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.
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U. Grenander, Pattern Theory: A Mathematical Study of Regular Structures, Oxford University Press, Oxford, 1993.
F. L. Bookstein, The Measurement of Biological Shape and Shape Change, vol. 24, Springer-Verlag: Lecture Notes in Biomathematics, New York, 1978.
F. L. Bookstein, Biometrics, biomathematics and the morphometric synthesis, Bulletin of Mathematical Biology, vol. 58, no. 2, pp. 313–365, 1996.
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G. Christensen, Deformable Shape Models for Anatomy, Ph.D. Dissertation, Department of Electrical Engineering, Sever Institute of Technology, Washington University, St. Louis, MO, Aug. 1994.
C. Broit, Optimal registration of deformed images, Dissertation, University of Pennsylvania, Philadelphia, Pennsylvania, 1983.
D. L. Snyder, Random Point Processes, John Wiley and Sons, 1975.
U. Grenander, Y. Chow, and D. M. Keenan, Hands: A Pattern Theoretic Study of Biological Shapes, Springer-Verlag, New York, 1991.
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.
B. Gidas, A renormalization group approach to image processing problems, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. PAMI-11, pp. 164–180, 1989.
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.
U. Grenander and M. I. Miller, Representations of knowledge in complex systems, J. R. Statist. Soc. B, vol. 56, no. 3, pp. 549–603, 1994.
L. Matejic, Group cascades for representing biological variability in medical images, Brown University Technical Report, 1996.
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.
E. Butkov, Mathematical Physics, Addison-Wesley, 1968.
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© Copyright 2003
American Mathematical Society
