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Probable and Improbable Faces

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Mathematical Progress in Expressive Image Synthesis I

Part of the book series: Mathematics for Industry ((MFI,volume 4))


The multivariate normal is widely used as the expected distribution of face shape. It has been used for face detection and tracking in computer vision, as a prior for facial animation editing in computer graphics, and as a model in psychological theory. In this contribution we consider the character of the multivariate normal in high dimensions, and show that these applications are not justified. While we provide limited evidence that facial proportions are not Gaussian, this is tangential to our conclusion: even if faces are truly “Gaussian”, maximum a posteriori and other applications and conclusions that assume that typical faces lie near the mean are not valid.

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  1. 1.

    In fact many results in statistics focus on the case where increasing amounts of data are available, i.e. \(n/d \rightarrow \infty \) with \(n\) the number of data points. In our problem we may have \(n/d\) finite and small, as in the case of a face model with several hundred training examples, each with 100 degrees of freedom.


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This research is partially supported by the Japan Science and Technology Agency, CREST project. JPL acknowledges a helpful discussion with Marcus Frean.

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Correspondence to J. P. Lewis .

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Appendix: Hyperellipsoidal Angle Calculation

Appendix: Hyperellipsoidal Angle Calculation

The interpolations in Fig. 8 start with a randomly chosen coefficient vector \(\mathbf {y}\) with \(y_i \sim N(0, \sqrt{\lambda _i}\)). This produces the first face. For the second face, we select a coefficient vector \(\mathbf {x}\) that has a specified Mahalanobis inner product with that of the first face, \(\mathbf {x}\varvec{\Lambda }^{-1} \mathbf {y}= c\) with \(c = -0.8\) for example. To find \(\mathbf {x}\) we solve a sequence of problems

$$\begin{aligned}&\mathbf {x}\leftarrow \arg \min _{\mathbf {x}} \quad (\mathbf {x}- \mathbf {r})^T \varvec{\Lambda }^{-1} (\mathbf {x}- \mathbf {r}) \quad +\quad \lambda (\mathbf {x}^T \varvec{\Lambda }^{-1} \mathbf {y}- c) \\&\mathbf {r}\leftarrow \frac{\mathbf {x}}{\mathbf {x}^T \varvec{\Lambda }^{-1} \mathbf {x}} \end{aligned}$$

with \(\mathbf {r}\) initialized to a random vector, in other words, find the vector that is closest to \(\mathbf {r}\) and has the desired Mahalanobis angle with \(\mathbf {y}\).

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Lewis, J.P., Mo, Z., Anjyo, K., Rhee, T. (2014). Probable and Improbable Faces. In: Anjyo, K. (eds) Mathematical Progress in Expressive Image Synthesis I. Mathematics for Industry, vol 4. Springer, Tokyo.

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