Discrete analytical Ridgelet transform

In this paper, we present a new implementation of the Ridgelet transform based on discrete analytical 2-D lines: the discrete analytical Ridgelet transform (DART). This transform uses the Fourier strategy for the computation of the associated discrete Radon transform. The innovative step of the DART is the construction of discrete analytical lines in the Fourier domain. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a DART adapted to a specific application. Indeed, the DART representation is not orthogonal it is associated with a flexible redundancy factor. The DART has a very simple forward/inverse algorithm that provides an exact reconstruction. We apply the DART and its extension to a local-DART (with smooth windowing) to the denoising of some images. These experimental results show that the simple thresholding of the DART coefficients is competitive or more effective than the classical denoising techniques.

[1]  Gabriele Steidl,et al.  A new linogram algorithm for computerized tomography , 2001 .

[2]  Jean-Pierre Reveillès.  Géométrie discrète, calcul en nombres entiers et algorithmique , 1991 .

[3]  D. Donoho.  Wavelet Shrinkage and W.V.D.: A 10-minute tour , 1997 .

[4]  P. Toft.  The Radon Transform - Theory and Implementation , 1996 .

[5]  Gabriele Steidl,et al.  Fast Fourier Transforms for Nonequispaced Data: A Tutorial , 2001 .

[6]  D. Donoho,et al.  Fast Slant Stack: a notion of Radon transform for data in a Cartesian grid which is rapidly computable, algebraically exact, geometrically faithful and invertible , 2003 .

[7]  D. Donoho,et al.  Translation-Invariant De-Noising , 1995 .

[8]  Jack Bresenham,et al.  Algorithm for computer control of a digital plotter , 1965, IBM Syst. J..

[9]  David L. Donoho,et al.  Orthonormal Ridgelets and Linear Singularities , 2000, SIAM J. Math. Anal..

[10]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[12]  Karsten Fourmont.  Non-Equispaced Fast Fourier Transforms with Applications to Tomography , 2003 .

[13]  Yoel Shkolnisky,et al.  3D discrete X-ray transform , 2004 .

[14]  Martin L. Brady,et al.  A Fast Discrete Approximation Algorithm for the Radon Transform , 1998, SIAM J. Comput..

[15]  Eric Andres,et al.  3D fast ridgelet transform , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[16]  Eric Andres,et al.  Ridgelet Transform Based on Reveillès Discrete Lines , 2002, DGCI.

[17]  J CandèsEmmanuel,et al.  New multiscale transforms, minimum total variation synthesis , 2002 .

[18]  A. Averbuch,et al.  3D Fourier based discrete Radon transform , 2003 .

[19]  A. G. Flesia,et al.  Digital Implementation of Ridgelet Packets , 2003 .

[20]  Minh N. Do,et al.  Discrete Ridgelet Transforms for Image Representation , 2001 .

[21]  Eric Andres,et al.  Discrete Analytical Hyperplanes , 1997, CVGIP Graph. Model. Image Process..

[22]  M. Do.  Directional multiresolution image representations , 2002 .

[23]  E. Candès,et al.  Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges , 2000 .

[24]  A. G. Flesia,et al.  Digital Ridgelet Transform Based on True Ridge Functions , 2003 .

[25]  Emmanuel J. Candès,et al.  The curvelet transform for image denoising , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[26]  Jeffrey A. Fessler,et al.  A min-max approach to the multidimensional nonuniform FFT: application to tomographic image reconstruction , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[27]  G. MallatS..  A Theory for Multiresolution Signal Decomposition , 1989 .

[28]  Emmanuel J. Candès,et al.  New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction , 2002, Signal Process..

[29]  Eric Andres,et al.  Discrete Ridgelet transform: application to the denoising of color images , 2002, CGIV.

[30]  E. Kolaczyk.  Wavelet Methods For The Inversion Of Certain Homogeneous Linear Operators In The Presence Of Noisy D , 1994 .

[31]  Jan Flusser,et al.  Image Representation Via a Finite Radon Transform , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[32]  Jos B. T. M. Roerdink,et al.  Data-parallel tomographic reconstruction: A comparison of filtered backprojection and direct Fourier reconstruction , 1998, Parallel Comput..