Abstract
There is an increasing demand to develop image processing tools for the filtering and analysis of matrix-valued data, so-called matrix fields. In the case of scalar-valued images parabolic partial differential equations (PDEs) are widely used to perform filtering and denoising processes. Especially interesting from a theoretical as well as from a practical point of view are PDEs with singular diffusivities describing processes like total variation (TV-) diffusion, mean curvature motion and its generalisation, the so-called self-snakes. In this contribution we propose a generic framework that allows us to find the matrix-valued counterparts of the equations mentioned above. In order to solve these novel matrix-valued PDEs successfully we develop truly matrix-valued analogs to numerical solution schemes of the scalar setting. Numerical experiments performed on both synthetic and real world data substantiate the effectiveness of our matrix-valued, singular diffusion filters.
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References
Alvarez, L., et al.: Axioms and fundamental equations in image processing. Archive for Rational Mechanics and Analysis 123, 199–257 (1993)
Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM Journal on Numerical Analysis 29, 845–866 (1992)
Andreu, F., et al.: Minimizing total variation flow. Differential and Integral Equations 14(3), 321–360 (2001)
Andreu, F., et al.: Qualitative properties of the total variation flow. Journal of Functional Analysis 188(2), 516–547 (2002)
Arsigny, V., Pennec, P.F.X., Ayache, N.: 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)
Bellettini, G., Caselles, V., Novaga, M.: The total variation flow in R N. Journal of Differential Equations 184(2), 475–525 (2002)
Brox, T., et al.: Nonlinear structure tensors. Image and Vision Computing 24(1), 41–55 (2006)
Burgeth, B., et al.: Morphology for matrix-data: Ordering versus PDE-based approach. Image and Vision Computing (2006)
Chefd’Hotel, C., 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)
Dibos, F., Koepfler, G.: Total variation minimization by the Fast Level Sets Transform. In: Proc. First IEEE Workshop on Variational and Level Set Methods in Computer Vision, Vancouver, Canada, July 2001, pp. 145–152. IEEE Computer Society Press, Los Alamitos (2001)
Feddern, C., et al.: Curvature-driven PDE methods for matrix-valued images. International Journal of Computer Vision 69(1), 91–103 (2006)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)
Keeling, S.L., Stollberger, R.: Nonlinear anisotropic diffusion filters for wide range edge sharpening. Inverse Problems 18, 175–190 (2002)
Pierpaoli, C., et al.: Diffusion tensor MR imaging of the human brain. Radiology 201(3), 637–648 (1996)
Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001)
Schultz, T., Burgeth, B., Weickert, J.: Flexible segmentation and smoothing of DT-MRI fields through a customizable structure tensor. In: Bebis, G., et al. (eds.) ISVC 2006. LNCS, vol. 4291, Springer, Heidelberg (2006)
Tsurkov, V.I.: An analytical model of edge protection under noise suppression by anisotropic diffusion. Journal of Computer and Systems Sciences International 39(3), 437–440 (2000)
Weickert, J.: Applications of nonlinear diffusion in image processing and computer vision. Acta Mathematica Universitatis Comenianae 70(1), 33–50 (2001)
Weickert, J., Brox, T.: Diffusion and regularization of vector- and matrix-valued images. In: Nashed, M.Z., Scherzer, O. (eds.) Inverse Problems, Image Analysis, and Medical Imaging. Contemporary Mathematics, vol. 313, pp. 251–268. AMS, Providence (2002)
Weickert, J., Hagen, H. (eds.): Visualization and Processing of Tensor Fields. Springer, Berlin (2006)
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Burgeth, B., Didas, S., Florack, L., Weickert, J. (2007). A Generic Approach to the Filtering of Matrix Fields with Singular PDEs. 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_48
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DOI: https://doi.org/10.1007/978-3-540-72823-8_48
Publisher Name: Springer, Berlin, Heidelberg
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