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The Contourlet Transform: An Efficient Directional Multiresolution Image Representation

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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).