Journal of Mathematical Imaging and Vision

, Volume 42, Issue 1, pp 64–75 | Cite as

Multiplicative Calculus in Biomedical Image Analysis

  • Luc FlorackEmail author
  • Hans van Assen
Open Access


We advocate the use of an alternative calculus in biomedical image analysis, known as multiplicative (a.k.a. non-Newtonian) calculus. It provides a natural framework in problems in which positive images or positive definite matrix fields and positivity preserving operators are of interest. Indeed, its merit lies in the fact that preservation of positivity under basic but important operations, such as differentiation, is manifest. In the case of positive scalar functions, or in general any set of positive definite functions with a commutative codomain, it is a convenient, albeit arguably redundant framework. However, in the increasingly important non-commutative case, such as encountered in diffusion tensor imaging and strain tensor analysis, multiplicative calculus complements standard calculus in a truly nontrivial way. The purpose of this article is to provide a condensed review of multiplicative calculus and to illustrate its potential use in biomedical image analysis.


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


  1. 1.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56(2), 411–421 (2006) CrossRefGoogle Scholar
  2. 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) Google Scholar
  3. 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) Google Scholar
  4. 4.
    Bashirov, A.E., Kurpinar, E.M., Özyapici, A.: Multiplicative calculus and its applications. J. Math. Anal. Appl. 337, 36–48 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bellman, R.: Introduction to Matrix Analysis, 2nd edn. Classics in Applied Mathematics, vol. 19. SIAM, Philadelphia (1997) CrossRefzbMATHGoogle Scholar
  6. 6.
    Buchheim, A.: On the theory of matrices. Proc. Lond. Math. Soc. 16, 63–82 (1884) CrossRefGoogle Scholar
  7. 7.
    Buchheim, A.: An extension of a theorem of professor Sylvester’s relating to matrices. Philos. Mag. 22(135), 173–174 (1886) Google Scholar
  8. 8.
    Burgeth, B., Didas, S., Florack, L., Weickert, J.: A generic approach to diffusion filtering of matrix-fields. Computing 81(2–3), 179–197 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 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) Google Scholar
  10. 10.
    Florack, L.M.J.: Image Structure, Computational Imaging and Vision Series, vol. 10. Kluwer Academic, Dordrecht (1997) Google Scholar
  11. 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 Google Scholar
  12. 12.
    Florack, L., van Assen, H.: A new methodology for multiscale myocardial deformation and strain analysis based on tagging MRI. Int. J. Biomed. Imaging (2010). doi: 10.1155/2010/341242. URL Google Scholar
  13. 13.
    Florack, L.M.J., Haar Romeny, B.M.T., Koenderink, J.J., Viergever, M.A.: Scale and the differential structure of images. Image Vis. Comput. 10(6), 376–388 (1992) CrossRefGoogle Scholar
  14. 14.
    Florack, L.M.J., Haar Romeny, B.M.T., Koenderink, J.J., Viergever, M.A.: Cartesian differential invariants in scale-space. J. Math. Imaging Vis. 3(4), 327–348 (1993) CrossRefGoogle Scholar
  15. 15.
    Florack, L.M.J., Haar Romeny, B.M.T., Koenderink, J.J., Viergever, M.A.: General intensity transformations and differential invariants. J. Math. Imaging Vis. 4(2), 171–187 (1994) CrossRefGoogle Scholar
  16. 16.
    Florack, L.M.J., Maas, R., Niessen, W.J.: Pseudo-linear scale-space theory. Int. J. Comput. Vis. 31(2–3), 247–259 (1999) CrossRefGoogle Scholar
  17. 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 Google Scholar
  18. 18.
    Fung, T.C.: Computation of the matrix exponential and its derivatives by scaling and squaring. Int. J. Numer. Methods Eng. 59, 1273–1286 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gantmacher, F.R.: The Theory of Matrices. American Mathematical Society, Providence (2001) Google Scholar
  20. 20.
    Gill, R.D., Johansen, S.: A survey of product-integration with a view toward application in survival analysis. Ann. Stat. 18, 1501–1555 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Grossman, M., Katz, R.: Non-Newtonian Calculus. Lee Press, Pigeon Cove (1972) zbMATHGoogle Scholar
  22. 22.
    Guenther, R.A.: Product integrals and sum integrals. Int. J. Math. Educ. Sci. Technol. 14(2), 243–249 (1983) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Haupt, P.: Continuum Mechanics and Theory of Materials. Springer, Berlin (2002) zbMATHGoogle Scholar
  24. 24.
    Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008) CrossRefzbMATHGoogle Scholar
  25. 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) Google Scholar
  26. 26.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, Mineola (1994) Google Scholar
  27. 27.
    Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20(4), 801–836 (1978) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006) CrossRefMathSciNetGoogle Scholar
  29. 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) Google Scholar
  30. 30.
    Rutz, A.K., Ryf, S., Plein, S., Boesiger, P., Kozerke, S.: Accelerated whole-heart 3D CSPAMM for myocardial motion quantification. Magn. Reson. Med. 59(4), 755–763 (2008) CrossRefGoogle Scholar
  31. 31.
    Rybaczuk, M., Kȩdzia, A., Zieliński, W.: The concept of physical and fractal dimension II. the differential calculus in dimensional spaces. Chaos Solitons Fractals 12, 2537–2552 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Ryf, S., Spiegel, M.A., Gerber, M.P.B.: Myocardial tagging with 3D–CSPAMM. J. Magn. Reson. Imaging 16(3), 320–325 (2002) CrossRefGoogle Scholar
  33. 33.
    Slavík, A.: Product Integration, Its History and Applications. Matfyzpress, Prague (2007) zbMATHGoogle Scholar
  34. 34.
    Stanley, D.: A multiplicative calculus. PRIMUS, Probl. Resour. Issues Math. Undergrad. Stud. IX(4), 310–326 (1999) MathSciNetGoogle Scholar
  35. 35.
    Volterra, V.: Sulle equazioni differenziali lineari. Rend. Acad. Lincei, Ser. 4 3, 393–396 (1887) Google Scholar
  36. 36.
    Zerhouni, E.A., Parish, D.M., Rogers, W.J., Yang, A., Shapiro, E.P.: Human heart: tagging with MR imaging—a method for noninvasive assessment of myocardial motion. Radiology 169(1), 59–63 (1988) Google Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of Mathematics & Computer Science and Department of Biomedical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Department of Biomedical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations