3-D Discrete Analytical Ridgelet Transform

Abstract : 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.
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IEEE Transactions on Image Processing, Institute of Electrical and Electronics Engineers, 2006, 15 (12), p. 3701-3714. 〈10.1109/TIP.2006.881936〉
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David Helbert, Philippe Carré, Éric Andrès. 3-D Discrete Analytical Ridgelet Transform. IEEE Transactions on Image Processing, Institute of Electrical and Electronics Engineers, 2006, 15 (12), p. 3701-3714. 〈10.1109/TIP.2006.881936〉. 〈hal-00331384〉

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