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Numerical Schemes for Linear and Non-Linear Enhancement of DW-MRI

Published online by Cambridge University Press:  28 May 2015

Eric Creusen ,
Remco Duits ,
Anna Vilanova  and
Luc Florack
Eric Creusen*
Affiliation:
Department of Mathematics and Computer Science, CASA Applied Analysis, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
Remco Duits*
Affiliation:
Department of Mathematics and Computer Science, CASA Applied Analysis, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands Department of Biomedical Engineering, BMIA Biomedical Image Analysis, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
Anna Vilanova
Affiliation:
Department of Biomedical Engineering, BMIA Biomedical Image Analysis, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
Luc Florack
Affiliation:
Department of Mathematics and Computer Science, CASA Applied Analysis, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands Department of Biomedical Engineering, BMIA Biomedical Image Analysis, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

We consider the linear and non-linear enhancement of diffusion weighted magnetic resonance images (DW-MRI) to use contextual information in denoising and inferring fiber crossings. We describe the space of DW-MRI images in a moving frame of reference, attached to fiber fragments which allows for convection-diffusion along the fibers. Because of this approach, our method is naturally able to handle crossings in data. We will perform experiments showing the ability of the enhancement to infer information about crossing structures, even in diffusion tensor images (DTI) which are incapable of representing crossings themselves. We will present a novel non-linear enhancement technique which performs better than linear methods in areas around ventricles, thereby eliminating the need for additional preprocessing steps to segment out the ventricles. We pay special attention to the details of implementation of the various numeric schemes.

Keywords

65M1078A48DTIDW-MRIscale spacesfinite differencesconvection-diffusionadaptive diffusionPerona-Malik diffusionLie groups
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 1 , February 2013 , pp. 138 - 168
Copyright
Copyright © Global Science Press Limited 2013

