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The Contourlet Transform: An Efficient Directional Multiresolution Image pdf
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The Contourlet Transform: An Efficient Directional Multiresolution Image Representation


.pdf   Contourlet Transform1111.pdf (Size: 1.03 MB / Downloads: 10)

Abstract

The limitations of commonly used separable ex-
tensions of one-dimensional transforms, such as the Fourier
and wavelet transforms, in capturing the geometry of image
edges are well known. In this paper, we pursue a “true” two-
dimensional transform that can capture the intrinsic geometrical
structure that is key in visual information. The main challenge
in exploring geometry in images comes from the discrete nature
of the data. Thus, unlike other approaches, such as curvelets,
that first develop a transform in the continuous domain and
then discretize for sampled data, our approach starts with a
discrete-domain construction and then studies its convergence
to an expansion in the continuous domain. Specifically, we
construct a discrete-domain multiresolution and multidirection
expansion using non-separable filter banks, in much the same way
that wavelets were derived from filter banks. This construction
results in a flexible multiresolution, local, and directional image
expansion using contour segments, and thus it is named the
contourlet transform. The discrete contourlet transform has a fast
iterated filter bank algorithm that requires an order N operations
for N -pixel images.

I NTRODUCTION

Efficient representation of visual information lies at the
heart of many image processing tasks, including compression,
denoising, feature extraction, and inverse problems. Efficiency
of a representation refers to the ability to capture significant
information about an object of interest using a small descrip-
tion. For image compression or content-based image retrieval,
the use of an efficient representation implies the compactness
of the compressed file or the index entry for each image
in the database.

I NTRODUCTION

Efficient representation of visual information lies at the
heart of many image processing tasks, including compression,
denoising, feature extraction, and inverse problems. Efficiency
of a representation refers to the ability to capture significant
information about an object of interest using a small descrip-
tion. For image compression or content-based image retrieval,
the use of an efficient representation implies the compactness
of the compressed file or the index entry for each image
in the database

Iterated directional filter banks

Bamberger and Smith [24] constructed a 2-D directional
filter bank (DFB) that can be maximally decimated while
achieving perfect reconstruction. The DFB is efficiently im-
plemented via an l-level binary tree decomposition that leads
to 2l subbands with wedge-shaped frequency partitioning as
shown in Figure 3(a). The original construction of the DFB in
[24] involves modulating the input image and using quincunx
filter banks with diamond-shaped filters [27]. To obtain the
desired frequency partition, a complicated tree expanding rule
has to be followed for finer directional subbands (e.g., see [28]
for details).
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#2
Effective rendering of visual information is at the heart of many image processing tasks, including compression, denoise removal, feature extraction, and reverse problems. The efficiency of a representation refers to the ability to capture meaningful information about an object of interest using a short description. For image compression or content-based image retrieval, using an efficient rendering implies the compactness of the compressed file or the index entry for each image in the database. For practical applications such efficient representation must be obtained through structured transformations and fast algorithms. For unidimensional, smooth, part-like signals as scanning lines for an image, wavelets have been established as the proper tool because they provide an optimal representation for these signals in a sense. In addition, the wavelet representation is susceptible to efficient algorithms; in particular, leads to rapid transformations and suitable tree data structures. These are the key reasons for the success of wavelets in many signal processing and communications applications; for example, the wavelet transformation was adopted as the transformation for the new standard of image compression, JPEG-2000. However, natural images are not simply stacks of straight 1-D scanning lines; the points of discontinuity (ie, the edges) are typically located along smooth curves (ie contours) due to the smooth boundaries of the physical objects. Thus, natural images contain intrinsic geometric structures that are key features in visual information. As a result of a detachable extension of 1-D bases, 2-D wavelets are good for isolating discontinuities at edge points, but they will not "see" smoothness along the contours. In addition, separable wavelets can capture only limited directional information - an important and unique feature of multidimensional signals. These disappointing behaviors indicate that the most powerful representations are necessary in the higher dimensions.
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