Multiplicative calculus in biomedical image analysis

Journal of Mathematical Imaging and Vision

Volumn 42, Issue 1, 2012, Pages 64-75

  Florack, Luc ^a   Van Assen, Hans ^a  

^a EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Cardiac strain tensor analysis; Diffusion tensor imaging; Multiplicative calculus; Non Newtonian calculus; Positivity

Indexed keywords

CARDIAC STRAIN; DIFFUSION TENSOR IMAGING; MULTIPLICATIVE CALCULUS; NON-NEWTONIAN; POSITIVITY;

CALCULATIONS; DIFFERENTIATION (CALCULUS); IMAGE ANALYSIS; MAGNETIC RESONANCE IMAGING; NON NEWTONIAN FLOW;

TENSORS;

EID: 84856247738     PISSN: 09249907     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10851-011-0275-1     Document Type: Article

References (36)

1. V. Arsigny P. Fillard X. Pennec N. Ayache 2006 Log-Euclidean metrics for fast and simple calculus on diffusion tensors Magn. Reson. Med. 56 2 411 421 10.1002/mrm.20965 (Pubitemid 44162789)
2. Assen, H.V., Florack, L., Suinesiaputra, A., Westenberg, J., ter Haar Romeny, B.: Purely evidence based multiscale cardiac tracking using optic flow. In: Miller, K., Paulsen, K.D., Young, A.A., Nielsen, P.M.F. (eds.) Proceedings of the MICCAI Workshop on Computational Biomechanics for Medicine II, Brisbane, Australia, October 29, 2007, pp. 84-93 (2007)
3. Assen, H.C.V., Florack, L.M.J., Simonis, F.F.J., Westenberg, J.J.M., Strijkers, G.J.: Cardiac strain and rotation analysis using multi-scale optical flow. In: Wittek, A., Nielsen, P.M.F. (eds.) Proceedings of the MICCAI Workshop on Computational Biomechanics for Medicine V, Beijing, China, September 24, 2010, pp. 89-100 (2010)
4. A.E. Bashirov E.M. Kurpinar A. Özyapici 2008 Multiplicative calculus and its applications J. Math. Anal. Appl. 337 36 48 10.1016/j.jmaa.2007.03.081 1129.26007 2356052
5. Bellman, R.: Introduction to Matrix Analysis, 2nd edn. Classics in Applied Mathematics, vol. 19. SIAM, Philadelphia (1997)
6. A. Buchheim 1884 On the theory of matrices Proc. Lond. Math. Soc. 16 63 82 10.1112/plms/s1-16.1.63
7. A. Buchheim 1886 An extension of a theorem of professor Sylvester's relating to matrices Philos. Mag. 22 135 173 174
8. B. Burgeth S. Didas L. Florack J. Weickert 2007 A generic approach to diffusion filtering of matrix-fields Computing 81 2-3 179 197 10.1007/s00607-007-0248-9 1158.35373 2354195 (Pubitemid 350178352)
9. Fillard, P., Pennec, X., Arsigny, V., Ayache, N.: Clinical DT-MRI estimation, smoothing, and fiber tracking with log-Euclidean metrics. IEEE Trans. Med. Imag. 26(11) (2007)
10. Florack, L.M.J.: Image Structure, Computational Imaging and Vision Series, vol. 10. Kluwer Academic, Dordrecht (1997)
11. Florack, L.M.J., Astola, L.J.: A multi-resolution framework for diffusion tensor images. In: Fernández, S.A., de Luis Garcia, R. (eds.) CVPR Workshop on Tensors in Image Processing and Computer Vision, Anchorage, Alaska, USA, June 24-26, 2008. IEEE Press, New York (2008). Digital proceedings
12. L. Florack H. van Assen 2010 A new methodology for multiscale myocardial deformation and strain analysis based on tagging MRI Int. J. Biomed. Imaging 10.1155/2010/341242 URL http://www.hindawi.com/journals/ijbi/2010/341242.html
13. L.M.J. Florack B.M.T. Haar Romeny J.J. Koenderink M.A. Viergever 1992 Scale and the differential structure of images Image Vis. Comput. 10 6 376 388 10.1016/0262-8856(92)90024-W
14. L.M.J. Florack B.M.T. Haar Romeny J.J. Koenderink M.A. Viergever 1993 Cartesian differential invariants in scale-space J. Math. Imaging Vis. 3 4 327 348 10.1007/BF01664793
15. L.M.J. Florack B.M.T. Haar Romeny J.J. Koenderink M.A. Viergever 1994 General intensity transformations and differential invariants J. Math. Imaging Vis. 4 2 171 187 10.1007/BF01249895
