over a convex polytope.
Maximizing such a capacity is a convex
problem.

On Coding and Segmentation of Multivariate Mixed Data, [Ma-Derksen-Hong-Wright, PAMI

LOSSY DATA COMPRESSION – A Greedy (Agglomerative) Algorithm
Objective: minimizing the overall coding length

Input: “Bottom-up” merge

while true do
choose two sets such
that
is minimal QuickTime™ and a

if
PNG decompressor

then
else break
endif
end
Output:
Segmentation of Multivariate Mixed Data via Lossy Coding and Compression, [Ma-Derksen-Hong-Wright, PAMI’07]

SIMULATIONS – Mixture of Almost Degenerate Gaussians
Noisy samples from two lines and one plane in <3
Given Data Segmentation Results

ε0 = 0.01

ε0 = 0.08


SIMULATIONS – “Phase Transition”
#group v.s. distortion Rate v.s. distortion

ε0 = 0.0
0.08 8

ice
cubes
steam Stability: the same segmentation
water for ε across 3 magnitudes!

0.08


SIMULATIONS – Comparison with EM

100 x d uniformly distributed random samples from each subspace, corrupte
with 4% noise. Classification rate averaged over 25 trials for each case.


SIMULATIONS – Comparison with EM

Segmenting three degenerate or non-degenerate Gaussian clusters for 50 tria


SIMULATIONS – Robustness with Outliers
35.8% outliers 45.6%

71.5% 73.6%


SIMULATIONS – Affine Subspaces with Outliers

35.8% outliers 45.6%

66.2% 69.1%


SIMULATIONS – Piecewise-Linear Approximation of Manifolds

Swiss roll Mobius strip Torus Klein bottle

SIMULATIONS – Summary

– The minimum coding length objective automatically addresses
the
model selection issue: the optimal solution is very stable and
robust.

– The segmentation/merging is physically meaningful (measured
in bits).
The results resemble phase transition in statistical physics.

– The greedy algorithm is scalable (polynomial in both K and N)
and
converges well when ε is not too small w.r.t. the sample
density.

Clustering from a Classification Perspective

Assumption: The training data
are drawn from a distribution

Goal: Construct a classifier
such that the misclassification
error

reaches minimum.

Solution: Knowing the distributions and
, the optimal classifier is the maximum a posteriori (MAP)
classifier:

Difficulties: How to learn the two distributions from samples?
(parametric, non-parametric, model selection, high-dimension, outliers…)

MINIMUM INCREMENTAL CODING LENGTH – Problem Formulation

Ideas: Using the lossy coding length

as a surrogate for the Shannon lossless coding length w.r.t. true
distributions.

Additional bits need to encode the test
sample with the jth training set is

Classification Criterion: Minimum Incremental Coding Length
(MICL)

MICL (“Michael”) – Asymptotic Properties

Theorem: As the number of samples goes to infinity, the MICL
criterion converges with probability one to the following criterion:

where
?

is the “number of effective
parameters” of the j-th model (class).

Theorem: The MICL classifier converges to the above asymptotic form
at the rate of for some constant .

Minimum Incremental Coding Length (MICL), [Wright and Ma et. a.., NIPS’07]

SIMULATIONS – Interpolation and Extrapolation via MICL

MICL

SVM

k-NN


SIMULATIONS – Improvement over MAP and RDA [Friedman1989]

Two Gaussians in
R2
isotropic (left)
anisotropic
(right)
(500 trials)

Three Gaussians in
Rn
dim = n
dim = n/2
dim = 1
(500 trials)


SIMULATIONS – Local and Kernel MICL
Local MICL (LMICL): Applying MICL locally to the k-nearest
neighbors of the test sample (frequencylist + Bayesianist).

Kernel MICL (KMICL): Incorporating MICL with a nonlinear kernel
naturally through the identity (“kernelized” RDA):

LMICL k- KMICL-RBF SVM-RBF
NN


CONCLUSIONS

Assumptions: Data are in a high-dimensional space but have
low-dimensional structures (subspaces or submanifolds).

Compression => Clustering & Classification:
– Minimum (incremental) coding length subject to distortion.
– Asymptotically optimal clustering and classification.
– Greedy clustering algorithm (bottom-up, agglomerative).
– MICL corroborates MAP, RDA, k-NN, and kernel methods.

