3-D Discrete Analytical Ridgelet Transform

David Helbert (1) , Philippe Carré (1) , Éric Andrès (1)

1 SIC - SIGNAL-IMAGE-COMMUNICATION

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.

Mots clés: 3-D ridgelet transform; discrete analytical objects; denoising; video; colour images

Domaines: Traitement des images [eess.IV]

David Helbert, Philippe Carré, Éric Andrès. 3-D Discrete Analytical Ridgelet Transform. IEEE Transactions on Image Processing, 2006, 15 (12), p. 3701-3714. ⟨10.1109/TIP.2006.881936⟩. ⟨hal-00331384⟩