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References

[1] Aganj, I., Lenglet, C. and Sapiro, G., ODF reconstruction in q-ball imaging with solid angle consideration, In Biomedical Imaging: From Nano to Macro, 2009. ISBI ‘09. IEEE International Symposium on, pages 1398–1401, 28 July 2009.Google Scholar
[2] Alexander, D. C., Barker, G. J. and Arridge, S. R., Detection and modeling of non-Gaussian apparent diffusion coefficient profiles in human brain data, Magnetic Resonance in Medicine, 48 (2002), pp. 331–340.Google Scholar
[3] Basser, P. J., Mattiello, J. and Lebihan, D., MR diffusion tensor spectroscopy and imaging, Biophys. J., 66 (1994), pp. 259–267.Google ScholarPubMed
[4] Bobylev, A. V. and Ohwada, T., The error of the splitting scheme for solving evolutionary equations, Appl. Math. Lett., 14 (2001), pp. 45–48.Google Scholar
[5] Burgeth, B., Pizarro, L., Didas, S. and Weickert, J, Coherence-enhancing diffusion for matrix fields, Locally Adaptive Filtering in Signal and Image Proc., to appear.Google Scholar
[6] Chirikjian, G. S. and Kyatkin, A. B., Engineering Applications of Noncommutitative Harmonic Analysis: with Emphasis on Rotation and Motion Groups, Boca Raton CRC Press, 2001.Google Scholar
[7] Creusen, E. J., Duits, R. and Dela Haije, T., Numerical schemes for linear and non-linear enhancement of DW-MRI, In A. M. Bruckstein, editor, Proceedings of the 3rd International Conference on Scale Space and Variational Methods in Computer Vision, LNCS 6667, 2012.Google Scholar
[8] Descoteaux, M., High Angular Resolution Diffusion MRI: From Local Estimation to Segmentation and Tractography. PhD thesis, Universite de Nice, 2008.Google Scholar
[9] Duits, R., Dela, T. HAIJE, Ghosh, A., Creusen, E. J., Vilanova, A. and Ter Haar Romeny, B., Fiber enhancement in diffusion-weighted MRI, In Bruckstein, A. M., editor, Proceedings of the 3rd International Conference on Scale Space and Variational Methods in Computer Vision, LNCS 6667, 2012.Google Scholar
[10] Duits, R. and Franken, E., Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images, Int. J. Comput. Vision, 40 (2010).Google Scholar
[11] Duits, R. and Van Almsick, M., The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group, Quart. Appl. Math., 66(2008), pp. 27–67.Google Scholar
[12] Eisenberg, Murray and Guy, Robert, A proof of the hairy ball theorem, The American Mathematical Monthly, 86(7) (1979), pp. 571–574.CrossRefGoogle Scholar
[13] Franken, E. M., Duits, R. and Ter Haar Romeny, B. M., Diffusion on the 3D euclidean motion group for enhancement of HARDI data, In Scale Space and Variational Methods in Computer Vision (Lecture Notes in Computer Science), Volume 5567, pages 820–831, Heidelberg, 2009. Springer-Verlag.Google Scholar
[14] Gerschgorin, S., Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fix.-Mat. Nauk., 7 (1931), pp. 749–754.Google Scholar
[15] Gur, Y., Pasternak, O. and Sochen, N., SPD tensors regularization via Iwasawa decomposition, In Florack, L., Duits, R., Jongbloed, G., van Lieshout, M.-C. and Davies, L., editors, Mathematical Methods for Signal and Image Analysis and Representation, Springer, 2012.Google Scholar
[16] Gur, Y. and Sochen, N., Fast invariant Riemannian DT-MRI regularization, In IEEE 11th International Conference on Computer Vision, 2007. ICCV 2007, pages 1–7, 2007.Google Scholar
[17] MICHAEL Heath, T., Scientific Computing: An Introductory Survey, McGraw-Hill, International second edition, 2005.Google Scholar
[18] Mumford, D., Elastica and computer vision, Algebraic Geometry and Its Applications, Springer-Verlag, pages 491–506, 1994.Google Scholar
[19] Pasternak, O., Assaf, Y, Intrator, N. and Sochen, N., Variational multiple-tensor fitting of fiber-ambiguous diffusion-weighted magnetic resonance imaging voxels, Magnetic Resonance Imaging, 26 (2008), pp. 1133–1144.Google Scholar
[20] Perona, P. and Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12(7) (1990).Google Scholar
[21] Polzehl, J., Mathematical Methods for Signal and Image Analysis and Representation, pages 71–89, Springer-Verlag, 2011/2012. in press.Google Scholar
[22] Prcčkovska, V., Rodrigues, P R., Duits, R., Ter Haar Romeny, B. M. and Vilanova, A., Extrapolating fiber crossings from DTI data. can we gain the same information as HARDI? In Workshop on Computational Diffusion MRI, MICCAI; Beijing, China, 2010.Google Scholar
[23] Rodrigues, P., Duits, R., Vilanova, A. and Ter Haar Romeny, B. M., Accelerated Diffusion Operators for Enhancing DW-MRI, In Eurographics Workshop on Visual Computing for Biology and Medicine, pages 49–56, Leipzig, Germany, 2010.Google Scholar
[24] Rosman, G., Dascal, L., Tai, X. C. and Kimmel, R., On semi-implicit splitting schemes for the Beltrami color image filtering, J. Math. Image. Vision, 40(2) (2011), pp. 199–213.Google Scholar
[25] Savadjiev, P., Perceptual Organisation in Diffusion MRI: Curves and Streamline Flows, PhD thesis, McGill University, 2010.Google Scholar
[26] Savadjiev, P., Campbell, J. S.W., Pike, G. B. and Siddiqi, K., 3D curve inference for diffusion MRI regularization andfibre tractography, Medical Image Anal., 10(5) (2006), pp. 799–813.Google Scholar
[27] Spira, A., Kimmel, R. and Sochen, N., A short-time Beltrami kernel for smoothing images and manifolds, IEEE Trans. Image Process., 16(6) (2007), pp. 1628–1636.CrossRefGoogle ScholarPubMed
[28] Stejskal, E. O. and Tanner, J. E., Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient, The Journal of Chemical Physics, 42(1) (1965).Google Scholar
[29] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5(3) (1968), pp. 506–517.Google Scholar
[30] Tabelow, K., Keller, S. S., Mohammadi, S., Kugel, H., Gerdes, J. S., Polzehl, J. and Deppe, M., Structural adaptive smoothing increases sensitivity of DTI to detect microstructure grey matter alterations, Poster at the 17th Annual Meeting of the Organization for Human Brain Mapping, June 26-30, 2011, Quéec City, Canada, 2011.Google Scholar
[31] Tabelow, K., Polzehl, J., Spokoiny, V. and Voss, H. U., Diffusion tensor imaging: Structural adaptive smoothing, NeuroImage, 39(4) (2008), pp. 1763–1773.Google Scholar
[31] Tuch, D. S., Reese, T. G., Wiegell, M. R., Makris, N., Belliveau, J. W. and Wedeen, V. J., High Angular Resolution Diffusion Imaging Reveals Intravoxel White Matter Fiber Heterogeneity.Google Scholar
[33] Weickert, J., Coherence-enhancing diffusion filtering, Int. J. Comput. Vision, 31(2) (1999), pp. 111–127.Google Scholar
[34] Yanenko, N. N., The Method of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables, Springer, 1971.Google Scholar