3-D Discrete Analytical Ridgelet Transform

Article English OPEN
Helbert , David ; Carré , Philippe ; Andrès , Éric (2006)
  • Publisher: Institute of Electrical and Electronics Engineers
  • Related identifiers: doi: 10.1109/TIP.2006.881936
  • Subject: denoising | colour images | [ INFO.INFO-TI ] Computer Science [cs]/Image Processing | 3-D ridgelet transform | video | discrete analytical objects
    arxiv: Mathematics::History and Overview

International audience; In this paper, we propose an implementation of the 3-D Ridgelet transform: the 3-D discrete analytical Ridgelet transform (3-D DART). This transform uses the Fourier strategy for the computation of the associated 3-D discrete Radon transform. The innovative step is the definition of a discrete 3-D transform with the discrete analytical geometry theory by the construction of 3-D discrete analytical lines in the Fourier domain. We propose two types of 3-D discrete lines: 3-D discrete radial lines going through the origin defined from their orthogonal projections and 3-D planes covered with 2-D discrete line segments. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a 3-D DART adapted to a specific application. Indeed, the 3-D DART representation is not orthogonal, It is associated with a flexible redundancy factor. The 3-D DART has a very simple forward/inverse algorithm that provides an exact reconstruction without any iterative method. In order to illustrate the potentiality of this new discrete transform, we apply the 3-D DART and its extension to the Local-DART (with smooth windowing) to the denoising of 3-D image and color video. These experimental results show that the simple thresholding of the 3-D DART coefficients is efficient.
  • References (48)
    48 references, page 1 of 5

    [1] E. Cande`s, “Ridgelets : Theory and applications,” Ph.D. dissertation, University of Stanford, August 1998.

    [2] M. Do, “Directional multiresolution image representations,” Ph.D. dissertation, Department of Communication Systems, Swiss Federal Institute of Technology Lausanne, November 2001.

    [3] A. Flesia, H. Hel-Or, A. Averbuch, E. Cande`s, R. Coifman, and D. Donoho, “Digital implementation of ridgelet packets,” in Beyond Wavelets, J. Stoeckler and G. V. Welland, Eds. Academic Press, 2002.

    [4] P. Carre´ and E. Andre`s, “Discrete Analytical Ridgelet Transform,” Signal Processing, vol. 84, no. 11, November 2004.

    [5] P. Querre, J. L. Starck, and V. Martinez, “Analysis of the galaxy distribution using multiscale methods,” in Astronomical Data Analysis Conference, J. Starck and Murtagh, Eds., vol. 4847. SPIE Conference, 2002.

    [6] J.-L. Starck, V. Martinez, D. Donoho, O. Levi, P. Querre, and E. Saar, “Analysis of the spatial distribution of galaxies by multiscale methods,” Eurasip Journal on Applied Signal Processing, special issues on applications of signal processing in Astrophysics and Cosmology, vol. 5, pp. 2455-2469, 2005.

    [7] D. Donoho and O. Levy, “Fast X-ray and beamlet transforms for three-dimensional data,” in Modern Signal Processing, C. U. Press, Ed. MSRI Publications, 2002, vol. 46, pp. 79-116.

    [8] A. Averbuch and Y. Shkolnisky, “3D Fourier based discrete Radon transform,” Applied and Computational Harmonic, vol. 15, pp. 33-69, 2003.

    [9] S. Mallat, “A theory for multiresolution signal decomposition : the wavelet transform,” IEEE Trans. on PAMI, vol. 11, no. 7, pp. 674-693, 1989.

    [10] F. Matus and J. Flusser, “Image representation via a finite Radon transform,” IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 15, no. 10, pp. 996-1006, 1993.

  • Metrics
    No metrics available
Share - Bookmark