3-D Discrete Analytical Ridgelet Transform

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 t...

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Bibliographic Details
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
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Summary: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
ISSN:1057-7149
1941-0042
DOI:10.1109/TIP.2006.881936