Abstract
We consider left-invariant diffusion processes on DTI data by embedding the data into the space \(\mathbb{R}^3\rtimes S^2\) of 3D positions and orientations. We then define and solve the diffusion equation in a moving frame of reference defined using left-invariant derivatives. The diffusion process is made adaptive to the data in order to do Perona-Malik-like edge preserving smoothing, which is necessary to handle fiber structures near regions of large isotropic diffusion such as the ventricles of the brain. The corresponding partial differential systems are solved using finite difference stencils. We include experiments both on synthetic data and on DTI-images of the brain.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alexander, D.C., Barker, G.J., Arridge, S.R.: Detection and modeling of non gaussian apparent diffusion coefficient profiles in human brain data. Magnetic Resonance in Medicine 48, 331–340 (2002)
Basser, P.J., Mattiello, J., Lebihan, D.: MR diffusion tensor spectroscopy and imaging. Biophysical Journal 66, 259–267 (1994)
Burgeth, B., Pizarro, L., Didas, S., Weickert, J.: Coherence-Enhancing Diffusion for Matrix Fields. In: Proc. of Locally Adaptive Filtering in Signal and Image (to appear)
Creusen, E.J.: Numerical schemes for linear and non-linear enhancement of HARDI data. Master’s thesis, Technische Universiteit Eindhoven, the Netherlands (2010)
Duits, R., Creusen, E., Ghosh, A., Dela Haije, T.: Diffusion, convection and erosion on ℝ3 \(\rtimes\) S2 and their application to the enhancement of crossing fibers. Technical report, TU/e, CASA, The Netherlands, Eindhoven, CASA-report nr.6 (January 2011), http://www.win.tue.nl/casa/research/casareports/2011.html
Duits, R., Dela Haije, T., Ghosh, A., Creusen, E.J., Vilanova, A., ter Haar Romeny, B.: Fiber enhancement in DW-MRI. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 1–13. Springer, Heidelberg (2011)
Duits, R., Franken, E.: Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images. CASA-report 18, Department of Mathematics and Computer Science, Technische Universiteit Eindhoven (May 2009), http://www.win.tue.nl/casa/research/casareports/2009.html
Duits, R., Franken, E.: Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images. International Journal of Computer Vision 40 (2010)
Franken, E.: Enhancement of crossing elongated structures in images. PhD thesis, Eindhoven University of Technology (2008)
Franken, E.M., Duits, R., ter Haar Romeny, B.M.: Diffusion on the 3D Euclidean motion group for enhancement of HARDI data. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 820–831. Springer, Heidelberg (2009)
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7) (1990)
Prčkovska, V., Rodrigues, P., Duits, R., Vilanova, A., ter Haar Romeny, B.M.: Extrapolating fiber crossings from DTI data. Can we infer similar fiber crossings as in HARDI? In: CDMRI 2010 Proc. MICCAI Workshop Computational Diffusion MRI, China (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Creusen, E.J., Duits, R., Dela Haije, T.C.J. (2012). Numerical Schemes for Linear and Non-linear Enhancement of DW-MRI. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2011. Lecture Notes in Computer Science, vol 6667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24785-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-24785-9_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24784-2
Online ISBN: 978-3-642-24785-9
eBook Packages: Computer ScienceComputer Science (R0)