Abstract
Images of faces, represented as high-dimensional pixel arrays, often belong to
a manifold of intrinsically low dimension. Face recognition, and computer vision
research in general, has witnessed a growing interest in techniques that capitalize on
this observation, and apply algebraic and statistical tools for extraction and analysis of
the underlying manifold. In this chapter we describe in roughly chronological order
techniques that identify, parameterize and analyze linear and nonlinear subspaces,
from the original Eigenfaces technique to the recently introduced Bayesian method
for probabilistic similarity analysis, and discuss comparative experimental evaluation
of some of these techniques. We also discuss practical issues related to the application
of subspace methods for varying pose, illumination and expression.
Images of faces, represented as high-dimensional pixel arrays, often belong
to a manifold of intrinsically low dimension. Face recognition, and computer
vision research in general, has witnessed a growing interest in techniques that
capitalize on this observation, and apply algebraic and statistical tools for extraction
and analysis of the underlying manifold. In this chapter we describe
in roughly chronological order techniques that identify, parameterize and analyze
linear and nonlinear subspaces, from the original Eigenfaces technique to
the recently introduced Bayesian method for probabilistic similarity analysis,
and discuss comparative experimental evaluation of some of these techniques.
We also discuss practical issues related to the application of subspace methods
for varying pose, illumination and expression.
1 Face Space and its Dimensionality
Computer analysis of face images deals with a visual signal (light re
ected o
the surface of a face) that is registered by a digital sensor as an array of pixel
values. The pixels may encode color or only intensity; In this chapter we will
assume the latter case, i.e. gray-level imagery. After proper normalization and resizing to a xed m-by-n size, the pixel array can be represented as a point (i.e. vector) in an mn-dimensional image space by simply writing its pixel values in a xed (typically raster) order. A critical issue in the analysis of
such multi-dimensional data is the dimensionality, the number of coordinates necessary to specify a data point. Below we discuss the factors aecting this number in the case of face images.
1.1 Image Space vs. Face Space
In order to specify an arbitrary image in the image space, one needs to specify every pixel value. Thus the \nominal" dimensionality of the space, dictated by the pixel representation, is mn - a very high number even for images of modest

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