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Principal Manifolds and Probabilistic Subspaces for Visual Recognition. Baback Moghaddam TPAMI, June 2002. John Galeotti Advanced Perception February 12, 2004. It’s all about subspaces. Traditional subspaces PCA ICA Kernel PCA (& neural network NLPCA) Probabilistic subspaces.

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Principal manifolds and probabilistic subspaces for visual recognition

Principal Manifolds and Probabilistic Subspaces for Visual Recognition

Baback Moghaddam

TPAMI, June 2002.

John Galeotti

Advanced Perception

February 12, 2004

It s all about subspaces
It’s all about subspaces Recognition

  • Traditional subspaces

    • PCA

    • ICA

    • Kernel PCA (& neural network NLPCA)

  • Probabilistic subspaces

Linear pca
Linear PCA Recognition

  • We already know this

  • Main properties

    • Approximate reconstruction

      x ≈ y

    • Orthonormality of the basis 


    • Decorrelated principal components

      E{yiyj}i≠j = 0

Linear ica
Linear ICA Recognition

  • Like PCA, but the components’ distribution is designed to be sub/super Gaussian  statistical independence

  • Main properties

    • Approximate reconstruction

      x ≈ Ay

    • Nonorthogonality of the basis A


    • Near factorization of the joint distribution P(y)

      P(y)≈ ∏ p(yi)

Nonlinear pca nlpca
Nonlinear PCA (NLPCA) Recognition

  • AKA principal curves

  • Essentially nonlinear regression

  • Finds a curved subspace passing “through the middle of the data”

Nonlinear pca nlpca1
Nonlinear PCA (NLPCA) Recognition

  • Main properties

    • Approximate reconstruction

      y = f(x)

    • Nonlinear projection

      x ≈ g(y)

    • No prior knowledge regarding joint distribution of the components (typical)

      P(y) = ?

  • Two main methods

    • Neural network encoder

    • Kernel PCA (KPCA)

Nlpca neural network encoder
NLPCA neural network encoder Recognition

  • Trained to match the output to the input

  • Uses a “bottleneck” layer to force a lower-dimensional representation

KPCA Recognition

  • Similar to kernel-based nonlinear SVM

  • Maps data to a higher dimensional space in which linear PCA is applied

    • Nonlinear input mapping

      (x): NL, N<L

    • Covariance is computed with dot-products

    • For economy, make (x) implicit

      k(xi,xj) = ( (xi) (xj) )

KPCA Recognition

  • Does not require nonlinear optimization

  • Is not subject to overfitting

  • Requires no prior knowledge of network architecture or number of dimensions

  • Requires the (unprincipled) selection of an “optimal” kernel and its parameters

Nearest neighbor recognition
Nearest-neighbor recognition Recognition

  • Find labeled image most similar to N-dim input vector using a suitable M-dim subspace

  • Similarity ex: S(I1,I2)  || ∆ ||-1, ∆ = I1 - I2

  • Observation: Two types of image variation

    • Critical: Images of different objects

    • Incidental: Images of same object under

      different lighting, surroundings, etc.

  • Problem: Preceding subspace projections do

    not help distinguish variation type

    when calculating similarity

Probabilistic similarity
Probabilistic similarity Recognition

  • Similarity based on probability that ∆ is characteristic of incidental variations

    • ∆ = image-difference vector (N-dim)

    • ΩI = incidental (intrapersonal) variations

    • ΩE = critical (extrapersonal) variations

Probabilistic similarity1
Probabilistic similarity Recognition

  • Likelihoods P(∆|Ω) estimated using subspace density estimation

  • Priors P(Ω) are set to reflect specific operating conditions (often uniform)

  • Two images are of the same object if P(ΩI|∆) > P(ΩE|∆)  S(∆) > 0.5

Subspace density estimation
Subspace density estimation Recognition

  • Necessary for each P(∆|Ω),Ω { ΩI, ΩE }

  • Perform PCA on training-sets of ∆ for each Ω

    • The covariance matrix (∑) will define a Gaussian

  • Two subspaces:

    • F = M-dimensional principal subspace of ∑

    • F = non-principal subspace orthogonal to F

  • yi = ∆ projected onto principal eigenvectors

  • i = ranked eigenvalues

    • Non-principal eigenvalues are typically unknown and are estimated by fitting a function of the form f -n to the known eigenvalues

Subspace density estimation1
Subspace density estimation Recognition

  • 2(∆) = PCA residual (reconstruction error)

  •  = density in non-principal subspace

    • ≈ average of (estimated) F eigenvalues

  • P(∆|Ω) is marginalized into each subspace

    • Marginal density is exact in F

    • Marginal density is approximate in F

Efficient similarity computation
Efficient similarity computation Recognition

  • After doing PCA, use a whitening transform to preprocess the labeled images into single coefficients for each of the principal subspaces:

    where  and V are matrices of the principal eigenvalues and eigenvectors of either ∑I or ∑E

  • At run time, apply the same whitening transform to the input image

Efficient similarity computation1
Efficient similarity computation Recognition

  • The whitening transform reduces the marginal Gaussian calculations in the principal subspaces F to simple Euclidean distances

  • The denominators are easy to precompute

Efficient similarity computation2
Efficient similarity computation Recognition

  • Further speedup can be gained by using a maximum likelihood (ML) rule instead of a maximum a posteriori (MAP) rule:

  • Typically, ML is only a few percent less accurate than MAP, but ML is twice as fast

    • In general, ΩE seems less important than ΩI

Similarity comparison
Similarity Comparison Recognition

Probabilistic Similarity

Eigenface (PCA) Similarity

Experiments Recognition

  • 21x12 low-res faces, aligned and normalized

  • 5-fold cross validation

    • ~ 140 unique individuals per subset

    • No overlap of individuals between subsets to test generalization performance

    • 80% of the data only determines subspace(s)

    • 20% of the data is divided into labeled images and query images for nearest-neighbor testing

  • Subspace dimensions = d = 20

    • Chosen so PCA ~ 80% accurate

Experiments Recognition

  • KPCA

    • Empirically tweaked Gaussian, polynomial, and sigmoidal kernels

    • Gaussian kernel performed the best, so it is used in the comparison

  • MAP

    • Even split of the 20 subspace dimensions

      • ME = MI = d/2 = 10 so that ME + MI = 20

Results Recognition

Recognition accuracy (percent)


Nearest Neighbor

(no subspace)

Results Recognition

Recognition accuracy vs subspace dimensionality

Note: data split 50/50 for

training/testing rather

than using CV

Conclusions Recognition

  • Bayesian matching outperforms all other tested methods and even achieves ≈ 90% accuracy with only 4 projections (2 for each class of variation)

  • Bayesian matching is an order of magnitude faster to train than KPCA

  • Bayesian superiority with higher resolution images verified in independent US Army FERIT tests

  • Wow!

  • You should use this 

My results
My results Recognition

  • 50% Accuracy

  • Why so bad?

    • I implemented all suggested approximations

    • Poor data--hand registered

    • Too little data

Note: data split 50/50 for

training/testing rather

than using CV

My results1
My results Recognition

  • My data

  • His data