Applications (Next Lectures):
– Video segmentation, motion segmentation (Vidal)
– Image representation & segmentation (Ma)
– Others: microarray clustering, recognition of faces and
handwritten digits (Ma)

FUTURE DIRECTIONS

Theory
– More complex structures: manifolds, systems, random
fields…
– Regularization (ridge, lasso, banding etc.)
– Sparse representation and subspace arrangements

Computation
– Global optimality (random techniques, convex
optimization…)
– Scalability: random sampling, approximation…

Future Application Domains
– Image/video/audio classification, indexing, and retrieval
– Hyper-spectral images and videos
– Biomedical images, microarrays
– Autonomous navigation, surveillance, and 3D mapping
– Identification of hybrid linear/nonlinear systems

REFERENCES & ACKNOWLEGMENT
References:
– Segmentation of Multivariate Mixed Data via Lossy Data
Compression, Yi Ma, Harm Derksen, Wei Hong, John Wright,
PAMI, 2007.
– Classification via Minimum Incremental Coding Length (MICL),
John Wright et. al., NIPS, 2007.
– Website: http://perception.csl.uiuc.edu/coding/home.htm

People:
– John Wright, PhD Student, ECE Department, University of Illinois
– Prof. Harm Derksen, Mathematics Department, University of
Michigan
– Allen Yang (UC Berkeley) and Wei Hong (Texas Instruments R&D)
– Zhoucheng Lin and Harry Shum, Microsoft Research Asia, China

Funding:
– ONR YIP N00014-05-1-0633
– NSF CAREER IIS-0347456, CCF-TF-0514955, CRS-EHS-0509151

11/2003

“The whole is more than the sum of its
parts.”
--
Aristotle
Questions, please?

Yi Ma, CVPR 2008

Part II
Applications of GPCA in Computer Vision
René Vidal

Applications to motion and and video
segmentation
René Vidal

3-D motion segmentation problem
• Given a set of point correspondences in multiple views, determine
– Number of motion models
– Motion model: affine, homography, fundamental matrix, trifocal tensor
– Segmentation: model to which each pixel belongs

• Mathematics of the problem depends on
– Number of frames (2, 3, multiple)
– Projection model (affine, perspective)
– Motion model (affine, translational, homography, fundamental matrix, etc.)
– 3-D structure (planar or not)

Taxonomy of problems
• 2-D Layered representation
– Probabilistic approaches: Jepson-Black’93, Ayer-Sawhney’95, Darrel-Pentland’95, Weiss-
Adelson’96, Weiss’97, Torr-Szeliski-Anandan’99
– Variational approaches: Cremers-Soatto ICCV’03
– Initialization: Wang-Adelson’94, Irani-Peleg’92, Shi-Malik‘98, Vidal-Singaraju’05-’06

• Multiple rigid motions in two perspective views
– Probabilistic approaches: Feng-Perona’98, Torr’98
– Particular cases: Izawa-Mase’92, Shashua-Levin’01, Sturm’02,
– Multibody fundamental matrix: Wolf-Shashua CVPR’01, Vidal et al. ECCV’02, CVPR’03, IJCV’06
– Motions of different types: Vidal-Ma-ECCV’04, Rao-Ma-ICCV’05

• Multiple rigid motions in three perspective views
– Multibody trifocal tensor: Hartley-Vidal-CVPR’04

• Multiple rigid motions in multiple affine views
– Factorization-based: Costeira-Kanade’98, Gear’98, Wu et al.’01, Kanatani’ et al.’01-02-04
– Algebraic: Yan-Pollefeys-ECCV’06, Vidal-Hartley-CVPR’04

• Multiple rigid motions in multiple perspective views
– Schindler et al. ECCV’06, Li et al. CVPR’07

A unified approach to motion segmentation
• Estimation of multiple motion models equivalent to
estimation of one multibody motion model

chicken-and-egg

– Eliminate feature clustering: multiplication

– Estimate a single multibody motion model: polynomial fitting

– Segment multibody motion model: polynomial differentiation

A unified approach to motion segmentation
• Applies to most motion models in computer vision

• All motion models can be segmented algebraically by
– Fitting multibody model: real or complex polynomial to all data
– Fitting individual model: differentiate polynomial at a data point

Segmentation of 3-D translational motions
• Multiple epipoles (translation)

• Epipolar constraint: plane in
– Plane normal = epipoles
– Data = epipolar lines

• Multibody epipolar constraint • Epipoles are derivatives of
at epipolar lines

Segmentation of 3-D translational motions

Single-body factorization
Structure = 3D surface
• Affine camera model

– p = point
– f = frame

Motion = camera position and orientation

• Motion of one rigid-body lives in a 4-D subspace
(Boult and Brown ’91,
Tomasi and Kanade ‘92)

