Abstract
Image registration is a widely used task in image analysis, having applications in various fields. Its classical formulation is usually given in the spatial domain. In this paper, a novel theoretical framework defined in the frequency domain is proposed for approaching the multidimensional image registration problem. The variational minimization of the joint energy functional is performed entirely in the frequency domain, leading to a simple formulation and design, and offering important computational savings if the multidimensional FFT algorithm is used. Therefore the proposed framework provides more efficient implementations of the most common registration methods than already existing approaches, adding simplicity to the variational image registration formulation and allowing for an easy extension to higher dimensions by using the multidimensional Fourier transform of discrete multidimensional signals. The new formulation also provides an interesting framework to design tailor-made regularization models apart from the classical, spatial domain based schemes. Simulation examples validate the theoretical results.
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Abel, J.: Medical image processing info 2004–2007. www.medical-image-processing.info
Amit, Y.: A nonlinear variational problem for image matching. SIAM J. Sci. Comput. 15(1), 207–224 (1994)
Bajcsy, R., Kovacic, S.: Multiresolution elastic matching. Comput. Vis. Graph. Image Process. 46(1), 1–21 (1989)
Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications. Wiley, New York (1977)
Braumann, U.-D., Kuska, J.-P.: Influence of the boundary conditions on the results of non-linear image registration. IEEE Int. Conf. Image Process. I, 1129–1132 (2005)
Bro-Nielsen, M., Gramkow, C.: Fast fluid registration of medical images. In: Lecture Notes in Computer Science, vol. 1131, pp. 267–276. Springer, Berlin (1996)
Broit, C.: Optimal registration of deformed images. Ph.D. thesis, University of Pennsylvania (1981)
Brown, L.G.: A survey of image registration techniques. ACM Comput. Surv. 24(4), 325–376 (1992)
Cachier, P., Bardinet, E., Dormont, D., Pennec, X., Ayache, N.: Iconic feature based nonrigid registration: the PASHA algorithm. Comput. Vis. Image Underst. 89, 272–298 (2003)
Christensen, G.E.: Deformable shape models for anatomy. Ph.D. thesis, Washington University (1994)
Clarenz, U., Droske, M., Henn, S., Rumpf, M., Witsch, K.: Computational methods for nonlinear image registration. Math. Method Regist. Appl. Med. Imaging Math. Ind. 10, 1–22 (2006)
D’Agostino, E., Maes, F., Vandermeulen, D., Suetens, P.: A viscous fluid model for multimodal non-rigid image registration using mutual information. Med. Image Anal. 7, 565–575 (2003)
D’Agostino, E., Maes, F., Vandermeulen, D., Suetens, P.: Validation of non-rigid image registration using mutual information. Technical Report, Katholieke Universiteit Leuven, March 2004
Davis, P.J.: Circulant Matrices. Wiley-Interscience, New York (1979)
Fischer, B., Modersitzki, J.: Fast inversion of matrices arising in image processing. Numer. Algorithms 22, 1–11 (1999)
Fischer, B., Modersitzki, J.: Fast diffusion registration. In: Nashed, M.Z., Scherzer, O. (eds.) Inverse Problems, Image Analysis, and Medical Imaging. Contemporary Mathematics, vol. 313, pp. 117–129. AMS (2002)
Fischer, B., Modersitzki, J.: Curvature based image registration. J. Math. Imaging Vis. 18(1), 81–85 (2003)
Fischer, B., Modersitzki, J.: Fast image registration—a variational approach. In: Psihoyios, G. (ed.) Proceedings of the International Conference on Numerical Analysis & Computational Mathematics, pp. 69–74. Wiley, New York (2003)
Fischer, B., Modersitzki, J.: FLIRT: a flexible image registration toolbox. In: Lecture Notes in Computer Science, vol. 2717, pp. 261–270. Springer, Berlin (2003)
Fischer, B., Modersitzki, J.: A unified approach to fast image registration and a new curvature based registration technique. Linear Algebra Appl. 308, 107–124 (2004)
Frigo, M., Johnson, S.G.: The design and implementation of FFTW3. Proc. IEEE 93(2), 216–231 (2005)
