Skip to main content

Comparative Evaluation of Symmetric SVD Algorithms for Real-Time Face and Eye Tracking

  • Chapter
  • First Online:

Abstract

Computation of singular value decomposition (SVD) has been a topic of concern by many numerical linear algebra researchers. Fast SVD has been a very effective tool in computer vision in a number of aspects, such as: face recognition, eye tracking etc. At the present state of the art fast and fixed-point power efficient SVD algorithm needs to be developed for real-time embedded computing. The work in this paper is the genesis of an attempt to build an on-board real-time face and eye tracking system for human drivers to detect loss of attention due to drowsiness or fatigue. A major function of this on-board system is quick customization. This is carried out when a new driver comes in. The face and eye images are recorded while instructing the driver for making specific poses. The eigen faces and eigen eyes are generated at several resolution levels and stored in the on-board computer. The discriminating eigen space of face and eyes are determined and stored in the on-board flash memory for detection and tracking of face and eyes and classification of eyes (open or closed) as well. Therefore, fast SVD of image covariance matrix at various levels of resolution needs to be carried out to generate the eigen database. As a preliminary step, we review the existing symmetric SVD algorithms and evaluate their feasibility for such an application. In this article, we compare the performance of (1) Jacobi’s, (2) Hestenes’, (3) Golub-Kahan, (4) Tridiagonalization and Symmetric QR iteration and (5) Tridiagonalization and Divide and Conquer algorithms. A case study has been demonstrated as an example.

Keywords

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Anderson E., Bai Z., Bischof C., Blackford S., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., McKenney A., Sorensen D.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 22 Aug (1999)

    Google Scholar 

  2. Brent, R., Luk, F.: The solution of singular value and symmetric eigenvalue problems on multiprocessor arrays. SIAM J. Sci. Stat. Comput. 6, 69–84 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bunch, J.R., Nielsen, C.P., Sorensen, D.C.: Rank-one modification of the symmetric eigenproblem. Numer. Math. 31, 31–48 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cilio, A.G.M., Corporaal, H.: Floating point to fixed point conversion of C code. Proceedings of CC 99, the 8th International Conference on Compiler Construction, Amsterdam, March 1999. Lecture Notes in Computer Science 1575, pp. 229–243. Springer, Berlin (1999).

    Google Scholar 

  5. Cuppen, J.J.M.: A divide and conquer method for the symmetric tridiagonal eigenproblem. Numer. Math. 36, 177–195 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Datta, B.N.: Numerical Linear Algebra and Applications. Brooks/Cole Publishing Company, Pacific Grove (1995)

    MATH  Google Scholar 

  7. Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)

    Book  MATH  Google Scholar 

  8. Demmel, J., Kahan, W.: Accurate singular values of bidiagonal matrices. SIAM J. Sci. Stat. Comput. 11(5), 873–912 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dinges, D.F., Maislin, G., Brewster, R.M., Krueger, G.P., Carroll, R.J.: Pilot Test of Fatigue Management Technologies: Final Report. Federal Motor Carrier Safety Administration, Washington (2004)

    Google Scholar 

  10. Dinges, D.F., Mallis, M., Maislin, G., Powell, J.W.: Final Report: Evaluation of Techniques for Ocular Measurement as an Index of Fatigue and as the Basis for Alertness Management (Report No. DOT HS 808 762). National Highway Traffic Safety Administration, Washington, DC (1998)

    Google Scholar 

  11. Golub, G., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. J. SIAM Numer. Anal., Ser. B 2(2), 205–224 (1965)

    Google Scholar 

  12. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Jhons Hopkins University Press, Baltimore and London (1996)

    Google Scholar 

  13. Gu, M., Eisenstat, S.C.: A divide-and-conquer algorithm for the symmetric tridiagonal eigen problem. Research Report YALEU/DCS/RR-932, 27 Nov 1992

    Google Scholar 

  14. Hestenes, M.R.: Inversion of matrices by biorthogonalization and related results. SIAM 6(1), 51–90 (1958)

    Google Scholar 

  15. Hjelms, E., Low, B.K.: Face detection: a survey. Comput. Vis. Image Underst. 83, 236–274 (2001)

    Article  Google Scholar 

  16. Keding, H., Willems, M., Coors, M., Meyr, H.: FRIDGE: A fixed-point design and simulation environment. In: Proceeding of the Design Automation and Test in Europe, 1998. pp. 429–435

    Google Scholar 

  17. Kim, S., Kum, K., Sung, W.: Fixed-point optimization utility for C and C++ based digital signal processing programs. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 45(11), 1455–1464 (1998)

    Google Scholar 

  18. Li, R.C.: Solving secular equations stably and efficiently. Computer Science Division Technical Report UCB//CSD-94-851, University of California, Berkeley, 94720, Dec 1994

    Google Scholar 

  19. Melman, A.: A numerical comparison of methods for solving secular equations. J. Comput. Appl. Math. 86, 237–249 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nikolić, Z., Nguyen, H.T., Frantz, G.: Design and implementation of numerical linear algebra algorithms on fixed point DSPs. EURASIP J. Adv. Signal Process. 2007 Article ID 87046 (2007) doi:10.1155/2007/87046

  21. Parlett, B.N.: The Symmetric Eigenvalue Problem. SIAM, Philadelphia (1998)

    Book  MATH  Google Scholar 

  22. Rutter, J.: A serial implementation of Cuppen’s divide and Conquer algorithm for the Symmetric Eigenvalue problem. http://www.netlib.org/lapack/lawnspdf/lawn69.pdf

  23. Sirovich, L., Kirby, M.: Low-dimensional procedure for the characterization of human faces. J. Opt. Soc. Am. A 4, 519 (1987)

    Google Scholar 

  24. Stewart, G.W.: Perturbation theory for the singular value decomposition. UMIACS-TR-90-124 CS-TR 2539, Sept 1990

    Google Scholar 

  25. Svensson, G.: A Block-Hestenes Method for the SVD, Technical Report. LiTH-MAT-R -1989-28, Department of Mathematics, Linköping University, Sept 1989. hem.passagen.se/-gohel/num/hest.ps

    Google Scholar 

  26. Szecówka, P.M., Malinowski, P.: CORDIC and SVD implementation in digital hardware. 17th International Conference Mixed Design of Integrated Circuits and Systems MIXDES 2010, 24–26 June 2010

    Google Scholar 

  27. Turk, M., Pentland, A.: Eigenfaces for recognition. J. Cogn. Neurosci. 3(1), 71–86 (1991)

    Google Scholar 

  28. Wang, H., Leray, P., Palicot, J.: A CORDIC-based dynamically reconfigurable FPGA architecture for signal processing algorithms, URSI 08. The XXIX General Assembly of the International Union of Radio Science, Chicago (2008)

    Google Scholar 

  29. Yang, M-H., Kriegman, D.J., Ahuja, N.: Detecting faces in images: a survey. IEEE Trans. Pattern Anal. Mach. Intell. 24(1), 34–58 (2001)

    Google Scholar 

  30. Zhou, B.B., Brent, R.P.: On parallel implementation of the one-sided Jacobi algorithm for singular value decompositions. http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=389182

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tapan Pradhan .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Pradhan, T., Routray, A., Kabi, B. (2013). Comparative Evaluation of Symmetric SVD Algorithms for Real-Time Face and Eye Tracking. In: Nielsen, F., Bhatia, R. (eds) Matrix Information Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30232-9_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-30232-9_13

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30231-2

  • Online ISBN: 978-3-642-30232-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics