Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A mathematical representation of biological variability in medical images


Author: Larisa Matejic
Journal: Quart. Appl. Math. 61 (2003), 1-16
MSC: Primary 92C55; Secondary 60H10, 62H35
DOI: https://doi.org/10.1090/qam/1955221
MathSciNet review: MR1955221
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Abstract: Medical image ensembles exhibit variability and it is the aim of computational anatomy to represent such variabilities mathematically and to exploit these knowledge representations by inference algorithms implemented through code. The variability is caused by several factors and our pattern theoretic approach rests on the assumption that they can be understood in terms of groups of transformations and probability measures on such groups. We shall arrange the similarity groups in a cascade, typically starting with the more rigid transformations and continuing with more flexible ones. Most importantly, however, we attach great significance to the physical and biological interpretation of the similarity groups.


References [Enhancements On Off] (What's this?)

  • [1] Julian Besag, Spatial interaction and the statistical analysis of lattice systems, J. Roy. Statist. Soc. Ser. B 36 (1974), 192–236. With discussion by D. R. Cox, A. G. Hawkes, P. Clifford, P. Whittle, K. Ord, R. Mead, J. M. Hammersley, and M. S. Bartlett and with a reply by the author. MR 0373208
  • [2] Ulf Grenander, General pattern theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. A mathematical study of regular structures; Oxford Science Publications. MR 1270904
  • [3] F. L. Bookstein, The Measurement of Biological Shape and Shape Change, vol. 24, Springer-Verlag: Lecture Notes in Biomathematics, New York, 1978.
  • [4] F. L. Bookstein, Biometrics, biomathematics and the morphometric synthesis, Bulletin of Mathematical Biology, vol. 58, no. 2, pp. 313-365, 1996.
  • [5] D. Terzopoulos and K. Waters, Analysis and synthesis of facial image sequences using physical and anatomical models, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 15, no. 6, pp. 569-579, 1993.
  • [6] R. Bajcsy, R. Lieberson, and M. Reivich, A computerized system for the elastic matching of deformed radiographic images to idealized atlas images, Journal of Computer Assisted Tomography, vol. 7, no. 4, pp. 618-625, 1983.
  • [7] Gary Edward Christensen, Deformable shape models for anatomy, ProQuest LLC, Ann Arbor, MI, 1994. Thesis (D.Sc.)–Washington University in St. Louis. MR 2691771
  • [8] C. Broit, Optimal registration of deformed images, Dissertation, University of Pennsylvania, Philadelphia, Pennsylvania, 1983.
  • [9] Donald L. Snyder, Random point processes, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1975. MR 0501325
  • [10] U. Grenander, Y. Chow, and D. M. Keenan, Hands, Research Notes in Neural Computing, vol. 2, Springer-Verlag, New York, 1991. A pattern-theoretic study of biological shapes. MR 1084371
  • [11] M. V. Ranganath, A. P. Dhawan, and N. Mullani, A multigrid expectation maximization algorithm for positron emission tomography, IEEE Trans. on Medical Imaging, vol. 7, pp. 273-278, 1988.
  • [12] B. Gidas, A renormalization group approach to image processing problems, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. PAMI-11, pp. 164-180, 1989.
  • [13] A. D. Sokal, Monte Carlo methods in statistical mechanics: Foundations and new algorithms, Cours de Troisieme cycle de la physique en suisse romande, June 1989.
  • [14] Ulf Grenander and Michael I. Miller, Representations of knowledge in complex systems, J. Roy. Statist. Soc. Ser. B 56 (1994), no. 4, 549–603. With discussion and a reply by the authors. MR 1293234
  • [15] Larisa Matejic, Group cascades for representing biological variability in medical images, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)–Brown University. MR 2696150
  • [16] F. L. Bookstein and W. D. K. Green, Edge information at landmarks in medical images, in Visualization in Biomedical Computing 1992, Richard A. Robb, Ed., 1992, pp. 242-258, SPIE 1808.
  • [17] E. Butkov, Mathematical Physics, Addison-Wesley, 1968.

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DOI: https://doi.org/10.1090/qam/1955221
Article copyright: © Copyright 2003 American Mathematical Society


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