Abstract
Tensors have been broadly used in mathematics and physics, since they are a generalization of scalars or vectors and allow to represent more complex properties. In this chapter we present an overview of some tensor applications, especially those focused on the image processing field. From a mathematical point of view, a lot of work has been developed about tensor calculus, which obviously is more complex than scalar or vectorial calculus. Moreover, tensors can represent the metric of a vector space, which is very useful in the field of differential geometry. In physics, tensors have been used to describe several magnitudes, such as the strain or stress of materials. In solid mechanics, tensors are used to define the generalized Hooke’s law, where a fourth order tensor relates the strain and stress tensors. In fluid dynamics, the velocity gradient tensor provides information about the vorticity and the strain of the fluids. Also an electromagnetic tensor is defined, that simplifies the notation of the Maxwell equations. But tensors are not constrained to physics and mathematics. They have been used, for instance, in medical imaging, where we can highlight two applications: the diffusion tensor image, which represents how molecules diffuse inside the tissues and is broadly used for brain imaging; and the tensorial elastography, which computes the strain and vorticity tensor to analyze the tissues properties. Tensors have also been used in computer vision to provide information about the local structure or to define anisotropic image filters.
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References
M. Raffel, C. Willert, and J. Kompenhans, Particle Image Velocimetry. A Practical Guide, Springer Verlag, 1998.
J.C.R. Hunt, “Vorticity and vortex dynamics in complex turbulent flows,” in Transactions Canadian Society for Mechanical Engineering (ISSN 0315-8977), 1987, vol. 11, pp. 21–35.
R. Haimes and D. Kenwright, “The velocity gradient tensor and fluid feature extraction,” in Proc. AIAA 14th Computational Fluid Dynamics Conference, 1999.
P. Basser, J. Mattiello, and D. Le Bihan, “MR diffusion tensor spectroscopy and imaging,” Biophysical Journal, vol. 66, pp. 259–267, 1994.
P.C. Sundgren, Q. Dong, D. Gómez-Hassan, S.K. Mukherji, P. Maly, and Welsh R., “Diffusion tensor imaging of the brain: review of clinical applications,” Neuroradiology, vol. 46, pp. 339–350, 2004.
A. Einsten, “Ber die von der molekularkinetischen theorie der wärme gefordete bewegung von in ruhenden flüssigkeiten suspendierten teilchen,” Annalen der Physik, , vol. 17, pp. 549–560, 1905.
E.O. Stejskal and T. E. Tanner, “Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient,” Journal of Chemical Physics, no. 42, pp. 288–292, 1965.
D LeBihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, and M. Laval-Jeantet, “MR imaging of intravocel incoherents motions: application to diffusion and perfusion in neurological disorders,” Radiology, vol. 161, pp. 401–407, 1986.
C. Pierpaoli and P.J. Basser, “Toward a quantitative assesment of diffusion anisotropy,” Magnetic Resonance in Medicine, vol. 36.
C.F. Westin, S. E. Maier, H. Mamata, F.A. Jolesz, and R. Kikinis, “Processing and visualization for diffusion tensor MRI,” Medical Image Analysis, vol. 6, pp. 93–108, 2002.
J. Weickert and H. Hagen, Eds., Visualization and Processing of Tensor Fields, part II. Diffusion Tensor Imaging, pp. 81–187, Springer, 2006.
Kindlmann G., “Superquadrics tensor glyphs,” in Proceedings IEEE TVCG/EG Symposium on Visualization, 2004, May 2004.
C.R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc., vol. 37, pp. 81–91, 1945.
J. Burbea and C.R. Rao, “Entropy differential metric, distance and divergence measures in probability spaces: A unified approach,” J. Multivariate Anal., vol. 12, pp. 575–596, 1982.
L.T. Skovgaard, “A Riemannian geometry of the multivariate normal model,” Tech. Rep. 81/3, Statistical Research Unit, Danish Medical Research Council, Danish Social Science Research Council, 1981.
M. Moakher, “A differential geometric approach to the geometric mean of symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl., vol. 26, no. 3, pp. 735–747, 2005.
C. Atkinson and A.F.S. Mitchell, “Rao’s distance measure,” Sankhya: The Indian Journal of Stats., vol. 43, no. A, pp. 345–365, 1981.
W. Förstner and B. Moonen, “A metric for covariance matrices,” Tech. Rep., Stuttgart University, Dept. of Geodesy and Geoinformatics, 1999.
D.F. Scollan, A. Holmes, R. Winslow, and J. Forder, “Histological validation of myocardial microstructure obtainde form diffusion tensor magnetic resonance imaging,” American Journal of Physiology, no. 275, pp. 2308–2318, 1998.
