Abstract
In this article we consider adaptive, PDE-driven morphological operations for 3D matrix fields arising e.g. in diffusion tensor magnetic resonance imaging (DT-MRI). The anisotropic evolution is steered by a matrix constructed from a structure tensor for matrix valued data. An important novelty is an intrinsically one-dimensional directional variant of the matrix-valued upwind schemes such as the Rouy-Tourin scheme. It enables our method to complete or enhance anisotropic structures effectively. A special advantage of our approach is that upwind schemes are utilised only in their basic one-dimensional version, hence avoiding grid effects and leading to an accurate algorithm. No higher dimensional variants of the schemes themselves are required. Experiments with synthetic and real-world data substantiate the gap-closing and line-completing properties of the proposed method.
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References
Alvarez, L., & Mazorra, L. (1994). Signal and image restoration using shock filters and anisotropic diffusion. SIAM Journal on Numerical Analysis, 31, 590–605.
Alvarez, L., Guichard, F., Lions, P.-L., & Morel, J.-M. (1993). Axioms and fundamental equations in image processing. Archive for Rational Mechanics and Analysis, 123, 199–257.
Arehart, A. B., Vincent, L., & Kimia, B. B. (1993). Mathematical morphology: The Hamilton–Jacobi connection. In Proc. fourth international conference on computer vision (pp. 215–219). Berlin, May 1993. New York: IEEE Computer Society Press.
Basser, P. J., Mattiello, J., & LeBihan, D. (1994). MR diffusion tensor spectroscopy and imaging. Biophysical Journal, 66, 259–267.
Bigün, J. (2006). Vision with direction. Berlin: Springer.
Bigün, J., Granlund, G. H., & Wiklund, J. (1991). Multidimensional orientation estimation with applications to texture analysis and optical flow. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(8), 775–790.
Boris, J. P., & Book, D. L. (1973). Flux corrected transport. I. SHASTA, a fluid transport algorithm that works. Journal of Computational Physics, 11(1), 38–69.
Boris, J. P., & Book, D. L. (1976). Flux corrected transport. III. Minimal error FCT algorithms. Journal of Computational Physics, 20, 397–431.
Boris, J. P., Book, D. L., & Hain, K. (1975). Flux corrected transport. II. Generalizations of the method. Journal of Computational Physics, 18, 248–283.
Breuß, M., & Weickert, J. (2006). A shock-capturing algorithm for the differential equations of dilation and erosion. Journal of Mathematical Imaging and Vision, 25(2), 187–201.
Breuß, M., Burgeth, B., & Weickert, J. (2007). Anisotropic continuous-scale morphology. In J. Martí, J. M. Benedí, A. M. Mendonça, & J. Serrat (Eds.), Lecture notes in computer science : Vol. 4478. Pattern recognition and image analysis (pp. 515–522). Berlin: Springer.
Brockett, R. W., & Maragos, P. (1994). Evolution equations for continuous-scale morphological filtering. IEEE Transactions on Signal Processing, 42, 3377–3386.
Brox, T., Weickert, J., Burgeth, B., & Mrázek, P. (2006). Nonlinear structure tensors. Image and Vision Computing, 24(1), 41–55.
Burgeth, B., Bruhn, A., Didas, S., Weickert, J., & Welk, M. (2007a). Morphology for matrix-data: Ordering versus PDE-based approach. Image and Vision Computing, 25(4), 496–511.
Burgeth, B., Bruhn, A., Papenberg, N., Welk, M., & Weickert, J. (2007b). Mathematical morphology for tensor data induced by the Loewner ordering in higher dimensions. Signal Processing, 87(2), 277–290.
Burgeth, B., Didas, S., Florack, L., & Weickert, J. (2007c). A generic approach to diffusion filtering of matrix-fields. Computing, 81, 179–197.
Burgeth, B., Breuß, M., Didas, S., & Weickert, J. (2009a). PDE-based morphology for matrix fields: Numerical solution schemes. In S. Aja-Fernández, R. de Luis García, D. Tao, & X. Li (Eds.), Advances in pattern recognition. Tensors in image processing and computer vision (pp. 125–150). London: Springer.
Burgeth, B., Breuß, M., Pizarro, L., & Weickert, J. (2009b). PDE-driven adaptive morphology for matrix fields. In X.-C. Tai et al. (Eds.), Lecture notes in computer science : Vol. 5567. Proc. of the second international conference on scale space and variational methods in computer vision (pp. 247–258). Berlin: Springer.
Burgeth, B., Didas, S., & Weickert, J. (2009c). A general structure tensor concept and coherence-enhancing diffusion filtering for matrix fields. In D. Laidlaw & J. Weickert (Eds.), Mathematics and visualization. Visualization and processing of tensor fields (pp. 305–323). Berlin: Springer.
Chefd’Hotel, C., Tschumperlé, D., Deriche, R., & Faugeras, O. (2002). Constrained flows of matrix-valued functions: Application to diffusion tensor regularization. In A. Heyden, G. Sparr, M. Nielsen, & P. Johansen (Eds.), Lecture notes in computer science : Vol. 2350. Computer vision—ECCV 2002 (pp. 251–265). Berlin: Springer.
Di Zenzo, S. (1986). A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing, 33, 116–125.
Feddern, C., Weickert, J., Burgeth, B., & Welk, M. (2006). Curvature-driven PDE methods for matrix-valued images. International Journal of Computer Vision, 69(1), 91–103.
Förstner, W., & Gülch, E. (1987). A fast operator for detection and precise location of distinct points, corners and centres of circular features. In Proc. ISPRS intercommission conference on fast processing of photogrammetric data (pp. 281–305). Interlaken, Switzerland, June 1987.
Gilboa, G., Sochen, N. A., & Zeevi, Y. Y. (2002). Regularized shock filters and complex diffusion. In A. Heyden, G. Sparr, M. Nielsen, & P. Johansen (Eds.), Lecture notes in computer science : Vol. 2350. Computer vision—ECCV 2002 (pp. 399–413). Berlin: Springer.
Guichard, F., & Morel, J.-M. (2003). A note on two classical enhancement filters and their associated PDE’s. International Journal of Computer Vision, 52(2/3), 153–160.
Horn, R. A., & Johnson, C. R. (1990). Matrix analysis. Cambridge: Cambridge University Press.
Kramer, H. P., & Bruckner, J. B. (1975). Iterations of a non-linear transformation for enhancement of digital images. Pattern Recognition, 7, 53–58.
Laidlaw, D., & Weickert, J. (Eds.) (2009). Visualization and processing of tensor fields. Berlin: Springer.
Matheron, G. (1967). Eléments pour une théorie des milieux poreux. Paris: Masson.
Matheron, G. (1975). Random sets and integral geometry. New York: Wiley.
Osher, S., & Fedkiw, R. P. (2002). Applied mathematical sciences : Vol. 153. Level set methods and dynamic implicit surfaces. New York: Springer.
Osher, S., & Rudin, L. I. (1990). Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis, 27, 919–940.
Osher, S., & Rudin, L. (1991). Shocks and other nonlinear filtering applied to image processing. In A. G. Tescher (Ed.), Proceedings of SPIE : Vol. 1567. Applications of digital image processing XIV (pp. 414–431). Bellingham: SPIE Press.
Osher, S., & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics, 79, 12–49.
Pizarro, L., Burgeth, B., Breuß, M., & Weickert, J. (2009). A directional Rouy-Tourin scheme for adaptive matrix-valued morphology. In M. H. F. Wilkinson & J. B. T. M. Roerdink (Eds.), Lecture notes in computer science : Vol. 5720. Proc. of the ninth international symposium on mathematical morphology, ISMM (pp. 250–260). Berlin: Springer.
Remaki, L., & Cheriet, M. (2003). Numerical schemes of shock filter models for image enhancement and restoration. Journal of Mathematical Imaging and Vision, 18(2), 153–160.
Rouy, E., & Tourin, A. (1992). A viscosity solutions approach to shape-from-shading. SIAM Journal on Numerical Analysis, 29, 867–884.
Sapiro, G., Kimmel, R., Shaked, D., Kimia, B. B., & Bruckstein, A. M. (1993). Implementing continuous-scale morphology via curve evolution. Pattern Recognition, 26, 1363–1372.
Schavemaker, J. G. M., Reinders, M. J. T., & van den Boomgaard, R. (1997). Image sharpening by morphological filtering. In Proc. 1997 IEEE workshop on nonlinear signal and image processing, Mackinac Island, MI, September 1997. www.ecn.purdue.edu/NSIP/.
Serra, J. (1967). Echantillonnage et estimation des phénomènes de transition minier. PhD thesis, University of Nancy, France.
Serra, J. (1982). Image analysis and mathematical morphology (Vol. 1). London: Academic Press.
Serra, J. (1988). Image analysis and mathematical morphology (Vol. 2). London: Academic Press.
Sethian, J. A. (1999). Level set methods and fast marching methods (2nd ed.). Cambridge: Cambridge University Press. Paperback edition.
Soille, P. (2003). Morphological image analysis (2nd ed.). Berlin: Springer.
van den Boomgaard, R. (1992). Mathematical morphology: Extensions towards computer vision. PhD thesis, University of Amsterdam, The Netherlands.
van den Boomgaard, R. (1999). Numerical solution schemes for continuous-scale morphology. In M. Nielsen, P. Johansen, O. F. Olsen, & J. Weickert (Eds.), Lecture notes in computer science : Vol. 1682. Scale-space theories in computer vision (pp. 199–210). Berlin: Springer.
van Vliet, L. J., Young, I. T., & Beckers, A. L. D. (1989). A nonlinear Laplace operator as edge detector in noisy images. Computer Vision, Graphics and Image Processing, 45(2), 167–195.
Weickert, J. (2003). Coherence-enhancing shock filters. In B. Michaelis & G. Krell (Eds.), Lecture notes in computer science : Vol. 2781. Pattern recognition (pp. 1–8). Berlin: Springer.
Weickert, J., & Brox, T. (2002). Diffusion and regularization of vector- and matrix-valued images. In M. Z. Nashed & O. Scherzer (Eds.), Contemporary mathematics : Vol. 313. Inverse problems, image analysis, and medical imaging (pp. 251–268). Providence: AMS.
Weickert, J., & Hagen, H. (Eds.) (2006). Visualization and processing of tensor fields. Berlin: Springer.
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Burgeth, B., Pizarro, L., Breuß, M. et al. Adaptive Continuous-Scale Morphology for Matrix Fields. Int J Comput Vis 92, 146–161 (2011). https://doi.org/10.1007/s11263-009-0311-4
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DOI: https://doi.org/10.1007/s11263-009-0311-4