Abstract
In this paper the Mojette transforms class is described. After recalling the birth of the Mojette transform, the Dirac Mojette transform is recalled with its basic properties. Generalizations to spline transform and to nD Mojette transform are also recalled. Applications of the Mojette transform demonstrate the power of frame description instead of basis in order to match different goals ranging from image coding, watermarking, discrete tomography, transmission and distributed storage. Finally, new insights for the future trends of the Mojette transform are sketched.
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Keywords
- Mathematical Morphology
- Projection Angle
- Multiple Description
- Multiple Description Code
- Discrete Tomography
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References
Aldroubi, A., Unser, M.: Families of wavelet transforms in connection with Shannon’s sampling theory and the Gabor transform. In: Chui, C.K. (ed.) Wavelets: A Tutorial in Theory and Applications, pp. 509–528. Academic Press, London (1992)
Del Lungo, A., Nivat, M., Pinzani, R.: The number of convex polyominoes reconstructible from their orthogonal projections. Discrete Math. 157(1-3), 65–78 (1996)
Dorst, L., van den Boomgaard, R.: Morphological signal processing and the slope transform. invited paper for Signal Processing 38, 79–98 (1994)
Dudgeon, D.E., Mersereau, R.M.: Multidimensional Digital Signal Processing. Prentice-Hall, Englewood Cliffs (1984)
Gardner, R.: Geometric Tomography. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1995)
Gritzmann, P., Nivat, M. (eds.): Discrete Tomography: Algorithms and Complexity, number 97042, Dagstuhl, Germany (January 1997)
Guédon, J.: Les problèmes d’échantillonnage dans la reconstruction d’images à partir de projections. PhD thesis, Université de Nantes (Novembre 1990)
Guédon, J., Barba, D., Burger, N.: Psychovisual image coding via an exact discrete radon transform. In: Wu, L.T. (ed.) VCIP 1995, Taipei, Taiwan, may, pp. 562–572 (1995); CORESA
Guédon, J., Bizais, Y.: Bandlimited and haar filtered back-projection reconstuction. IEEE transaction on medical imaging 13(3), 430–440 (1994)
Guédon, J., Normand, N.: Spline mojette transform. application in tomography and communication. In: EUSIPCO (September 2002)
Herman, G.T., Altschuler, M.D.: Image Reconstruction from Projections - Implementation and Applications. Topics in Applied Physics. Springer, New York (1979)
Katz, M.: Questions of uniqueness and resolution in reconstruction from projections. Lecture Notes in Biomathematics. Springer, New York (1978)
Kuba, A.: The reconstruction of two-directionally connected binary patterns from their two orthogonal projections. Comput. Vision, Graphics. Image Process (27), 249–265 (1984)
Matus, F., Flusser, J.: Image representation via a finite radon transform. IEEE transaction on pattern analysis and machine intelligence 15(10), 996–1006 (1993)
Normand, N., Guédon, J.: La transformée mojette: une représentation redondante pour l’image. Comptes-Rendus de l’Académie des Sciences, 123–126 (January 1998)
Ozarow, L.: On a source coding problem with two channels and three receivers. Bell Sys. Tech. Journal 59, 1909–1921 (1980)
Parrein, B., Normand, N., Guédon, J.: Multiple description coding using exact discrete radon transform. In: Data Compression Conference, Snowbird, March 2001, p. 508. IEEE, Los Alamitos (2001)
Parrein, B., Normand, N., Guédon, J.: Multimedia forward error correcting codes for wireless lan. Annals of telecommunications (3-4), 448–463 (2003)
Philippé, O., Guédon, J.: Correlation properties of the mojette representation for non-exact image reconstruction. In: Verlag, I.-F. (ed.) Proc. Picture Coding Symposium 1997, Berlin, September 1997, pp. 237–241 (1997)
Radon, J.: Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten. Ber. Ver. Sächs. Akad. Wiss. Leipzig, Math-Phys. Kl. 69, 262–277 (1917) In German; An english translation can be found in S. R. Deans: The Radon Transform and Some of Its Applications, app. A
Souhard, B., Chatellier, C., Olivier, C.: Simulation d’une chaîne de communication adaptée à la transmission d’images fixes sur canal réel. In: CORESA, Lyon (January 2003)
Svalbe, I., Kingston, A.: Farey sequences and discrete radon transform projection angles. In: IWCIA 2003, Palermo (May 2003)
Unser, M., Aldroubi, A., Eden, M.: Polynomial spline signal approximations: Filter design and asymptotic equivalence with shannon’s sampling theorem. IEEE Transaction on Information theory 38, 95–103 (1992)
Unser, M., Aldroubi, A., Eden, M.: B-Spline signal processing: Part I - Theory. IEEE Trans. Signal Process 41(2), 821–833 (1993)
Unser, M., Aldroubi, A., Eden, M.: B-Spline signal processing: Part II - Efficient design and applications. IEEE Trans. Signal Process 41(2), 834–848 (1993)
Verbert, P., Guédon, J.: An exact discrete backprojector operator. In: EUSIPCO, Toulouse (2002)
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Guédon, J., Normand, N. (2005). The Mojette Transform: The First Ten Years. In: Andres, E., Damiand, G., Lienhardt, P. (eds) Discrete Geometry for Computer Imagery. DGCI 2005. Lecture Notes in Computer Science, vol 3429. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31965-8_8
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DOI: https://doi.org/10.1007/978-3-540-31965-8_8
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