1. EE 290A Generalized Principal Component Analysis
René Vidal
Center for Imaging Science
Institute for Computational Medicine
Johns Hopkins University
Title: Segmentation of Dynamic Scenes and Textures
Abstract: Dynamic scenes are video sequences containing multiple objects moving in front of dynamic backgrounds, e.g. a bird floating on water. One can model such scenes as the output of a collection of dynamical models exhibiting discontinuous behavior both in space, due to the presence of multiple moving objects, and in time, due to the appearance and disappearance of objects. Segmentation of dynamic scenes is then equivalent to the identification of this mixture of dynamical models from the image data. Unfortunately, although the identification of a single dynamical model is a well understood problem, the identification of multiple hybrid dynamical models is not. Even in the case of static data, e.g. a set of points living in multiple subspaces, data segmentation is usually thought of as a "chicken-and-egg" problem. This is because in order to estimate a mixture of models one needs to first segment the data and in order to segment the data one needs to know the model parameters. Therefore, static data segmentation is usually solved by alternating data clustering and model fitting using, e.g., the Expectation Maximization (EM) algorithm.
Our recent work on Generalized Principal Component Analysis (GPCA) has shown that the "chicken-and-egg" dilemma can be tackled using algebraic geometric techniques. In the case of data living in a collection of (static) subspaces, one can segment the data by fitting a set of polynomials to all data points (without first clustering the data) and then differentiating these polynomials to obtain the model parameters for each group. In this talk, we will present ongoing work addressing the extension of GPCA to time-series data living in a collection of multiple moving subspaces. The approach combines classical GPCA with newly developed recursive hybrid system identification algorithms. We will also present applications of DGPCA in image/video segmentation, 3-D motion segmentation, dynamic texture segmentation, and heart motion analysis.

2. 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)
Applications: data compression, regression, computer vision (eigenfaces), pattern recognition, genomics
As we all know, Principal Component Analysis problem refers to the problem of estimating a SINGLE subspace from sample data points. Although there are various ways of solving PCA, a simple solution consist of building a matrix with all the data points, computing its SVD, and then extracting a basis for the subspace from the columns of the U matrix, and the dimension of the subspace from the rank of the U matrix.
There is no question that PCA is one of the most popular techniques for dimensionality reduction in various engineering disciplines. In computer vision, in particular, a successful application has been found in face recognition under the name of eigenfaces.
Basis for S

3. 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)
There have been various attempts to generalize PCA in different directions.
For example, Probabilistic PCA considers the case of noisy data and tries to estimate THE subspace in a maximum likelihood sense. For the case of noise in the exponential family, Collins has shown that this can be done using convex optimization techniques.
Other extensions considered the case of data lying on manifold, the so-called NonLinear PCA or Kernel PCA. This problem is usually solved by embedding the data into a higher dimensional space and then assuming that the embedded data DOES live in a linear subspace. Of course the correct embedding to use depends on the problem at hand, and learning the embedding is a current a topic of research in the machine learning community.
A third extension considers the case of identifying multiple subspaces at the same time. It is this case extension the one I will talk about in this talk under the name of Generalized PCA.

4. Generalized Principal Component Analysis
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
Given a set of data points lying on a collection of linear subspaces, without knowing which point corresponds to which subspace and without even knowing the number of subspaces, estimate the number of subspaces, a basis for each subspace and the segmentation of the data.
As stated, the GPCA problem is one of those challenging CHICKEN-AND-EGG problems in the following sense. If we knew the segmentation of the data points, and hence the number of subspaces, then we could solve the problem by applying standard PCA to each group. On the other hand, if we had a basis for each subspace, we could easily assign each point to each cluster. The main challenge is that we neither know the segmentation of the data points nor a basis for each subspace, and the questions is how to estimate everything from the data only.
Previous geometric approaches to GPCA have been proposed in the context of motion segmentation, and under the assumption that the subspaces do NOT intersect. Such approaches are based on first clustering the data, using for example K-means or spectral clustering techniques, and then applying standard PCA to each group.
Statistical methods, on the other hand, model the data as a mixture model whose parameters are the mixing proportions and the basis for each subspace. The estimation of the mixture is a (usually non-convex) optimization problem that can be solved with various techniques. One of the most popular ones is the expectation maximization algorithm (EM) which alternates between the clustering of the data and the estimation of the subspaces.
Unfortunately, the convergence of such iterative algorithms depends on good initialization. At present, initialization is mostly done either at random, using another iterative algorithm such as K-means, or else in an ad-hoc fashion depending on the particular application.
It is here where our main motivation comes into the picture. We would like to know if we can find a way of initializing iterative algorithms in an algebraic fashion. For example, if you look at the two lines in this picture, there has to be a way of estimating those two lines directly from data in a one shot solution, at least in the ABSENCE of noise.

5. 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)
Given a set of data points lying on a collection of linear subspaces, without knowing which point corresponds to which subspace and without even knowing the number of subspaces, estimate the number of subspaces, a basis for each subspace and the segmentation of the data.
As stated, the GPCA problem is one of those challenging CHICKEN-AND-EGG problems in the following sense. If we knew the segmentation of the data points, and hence the number of subspaces, then we could solve the problem by applying standard PCA to each group. On the other hand, if we had a basis for each subspace, we could easily assign each point to each cluster. The main challenge is that we neither know the segmentation of the data points nor a basis for each subspace, and the questions is how to estimate everything from the data only.
Previous geometric approaches to GPCA have been proposed in the context of motion segmentation, and under the assumption that the subspaces do NOT intersect. Such approaches are based on first clustering the data, using for example K-means or spectral clustering techniques, and then applying standard PCA to each group.
Statistical methods, on the other hand, model the data as a mixture model whose parameters are the mixing proportions and the basis for each subspace. The estimation of the mixture is a (usually non-convex) optimization problem that can be solved with various techniques. One of the most popular ones is the expectation maximization algorithm (EM) which alternates between the clustering of the data and the estimation of the subspaces.
Unfortunately, the convergence of such iterative algorithms depends on good initialization. At present, initialization is mostly done either at random, using another iterative algorithm such as K-means, or else in an ad-hoc fashion depending on the particular application.
It is here where our main motivation comes into the picture. We would like to know if we can find a way of initializing iterative algorithms in an algebraic fashion. For example, if you look at the two lines in this picture, there has to be a way of estimating those two lines directly from data in a one shot solution, at least in the ABSENCE of noise.

