Abstract
Estimation of local spatial structure has a long history and numerous analysis tools have been developed. A concept that is widely recognized as fundamental in the analysis is the structure tensor. However, precisely what it is taken to mean varies within the research community. We present a new method for structure tensor estimation which is a generalization of many of it’s predecessors. The method uses filter sets having Fourier directional responses being monomials of the normalized frequency vector, one odd order sub-set and one even order sub-set. It is shown that such filter sets allow for a particularly simple way of attaining phase invariant, positive semi-definite, local structure tensor estimates. We continue to compare a number of known structure tensor algorithms by formulating them in monomial filter set terms. In conclusion we show how higher order tensors can be estimated using a generalization of the same simple formulation.
Chapter PDF
Similar content being viewed by others
References
Riesz, M.: Sur les fonctions conjuge’es. Math. Zeit. 27, 218–244 (1927)
Zernike, F.: Diffraction theory of the cut procedure and its improved form, the phase contrast method. Physica 1, 689–704 (1934)
Gabor, D.: Theory of communication. J. Inst. Elec. Eng. 93(26), 429–457 (1946)
Hu, M.K.: Visual pattern recognition by moment invariants. IRE Transactions on Information Theory, IT-8(2), 179–187 (1962)
Roberts, L.G.: Machine Perception of three-dimensional Solid. In: Tippell, J.T. (ed.) Optical and Electro-Optical Information Processing, pp. 159–197. MIT Press, Cambridge (1965)
Granlund, G.H.: In search of a general picture processing operator. Computer Graphics and Image Processing 8(2), 155–178 (1978)
Danielsson, P.E.: Rotation invariant operators with directional response. In: Proceedings 5’th Int. Conf. on Pattern Recognition, Miami Beach, Florida (1980)
Knutsson, H., Wilson, R.G., Granlund, G.H.: Anisotropic filtering operations for image enhancement and their relation to the visual system. In: IEEE Computer Society Conference on Pattern Recognition and Image Processing, Dallas, Texas (August 1981)
Knutsson, H.: Filtering and Reconstruction in Image Processing. PhD thesis, Linköping University, Sweden, Diss. No. 88 (1982)
Knutsson, H., Granlund, G.H.: Texture analysis using two-dimensional quadrature filters. In: IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management - CAPAIDM, Pasadena (October 1983)
Knutsson, H.: Producing a continuous and distance preserving 5-D vector representation of 3-D orientation. In: IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management - CAPAIDM, pp. 175–182, Miami Beach, Florida, November 1985. IEEE. Report LiTH–ISY–I–0843, Linköping University, Sweden (1986)
Knutsson, H.: A tensor representation of 3-D structures. In: 5th IEEE-ASSP and EURASIP Workshop on Multidimensional Signal Processing, Noordwijkerhout, The Netherlands (September 1987), poster presentation
Bigün, J., Granlund, G.H.: Optimal orientation detection of linear symmetry. In: IEEE First International Conference on Computer Vision, London, Great Britain, pp. 433–438 (June 1987)
Lenz, R.: Rotation-invariant operators and scale space filtering. Pattern Recognition Letters 6, 151–154 (1987)
Forstner, W., Gulch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data, pp. 281–305 (1987)
Knutsson, H.: Representing local structure using tensors. In: The 6th Scandinavian Conference on Image Analysis, Oulu, Finland, pp. 244–251, (June 1989); Report LiTH–ISY–I–1019, Computer Vision Laboratory, Linköping University, Sweden
Knutsson, H., Bårman, H., Haglund, L.: Robust orientation estimation in 2D, 3D and 4D using tensors. In: Proceedings of Second International Conference on Automation, Robotics and Computer Vision, ICARCV 1992, Singapore (September 1992)
Granlund, G.H., Knutsson, H.: Signal Processing for Computer Vision. Kluwer Academic Publishers, Dordrecht (1995) ISBN 0-7923-9530-1
Farnebäck, G.: Fast and accurate motion estimation using orientation tensors and parametric motion models. In: Proceedings of 15th International Conference on Pattern Recognition, vol. 1, pp. 135–139. IAPR, Barcelona (2000)
Felsberg, M., Sommer, G.: The monogenic signal. IEEE Transactions on Signal Processing 49(12), 3136–3144 (2001)
Johansson, B., Farnebäck, G.: A theoretical comparison of different orientation tensors. In: Proceedings SSAB 2002 Symposium on Image Analysis, pp. 69–73. SSAB, Lund (2002)
Knutsson, H., Andersson, M.: Loglets: Generalized quadrature and phase for local spatio-temporal structure estimation. In: Proceedings of the Scandinavian Conference on Image Analysis (SCIA) (June 2003)
Köthe, U.: Inegrated edge and junction detection with the boundary tensor. In: Proceedings of Ninth IEEE International Conference on Computer Vision, ICCV (2003)
Knutsson, H., Andersson, M.: Implications of invariance and uncertainty for local structure analysis filter sets. Signal Processing: Image Communications 20(6), 569–581 (2005)
Nordberg, K.: A fourth order tensor for representation of orientation and position of oriented segments. Other academic, Linköping University, Department of Electrical Engineering, Sweden, diva2:288343 (2004)
Nordberg, K., Farnebäck, G.: Estimation of orientation tensors for simple signals by means of second-order filters. Signal Processing: Image Communication 20(6), 582–594 (2005)
Köthe, U., Felsberg, M.: Riesz-transforms versus derivatives: On the relationship between the boundary tensor and the energy tensor. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 179–191. Springer, Heidelberg (2005)
Felsberg, M., Jonsson, E.: Energy tensors: Quadratic, phase invariant image operators. In: Kropatsch, W.G., Sablatnig, R., Hanbury, A. (eds.) DAGM 2005. LNCS, vol. 3663, pp. 493–500. Springer, Heidelberg (2005)
Felsberg, M., Köthe, U.: GET: The Connection Between Monogenic Scale-Space and Gaussian Derivatives. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 192–203. Springer, Heidelberg (2005)
Herberthson, M., Brun, A., Knutsson, H.: Representing pairs of orientations in the plane. In: Ersbøll, B.K., Pedersen, K.S. (eds.) SCIA 2007. LNCS, vol. 4522, pp. 661–670. Springer, Heidelberg (2007)
Barmpoutis, A., Vemuri, B.C., Forder, J.R.: Registration of high angular resolution diffusion MRI images using 4 order tensors. In: Ayache, N., Ourselin, S., Maeder, A. (eds.) MICCAI 2007, Part I. LNCS, vol. 4791, pp. 908–915. Springer, Heidelberg (2007)
Wang, Q., Ronneberger, O., Burkhardt, H.: Fourier analysis in polar and spherical coordinates. Technical Report Internal Report 1/08, IIF-LMB, Computer Science Department, University of Freiburg (2008)
Westin, C.-F., Knutsson, H.: Representation and Estimation of Tensors-Pairs. In: Visualization and Processing of Tensor Fields: Proceedings of the Dagstuhl Workshop (2010) submitted
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Knutsson, H., Westin, CF., Andersson, M. (2011). Representing Local Structure Using Tensors II. In: Heyden, A., Kahl, F. (eds) Image Analysis. SCIA 2011. Lecture Notes in Computer Science, vol 6688. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21227-7_51
Download citation
DOI: https://doi.org/10.1007/978-3-642-21227-7_51
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21226-0
Online ISBN: 978-3-642-21227-7
eBook Packages: Computer ScienceComputer Science (R0)