Medical Image Registration and Interpolation by Optical Flow with Maximal Rigidity

In this paper a variational method for registering or mapping like points in medical images is proposed and analyzed. The proposed variational principle penalizes a departure from rigidity and thereby provides a natural generalization of strictly rigid registration techniques used widely in medical contexts. Difficulties with finite displacements are elucidated, and alternative infinitesimal displacements are developed for an optical flow formulation which also permits image interpolation. The variational penalty against non-rigid flows provides sufficient regularization for a well-posed minimization and yet does not rule out irregular registrations corresponding to an object excision. Image similarity is measured by penalizing the variation of intensity along optical flow trajectories. The approach proposed here is also independent of the order in which images are taken. For computations, a lumped finite element Eulerian discretization is used to solve for the optical flow. Also, a Lagrangian integration of the intensity along optical flow trajectories has the advantage of prohibiting diffusion among trajectories which would otherwise blur interpolated images. The subtle aspects of the methods developed are illustrated in terms of simple examples, and the approach is finally applied to the registration of magnetic resonance images.

[1]  Kazufumi Ito,et al.  Optimal Control Formulation for Determining Optical Flow , 2002, SIAM J. Sci. Comput..

[2]  Hector O. Fattorini,et al.  Infinite Dimensional Optimization and Control Theory: References , 1999 .

[3]  Jean-Philippe Thirion,et al.  Image matching as a diffusion process: an analogy with Maxwell's demons , 1998, Medical Image Anal..

[4]  Josien P. W. Pluim,et al.  Image Registration , 2003, IEEE Trans. Medical Imaging.

[5]  Paul A. Viola,et al.  Multi-modal volume registration by maximization of mutual information , 1996, Medical Image Anal..

[6]  G. M. Troianiello,et al.  Elliptic Differential Equations and Obstacle Problems , 1987 .

[7]  Colin Studholme,et al.  An overlap invariant entropy measure of 3D medical image alignment , 1999, Pattern Recognit..

[8]  Paul A. Viola,et al.  Alignment by Maximization of Mutual Information , 1995, Proceedings of IEEE International Conference on Computer Vision.

[9]  L. Rudin,et al.  Feature-oriented image enhancement using shock filters , 1990 .

[10]  Jan Modersitzki,et al.  Fast inversion of matrices arising in image processing , 1999, Numerical Algorithms.

[11]  Jan Modersitzki,et al.  Numerical Methods for Image Registration , 2004 .

[12]  Curtis R. Vogel,et al.  Iterative Methods for Total Variation Denoising , 1996, SIAM J. Sci. Comput..

[13]  Philippe G. Ciarlet,et al.  Mathematical elasticity. volume II, Theory of plates , 1997 .

[14]  Gerald Q. Maguire,et al.  Comparison and evaluation of retrospective intermodality brain image registration techniques. , 1997, Journal of computer assisted tomography.

[15]  Jan Modersitzki,et al.  Curvature Based Image Registration , 2004, Journal of Mathematical Imaging and Vision.

[16]  Karl Zilles,et al.  Elastische Anpassung in der digitalen Bildverarbeitung auf mehreren Auflösungsstufen mit Hilfe von Mehrgitterverfahren , 1997, DAGM-Symposium.

[17]  Daniel Rueckert,et al.  Non-rigid registration using higher-order mutual information , 2000, Medical Imaging.

[18]  Guy Marchal,et al.  Multimodality image registration by maximization of mutual information , 1997, IEEE Transactions on Medical Imaging.

[19]  Michael E. Taylor,et al.  Partial Differential Equations , 1996 .

[20]  Arthur W. Toga,et al.  Brain Image Analysis and Atlas Construction , 2000 .

[21]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[22]  Berthold K. P. Horn,et al.  Determining Optical Flow , 1981, Other Conferences.

[23]  Daniel Rueckert,et al.  Nonrigid registration using free-form deformations: application to breast MR images , 1999, IEEE Transactions on Medical Imaging.

[24]  Laurent D. Cohen,et al.  Image Registration, Optical Flow and Local Rigidity , 2001, Journal of Mathematical Imaging and Vision.

[25]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[26]  Michel Barlaud,et al.  Deterministic edge-preserving regularization in computed imaging , 1997, IEEE Trans. Image Process..

[27]  P. Ciarlet,et al.  Mathematical elasticity, volume I: Three-dimensional elasticity , 1989 .

[28]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[29]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[30]  Jay B. West,et al.  Predicting error in rigid-body point-based registration , 1998, IEEE Transactions on Medical Imaging.

[31]  Gary E. Christensen,et al.  Consistent image registration , 2001, IEEE Transactions on Medical Imaging.

[32]  Stephen L. Keeling,et al.  A variational approach to magnetic resonance coil sensitivity estimation , 2004, Appl. Math. Comput..

[33]  Karl Rohr,et al.  Parameter-Free Elastic Deformation Approach for 2D and 3D Registration Using Prescribed Displacements , 1999, Journal of Mathematical Imaging and Vision.

[34]  Ron Kikinis,et al.  Mass Preserving Mappings and Image Registration , 2001, MICCAI.

[35]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[36]  David J. Hawkes,et al.  Deformations Incorporating Rigid Structures , 1996, Comput. Vis. Image Underst..

[37]  A. Jamiołkowski.  Book reviewApplications of Lie groups to differential equations : Peter J. Olver (School of Mathematics, University of Minnesota, Minneapolis, U.S.A): Graduate Texts in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, XXVI+497pp. , 1989 .

[38]  D. R. J. Chillingworth,et al.  Differential topology with a view to applications , 1976 .