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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))

Abstract

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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Notes

  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.

References

  1. Anjyo K, Todo H, Lewis J (2012) A practical approach to direct manipulation blendshapes. J Graph Tools 16(3):160–176

    Google Scholar 

  2. Blanz T, Vetter T (1999) A morphable model for the synthesis of 3d faces. In: Proceedings of ACM SIGGRAPH, pp 187–194

    Google Scholar 

  3. Cootes TF, Edwards GJ, Taylor CJ (1998) Active appearance models. In: Burkhardt H, Neumann B (eds) ECCV’98: computer vision. Proceedings of the 5th European conference on computer vision, Volume II. Lecture notes in computer science vol 1407, Springer, Berlin

    Google Scholar 

  4. Lewis J, Anjyo K (2010) Direct manipulation blendshapes. Comput Graph Appl (special issue: Digital Human Faces) 30(4):42–50

    Article  Google Scholar 

  5. Li H, Yu J, Ye Y, Bregler C (2013) Realtime facial animation with on-the-fly correctives. ACM Trans Graph 42:1–10

    Google Scholar 

  6. MacKay DJ (1996) Hyperparameters: Optimize, or integrate out? In: Heidbreder G (ed) Maximum entropy and Bayesian methods. Springer, New York, pp 43–59

    Google Scholar 

  7. Matthews I, Xiao J, Baker S (2006) On the dimensionality of deformable face models. CMU-RI-TR-06-12

    Google Scholar 

  8. Meytlis M, Sirovich L (2007) On the dimensionality of face space. IEEE Trans Pattern Anal Mach Intell 29(7):1262–1267

    Article  Google Scholar 

  9. Mo Z, Lewis J, Neumann U (2004) Face inpainting with local linear representations. In: BMVC, BMVA, pp 347–356

    Google Scholar 

  10. Patel A, Smith W (2009) 3D morphable face models revisited. In: Computer vision and pattern recognition (CVPR), IEEE Computer Society, Los Alamitos, CA, USA, pp 1327–1334

    Google Scholar 

  11. Penev PS, Sirovich L (2000) The global dimensionality of face space. In: Proceedings of 4th international conference automatic face and gesture recognition, pp 264–270

    Google Scholar 

  12. Phillips PJ, Wechsler H, Huang J, Rauss P (1998) The feret database and evaluation procedure for face recognition algorithms. Image Vis Comput J 16(5):295–306

    Article  Google Scholar 

  13. Seo J, Irving G, Lewis JP, Noh J (2011) Compression and direct manipulation of complex blendshape models. ACM Trans Graph 30(6):164:1–164:10

    Article  Google Scholar 

  14. Valentine T (2012) Face-space models of face recognition. In: Wenger M, Townsend J (eds) Computational, geometric, and process perspectives on facial cognition: contexts and challenges, Scientific Psychology Series. Taylor & Francis, Oxford

    Google Scholar 

  15. Vlasic D, Brand M, Pfister H, Popovic J (2005) Face transfer with multilinear models. ACM Trans Graph 24(3):426–433

    Article  Google Scholar 

  16. Wang J (2011) Geometric structure of high-dimensional data and dimensionality reduction. Springer-Verlag, Berlin, Heidelberg

    Google Scholar 

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Acknowledgments

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. https://doi.org/10.1007/978-4-431-55007-5_4

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  • DOI: https://doi.org/10.1007/978-4-431-55007-5_4

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