Techniques for Face Motion & Expression Analysis on Monocular Images - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 211

, (2)

T

Λ = Φ ΣΦ

Λ

the diagonal matrix of eigenvalues and Φ the
matrix of eigenvectors. The vector base obtained is optimal in terms of
compactness (we can easily isolate vectors of low energy) and parametric (each
eigenvector is orthogonal to the others, creating a parametric eigenspace).
Elements of one class, that is, a vector whose dimension is M, can be represented
by the linear combination of the M eigenvectors obtained for this class. The
Principal Component Analysis (PCA) technique states that the same object can
be reconstructed by only combining the N<M eigenvectors of greatest energy,
also called principal components. It also says that we will minimize the error
difference when performing the approximation if the linear coefficients for the
combination are obtained from projecting the class vector onto the sub-space of
principal components.
This theory is only applicable to objects that can be represented by vectors.
Images have this property, therefore, this theory is easily extended to image
processing and generally used to model the variability of 2D objects on images
like, for example, faces.
Very often PCA is utilized to analyze and identify features of the face. It
introduces some restrictions. One of them is the need for one training stage
previous to the analysis, during which the base of principal component vectors,
in this case images, must be generated. It also forces all images being analyzed
to be the same size. Using PCA in face analysis has lead to the appearance of
concepts like Eigenfaces (Turk & Pentland, 1991), utilized for face recognition,
or Eigenfeatures (Pentland, Mohaddam & Starner, 1994) used to study more
concrete areas of faces robustly.
The topic Face Image Analysis by Unsupervised Learning (Bartlett, 2001) is
a complete study of the strengths and weaknesses of methods based on
Independent Component Analysis (ICA) in contrast with PCA. It also includes
a full explanation of concepts like Eigenactions and describes recent approaches
in facial image analysis.

being

Σ

the covariance matrix,

Active contour models — Snakes

Active contour models, generally called snakes, are geometric curves that
approximate the contours of an image by minimizing an energy function. Snakes
are used to track moving contours within video sequences because they have the
property of deforming themselves to stick onto a contour that evolves along the
time.