16. L.M.J. Florack R. Maas W.J. Niessen 1999 Pseudo-linear scale-space theory Int. J. Comput. Vis. 31 2-3 247 259 10.1023/A:1008026217765
17. Florack, L., van Assen, H., Suinesiaputra, A.: Dense multiscale motion extraction from cardiac cine MR tagging using HARP technology. In: Niessen, W., Westin, C.F., Nielsen, M. (eds.) Proceedings of the 8th IEEE Computer Society Workshop on Mathematical Methods in Biomedical Image Analysis, Held in Conjunction with the IEEE International Conference on Computer Vision, Rio de Janeiro, Brazil, October 14-20, 2007 (2007). Digital proceedings by Omnipress
18. T.C. Fung 2004 Computation of the matrix exponential and its derivatives by scaling and squaring Int. J. Numer. Methods Eng. 59 1273 1286 10.1002/nme.909 1048.65040 2036386 (Pubitemid 38335657)
19. Gantmacher, F.R.: The Theory of Matrices. American Mathematical Society, Providence (2001)
20. R.D. Gill S. Johansen 1990 A survey of product-integration with a view toward application in survival analysis Ann. Stat. 18 1501 1555 10.1214/aos/1176347865 0718.60087 1074422
21. Grossman, M., Katz, R.: Non-Newtonian Calculus. Lee Press, Pigeon Cove (1972)
22. R.A. Guenther 1983 Product integrals and sum integrals Int. J. Math. Educ. Sci. Technol. 14 2 243 249 10.1080/0020739830140215 0523.26008 699203
23. Haupt, P.: Continuum Mechanics and Theory of Materials. Springer, Berlin (2002)
24. Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)
25. Lenglet, C., Deriche, R., Faugeras, O.: Inferring white matter geometry from diffusion tensor MRI: Application to connectivity mapping. In: Pajdla, T., Matas, J. (eds.) Proceedings of the Eighth European Conference on Computer Vision, Prague, Czech Republic, May 2004. Lecture Notes in Computer Science, vol. 3021-3024, pp. 127-140. Springer, Berlin (2004)
26. Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, Mineola (1994)
27. C. Moler C. Van Loan 1978 Nineteen dubious ways to compute the exponential of a matrix SIAM Rev. 20 4 801 836 10.1137/1020098 0395.65012 508383
28. X. Pennec P. Fillard N. Ayache 2006 A Riemannian framework for tensor computing Int. J. Comput. Vis. 66 1 41 66 10.1007/s11263-005-3222-z 2288028 (Pubitemid 43004671)
29. Prados, E., Soatto, S., Lenglet, C., Pons, J.P., Wotawa, N., Deriche, R., Faugeras, O.: Control theory and fast marching techniques for brain connectivity mapping. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York, USA, June 2006, vol. 1, pp. 1076-1083. IEEE Computer Society, Los Alamitos (2006) (Pubitemid 44931441)
30. A.K. Rutz S. Ryf S. Plein P. Boesiger S. Kozerke 2008 Accelerated whole-heart 3D CSPAMM for myocardial motion quantification Magn. Reson. Med. 59 4 755 763 10.1002/mrm.21363 (Pubitemid 351501940)
31. M. Rybaczuk A. Kȩdzia W. Zieliński 2001 The concept of physical and fractal dimension II. the differential calculus in dimensional spaces Chaos Solitons Fractals 12 2537 2552 10.1016/S0960-0779(00)00231-9 0994.28003 1851077 (Pubitemid 32594576)
32. S. Ryf M.A. Spiegel M.P.B. Gerber 2002 Myocardial tagging with 3D-CSPAMM J. Magn. Reson. Imaging 16 3 320 325 10.1002/jmri.10145 (Pubitemid 34920432)
33. Slavík, A.: Product Integration, Its History and Applications. Matfyzpress, Prague (2007)
34. D. Stanley 1999 A multiplicative calculus PRIMUS, Probl. Resour. Issues Math. Undergrad. Stud. IX 4 310 326 1539021
35. V. Volterra 1887 Sulle equazioni differenziali lineari Rend. Acad. Lincei, Ser. 4 3 393 396
36. E.A. Zerhouni D.M. Parish W.J. Rogers A. Yang E.P. Shapiro 1988 Human heart: tagging with MR imaging-a method for noninvasive assessment of myocardial motion Radiology 169 1 59 63