Advertisement

Bayes Reconstruction of Missing Teeth

  • Jon Sporring
  • Katrine Hommelhoff Jensen
Article

Abstract

We propose a method for restoring the surface of tooth crowns in a 3D model of a human denture, so that the pose and anatomical features of the tooth will work well for chewing. This is achieved by including information about the position and anatomy of the other teeth in the mouth. Our system contains two major parts: A statistical model of a selection of tooth shapes and a reconstruction of missing data.

We use a training set consisting of 3D scans of dental cast models obtained with a laser scanner, and we have build a model of the shape variability of the teeth, their neighbors, and their antagonists, using the eigenstructure of the covariance matrix, also known as Principle Component Analysis (PCA). PCA is equivalent to fitting a multivariate Gaussian distribution to the data and the principle directions constitute a linear model for stochastic data and is used both for a data reduction or equivalently noise elimination and for data analysis. However for small sets of high dimensional data, the log-likelihood estimator for the covariance matrix is often far from convergence, and therefore reliable models must be obtained by use of prior information. We propose a natural and intrinsic regularization of the log-likelihood estimate based on differential geometrical properties of teeth surfaces, and we show general conditions under which this may be considered a Bayes prior.

Finally we use Bayes method to propose the reconstruction of missing data, for e.g. finding the most probable shape of a missing tooth based on the best match with our shape model on the known data, and we superior improved reconstructions of our full system.

Keywords

Principle component analysis Bayes method Missing data Reconstruction of teeth 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Modgil, S., Hutton, T.J., Hammond, P., Davenport, J.C.: Combining biometric and symbolic models for customised, automated prosthesis design. Artif. Intell. Med. 25, 227–245 (2002) CrossRefGoogle Scholar
  2. 2.
    Gürke, S.: Restoration of teeth by geometrically deformable models (1997). http://citeseer.comp.nus.edu.sg/gurke97restoration.html
  3. 3.
    Hayashi, T., Tsuchida, J., Kato, K.: Semi-automatic design of tooth crown using a 3-D dental CAD system, Vocs-1B. In: Proceedings of the 22nd Annual EMBS International Conference, Chicago IL, USA, July 2000, pp. 565–566 (2000) Google Scholar
  4. 4.
    Blanz, V., Mehl, A., Veter, T., Seidel, H.P.: A statistical method for robust 3D surface reconstruction from sparse data. In: Int. Symp. on 3D Data Processing, Visualization and Transmission, Thessaloniki, Greece (2004) Google Scholar
  5. 5.
    Hommelhoff Jensen, K., Sporring, J.: Reconstructing teeth with bite information. In: Ersbøll, B., Pedersen, K.S. (eds.) Proceedings of the Scandinavian Conference on Image Analysis (SCIA ’07). Lecture Notes in Computer Science, vol. 4522, pp. 102–111. Springer, Berlin (2007) Google Scholar
  6. 6.
    Pearson, K.: On lines and planes of closest fit to systems of points in space. Philos. Mag. 6(2), 559–572 (1901) Google Scholar
  7. 7.
    Hotelling, H.: Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24, 417–441 and 498–520 (1933). CrossRefGoogle Scholar
  8. 8.
    Bookstein, F.L.: Shape and the information in medical images: A decade of morphometric synthesis. Comput. Vis. Image Underst. 66(2), 97–118 (1997) CrossRefGoogle Scholar
  9. 9.
    Cootes, T.F., Taylor, C.J.: A mixture model for representing shape variation. Image Vis. Comput. 17, 567–573 (1999) CrossRefGoogle Scholar
  10. 10.
    Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, New York (1998) MATHGoogle Scholar
  11. 11.
    Cootes, T.F., Taylor, C.J., Cooper, D.H., Graham, J.: Active shape models—their training and application. Comput. Vis. Image Underst. 61(1) (1995) Google Scholar
  12. 12.
    Cootes, T.F., Taylor, C.J.: Statistical models of appearance for computer vision. Technical Report, University of Manchester to 1.8cm(March 2004). http://www.isbe.man.ac.uk/~bim/Models/app_models.pdf
  13. 13.
    de Bruijne, M., Lund, M.T., Tankó, L.B., Pettersen, P.C., Nielsen, M.: Quantitative vertebral morphometry using neighbor-conditional shape models. Med. Image Anal. (2007) Google Scholar
  14. 14.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999) MATHGoogle Scholar
  15. 15.
    Pizer, S., Thall, A., Chen, D.: M-reps: A new object representation for graphics. Technical Report, University of North Carolina (1999) Google Scholar
  16. 16.
    Hutton, T.J., Buxton, B.F., Hammond, P.: Dense surface point distribution models of the human face. In: IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA), p. 153 (2001) Google Scholar
  17. 17.
    Blanz, V., Vetter, T.: A morphable model for the synthesis of 3d faces. In: Proc. of SIGGRAPH ’99, Los Angeles, August 1999, pp. 187–194 (1999) Google Scholar
  18. 18.
    Turk, G., O’Brien, J.F.: Variational implicit surfaces. Technical Report, Georgia Institute of Technology (May 1999). Tech Report GIT-GVU-99-15 Google Scholar
  19. 19.
    Wang, Y., Staib, L.H.: Boundary finding with correspondence using statistical shape models. Proc. IEEE Conf. Comput. Vis. Pattern Recognit., pp. 338–345 (1998) Google Scholar
  20. 20.
    Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, New York (1988) MATHGoogle Scholar
  21. 21.
    Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1949) MATHGoogle Scholar
  22. 22.
    Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 3rd edn. Wiley, New York (2003) MATHGoogle Scholar
  23. 23.
    Rissanen, J.: Stochastic Complexity in Statistical Inquiry. World Scientific, Singapore (1989) MATHGoogle Scholar
  24. 24.
    Gerschgorin, S.: Über die abgrenzung der eignewerte einer matrix. Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 7, 749–754 (1931) Google Scholar
  25. 25.
    Gershgorin circle theorem. http://en.wikipedia.org/wiki/Gershgorin_circle_theorem, September 8 2007
  26. 26.
    Blanz, V., Vetter, T.: Reconstructing the complete 3d shape of faces from partial information. Technical Report, University of Freiburg (2001). Computer Graphics Technical Report No. 1 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations