Abstract
The Mojette transform is an entirely discrete form of the Radon transform developed in 1995. It is exactly invertible with both the forward and inverse transforms requiring only the addition operation. Over the last 10 years it has found many applications including image watermarking and encryption, tomographic reconstruction, robust data transmission and distributed data storage. This paper presents an elegant and efficient algorithm to directly apply the inverse Mojette transform. The method is derived from the inter-dependance of the “rational” projection vectors (p i ,q i ) which define the direction of projection over the parallel set of lines b = p i l – q i k. Projection values are acquired by summing the value of image pixels, f(k,l), centered on these lines. The new inversion is up to 5 times faster than previously proposed methods and solves the redundancy issues of these methods.
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References
Guédon, J., Barba, D., Burger, N.: Psychovisual image coding via an exact discrete Radon transform. In: Wu, L. (ed.) Proc. Visual Communications and Image Processing 1995 (VCIP 1995), Taipei, Taiwan, CORESA, pp. 562–572 (1995)
Guédon, J., Normand, N.: The Mojette transform: applications in image analysis and coding. The International Society for Optical Engineering 3024(2), 873–884 (1997)
Autrusseau, F., Guédon, J.: Image watermarking for copyright protection and data hiding via the Mojette transform. In: Sonka, M., Fitzpatrick, J. (eds.) Proc. SPIE: Security and Watermarking of Multimedia Contents IV, San Jose, CA, vol. 4675, pp. 378–386 (2002)
Autrusseau, F., Guédon, J., Bizais, Y.: Mojette cryptomarking scheme for medical images. In: Sonka, M., Fitzpatrick, J. (eds.) Proc. SPIE: Medical Imaging 2003: Image Processing, vol. 5032, pp. 958–965 (2003)
Autrusseau, F., Parrein, B., Servières, M.: Lossless compression based on a discrete and exact Radon transform: A preliminary study. In: Proc. IEEE International Conference on Acoustics, Speech and Signal Processing 2006 (ICASSP), Toulouse, France (2006)
Servières, M., Guédon, J., Normand, N.: A discrete tomography approach to PET reconstruction. In: Proc. 7th International Conference on Fully 3D Reconstruction in Radiology and Nuclear Medicine, Saint-Malo, France (2003)
Servières, M., Normand, N., Guédon, J., Bizais, Y.: The Mojette transform: Discrete angles for tomography. In: Herman, G., Kuba, A. (eds.) Proc. of Workshop on Discrete Tomography and its Applications, New York City. Elsevier Science Publishers, Amsterdam (2005)
Parrein, B., Guédon, J., Normand, N.: Multimedia forward error correcting codes for wireless LAN. Annals of Telecommunications 3(4), 448–463 (2003)
Guédon, J., Parrein, B., Normand, N.: Internet distributed image information system. Integrated Computer-Aided Engineering 8, 205–214 (2001)
Guédon, J.-P., Normand, N.: The mojette transform: The first ten years. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 79–91. Springer, Heidelberg (2005)
Katz, M.: Questions of uniqueness and resolution in reconstruction from projections. Springer, Heidelberg (1977)
Normand, N., Guédon, J., Philippé, O., Barba, D.: Controlled redundancy for image coding and high-speed transmission. In: Ansari, R., Smith, M. (eds.) Proc. SPIE Visual Communications and Image Processing 1996, vol. 2727, pp. 1070–1081 (1996)
Servières, M., Idier, J., Normand, N., Guédon, J.: Conjugate gradient Mojette reconstruction. In: Fitzpatrick, J., Reinhardt, J. (eds.) Proc. SPIE Medical Imaging 2005: Image Processing, vol. 5747, pp. 2067–2074 (2005)
Verbert, P., Ricordel, V., Guédon, J.: Analysis of the Mojette transform projections for an efficient coding. In: Proc. International Workshop on Image Analysis for Multimedia Interactive Services (WIAMIS), Barcelona, Spain (2004)
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Normand, N., Kingston, A., Évenou, P. (2006). A Geometry Driven Reconstruction Algorithm for the Mojette Transform. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds) Discrete Geometry for Computer Imagery. DGCI 2006. Lecture Notes in Computer Science, vol 4245. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11907350_11
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DOI: https://doi.org/10.1007/11907350_11
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