– P = #points
– F = #frames

Multi-body factorization
• Given n rigid motions

• Motion segmentation is obtained from
– Leading singular vector of (Boult and Brown ’91)
– Shape interaction matrix (Costeira & Kanade ’95, Gear ’94)

– Number of motions (if fully-dimensional)

• Motion subspaces need to be independent (Kanatani ’01)

Multi-body factorization
• Sensitive to noise

– Kanatani (ICCV ’01): use model selection to scale Q
– Wu et al. (CVPR’01): project data onto subspaces and iterate
• Fails with partially dependent motions
– Zelnik-Manor and Irani (CVPR’03)
• Build similarity matrix from normalized Q
• Apply spectral clustering to similarity matrix
– Yan and Pollefeys (ECCV’06)
• Local subspace estimation + spectral clustering
– Kanatani (ECCV’04)
• Assume degeneracy is known: pure translation in the image
• Segment data by multi-stage optimization (multiple EM problems)

• Cannot handle missing data
– Gruber and Weiss (CVPR’04)
• Expectation Maximization

PowerFactorization+GPCA
• A motion segmentation algorithm that
– Is provably correct with perfect data
– Handles both independent and degenerate motions
– Handles both complete and incomplete data

• Project trajectories onto a 5-D subspace of
– Complete data: PCA or SVD
– Incomplete data: PowerFactorization

• Cluster projected subspaces using GPCA
– Handles both independent and degenerate motions
– Non-iterative: can be used to initialize EM

Projection onto a 5-D subspace
• Motion of one rigid-body lives in
4-D subspace of Motion 1

• By projecting onto a 5-D
subspace of Motion 2
– Number and dimensions of
subspaces are preserved
– Motion segmentation is
equivalent to clustering
subspaces of dimension
2, 3 or 4 in
– Minimum #frames = 3
(CK needs a minimum of 2n
frames for n motions)
• What projection to use?
– Can remove outliers by robustly
– PCA: 5 principal components
fitting the 5-D subspace using
Robust SVD (DeLaTorre-Black) – RPCA: with outliers

Projection onto a 5-D subspace
PowerFactorization algorithm: Given , factor it as

• Complete data • Incomplete data

– Given A solve for B

– Orthonormalize B
Linear problem
– Given B solve for A

– Iterate
• It diverges in some cases
• Converges to rank-r
approximation with rate • Works well with up to 30% of
missing data

Motion segmentation using GPCA
• Apply polynomial embedding to 5-D points
Veronese map

Hopkins 155 motion segmentation database
• Collected 155 sequences
– 120 with 2 motions
– 35 with 3 motions

• Types of sequences
– Checkerboard sequences: mostly full
dimensional and independent motions
– Traffic sequences: mostly degenerate (linear,
planar) and partially dependent motions
– Articulated sequences: mostly full dimensional
and partially dependent motions

• Point correspondences
– In few cases, provided by Kanatani & Pollefeys
– In most cases, extracted semi-automatically
with OpenCV

Experimental results: Hopkins 155 database
• 2 motions, 120 sequences, 266 points, 30 frames

Experimental results: Hopkins 155 database
• 3 motions, 35 sequences, 398 points, 29 frames

Experimental results: missing data sequences

• There is no clear correlation between amount of missing data and
percentage of misclassification
• This could be because convergence of PF depends more on “where”
missing data is located than on “how much” missing data there is

Conclusions
• For two motions
– Algebraic methods (GPCA and LSA) are more accurate than
statistical methods (RANSAC and MSL)
– LSA performs better on full and independent sequences, while
GPCA performs better on degenerate and partially dependent
– LSA is sensitive to dimension of projection: d=4n better than d=5
– MSL is very slow, RANSAC and GPCA are fast

• For three motions
– GPCA is not very accurate, but is very fast
– MSL is the most accurate, but it is very slow
– LSA is almost as accurate as MSL and almost as fast as GPCA

Segmentation of Dynamic Textures
René Vidal

Modeling a dynamic texture: fixed boundary
• Examples of dynamic textures:

• Model temporal evolution as the output of a linear
dynamical system (LDS): Soatto et al. ‘01
dynamics
zt+1 = Azt + vt
images yt = Czt + wt
appearance

Segmenting non-moving dynamic textures
• One dynamic texture lives in the observability subspace
zt+1 = Azt + vt
yt = Czt + wt

• Multiple textures live in multiple subspaces

water

steam

• Cluster the data using GPCA

Segmenting moving dynamic textures

Segmenting moving dynamic textures

Ocean-bird

Level-set intensity-based segmentation
• Chan-Vese energy functional

• Implicit methods
– Represent C as the zero level set
of an implicit function , i.e.
C = {(x, y) : (x, y) = 0}

• Solution
– The solution to the gradient descent algorithm for is given by

– c1 and c2 are the mean intensities inside and outside the contour C.