Goshtasby, A.: Registration of images with geometric distortions. IEEE Trans. Geosci. Remote Sens. 26, 60–64 (1988)
Haber, E., Modersitzki, J.: COFIR: coarse and fine image registration. In: SIAM Real-Time PDE-Constrained Optimization, pp. 37–49 (2007)
Hajnal, J., Hill, D., Hawkes, D.: Medical Image Registration. CRC Press, Boca Raton (2001)
Hata, N., Dohi, T., Warfield, S., Wells, W., Kikinis, R., Jolesz, F.A.: Multimodality deformable registration of pre- and intraoperative images for MRI-guided brain surgery. In: Lecture Notes in Computer Science, vol. 1496, pp. 1067–1074. Springer, Berlin (1998)
Henn, S.: A multigrid method for a fourth-order diffusion equation with application to image processing. SIAM J. Sci. Comput. 27(3), 831–849 (2005)
Henn, S.: A full curvature based algorithm for image registration. J. Math. Imaging Vis. 24(2), 195–208 (2006)
Henn, S.: A translation and rotation invariant Gauss-Newton like scheme for image registration. BIT Numer. Math. 46, 325–344 (2006)
Henn, S., Witsch, K.: Multi-modal image registration using a variational approach. SIAM J. Sci. Comput. 23(4), 1429–1447 (2004)
Henn, S., Witsch, K.: Image registration based on multiscale energy information. Multiscale Model. Simul. 4(2), 584–609 (2005)
Hermosillo, G., Chefd’Hotel, C., Faugeras, O.: A variational approach to multi-modal image matching. Technical Report 4117, INRIA, February 2001
Lester, H., Arridge, S.: A survey of hierarchical non-linear medical image registration. Pattern Recognit. 32, 129–149 (1999)
Maes, F., Vandermeulen, D., Suetens, P.: Medical image registration using mutual information. Proc. IEEE 91, 1699–1722 (2003)
Maintz, J., Viergever, M.: A survey of medical image registration. Med. Image Anal. 2(1), 1–36 (1998)
Miller, M.I., Christensen, G.E., Amit, Y., Grenader, U.: Mathematical textbook of deformable neuroanatomies. Proc. Natl. Acad. Sci. USA 24, 11944–11948 (1993)
Modersitzki, J.: Numerical Methods for Image Registration. Oxford University Press, London (2004)
Oppenheim, A.V., Schafer, R.: Discrete-Time Signal Processing, 2nd edn. Prentice-Hall, Upper Saddle River (1999)
Papenberg, N., Schumacher, H., Heldmann, S., Wirtz, S., Bommersheim, S., Ens, K., Modersitzki, J., Fischer, B.: A fast and flexible image registration toolbox: design and implementation of the general approach. In: Bildverarbeitung fur die Medizin, pp. 106–110 (2007)
Research Center Caesar: Julius software development framework (1999–2006). www.julius.caesar.de
Roche, A., Malandain, G., Pennec, X., Ayache, N.: The correlation ratio as a new similarity measure for multimodal image registration. In: Lecture Notes in Computer Science, vol. 1496, pp. 1115–1124. Springer, Berlin (1998)
Rohr, K.: Landmark-based image analysis: using geometric and intensity models. In: Computational Imaging and Vision Series, p. 21. Kluwer Academic, Dordrecht (2001)
Thirion, J.-P.: Image matching as a diffusion process: an analogy with Maxwell’s demons. Med. Image Anal. 2(3), 243–260 (1998)
Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-posed Problems. Winston and Sons, Washington (1977)
Wang, Y., Staib, L.H.: Elastic model based non-rigid registration incorporating statistical shape information. In: Lecture Notes in Computer Science, vol. 1496, pp. 1162–1173. Springer, Berlin (1998)
Zhang, Z., Jiang, Y., Tsui, H.: Consistent multi-modal non-rigid registration based on a variational approach. Pattern Recognit. Lett. 27, 715–725 (2006)
Zitová, B., Flusser, J.: Image registration methods: a survey. Image Vis. Comput. 21, 997–1000 (2003)
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This work is partially supported by the Spanish Ministerio de Educación y Ciencia, under grant TEC2006-13338/TCM, and by Fundación Séneca, project 03122/PI/05.
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Larrey-Ruiz, J., Verdú-Monedero, R. & Morales-Sánchez, J. A Fourier Domain Framework for Variational Image Registration. J Math Imaging Vis 32, 57–72 (2008). https://doi.org/10.1007/s10851-008-0075-4
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DOI: https://doi.org/10.1007/s10851-008-0075-4