Complex Wavelet Domain Image Fusion Based on Fractional Lower Order Moments

Alin Achim
University of Bristol, Department of Electrical and Electron

Nishan Canagarajah
University of Bristol, Department of Electrical and Electronic Engineering

David Bull
University of Bristol, Department of Electrical and Electronic Engineering

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     Last modified: June 26, 2005

Abstract
After more than ten years since its inception as an image fusion algorithm, the method proposed by Burt and Kolczinsky remains one of the most effective, yet simple and easy to implement. Their method essentially consists in calculating a normalised correlation (match measure) between the two images’ subbands over a small local area. Then, the fused coefficient is calculated from this measure and the local variance (salience measure) via a weighted average of the two images’ coefficients.
In this work, we propose a generalization of the above method for the case when the data to be fused exhibit heavy tails with no convergent second- or higher-order moments. Specifically, our developments are based on recently published results showing that wavelet decomposition coefficients of images are best modelled by symmetric alpha-stable (SαS) distributions, a family of heavy-tailed densities. Unfortunately, only moments of orders less then α (0< α<2) can be defined for the general alpha-stable family members. Consequently, as measure of salience we employ the dispersion of the alpha-stable distribution computed in a neighbourhood around the reference coefficients. Also, due to the lack of finite variance, covariances do not exist either on the space of SαS random variables. Instead, quantities like covariations or codifferences, which under certain circumstances play analogous roles for SαS random variables to the one played by covariance for Gaussian random variables have been introduced. Therefore, we propose the use of symmetrized and normalised versions of these quantities, which enable us to define a new match measure for α-stable random vectors.
In our implementation we made use of the dual-tree complex wavelet transform that has been shown to offer near shift invariance and improved directional selectivity compared to the standard wavelet transform.
Simulation results show that our method achieves better performance in comparison with previously proposed pixel-level fusion approaches.

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