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Quadratic variation of deformations

Part of the Lecture Notes in Computer Science book series (LNCS,volume 1230)

Abstract

Hitherto no constitutive formalism of deformations provides a parameterization for the visually obvious features of their transformation grids. This paper notes a property of the thin-plate spline that one may exploit to this end. The bending energy that is minimized by the spline, usually expressed in matrix form, is also the double integral of the output of a nonlinear differential operator, the quadratic variation (sum of squared second partial derivatives of displacement), over the whole picture plane. Displaying this integrand as a scalar field over the medical image or template may prove a helpful guide to the interesting regions of a deformation, and the peaks of this field localize and orient a promising set of features for simplistically parameterized deformations that approximate the original.

Keywords

  • Corpus Callosum
  • Displacement Field
  • Quadratic Variation
  • Shape Space
  • Picture Plane

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Grenander, U., and M. Miller. Representations of knowledge in complex systems. Journal of the Royal Statistical Society B56:549–603, 1994.

    Google Scholar 

  2. Davatzikos, C., M. Vaillant, S. M. Resnick, J. L. Prince, S. Letovsky, and R. N. Bryan. A computerized approach for morphological analysis of the corpus callosum. Journal of Computer Assisted Tomography 20:88–97, 1996.

    PubMed  Google Scholar 

  3. Thompson, P., and A. Toga. A surface-based technique for warping threedimensional images of the brain. I.E.E.E. Transactions on Medical Imaging 15:402–417, 1996.

    Google Scholar 

  4. Collins, D. L., T. Peters, and A. Evans. An automated 3d non-linear image deformation procedure for determination of gross morphometric variability in human brain. Pp. 180–190 in R. Robb, ed., Visualization in Biomedical Computing. SPIE Proceedings, vol. 2359, 1994.

    Google Scholar 

  5. Cutting, C., D. Dean, F. Bookstein, B. Haddad, D. Khorramabadi, F. Zonnefeld, and J. McCarthy. A three-dimensional smooth surface analysis of untreated Crouzon's disease in the adult. Journal of Craniofacial Surgery 6:444–453, 1995.

    PubMed  Google Scholar 

  6. Joshi, S. C., M. Miller, G. Christensen, A. Banerjee, T. Coogan, and U. Grenander. Hierarchical brain mapping via a generalized dirichlet solution for mapping brain manifolds. Pp. 278–289 in R. Melter et al., eds., Vision Geometry IV. SPIE Proceedings, vol. 2573, 1995.

    Google Scholar 

  7. Bookstein, F. L. Landmark methods for forms without landmarks. Computer Vision and Image Understanding, in press, 1997.

    Google Scholar 

  8. Green, W. D. K. The thin-plate spline and images with curving features. Pp. 79–87 in K. V. Mardia et al., eds., Proceedings in Image Fusion and Shape Variability Techniques. Leeds University Press, 1996.

    Google Scholar 

  9. Kendall, D. G. 1984. Shape-manifolds, procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society 16:81–121.

    Google Scholar 

  10. Bookstein, F. L. Biometrics, biomathematics, and the morphometric synthesis. Bulletin of Mathematical Biology 58:313–365, 1996.

    PubMed  Google Scholar 

  11. Bookstein, F. L. Morphometric Tools for Landmark Data. Cambridge University Press, 1991.

    Google Scholar 

  12. DeQuardo, J. R., F. Bookstein, W. D. K. Green, J. Brumberg, and R. Tandon. Spatial relationships of neuroanatomic landmarks in schizophrenia. Psychiatry Research: Neuroimaging 67:81–95, 1996.

    Google Scholar 

  13. Bookstein, F. L. Biometrics and brain maps: the promise of the Morphometric Synthesis. Pp. 203–254 in S. Koslow and M. Huerta, eds., Neuroinformatics: An Overview of the Human Brain Project. Progress in Neuroinformatics, vol. 1. Hillsdale, NJ: Lawrence Erlbaum, 1996.

    Google Scholar 

  14. Bookstein, F. L., and W. D. K. Green. Edgewarp: A flexible program package for biometric image warping in two dimensions. Pp. 135–147 in R. Robb, ed., Visualization in Biomedical Computing 1994. S.P.I.E. Proceedings, vol. 2359.

    Google Scholar 

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© 1997 Springer-Verlag Berlin Heidelberg

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Bookstein, F.L. (1997). Quadratic variation of deformations. In: Duncan, J., Gindi, G. (eds) Information Processing in Medical Imaging. IPMI 1997. Lecture Notes in Computer Science, vol 1230. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63046-5_2

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  • DOI: https://doi.org/10.1007/3-540-63046-5_2

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