Summary
Structure tensors are a common tool for orientation estimation in image processing and computer vision. We present a generalization of the traditional second-order model to a higher-order structure tensor (HOST), which is able to model more than one significant orientation, as found in corners, junctions, and multichannel images. We provide a theoretical analysis and a number of mathematical tools that facilitate practical use of the HOST, visualize it using a novel glyph for higher-order tensors, and demonstrate how it can be applied in an improved integrated edge, corner, and junction detector.
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
Förstner, W., Gülch, E. A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: International Society for Photogrammetry and Remote Sensing (ISPRS) Intercomission Conference on Fast Processing of Photogrammetric Data, Interlaken (1987) pp. 281–305
Bigün, J., Granlund, G., Wiklund, J. Multidimensional orientation estimation with applications to texture analysis and optical fbw. IEEE Transactions on Pattern Analysis and Machine Intelligence 13 (1991) 775–790
Weickert, J. Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)
Rousson, M., Brox, T., Deriche, R. Active unsupervised texture segmentation on a diffusion based feature space. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Madison, Wisconsin, USA (2003) pp. 699–706
Tschumperlé, D., Deriche, R. Vector-valued image regularization with PDE's: A common framework for different applications. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2003). (2003) pp. 651–656
Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.P. Image compression with anisotropic diffusion. Journal of Mathematical Imaging and Vision 31 (2008) 255–269
Di Zenzo, S. A note on the gradient of a multi-image. Computer Vision, Graphics, and Image Processing 33 (1986) 116–125
Bigün, J., Bigün, T., Nilsson, K. Recognition by symmetry derivatives and the generalized structure tensor. IEEE Transactions on Pattern Analysis and Machine Intelligence 26 (2004) 1590–1605
Weickert, J., Brox, T. Diffusion and regularization of vector- and matrix-valued images. In: Nashed, M., Scherzer, O., eds., Inverse Problems, Image Analysis, and Medical Imaging. Volume 313 of Contemporary Mathematics. AMS, Providence (2002) pp. 251–268
Brox, T., Weickert, J., Burgeth, B., Mrázek, P. Nonlinear structure tensors. Image and Vision Computing 24 (2006) 41–55
Arseneau, S., Cooperstock, J.R. An improved representation of junctions through asymmetric tensor diffusion. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Nefian, A.V., Gopi, M., Pascucci, V., Zara, J., Molineros, J., Theisel, H., Malzbender, T., eds., Advances in Visual Computing. Volume 4291 of Lecture Notes in Computer Science, Springer (2006) pp. 363–372
Herberthson, M., Brun, A., Knutsson, H. Pairs of orientations in the plane. In: Proceedings of the SSBA Symposium on Image Analysis, Umeå, Sweden, SSBA (2006) pp. 97–100
Brox, T., Weickert, J. A TV fbw based local scale measure for texture discrimination. In: Pajdla, T., Matas, J., eds., Proc. 8th European Conference on Computer Vision (ECCV'04). Volume 3022 of LNCS., Springer (2005) pp. 27–34
Özarslan, E., Mareci, T. Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magnetic Resonance in Medicine 50 (2003) 955–965
Hlawitschka, M., Scheuermann, G. HOT-lines: Tracking lines in higher order tensor fields. In: Silva, C, Gröller, E., Rushmeier, H., eds., Proceedings of IEEE Visualization 2005, Minneapolis, MN, USA (2005) pp. 27–34
Feddern, C., Weickert, J., Burgeth, B. Level-set methods for tensor-valued images. In: Faugeras, O.D., Paragios, N., eds., Proc. Second IEEE Workshop on Geometric and Level Set Methods in Computer Vision, Nice, France (2003) pp. 65–72
Özarslan, E., Vemuri, B.C., Mareci, T.H. Generalized scalar measures for diffusion MRI using trace, variance, and entropy. Magnetic Resonance in Medicine 53 (2005) 866–876
Köthe, U. Edge and junction detection with an improved structure tensor. In: Michaelis, B., Krell, G., eds., Pattern Recognition. 25th DAGM Symposium. Volume 2781 of Lecture Notes in Computer Science, Springer (2003) pp. 25–32
Hitchcock F.L. The expression of a tensor or a polyadic as a sum of products. Journal of Mathematics and Physics 6 (1927) 164–189
Hitchcock F.L. Multiple invariants and generalized rank of a p-way matrix or tensor. Journal of Mathematics and Physics 7 (1927) 39–79
Comon P., Mourrain B. Decomposition of quantics in sums of powers of linear forms. Signal Processing 53 (1996) 96–107
Comon, P., Golub, G., Lim, L.H., Mourrain, B. Symmetric tensors and symmetric tensor rank. Technical Report SCCM-06-02, Stanford Scientific Computing and Computational Mathematics (SCCM) (2006)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P. Numerical Recipes in C++: The Art of Scientific Computing, Second Edition. Cambridge University Press (2002)
Gentleman W. An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients. The Computer Journal 12 (1969) 160–164
Newbery A. Error analysis for Fourier series evaluation. Mathematics of Computation 27 (1973) 639–644
Acknowledgments
We thank Holger Theisel, who is with the University of Magdeburg, for discussions at all stages of this project. Discussions with Torsten Langer, who is with the MPI Informatik, helped in developing parts of the “mathematical toolbox” in Sect. 4.
Our implementation uses the CImg library by David Tschumperlé, available from http://cimg.sf.net/.
This research has partially been funded by the Max Planck Center for Visual Computing and Communication (MPC-VCC).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Schultz, T., Weickert, J., Seidel, HP. (2009). A Higher-Order Structure Tensor. In: Laidlaw, D., Weickert, J. (eds) Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88378-4_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-88378-4_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-88377-7
Online ISBN: 978-3-540-88378-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)