Abstract
Diffusion tensor magnetic resonance imaging (DT-MRI) can provide the fundamental information required to visualize structural connectivity. However, this high-dimensional data can be rather noisy and requires restoration. In this paper, we present a novel unified formulation involving a variational principle for simultaneous smoothing and estimation of the diffusion tensor field from DT-MRI. This tensor field is estimated directly from the measurements using a combination of L p smoothness and positive definiteness constraints respectively. The data term we employ is the Stejskal-Tanner equation instead of its linearized version as usually employed in the published literature. In addition, we impose the positive definite constraint via the Cholesky decomposition of the tensors in the field. Our unified variational principle is discretized and solved numerically using the limited memory quasi-Newton method. Algorithm performance is depicted via both synthetic and real data experiments.
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.
References
Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion. ii. SIAM J. Numer. Anal. 29(3), 845–866 (1992)
Basser, P.J., Mattiello, J., Lebihan, D.: Estimation of the Effective Self- Diffusion Tensor from the NMR Spin Echo. J. Magn. Reson B 103, 247–254 (1994)
Basser, P.J., Pierpaoli, C.: Microstructural and Physiological Features of Tissue Elucidated by Quantitative-Diffusion-Tensor MRI. J. Magn. Reson B 111, 209–219 (1996)
Blomgren, P., Chan, T.F., Mulet, P.: Extensions to Total Variation Denoising., Tech. Rep.97-42, UCLA (September 1997)
Chefd’hotel, C., Tschumperle’, D., Deriche, R., Faugeras, O.D.: Constrained Flows of Matrix-Valued Functions: Application to Diffusion Tensor Regularization. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 251–265. Springer, Heidelberg (2002)
Caselles, V., Morel, J.M., Sapiro, G., Tannenbaum, A.: IEEE TIP. special issue on PDEs and geometry-driven diffusion in image processing and analysis 7(3) (1998)
Chan, T.F., Golub, G., Mulet, P.: A nonlinear primal-dual method for TVbased image restoration. In: Berger, M., et al. (eds.) Proc. 12th Int. Conf. Analysis and Optimization of Systems: Images, Wavelets, and PDE’s, Paris, France, 26-28, 1996, vol. 219, pp. 241–252 (June 1996)
Conturo, T.E., Lori, N.F., Cull, T.S., Akbudak, E., Snyder, A.Z., Shimony, J.S., McKinstry, R.C., Burton, H., Raichle, M.E.: Tracking neuronal fiber pathways in the living human brain. Proc. Natl. Acad. Sci. USA 96, 10422–10427 (1999)
Coulon, O., Alexander, D.C., Arridge, S.R.: A Regularization Scheme for Diffusion Tensor Magnetic Resonance Images. In: Insana, M.F., Leahy, R.M. (eds.) IPMI 2001. LNCS, vol. 2082, pp. 92–105. Springer, Heidelberg (2001)
Evans, L.C.: Partial Differential Equations. In: Graduate Studies in Mathematics. American Mathematical Society, Providence (1997)
Golub, G.H., van Loan, CF.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore (1989)
Jones, D.K., Simmons, A., Williams, S.C.R., Horsfield, M.A.: Non-invasive assessment of axonal fiber connectivity in the human brain via diffusion tensor MRI. Magn. Reson. Med. 42, 37–41 (1999)
Kimmel, R., Malladi, R., Sochen, N.A.: Images as Embedded Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images. In: IJCV, vol. 39(2), pp. 111–129 (2000)
McGraw, T.E., Vemuri, B.C., Chen, Y., Rao, M., Mareci, T.: LIC for visualization of fiber tract maps. In: Dohi, T., Kikinis, R. (eds.) MICCAI 2002. LNCS, vol. 2489, p. 615. Springer, Heidelberg (2002)
Nocedal, J., Wright, S.J.: Num. Optimization. Springer, Heidelberg (2000)
Pang, A., Smith, K.: Spray Rendering: Visualization Using Smart Particles. In: IEEE Visualization 1993 Conference Proceedings, pp. 283–290 (1993)
Parker, G.J.M., Schnabel, J.A., Symms, M.R., Werring, D.J., Baker, G.J.: Nonlinear smoothing for reduction of systematic and random errors in diffusion tensor imaging. Magn. Reson. Imag. 11, 702–710 (2000)
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE TPAMI 12(7), 629–639 (1990)
Perona, P.: Orientation diffusions. IEEE TIP 7(3), 457–467 (1998)
Poupon, C., Mangin, J.F., Clark, C.A., Frouin, V., Regis, J., Le Bihan, D., Block, I.: Towards inference of human brain connectivity from MR diffusion tensor data. Med. Image Anal. 5, 1–15 (2001)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Tang, B., Sapiro, G., Caselles, V.: Diffusion of General Data on Non-Flat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case. In: IJCV, vol. 36(2), pp. 149–161 (2000)
Tschumperle, D., Deriche, R.: Regularization of orthonormal vector sets using coupled PDE’s. In: Proceedings of IEEE Workshop on Variational and Level Set Methods in Computer Vision, pp. 3–10 (July 2001)
Vemuri, B.C., Chen, Y., Rao, M., McGraw, T., Wang, Z., Mareci, T.: Fiber Tract Mapping from Diffusion Tensor MRI. In: Proceedings of IEEE Workshop on Variational and Level Set Methods in Computer Vision, pp. 81–88 (July 2001)
Weickert, J.: A review of nonlinear diffusion filtering. In: ter Haar Romeny, B.M., Florack, L.M.J., Viergever, M.A. (eds.) Scale-Space 1997. LNCS, vol. 1252, pp. 3–28. Springer, Heidelberg (1997)
Westin, C.F., Maier, S.E., Mamata, H., Nabavi, A., Jolesz, F.A., Kikinis, R.: Processing and visualization for diffusion tensor MRI. Med. Image Anal. 6, 93–108 (2002)
Zhang, S., Demiralp, C., Laidlaw, D.: Visualizing diffusion tensor MRI using stream tubes and stream surfaces. In: IEEE Trans. on Visualization and Computer Graphics (2003) (in press)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wang, Z., Vemuri, B.C., Chen, Y. (2003). Diffusion Tensor MR Image Restoration. In: Rangarajan, A., Figueiredo, M., Zerubia, J. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2003. Lecture Notes in Computer Science, vol 2683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45063-4_27
Download citation
DOI: https://doi.org/10.1007/978-3-540-45063-4_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40498-9
Online ISBN: 978-3-540-45063-4
eBook Packages: Springer Book Archive