# Data fusion using 2d discrete wavelet transform (DWT)

I am working on a project that employs a linear array sensor that provides data from the same object at two different energies. Collected in time, I end up with two images (16-bit sensor values, MxM image), call one HIGH the other LOW. My engineering background is Material Science and metallurgy, so I am coming up to speed on DSP, image analysis, and all of the bits that bring all of that together.

I have been researching "image fusion" and have been working from Gonzalez's fabulous Digital Image Processing (3rd ed.) text. I have developed an understanding of DWT and have successfully written code to replicate the 3-scale DWT transforms in the text (major accomplishment for me).

Now, I am trying to understand how to "fuse" the DWT coefficients from the LOW and HIGH images. From the IEEE Xplore database, I pulled several papers on the topic; one of the more useful is by Zhiyu Chen, et.al. "A Combinational Approach to the Fusion, De-noising and Enhancement of Dual-Energy X-Ray Luggage Images".

Succinctly, they recommend "averaging the corresponding approximation coefficients of L and H" (sigma)... and "summing the corresponding detail coefficients of L and H" (psi). But, that is all of what is said on the matter.

I have several other sources, but they must presume some obvious bit of understanding that I am lacking!

My question is, in order to successfully fuse the HIGH and LOW images: what does this mean? In the case of a 3-scale DWT, do I average or sum the 3-rd scale approximation/detail coefficients? Do I somehow average across each scale, despite the fact that the size of the coefficient matrices are different by powers of 2?

• Are the LOW and HIGH images overlapping? In other words, does each "pixel" value in the HIGH image and the LOW image correspond to measuring the same thing? – Mustafa Jul 31 '13 at 17:19
• @Mustafa Yes, each pixel in the LOW image is in 1-to-1 correlation to the HIGH image. Just to be explicit: pixel m,n in LOW corresponds to m,n in HIGH – jjwebster Jul 31 '13 at 18:32
• You won't have to average across scale. If the paper explicitly mentioned averaging the approximation coefficients and summing the detail coefficients.. why not try that? What I mean is, for a given scale, take the approximation coefficients from LOW and average with the approximation coefficients from HIGH. For that same scale, take the detail coefficients from LOW and add to the detail coefficient from HIGH. Intuitively it makes sense, because the approximation coefficients are responsible for gross structure so you want to preserve energy by averaging. – Mustafa Jul 31 '13 at 20:41
• It looks like Chen's PhD thesis is available online. Particularly, see Appendix A.2 ("Wavelet-based image fusion") on page 174. Perhaps it will help. – Peter K. Jul 31 '13 at 20:46
• @Mustafa I think I'm confused about the "order of operation". Help me understand: I perform the first decomposition on each image, yielding the approximation and the H, V, and D coefficients. I average the approximation from each, and sum the detail from each. Then, I perform the second decomposition... using which approximation? The "original" approximation from each? At the end of N-scales, I use the final approximation and reconstruct back through the tree, using the averages/sums from each scale? I've tried that, but the results were not what I expect (could be my algo, though). – jjwebster Jul 31 '13 at 23:13