6. 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
We are therefore interested in finding an analytic solution to the above segmentation problem. In particular we would like to know if we can estimate all the subspaces simultaneously from all the data without having to iterate between data clustering and subspace estimation. Furthermore we would like to know if we can do so in an ANALYTIC fashion, and even more, we would like to understand under what conditions we can do it in closed form and whether there is a formula for the number of subspaces.
It turns out that all these questions can be answered using tools from algebraic geometry. Since at first data segmentation and algebraic geometry may seem to be totally disconnected, let me first tell you what the connection is. We will model the number of subspaces as the degree of a polynomial and the subspace parameters (basis) as the roots of a polynomial, or more precisely as the factors of the polynomial.
With this algebraic geometric algebraic geometric interpretation in mind, we will be able to prove the following the GPCA problem.
First, there are some geometric constraints that do not depend on the segmentation of the data. Therefore, the chicken-and-egg dilemma can be resolved since the segmentation of the dada can be algebraically eliminated.
Second, one can prove that the problem has a unique solution, which can be obtained in closed form if and only if the number of subspaces is less than or equal to four
And third, the solution can be computed using linear algebraic techniques.
In the presence of noise, one can use the algebraic solution to GPCA to initialize any iterative algorithm, for example EM. In the case of zero-mean Gaussian noise, we will show that the E-step can be algebraically eliminated, since we will derive an objective function that depends on the motion parameters only.

7. 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
One of the reasons we are interested in GPCA is because there are various problems in computer vision that have to do with the simultaneous estimation of multiple models from visual data.
Consider for example segmenting an image into different regions based on intensity, texture of motion information. Consider also the recognition of various static and dynamic processes such as human faces or human gaits from visual data.
Although in this talk I will only consider the first class of problems, it turns out that, at least from a mathematical perspective, all the above problems can be converted into following generalization of principal component analysis, which we conveniently refer to as GPCA

8. Introductory example: algebraic clustering in 1D
A -> c
Number of groups?

9. Introductory example: algebraic clustering in 1D
How to compute n, c, b’s?
Number of clusters
Cluster centers
Solution is unique if
Solution is closed form if
A -> c

10. Introductory example: algebraic clustering in 2D
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

11. Representing one subspace
One plane
One line
One subspace can be represented with
Set of linear equations
Set of polynomials of degree 1

12. Representing n subspaces
Two planes
One plane and one line
Plane:
Line:
A union of n subspaces can be represented with a set of homogeneous polynomials of degree n
De Morgan’s rule

13. Fitting polynomials to data points
Polynomials can be written linearly in terms of the vector of coefficients by using polynomial embedding
Coefficients of the polynomials can be computed from nullspace of embedded data
Solve using least squares
N = #data points
Veronese map

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

15. Finding a basis for each subspace
To learn a mixture of subspaces we just need one positive example per class
Polynomial Differentiation (GPCA-PDA) [CVPR’04]

16. 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?

17. 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

18. 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

19. 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

20. 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

21. An example
Can compute polynomials from
Can compute normals from

22. Dealing with high-dimensional data
Minimum number of points
K = dimension of ambient space
n = number of subspaces
In practice the dimension of each subspace ki is much smaller than K
Number and dimension of the subspaces is preserved by a linear projection onto a subspace of dimension
Can remove outliers by robustly fitting the subspace
Open problem: how to choose projection?
PCA?
Subspace 1
Subspace 2

23. GPCA with 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

24. Comparison of PFA, PDA, K-sub, EM

25. 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
In this paper, we have presented a new way of looking at segmentation problems that is based on algebraic geometric techniques. In essence we showed that segmentation is equivalent to polynomial factorization and presented an algebraic solution for the case of mixtures of subspaces, which is a natural generalization of the well known Principal Component Analysis (PCA).
Of course, as this theory is fairly new, there are still a lot of open problems. For example, how to do polynomial factorization in the presence of noise, or how to deal with outliers. We are currently working on such problems and expect to report our results in the near future. Also, we are currently working on generalizing the ideas in this paper to the cases of subspaces of different dimensions and to other types of data.
In fact, I cannot resist doing some advertising for my talk tomorrow, where I will talk about the segmentation of multiple rigid motions, which corresponds to the case of bilinear data.
Finally, before taking your questions, I would like to say that I think it is time that we start thinking about putting segmentation problems in a theoretical footing. I think this can be done by building a mathematical theory of segmentation that not only incorporates well known statistical modeling techniques, but also the geometric constraints inherent in various applications in computer vision. We hope that by combining the algebraic geometric approach to segmentation presented in this paper with the more standard statistical approaches, we should be able to build a new mathematical theory of segmentation in the followi should be able to build such a theory, which a nice mathematical theory of segmentation. Such a theory should find applications in various disciplines, including of course lots of segmentation and recognition problems in computer vision.

26. Thank You! For more information, Vision, Dynamics and Learning Lab @
Johns Hopkins University
Thank You!