The contourlet transform: an efficient directional multiresolution image representation

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Do, M.N.; Vetterli, M.: The contourlet transform: an efficient directional multiresolution image representation. Image Processing, IEEE Transactions on , vol.14, no.12, pp.2091-2106, Dec. 2005

DOI

http://dx.doi.org/10.1109/TIP.2005.859376

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 nonseparable 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 continuous-domain 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.

Extended Abstract

Bibtex

@ARTICLE{1532309,
author={M. N. Do and M. Vetterli},
journal={IEEE Transactions on Image Processing},
title={The contourlet transform: an efficient directional multiresolution image representation},
year={2005},
volume={14},
number={12},
pages={2091-2106},
keywords={channel bank filters;image reconstruction;image representation;image resolution;wavelet transforms;Fourier transform;contour segments;contourlet transform;curvelets;directional multiresolution analysis;discrete-domain construction;discrete-domain multiresolution;image edges;image expansion;multidirection expansion;multiresolution image representation;nonseparable filter banks;wavelet transform;Convergence;Discrete Fourier transforms;Discrete transforms;Discrete wavelet transforms;Filter bank;Fourier transforms;Geometry;Image representation;Image resolution;Wavelet transforms;Contourlets;contours;filter banks;geometric image processing;multidirection;multiresolution;sparse representation;wavelets;Algorithms;Artificial Intelligence;Computer Graphics;Image Enhancement;Image Interpretation, Computer-Assisted;Information Storage and Retrieval;Numerical Analysis, Computer-Assisted;Signal Processing, Computer-Assisted},
doi={10.1109/TIP.2005.859376},
url={http://dx.doi.org/10.1109/TIP.2005.859376 http://de.evo-art.org/index.php?title=The_contourlet_transform:_an_efficient_directional_multiresolution_image_representation },
ISSN={1057-7149},
month={Dec} 
}

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