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Medical Image Registration and Interpolation by Optical Flow with Maximal Rigidity

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Abstract

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.

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Authors and Affiliations

  1. Institut für Mathematik, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010, Graz, Austria

    Stephen L. Keeling & Wolfgang Ring

Authors
  1. Stephen L. Keeling
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  2. Wolfgang Ring
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Correspondence to Stephen L. Keeling or Wolfgang Ring.

Additional information

Supported by the Fonds zur Förderung der wissenschaftliche Forschung under SFB 03, “Optimierung und Kontrolle”.

Stephen Keeling was born in 1956 in Louisville, KY, USA. He received the B.S. in Biology and Chemistry from Eastern Kentucky University in 1978, the M.S. in Biomedical Engineering (bioelectric phenomena) from Case Western Reserve University in 1981, and the Ph.D. in Mathematics (numerical analysis for PDEs) from the University of Tennessee in 1986. His postdoctoral research at ICASE and at Vanderbilt University focused on active noise control until 1989. From 1989 until 1998 he worked as Senior and then Principal Scientist in the CFD Group at the Arnold Engineering Development Center specializing in flow control and imaging. From 1998 until 2001 he worked with the mathematics and radiology faculties of the University of Graz as a Research Associate of the Special Research Center on Optimization and Control. Since 2001 he is Assistant Professor at the Institute of Mathematics of the University of Graz. His research interests include early vision problems in medical imaging with emphasis on MRI applications.

Wolfgang Ring was born on November 6, 1965 in Zeltweg, Austria. He received his Masters Degree from the University of Graz in 1991 and his Ph.D. in Mathematics from the Technical University of Graz in 1994. He worked as an Assistant Professor at the Institute of Mathematics at the Technical University of Graz between 1993 and 1996. Since 1997 he is Assistant Professor at the Institute of Mathematics at the University of Graz. His main scientific interests are geometric inverse problems, mathematical imaging and optimal shape design.

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Keeling, S.L., Ring, W. Medical Image Registration and Interpolation by Optical Flow with Maximal Rigidity. J Math Imaging Vis 23, 47–65 (2005). https://doi.org/10.1007/s10851-005-4967-2

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