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Introduction :
Recently multiresolution analysis is one of the acceptable methods to analyse the Remotely_sensed images. A new method baesd on discrete wavelet transform is proposed here. In this paper first we introduce standard methods to fuse images then we describe the proposed method and related algorithm to do discrete wavelet transform. Qualifed and quantified results of this method are compared with the other methods. At last, some important conclusions are stated briefly.

Standard fusion methods :

1-Principle Component method (PC)
To do this approach first we must get the principle components of the multispecral image (band1, band2, band3 or R,G, B). After that the first principle component which contains the most information of the image is substituted by the panchromatic image. At last the inverse PC transform is done to get the new R,G,B bands of multispectral image from the principle components.

2-Brovey method
The fused R, G, B bands are as follows:


3-HIS (LHS) method
This method is based on the transformation of RGB multispectral channels into HIS (Hue-Intensity-Saturation). The Intensity component is the most important component and can be defined as follows:


In this method, we substitute the Intensity component by the panchromatic image and then the inverse transformation (HIS to RGB) is done. To compare the results, all these methods were used in addition the proposed method based on wavelet decomposition.

Wavelet-based image fusion
Wavelet decompostion: wavelet decomposition is used for image processing very much. Wavelet transform produce the images in different resolution. Wavelet represention refers to both spatial and frequency space. It can show a good position of a function (here this function is the image) in spatial and frequency space.

There are different approaches to do wavelet decomposition. One of them is Mallat algorithm which can use wavelet function such as "Daubechies functions (db1, db2 ,…)" Here we use "a trous" algorithm, which uses dyadic wavelet to merge non_dyadic data in a simple and efficient procedure. In this algorithm for the discrete wavelet transform we must do the successive convolution with a filter.


To convolve the image and the filter we can use two methods:

• Using convolution theory: Transform both the image and filter into the frequency space by using (fast) 2D Fourior transform (FT or FFT). Then multiply both of them together and finally do the inverse 2D Fourior transform (IFT) to get the convolved image.
• To use convolution function directly.In each step we get a version of the image ,…. The wavelet coefficient is defined as follows:

So if we decompose an image I into wavelet coefficients, then we can write:
in which I_r is a residual image. In this approach all wavelet planes have the same number of pixels as the original image. There are two approaches for image fusion based on wavelet decomposition: