From financial crisis to image processing: Ignore Topology At Your Own Risk.
Very interesting article in Wired Recipe for Disaster: The Formula That Killed Wall Street . I’m not a statistician, but I’ll try to explain it. The gist of the article is that in the heart of the current financial crisis is the David X. Li formula, which use “Gaussian copula function” for risk estimation. The idea of formula is that if we have to estimate joint probability of two random events, it could be done with simple formula, which use only probability distributions of each event as if they were independent and a single parameter – statistical correlation. So what bankers did – instead of looking into relationships and connections between events they just did calculate one single statistical parameter and used it for risk estimation. Even more – they applied the same formula to the results of those relatively simple calculations and build pyramids of estimations, each next step applying the same simple formula to results of the previous step. As a result, an extremely complex behavior was reduced to the simple linear model, which had little in common with reality.
And now – the illustration from wiki, what exactly this single parameter – correlation is:
Here are several two-variable distributions and their correlation coefficients. It could be seen that for linear relationships correlation capture dependence of variables perfectly (middle raw). For upper row – normal distributions – it capture the essence of dependency. We can say something about other variable if we know one variable and correlation in that case. For complex shapes – lower row – correlation is zero for each. Each of the lower shapes will be represented as the upper central shape (fuzzy ball) with correlation. Correlation capture nil information about how one variable depend on another for the lower shapes. Correlation allow representation of any shape only as fuzzy ellipse. Li’s formula reduce dimensionality. The thing is, dimensionality – topological property, and you don’t mess with topological properties easily. Imagine bankers using fuzzy ball instead of ring for risk estimation…
Now to the image processing. Most of feature detection in image processing is done for grayscale image. Original image is usually RGB, but before features extraction it converted to grayscale.
However the original image is colored, why not to use colors for feature detection ? For example detect features in each color channel separately?
The thing is, the pictures in each color channel are very similar.
The extraction of blobs in each channel in most cases will triple the job without gaining of significant new information – all the channels will give about the same blobs.
Nevertheless it’s obvious, there is some nontrivial information about the image, encoded in colors.
Why blob detection for each color don’t give access to it ?
The reason is the same as for current financial crisis – dimensionality. Treating each color channel separately we replace five-dimensional RGB+coordinates space with three three-dimensional color+coordinates spaces. Relationships between color channels are lost. Topology of color structure is lost.
To actually use color information, statistical relationships between colors of the image should be explored – something like three dimensional color bins histogram, essentially converting image from RGB to indexed color.
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