The Contourlet Transform: An Efficient Directional Multiresolution Image Representation

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The Contourlet Transform: An Efficient Directional Multiresolution Image Representation

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by Minh N. Do , Martin Vetterli

Venue:	IEEE TRANSACTIONS ON IMAGE PROCESSING
Citations:	513 - 20 self

BibTeX

@MISC{Do_thecontourlet,
    author = {Minh N. Do and Martin Vetterli},
    title = {The Contourlet Transform: An Efficient Directional Multiresolution Image Representation},
    year = {}
}

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Abstract

The limitations of commonly used separable extensions 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. Furthermore, we establish a precise link between the developed filter bank and the associated continuousdomain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications.

Keyphrases

contourlet transform ieee transaction image processing efficient directional multiresolution image representation filter bank continuous domain order operation index term sparse representation true twodimensional transform piecewise smooth function non-separable filter bank discrete nature intrinsic geometrical structure main challenge separable extension wavelet transforms image edge one-dimensional transforms visual information several image processing application discrete-domain multiresolution directional image expansion flexible multiresolution parabolic scaling developed filter bank discrete contourlet transform multidirection expansion discrete-domain construction directional multiresolution analysis framework precise link numerical experiment contour segment iterated filter bank optimal approximation rate sufficient directional vanishing moment n-pixel image construction result continuousdomain contourlet expansion differentiable curve

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