Dynamics & intensity-based energy
• We represent the intensities of the pixels in the images as
the output of a mixture of AR models of order p

• We propose the following spatial-temporal extension of the
Chan-Vese energy functional

where

Variational segmentation of dynamic textures
• Given the ARX parameters, we can solve for the implicit
function by solving the PDE

• Given the implicit function , we can solve for the ARX
parameters of the jth region by solving the linear system

• Fixed boundary segmentation results and comparison

Ocean-smoke Ocean-dynamics Ocean-appearance

• Moving boundary segmentation results and comparison

Ocean-fire

• Results on a real sequence

Raccoon on River

Temporal video segmentation
• Segmenting N=30 frames of a
sequence containing n=3
scenes
– Host
– Guest
– Both

• Image intensities are output of
linear system

dynamics
xt+1 = Axt +vt
• y =C
Apply GPCA totfit n=3 xt +wt
images
observability subspaces
appearance

Temporal video segmentation
• Segmenting N=60 frames of a
sequence containing n=3
scenes
– Burning wheel
– Burnt car with people
– Burning car

• Image intensities are output of linear
system
dynamics
xt+1 = Axt +vt
yt = Cxt +wt
images
• Apply GPCA to fit n=3 appearance
observability
subspaces

Conclusions
• Many problems in computer vision can be posed as subspace
clustering problems
– Temporal video segmentation
– 2-D and 3-D motion segmentation
– Dynamic texture segmentation
– Nonrigid motion segmentation

• These problems can be solved using GPCA: an algorithm for clustering
subspaces
– Deals with unknown and possibly different dimensions
– Deals with arbitrary intersections among the subspaces

• GPCA is based on
– Projecting data onto a low-dimensional subspace
– Recursively fitting polynomials to projected subspaces
– Differentiating polynomials to obtain a basis

for Image Representation & Segmentation

Yi Ma
Control & Decision, Coordinated Science Laboratory
Image Formation & Processing Group, Beckman
Department of Electrical & Computer Engineering

University of Illinois at Urbana-Champaign

INTRODUCTION

GPCA FOR LOSSY IMAGE REPRESENTATION

IMAGE SEGMENTATION VIA LOSSY COMPRESSION

OTHER APPLICATIONS

CONCLUSIONS AND FUTURE DIRECTIONS

Introduction – Image Representation via Linear Transformations

better
representations?

pixel-based representation
three matrixes of RGB-values

a more compact
linear transformation representation

Introduction
Fixed Orthogonal Bases (representation, approximation, compression)
- Discrete Fourier transform (DFT) or discrete cosine transform (DCT)
(Ahmed ’74): JPEG.
- Wavelets (multi-resolution) (Daubechies’88, Mallat’92): JPEG-2000.
- Curvelets and contourlets (Candes & Donoho’99, Do & Veterlli’00)

Discrete Fourier transform (DFT)
6.25% coefficients.

Wavelet transform

Unorthogonal Bases (for redundant representations)
- Extended lapped transforms, frames, sparse representations (Lp
geometry)…

Introduction

Adaptive Bases (optimal if imagery data are uni-modal)
- Karhunen-Loeve transform (KLT), also known as PCA (Pearson’1901,
Hotelling’33, Jolliffe’86)

stack

adaptive bases

Introduction – Principal Component Analysis (PCA)
Dimensionality Reduction

Find a low-dimensional representation (model) for high-dimensional data.
Principal Component Analysis (Pearson’1901, Hotelling’1933, Eckart &
Young’1936) or Karhunen-Loeve transform (KLT).

