Abstract
In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from diffusion weighted imaging (DWI). The constrained variational principle involves the minimization of a regularization term in an L p norm, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The original nonlinear form leads to a more accurate (when compared to the linearized form) estimated tensor field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a possibly known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion tensor, it is expressed in terms of cholesky factors and estimated. variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Both synthetic and real data experiments are shown to depict the performance of the tensor field estimation algorithm. Fiber tracts in a rat brain are then mapped using a particle system based visualization technique.
This research was supported in part by the NIH grant NS42075.
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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., series 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., series 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 TV-based image restoration. In: Berger, M., et al. (eds.) Proc. 12th Int. Conf. Analysis and Optimization of Systems: Images, Wavelets, and PDE’s, Paris, France, June 26–28, vol. (219), pp. 241–252 (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. Graduate Studies in Mathematics. American Mathematical Society, Providence (1997)
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. IJCV 39(2), 111–129 (2000)
Lindgren, B.W.: Statistical Theory. Chapman & Hall/CRC (1993)
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. IJCV 36(2), 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, July 2001, pp. 3–10 (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, July 2001, pp. 81–88 (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)
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Wang, Z., Vemuri, B.C., Chen, Y., Mareci, T. (2003). A Constrained Variational Principle for Direct Estimation and Smoothing of the Diffusion Tensor Field from DWI. In: Taylor, C., Noble, J.A. (eds) Information Processing in Medical Imaging. IPMI 2003. Lecture Notes in Computer Science, vol 2732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45087-0_55
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DOI: https://doi.org/10.1007/978-3-540-45087-0_55
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