3-D Discrete Analytical Ridgelet Transform

• Published in: IEEE transactions on image processing 2006-12, Vol.15 (12), p.3701-3714
• Main Authors: Helbert, D., Carre, P., Andres, E.
• Format: Article
• Language: English
• Subjects: 3-D Ridgelet transform; Algorithms; Applied sciences; Color images; Computer Science; denoising; Detection, estimation, filtering, equalization, prediction; discrete analytical objects; Discrete Fourier transforms; Discrete transforms; Exact sciences and technology; Fourier analysis; Fourier transforms; Geometry; Image analysis; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Image Processing; Image reconstruction; Imaging, Three-Dimensional - methods; Information Storage and Retrieval - methods; Information, signal and communications theory; Iterative algorithms; Iterative methods; Mathematical analysis; Miscellaneous; Noise reduction; Numerical Analysis, Computer-Assisted; Reconstruction; Redundancy; Signal and communications theory; Signal processing; Signal Processing, Computer-Assisted; Signal, noise; Studies; Telecommunications and information theory; Transforms; video; Wavelet analysis
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2006.881936

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