E. W. Hsu and Setton L. A., “Diffusion tensor microscopy of the intervertebral disc annulus fibrosus,” Magnetic Resonance in Medicine, no. 41, pp. 992–999, 1999.
P. Hagmann, L. Jonasson, P. Maeder, J. Thiran, V. Wedeen, and R. Meuli, ,“Understanding diffusion MRI imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond,” Radiographics, pp. S205–S223.
H. Jiang, P.C. Van Zijl, J. Kim, G.D. Pearlson, and S. Mori, “DTIstudio: Resource program for diffusion tensor computation and fiber bundle tracking,” Comput. Methods Programs Biomed., vol. 81, no. 2, pp. 106–116, 2006.
S. Mori and P.C.M. van Zijl, “Fiber tracking: principles and strategies -a technical review,” NMR in Biomedicine, vol. 15, no. 7-8, pp. 468–480, 2002.
T. E. Conturo, N. F. Lori, T. S. Cull, E. Akbudak, A. Z. Snyder, J. S. Shimony, R. C. Mckinstry, H. Burton, and M. E. Raichle, “Tracking neuronal fiber pathways in the living human brain,” in Proc. Natl. Acad. Sci. USA, August 1999, pp. 10422–10427.
P.J. Basser, S. Pajevic, C. Pierpaoli, J. Duda, and A. Aldroubi, “In vivo fiber tractography using DT-MRI data,” Mag. Res. in Med., vol. 44, pp. 625–632, 2000.
M. Lazar, D.M.Weinstein, J.S.Tsuruda, K.M.Hasan, K. Arfanakis, M.E. Meyerand, B. Badie, H.A.Rowley, V.Haughton, A. Field, and A. L.Alexander, “White matter tractography using diffusion tensor deflection,” Human Brain Mapping, vol. 18, pp. 306–321, 2003.
P. Hagmann, J.-P. Thiran, L. Jonasson, P. Vandergheynst, S. Clarke, P. Maeder, and R. Meulib, “DTI mapping of human brain connectivity: Statistical fibre tracking and virtual dissection,” NeuroImage, vol. 19, pp. 545–554, 2003.
S. Mori, B.J. Crain, V.P. Chacko, and P.C. van Zijl., “Three dimensional tracking of axonal projections in th brain by magnetic resonance imaging,” Ann. Neurol., vol. 45, no. 2, pp. 265–269, 1999.
C. Pierpaoli, P. Jezzard, PJ. Basser, A. Barnett, and G. Di Chiro, “Diffusion tensor MR imaging of the human brain,” Radiology, vol. 201.
E. von dem Hagen and R. Henkelman, “Orientational diffusion reflects fiber structure within a voxel,” Magn. Reson. Med., vol. 48.
D. Jones, “Determining and visualizing uncertainty in estimates of fiber orientation from diffusion tensor MRI,” Magn. Reson. Med., vol. 49.
D. Jones, A. Simmons, S. Williams, and M. Horsfield, ,” .
S. Mori, B. Crain, V. Chacko, and P. Van Zijl, “Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging,” Ann. Neurol., vol. 45.
C. Tench, P. Morgan, M. Wilson, and L. Blumhardt, “White matter mapping using diffusion tensor MRI,” Magn. Reson. Med., vol. 47.
J. Ophir, I. Céspedes, B. Garra, H. Ponnekanti, Y. Huang, and N. Maklad, “Elastography: A quantitative method for imaging the elasticity of biological tissues,” Ultrasound Imaging, vol. 13, pp. 111-134, 1991.
B.S. Garra, I. Céspedes, J. Ophir, S. Spratt, R. A. Zuurbier, C. M. Magnant, and M. F. Pennanen, “Elastography of breast lesions: initial clinical results,” Radiology, vol. 202, pp. 79-86, 1997.
R. L Maurice, M. Daronat, J. Ohayon, E. Stoyanova1, F. S. Foster, and G. Cloutier, “Non-invasive high-frequency vascular ultrasound elastography,” Phys. Med. Biol., no. 50, pp. 1611–1628, 2005.
D. Sosa-Cabrera, M.A. Rodriguez-Florido, E. Suarez-Santana, and J. Ruiz-Alzola, Tensor Elastography: A New Approach for Visualizing the Elastic Properties of the Tissue, 2006.
A. Neeman, B. Jeremic, and A. Pang, “Visualizing tensor fields in geomechanics,” in IEEE Visualization, 2005, pp. 329–343.
