03-05-2011, 04:56 PM
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 colour video. These experimental results show that the simple
thresholding of the 3-D DART coefficients is efficient.
Index Terms
3-D ridgelet transform, discrete analytical objects, denoising, video, colour images
I. INTRODUCTION
Ateam from University of Stanford has recently developed an alternative system of multiresolution analysis, called ridgelet
transform, specifically designed to efficiently represent edges in images [1]. The ridgelet transform can be computed by
performing a wavelet analysis in the Radon domain. However, most of the work done with ridgelets has been theoretical in
nature. To our knowledge, we can only find in literature three main implementations for the 2-D discrete ridgelet decomposition
[2]–[4]. This paper presents an extension to 3-D of our approach proposed in [4] that aims at representing linear singularities
with a discrete ridgelet transform based on discrete analytical objects: the 3-D Discrete Analytical ridgelet Transform (3-D
DART). The idea behind the 3-D associated discrete Radon transform is to define each Radon projection by a 3-D discrete
analytical line in the Fourier domain. There are several advantages in using discrete analytical lines:
• They offer a theoretical framework for the definition of the 3-D discrete Radon projections.
• This solution allows us to have different ridgelet decompositions according to the arithmetical thickness of the 3-D discrete
lines (control of the representation redundancy factor).
• The 3-D DART has a very simple forward/inverse algorithm (It is an important quality for the 3-D computation).
• The simple straightforward approach ensures an exact reconstruction without any interpolation or iterative process.
In section II, we will present the ridgelet transform. In section III, we will define the 3-D discrete analytical Radon transform
with two geometrical approaches. In section IV, we will present the 3-D Discrete Analytical ridgelet Transform. In order to
illustrate the performances of the 3-D DART, we have applied our transform and its local extension to the denoising of some
3-D images in section V and colour videos in section VI.
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