Abstract
We present a novel approach for the derivation of PDE modeling curvature-driven flows for matrix-valued data. This approach is based on the Riemannian geometry of the manifold of symmetric positive-definite matrices \(\mathcal{P}(n)\). The differential geometric attributes of \(\mathcal{P}(n)\) −such as the bi-invariant metric, the covariant derivative and the Christoffel symbols− allow us to extend scalar-valued mean curvature and snakes methods to the tensor data setting. Since the data live on \(\mathcal{P}(n)\), these methods have the natural property of preserving positive definiteness of the initial data. Experiments on three-dimensional real DT-MRI data show that the proposed methods are highly robust.
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. Num. Anal. 29, 845–866 (1992)
Alvarez, L., Morel, J.-M.: Morphological Approach to Multiscale Analysis: From Principles to Equations. Kluwer Academic Publishers, Dordrecht (1994)
Arsigny, V., et al.: Fast and simple calculus on tensors in the Log-Euclidean framework. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3749, pp. 115–122. Springer, Heidelberg (2005)
Basser, P., Matiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophysical J. 66, 259–267 (1994)
Blomgren, P., Chan, T.: Color TV: Total variation methods for restoration of vector valued images. IEEE Trans. Image Process. 7, 304–378 (1998)
Faugeras, O., et al.: Constrained Flows of Matrix-Valued Functions: Application to Diffusion Tensor Regularization. In: Heyden, A., et al. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 251–265. Springer, Heidelberg (2002)
Cumani, A.: Edge detection in multispectral images. Computer Vision Graphics and Image Processing: Graphical Models and Image Processing 53, 40–51 (1991)
Deriche, R., et al.: Variational approaches to the estimation, regularization and segmentation of diffusion tensor images. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Mathematical Models in Computer Vision: The Handbook, Springer, Heidelberg (2005)
Feddern, C., Burgeth, J.W.B., Welk, M.: Curvature-driven PDE methods for matrix-valued images. Int. J. Comput. Vision 69, 93–107 (2006)
Henderson, H., Searle, S.: Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics. Canad. J. Statist. 7, 65–81 (1979)
Jost, J.: Riemannian Geometry and Geometric Analysis, 2nd edn. Springer, Heidelberg (1998)
Kimmel, R., Malladi, R., Sochen, N.: Images as embedded maps and minimal surfaces: movies, color, texture, and volumetric medical images. Int. J. Compt. Vision 39, 111–129 (2000)
Lee, H.-C., Cok, D.: Detecting boundaries in a vector field. IEEE Trans. Signal Proc. 39, 1181–1194 (1991)
Lenglet, C., et al.: Statistics on the manifold of multivariate normal distributions: Theory and application to diffusion tensor MRI processing. J. Mathematical Imaging and Vision 25, 423–444 (2006)
Magnus, J., Neudecker, H.: The elimination matrix: some lemmas and applications. SIAM J. Alg. Disc. Meth. 1, 422–449 (1980)
Marquina, A., Osher, S.: Explicit algoritms for a new time dependent model based on level set motion for non-linear deblurring and noise removal. SIAM J. Sci. Compt. 22, 378–405 (2000)
Moakher, M., Batchelor, P.: The symmetric space of positive definite tensors: From geometry to applications and visualization. In: Weickert, J., Hagen, H. (eds.) Visualization and Processing of Tensor Fields, pp. 285–298. Springer, Berlin (2006)
Neviata, R.: A color edge detector and its use in scene segmentation. IEEE Trans. Syst. Man. Cybern. 7, 820–826 (1977)
Osher, S., Rudin, L.: Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27, 919–940 (1990)
Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Compt. Vision 66, 41–66 (2006)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Sapiro, G.: Color snakes.Tech. Rep. HPL-95-113, Hewlett Packard Computer Peripherals Laboratory (1995)
Sapiro, G.: Vector-valued active contours. In: Proceedings of Computer Vision and Pattern Recognition, pp. 520–525 (1996)
Sapiro, G., Ringach, D.: Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Image Process. 5, 1582–1586 (1996)
Sochen, N.A., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Trans. Image Process. 7, 310–318 (1998)
Wang, Z., et al.: A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. IEEE Trans. Medical Imaging 23, 930–939 (2004)
Weickert, J., Hagen, H. (eds.): Visualization and Processing of Tensor Fields. Springer, Berlin (2006)
Zéraï, M., Moakher, M.: The Riemannian geometry of the space of positive-definite matrices and its application to the regularization of diffusion tensor MRI data. Submitted to: J. Mathematical Imaging and Vision (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Zéraï, M., Moakher, M. (2007). Riemannian Curvature-Driven Flows for Tensor-Valued Data. In: Sgallari, F., Murli, A., Paragios, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer Science, vol 4485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72823-8_51
Download citation
DOI: https://doi.org/10.1007/978-3-540-72823-8_51
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72822-1
Online ISBN: 978-3-540-72823-8
eBook Packages: Computer ScienceComputer Science (R0)