B. Wuensche, “The visualization of 3d stress and strain tensor fields,” in Proc. of the 3rd New Zealand Comp. Science Research Student Conf., Apr. 1999, vol. 3, pp. 109–116.
P. Selskog, E. Heiberg, T. Ebbers, L. Wigstrom, and M. Karlsson, “Kinematics of the heart: strain-rate imaging from time-resolved three-dimensional phase contrast MRI,” IEEE Trans. Med. Imaging, vol. 21, no. 1, pp. 1105-1109, 2002.
D. Sosa-Cabrera, Novel Processing Schemes and Visualization Methods for Elasticity Imaging, Phd dissertation, University of Las Palmas de GC, 2008.
M.C. Morrone and R.A. Owens, “Feature detection from local energy,” Pattern Recognition Letters, vol. 6, pp. 303–313, 1987.
H. Knutsson, “Representing local structure using tensors,” in 6th Scandinavian Conference on Image Analysis. Oulu, Finland, 1989, pp. 244–251.
U. Köthe, “Integrated edge and junction detection with the boundary tensor,” in 9th Intl. Conf. on Computer Vision, Nice, 2003, IEEE Computer Society, vol. 1, pp. 424–431.
C.-F. Westin, S.E. Maier, H. Mamata, A. Nabavi, F.-A. Jolesz, and R. Kikinis, “Processing and visualization for diffusion tensor MRI,” Medical Image Analysis, vol. 6, no. 2, pp. 93–108, Jun. 2002.
J. Bigün and G.-H. Granlund, “Optimal orientation detection of linear symmetry.,” in IEEE First International Conference on Computer Vision, London, UK, Jun. 1987, pp. 433–438.
W. Förstner and E. Gülch, “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, Interlaken, Switzerland, June 1987, pp. 281–305.
M. Kass and A. Witkin, “Analyzing oriented patterns,” Computer Vision Graphics and Image Processing, vol. 37, pp. 362–385, 1987.
R. San Jose, Local Structure Tensor for Multidimensional Signal Processing. Applications to Medical Image Analysis, Ph.D. thesis, Universidad de Valladolid, E.T.S. Ingenieros de Telecomunicación, February 2005.
A.-R. Rao and B.-G. Schunck, “Computing oriented texture fields”, CVGIP: Graphical Models and Image Processing, vol. 53, no. 2, pp. 157–185, 1991.
R. Mester and M. Mühlich, “Improving motion and orientation estimation using an equilibrated total least squares approach,” in Proc. IEEE International Conference on Image Processing (ICIP 2001), Thessaloniki, Greece, October 2001, pp. 640–643.
H. Knutsson and M. Andersson, “What’s so good about quadrature filters?”, in 2003 IEEE International Conference on Image Processing, 2003.
H. Knutsson and M. Andersson, “Loglets: Generalized quadrature and phase for local spatio-temporal structure estimation,” in Proc. of SCIA03. LNCS., J. Bigün and T. Gustavsson, Eds., 2003, vol. 2749, pp. 741–748, Springer.
B. Rieger and L. Vlieet, “A systematic approach to nd orientation representation,” Image and Vision Computing, vol. 22, no. 6, pp. 453–459, 2004.
J. Ruiz-Alzola, R. Kikinis, and C.-F. Westin, “Detection of point landmarks in multidimensional tensor data,” Signal Processing, vol. 81, no. 10, Oct. 2001.
G.-H. Granlund and H. Knutsson, Signal Processing for Computer Vision, Kluwer Academic Publishers, 1995.
T. Brox, J. Weickert, B. Burgeth, and P. Mrázek, “Nonlinear structure tensors,” Image and Vision Computing, vol. 24, no. 1, pp. 41–55, January 2006.
J. Weickert and T. Brox, “Diffusion and regularization of vector- and matrix-valued images,” Inverse Problems, Image Analysis and Medical Imaging. Contemporary Mathematics, vol. 313, pp. 251–268, 2002.
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Applied Mathematical Sciences, vol. 147. Springer-Verlang, 2002.
J. Weickert, A Review of Nonlinear Diffusion Filtering, vol. 1252 of Lecture Notes in Computer Science - Scale Space Theory in Computer Science, Springer, Berlin, 1997.
J. Weickert, “Coherence enhancing diffusion filtering,” Int. J. of Computer Vision, vol. 31, pp. 111–127, 1999.
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Cammoun, L. et al. (2009). A Review of Tensors and Tensor Signal Processing. In: Aja-Fernández, S., de Luis García, R., Tao, D., Li, X. (eds) Tensors in Image Processing and Computer Vision. Advances in Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-84882-299-3_1
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