Abstract
In recent years, new nonlinear partial differential equation (PDE) based approaches have become popular for solving image processing problems. Although the outcome of these methods is often very promising, their actual realization is in general computationally intensive. Therefore, accurate and efficient schemes are needed. The aim of this paper is twofold. First, we will show that the three dimensional alignment problem of a histological data set of the human brain may be phrased in terms of a nonlinear PDE. Second, we will devise a fast direct solution technique for the associated structured large systems of linear equations. In addition, the potential of the derived method is demonstrated on real-life data.
Similar content being viewed by others
References
Y. Amit, U. Grenander and M. Piccioni, Structural image restoration through deformable templates, J. Amer. Statist. Assoc. 86(414) (1991) 376–387.
R. Bajcsy and S. Kovačič, Toward an individualized brain atlas elastic matching, MS-CIS-86-71 Grasp Lap 76, Department of Computer and Information Science, Moore School, University of Philadelphia (1986).
F.L. Bookstein, A statistical method for biological shape comparisons, J. Theor. Biol. 107 (1984) 475–520.
F.L. Bookstein, Size and shape spaces for landmark data in two dimensions, Statist. Sci. 1 (1986) 181–242.
M. Bro-Nielsen, Medical image registration and surgery simulation, Ph.D. thesis, IMM, Technical University of Denmark (1996).
G.E. Christensen, Deformable shape models for anatomy, Ph.D. thesis, Sever Institute of Technology, Washington University (1994).
P.J. Davis, Circulant Matrices (Chelsea, New York, 1979).
G.H. Golub and C.F. van Loan, Matrix Computations (Johns Hopkins Univ. Press, Baltimore, MD, 2nd ed., 1989).
M.E. Gurtin, An Introduction to Continuum Mechanics (Academic Press, Orlando, 1981).
MATLAB User's Guide (MathWorks, Natick, MA, 1992).
T. Schormann, S. Henn and K. Zilles, A new approach to fast elastic alignment with applications to human brains, in: Lecture Notes in Computer Sciences, Vol. 1131 (Springer, 1996) pp. 337–342.
J.-P. Thirion, Non-rigid matching using demons, in: IEEE, Correspondence on Computer Vision and Pattern Recognition (1996) pp. 245–251.
Rights and permissions
About this article
Cite this article
Fischer, B., Modersitzki, J. Fast inversion of matrices arising in image processing. Numerical Algorithms 22, 1–11 (1999). https://doi.org/10.1023/A:1019194421221
Issue Date:
DOI: https://doi.org/10.1023/A:1019194421221