ELASTIC REGISTRATION OF EDGE SETS BY MEANS OF DIFFUSE SURFACES - With an Application to Embedding Purkinje Fiber Networks

Stefan Fürtinger, Stephen Keeling, Gernot Plank, Anton J. Prassl

Abstract

In this work, edge sets are mapped one to the other by representing these zero area sets as diffuse images which have positive measure supports that can be registered elastically. The driving application for this work is to map a Purkinje fiber network in the epicardium of one heart to the epicardium of another heart. The approach is to register sufficiently accurate diffuse surface representations of two epicardia and then to apply the resulting transformation to the points of the Purkinje fiber network. To create a diffuse image from a given edge set, a region growing method is used to approximate diffusion of brightness from an edge set to a given point. To be minimized is the sum of squared differences of the registered diffuse images along with a linear elastic penalty for the registration. A Newton iteration is employed to solve the optimality system, and the degree of diffusion is larger in initial iterations while smaller in later iterations so that a desired local minimum is selected by means of vanishing diffusion. Favorable results are shown for registering highly detailed rabbit heart models.

References

  1. Dennis, J. and Schnabel, R. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ,.
  2. Droske, M. and Ring, W. (2006). A mumford-shah levelset approach for geometric image registration. SIAM Journal of Applied Mathematics, 66(6):2127-2148.
  3. Fitzpatrick, J., Hill, D., and Maurer, C. (2000). Image Registration, Medical Image Processing, volume 2, chapter 8 of the Handbook of Medical Imaging. SPIE Press.
  4. Fuchs, M., Jüttler, B., Scherzer, O., and Yang, H. (2009). Shape metrics based on elastic deformations. J. Math. Imaging Vis., 35(1):86-102.
  5. Hill, W. and Baldock, R. A. (2006). The constrained distance transform: Interactive atlas registration with large deformations through constrained distances. In DEFORM'06 - Workshop on Image Registration in Deformable Environments.
  6. Huelsing, D. J., Spitzer, K. W., Cordeiro, J. M., and Pollard, A. E. (1998). Conduction between isolated rabbit purkinje and ventricular myocytes coupled by a variable resistance. Am J Physiol, 274(4 Pt 2):H1163- H1173.
  7. Keeling, S. and Ring, W. (2005.). Medical image registration and interpolation by optical flow with maximal rigidity. Journal of Mathematical Imaging and Vision, 23(1):47-65.
  8. Keeling, S. L. (2007). Generalized rigid and generalized affine image registration and interpolation by geometric multigrid. Journal of Mathematical Imaging and Vision, 29:163-183.
  9. Knauer, C., Kriegel, K., and Stehn, F. (2009). Minimizing the weighted directed hausdorff distance between colored point sets under translations and rigid motions. In FAW 7809: Proceedings of the 3d International Workshop on Frontiers in Algorithmics, pages 108-119, Berlin, Heidelberg. Springer-Verlag.
  10. Modersitzki, J. (2004). Numerical Methods for Image Registration. Oxford Science Publications.
  11. Mumford, D. and Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42(5):577-685.
  12. Nocedal, J. and Wright, S. (2000). Numerical Optimization. Springer.
  13. Ortega, J. M. (1968). The newton-kantorovich theorem. The American Mathematical Monthly, 75(6):658- 660.
  14. Paragios, N. and Ramesh, M. R. V. (2002). Matching distance functions: A shape-to-area variational approach for global-to-local registration. In Proceedings of the 7th European Conference on Computer Vision-Part II.
  15. Peckar, W., Schnörr, C., Rohr, K., and Stiehl, H. S. (1999). Parameter-free elastic deformation approach for 2d and 3d registration using prescribed displacements. J. Math. Imaging Vis., 10(2):143-162.
  16. Plank, G., Burton, R. A., Hales, P., Bishop, M., Mansoori, T., Bernabeu, M. O., Garny, A., Prassl, A. J., Bollensdorff, C., Mason, F., Mahmood, F., Rodriguez, B., Grau, V., Schneider, J. E., Gavaghan, D., and Kohl, P. (2009). Generation of histo-anatomically representative models of the individual heart: tools and application. Philos Transact A Math Phys Eng Sci, 367(1896):2257-92.
  17. Prassl, A. J., Kickinger, F., Ahammer, H., Grau, V., Schneider, J. E., Hofer, E., Vigmond, E. J., Trayanova, N. A., and Plank, G. (2009). Automatically generated, anatomically accurate meshes for cardiac electrophysiology problems. IEEE Trans Biomed Eng, 56(5):1318-30.
  18. Sim, D.-G., Kwon, O.-K., and Park, R.-H. (1999). Object matching algorithms using robust hausdorff distance measures. IEEE Transactions on Image Processing, 8(3):425-429.
  19. Tranum-Jensen, J., Wilde, A. A., Vermeulen, J. T., and Janse, M. J. (1991). Morphology of electrophysiologically identified junctions between purkinje fibers and ventricular muscle in rabbit and pig hearts. Circ Res, 69(2):429-437.
  20. Vetter, F. and McCulloch, A. (1998). Three-dimensional analysis of regional cardiac function: a model of rabbit ventricular anatomy. Prog Biophys Mol Biol, 69(2- 3):157-83.
  21. Vigmond, E. J. and Clements, C. (2007). Construction of a computer model to investigate sawtooth effects in the purkinje system. IEEE Trans Biomed Eng, 54(3):389- 399.
  22. Yang, S., Kohler, D., Teller, K., Cremer, T., Le Baccon, P., Heard, E., Eils, R., and Rohr, K. (2008). Nonrigid registration of 3-d multichannel microscopy images of cell nuclei. IEEE Transactions on Image Processing, 17:493-499.
  23. Zhao, C., Shi, W., and Deng, Y. (2005). A new hausdorff distance for image matching. Pattern Recognition Letters, 26(5):581-586.
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Paper Citation


in Harvard Style

Fürtinger S., Keeling S., Plank G. and J. Prassl A. (2011). ELASTIC REGISTRATION OF EDGE SETS BY MEANS OF DIFFUSE SURFACES - With an Application to Embedding Purkinje Fiber Networks . In Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2011) ISBN 978-989-8425-47-8, pages 12-21. DOI: 10.5220/0003322200120021


in Bibtex Style

@conference{visapp11,
author={Stefan Fürtinger and Stephen Keeling and Gernot Plank and Anton J. Prassl},
title={ELASTIC REGISTRATION OF EDGE SETS BY MEANS OF DIFFUSE SURFACES - With an Application to Embedding Purkinje Fiber Networks},
booktitle={Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2011)},
year={2011},
pages={12-21},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0003322200120021},
isbn={978-989-8425-47-8},
}


in EndNote Style

TY - CONF
JO - Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2011)
TI - ELASTIC REGISTRATION OF EDGE SETS BY MEANS OF DIFFUSE SURFACES - With an Application to Embedding Purkinje Fiber Networks
SN - 978-989-8425-47-8
AU - Fürtinger S.
AU - Keeling S.
AU - Plank G.
AU - J. Prassl A.
PY - 2011
SP - 12
EP - 21
DO - 10.5220/0003322200120021