# Need a 2 parameter ranking system for image compression

I'm working on a project in which specific bit planes of an image are extracted and compressed. This process is repeated for every combination of bit planes possible.For an image, I have calculated the compression ratio and the Structural Similarity Index(SSIM).

I need a method to allot ranks when compressed with different combinations of bit planes.

My progress: I tried to give equal weight to the compression ratio and the SSIM value.

score = 0.5*ssim + 0.5*compression_ratio;


This equation gave me a value between 0 and 1. The problem with this equation is it gives an equal score for multiple combinations and few combinations are extreme opposites on the scale. For example, i got equal scores when I compressed an image using the {1,2,3,5,7,8} bit planes(High SSIM value) and only the 3rd bit plane(High compression ratio).

I need help in balancing the parameters and getting an appropriate score. The score doesn't need to give equal weights like I have given.

1. score depends on two parameters right now, but it is clearly unable to resolve the differences between different schemes. You can add a third parameter which captures the bit planes you are using. This could be either via:
1. Normalising the number of bit planes used (e.g. $\frac{3}{8} = 0.375$) but this might still be not enough as a simple reshuffling of the bitplanes can show;
2. Indexing your score by the number that results from the bitplanes used. So, if you used bitplanes 0,3,5, that would be 2^0 + 2^3 + 2^5, or simply a tuple of $(0,3,5)$ so that implicitly you have the number of bitplanes and which ones were used. This would give you a lookup of (bitplanes_used, score).
2. Now, the danger you are running here is formulating some $score = f(\Theta)$, where $f$ is a function whose parameter set $\Theta$ keeps growing, more than the 2 you specify. What if we include the ratio of high frequencies to low frequencies in there too? What if we add an entropy related term?...and so on. Ultimately, you might end up with a very large model that is still not informative.
• A common thing to do in this case is to penalise large models. This is an extra factor that increases with the number of parameters used. For example, in your case, you could say something like "If two bitplane combinations give me the same score, then I will select the combination that uses fewer bitplanes than the other"