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Handbook of Face Recognition pp 19–49Cite as

Face Recognition in Subspaces

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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 chronologic 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. We also discuss comparative experimental evaluation of some of these techniques as well as practical issues related to the application of subspace methods for varying pose, illumination, and expression.

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Notes

  1. 1.

    A singular value of a matrix X is the square root of an eigenvalue of XX T.

  2. 2.

    For comparison, note that the objective of PCA can be seen as maximizing the total scatter across all the images in the database.

  3. 3.

    A number of algorithms exist; most notable are Jade [5], InfoMax, and FastICA [16].

  4. 4.

    These eigenfaces are linear combination of the original images, which under the assumptions of ICA should not affect the resulting decomposition.

  5. 5.

    This also provides an estimate of the parameters (e.g., illumination) for the input image.

  6. 6.

    The class of functions attainable by this neural network restricts the projection function f() to be smooth and differentiable, and hence suboptimal in some cases [22].

  7. 7.

    However, computing Σ K in (2.21) requires “centering” the data by computing the mean of Ψ(x i). Because there is no explicit computation of Ψ(x i ), the equivalent must be carried out when computing the kernel matrix K. For details on “centering” K, see Schölkopf et al. [32].

  8. 8.

    In practice, k I >k E often works just as well. In fact, as k E →0, one obtains a maximum-likelihood similarity S=P(Δ∣Ω I ) with k I =k, which for this data set is only a few percent less accurate than MAP [26].

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Acknowledgements

We thank M.S. Bartlett and M.A.O. Vasilescu for kind permission to use figures from their published work and for their comments. We also acknowledge all who contributed to the research described in this chapter.

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Authors and Affiliations

  1. Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, 02139, USA

    Gregory Shakhnarovich

  2. Mitsubishi Electric Research Labs, Cambridge, MA, 02139, USA

    Baback Moghaddam

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Correspondence to Gregory Shakhnarovich .

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  1. Institute of Automation, Center Biometrics Research & Security, Chinese Academy of Science, Room 1227, Zhongguancun 95, Beijing Donglu, 100080, China, People's Republic

    Stan Z. Li

  2. Dept. Computer Science & Engineering, Michigan State University, East Lansing, 48824-1226, Michigan, USA

    Anil K. Jain

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Shakhnarovich, G., Moghaddam, B. (2011). Face Recognition in Subspaces. In: Li, S., Jain, A. (eds) Handbook of Face Recognition. Springer, London. https://doi.org/10.1007/978-0-85729-932-1_2

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  • DOI: https://doi.org/10.1007/978-0-85729-932-1_2

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