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1-Substitution method (SUBRGB) :
In this method after getting the wavelet coefficients of multispectral and panchromatic images, we substitute some wavelet coefficients of multispectral image by some wavelet coefficients of panchromatic image. At last we do the inverse wavelet transform.

2-Additive method :
this approach can be done in two ways.

• AWRGB(Using the of R,G,B components): in this approach we first only produce the wavelet planes of the panchromatic image and add them to R,G,B bands directly.
• AWI(Using the Intensity component):here after getting the wavelet coefficients of the panchromatic image we add them into the Intensity component whish is extracted from R,G,B bands. After that we transform the HIS component (with a new Intensity) into new R,G,B.

There are different ways to get the Intensity-Hue-Saturation components from R,G,B bands. The algorithm we used here to get the Intensity componet is called "Hexcone" .This method is as follow :

Max = Maximum (R,G,B)
Min = Minimum (R,G,B)
Delta = Max-Min
Intensity = Max
If (Max # 0) then "Satuation = Delta/Max "
If (Max = 0) then "Saturation = 0"
If (Saturation = 0) then " Hue=0"
If (R = Max) then "Hue = (G-B ) /Delta" (Between yellow and magenta)
If (G = Max) then "Hue = 2 + (B-R ) /Delta" (Between cyan and yellow)
If (B = Max) then "Hue = 4 + (R-G ) /Delta" (Between magenta and cyan)
Hue = Hue * 60 (Convert Hue to degree)
If (Hue < 0) then "Hue = Hue+360" (Hue must be positive)
If (Hue >= 360) then "Hue = Hue - 360"

to scale the Hue and Saturation between 0 and 255 :
Hue = Hue * (255/360)
Saturation = Saturation *255
To do inverse transform (RGB to HIS) :
Hue = Hue * (360/255)
Saturation = Saturation /255
If (Saturation =0) then "(R,G,B)=(1,1,1)"
If (Saturation > 0) then "Hue = Hue / 60"
J = floor (Hue)
F = Hue - J
P = Intensity * (1 - Saturation )
Q = Intensity * (1 - ( Saturation * F))
T = Intensity * (1 - (Saturation * (1-F)))
If (J = 0) then " (R,G,B) = (Intensity , T , P)"
If (J = 1) then " (R,G,B) = (Q , I , P)"
If (J = 2) then " (R,G,B) = (P , Intensity , T )"
If (J = 3) then " (R,G,B) = (P , Q , T)"
If (J = 4) then " (R,G,B) = (T , P , Intensity)"
If (J = 5) then " (R,G,B) = (Intensity , P ,T)"

In the substitution method the wavelet coefficients of the multispectral image are discarded and substituted by the wavelet coefficients of panchromatic image completly but in the additive method most of the spatial information of both of images (panchromatic and multispectral) is preserved. The difference between two additive methods is that in the first method (AWRGB) the panchromatic wavelet coefficients are added in the same amount to the RGB bands while in the second case high resolution information only is added to Intensity component that leads to preserve information in the better manner.

Results:
Here we use the panchromatic and multispectral bands of the LANDSAT ETM satellite at the spatial resolutions of 15m and 30m. The methods we used in this paper are:

1-Additive wavelet method contains two approaches:

• AWRGB (using RGB components)
  
AWI (using Intensity component)

2- Substitution wavelet method (SUBRGB) the standard methods used here are :

3- HIS (Hue-Intensity-Saturation) method

4- Brovey method

5- PC(Principle component) method

To do this we used MATLAB (version : 6.1) and the ERDAS IMAGINE 8.4 software (image processing software).

Since there isn't any multispectral image at resolution 15m to be compared with the results, we used an inferior level of the images. So we reduced the spatial resolution of the panchromatic and multispectral bands into 30m and 60m respectively. Therefore the merged multispectral image has spatal resolution 30m.

The steps of using above different methods are as follow :

1. to register the multispectral image to the same size of panchromatic image . To do that we selected about 30 points in both images and registeration was done within subpixel RMSE(Root Mean Square Error). This step is very important and need to be done very precisely because the wavelet-based image fusion method is very sensitive to that. If there is a little displacement between two images, the resulted images will have bad quality. To do the registation, the PCI V8.2.0 software(OrthoEngine module) was used.
2. To do histogram matching between two images. Since the panchromatic and multispectral images used here ,belong to one sensor, the atmosphere and illumination conditions for both images are nearly the same. So this step can be omitted.
3. Up to here the panchroamatic and multispectral images have been perpered to be merged.
4. All fusion methods stated above can be used now.
   
5. Accurscy assessment : to quantify the behavior of the fusion methods used here, we computed the correlation between the different solutions and the original multispectral image. The correlation coefficient is computed as follows :