Basis for S SVD
Variations of PCA
– Nonlinear Kernel PCA (Scholkopf-Smola-Muller’98)
– Probabilistic PCA (Tipping-Bishop’99, Collins et.al’01)
– Higher-Order SVD (HOSVD) (Tucker’66, Davis’02)
– Independent Component Analysis (Hyvarinen-Karhunen-Oja’01)

Hybrid Linear Models – Multi-Modal Characteristics

Distribution of the first three principal components of
the Baboon image: A clear multi-modal distribution

Hybrid Linear Models – Multi-Modal Characteristics

Vector Quantization (VQ)
- multiple 0-dimensional affine subspaces (i.e. cluster means)
- existing clustering algorithms are iterative (EM, K-means)

Hybrid Linear Models – Versus Linear Models
A single linear model
Linear
stack

Hybrid linear models

Hybrid linear
stack

Hybrid Linear Models – Characteristics of Natural Images
Multivariate Hybrid Hierarchical High-dimensio
1D 2D (multi-modal) (multi-scale) (vector-value
Fourier
(DCT) X X

Wavelets X
X
Curvelets X

Random fields X X X

PCA/KLT X X X

VQ X X X X

Hybrid linear X X X X X

We need a new & simple paradigm to effectively account for all
these characteristics simultaneously.

Hybrid Linear Models – Subspace Estimation and Segmentation

Hybrid Linear Models (or Subspace
Arrangements)
– the number of subspaces is
unknown
– the dimensions of the
subspaces are unknown
– the basis of the subspaces are
unknown
– the segmentation of the data
points is unknown

“Chicken-and-Egg” Coupling
– Given segmentation, estimate subspaces
– Given subspaces, segment the data

Hybrid Linear Models – Recursive GPCA (an Example)

Hybrid Linear Models – Effective Dimension
Model Selection (for Noisy Data)
Model complexity;
Data fidelity;

Number of
subspaces

Total Dimension Number of
number of of each points in each
points subspace subspace

Model selection criterion: minimizing effective dimension
subject to a given error tolerance (or PSNR)

Hybrid Linear Models – Simulation Results (5% Noise)

ED=3

ED=2.0067

ED=1.6717

Hybrid Linear Models – Subspaces of the Barbara Image

Hybrid Linear Models – Lossy Image Representation (Baboon)

GPCA

Original PCA (8x8)

DCT (JPEG)

Harr
Wavelet GPCA (8x8)

Multi-Scale Implementation – Algorithm Diagram
Diagram for a level-3 implementation of hybrid linear models
for image representation

Multi-Scale Hybrid Linear Models for Lossy Image Representation, [Hong-Wright-Ma, TIP

Multi-Scale Implementation – The Baboon Image

The Baboon image

downsample
by two twice

segmentation of
2 by 2 blocks


Multi-Scale Implementation – Comparison with Other Methods

The Baboon image


Multi-Scale Implementation – Image Approximation

Comparison with level-3 wavelet (7.5% coefficients)

Level-3 bior-4.4 wavelets Level-3 hybrid linear model
PSNR=23.94 PSNR=24.64

Multi-Scale Implementation – Block Size Effect

The Baboon image

Some problems with the multi-scale hybrid linear model:
1. has minor block effect;
2. is computationally more costly (than Fourier, wavelets, PCA);
3. does not fully exploit spatial smoothness as wavelets.


Multi-Scale Implementation – The Wavelet Domain

The Baboon image HL

LH HH

segmentation
at each scale


Multi-Scale Implementation – Wavelets v.s. Hybrid Linear Wavelets

The Baboon image

Advantages of the hybrid linear model in wavelet domain:
1. eliminates block effect;
2. is computationally less costly (than in the spatial domain);
3. achieves higher PSNR.


Multi-Scale Implementation – Visual Comparison
Comparison among several models (7.5% coefficients)

Original Wavelets
Image PSNR=23.94

Hybrid model Hybrid model
in spatial in wavelet
domain domain
PSNR=24.64 PSNR=24.88


Image Segmentation – via Lossy Data Compression

stack

QuickTime™ and a
PNG decompressor

APPLICATIONS – Texture-Based Image Segmentation
Naïve approach:
– Take a 7x7 Gaussian window around every
pixel.
– Stack these windows as vectors.
– Clustering the vectors using our algorithm.

A few results:


APPLICATIONS – Distribution of Texture Features
Question: why does such a simple algorithm work at all?

Answer: Compression (MDL/MCL) is well suited to mid-level texture
segmentation.

Using a single representation (e.g. windows, filterbank responses) for textures
different complexity ⇒ redundancy and degeneracy, which can be exploited fo
clustering / compression.

QuickTime™ and a
TIFF (LZW) decompressor

Above: singular values of feature vectors from two different
segments of the image at left.

APPLICATIONS – Compression-based Texture Merging (CTM)

Problem with the naïve
approach: QuickTime™ and a
TIFF (LZW) decompressor
QuickTime™ and a QuickTime™ and a are needed to see this picture.
Strong edges, segment boundaries
TIFF (LZW) decompressor TIFF (LZW) decompressor
are needed to see this picture. are needed to see this picture.

Solution:
Low-level, edge-preserving over-segmentation into small homogeneous
regions.
Simple features: stacked Gaussian windows (7x7 in our experiments).
Merge adjacent regions to minimize coding length (“compress” the features).

APPLICATIONS – Hierarchical Image Segmentation via CTM

ε = 0.1 ε = 0.2 ε = 0.4
Lossy coding with varying distortion ε => hierarchy of
segmentations

APPLICATIONS – CTM: Qualitative Results

APPLICATIONS – CTM: Quantitative Evaluation and Comparison
Berkeley Image Segmentation Database

PRI: Probabilistic Rand Index [Pantofaru 2005]
VoI: Variation of Information [Meila 2005]
GCE: Global Consistency Error [Martin 2001]
BDE: Boundary Displacement Error [Freixenet 2002]

Unsupervised Segmentation of Natural Images via Lossy Data Compression, CVIU, 200

Other Applications: Multiple Motion Segmentation (on Hopkins155)

QuickTime™ and a QuickTime™ and a
Cinepak decompressor Cinepak decompressor
are needed to see this picture. are needed to see this picture.

Two Motions: MSL 4.14%, LSA 3.45%, ALC 2.40%, and work with up to 25% outliers.
Three Motions: MSL 8.32%, LSA 9.73%, ALC 6.26%.

Shankar Rao, Roberton Tron, Rene Vidal, and Yi Ma, to appear in CVPR’08

Other Applications – Clustering of Microarray Data


Other Applications – Supervised Classification

Premises: Data lie on an
arrangement of subspaces

Unsupervised Clustering Supervised Classification
– Generalized PCA – Sparse Representation

Other Applications – Robust Face Recognition

Robust Face Recognition via Sparse Representation, to appear in PAMI 2008

Other Applications: Robust Motion Segmentation (on Hopkins155)

Dealing with incomplete or mistracked features with dataset 80%
corrupted!
Shankar Rao, Roberto Tron, Rene Vidal, and Yi Ma, to appear in CVPR’08

Three Measures of Sparsity: Bits, L_0 and L1-Norm

Reason: High-dimensional data, like images, do have compact,
compressible, sparse structures, in terms of their geometry,
statistics, and semantics.

Conclusions

Most imagery data are high-dimensional, statistically or
geometrically heterogeneous, and have multi-scale
structures.

Imagery data require hybrid models that can adaptively
represent different subsets of the data with different
(sparse) linear models.

Mathematically, it is possible to estimate and segment
hybrid (linear) models non-iteratively. GPCA offers one such
method.

Hybrid models lead to new paradigms, new principles, and
new applications for image representation, compression,
and segmentation.

Future Directions
Mathematical Theory
– Subspace arrangements (algebraic properties).
– Extension of GPCA to more complex algebraic varieties (e.g.,
hybrid multilinear, high-order tensors).
– Representation & approximation of vector-valued functions.

Computation & Algorithm Development
– Efficiency, noise sensitivity, outlier elimination.
– Other ways to combine with wavelets and curvelets.

Applications to Other Data
– Medical imaging (ultra-sonic, MRI, diffusion tensor…)
– Satellite hyper-spectral imaging.
– Audio, video, faces, and digits.
– Sensor networks (location, temperature, pressure, RFID…)
– Bioinformatics (gene expression data…)

Acknowledgement

People
– Wei Hong, Allen Yang, John Wright, University of Illinois
– Rene Vidal of Biomedical Engineering Dept., Johns Hopkins
University
– Kun Huang of Biomedical & Informatics Science Dept., Ohio-
State University

Funding
– Research Board, University of Illinois at Urbana-Champaign
– National Science Foundation (NSF CAREER IIS-0347456)
– Office of Naval Research (ONR YIP N000140510633)
– National Science Foundation (NSF CRS-EHS0509151)
– National Science Foundation (NSF CCF-TF0514955)

Generalized Principal Component Analysis:
Modeling and Segmentation of Multivariate Mixed
Data

Rene Vidal, Yi Ma, and Shankar Sastry
Springer-Verlag, to appear

Thank You!

Yi Ma, CVPR 2008

CVPR2008 tutorial generalized pca

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