CBS’s primetime show “Numb3ers” is the story of 2 brothers, one a hardened FBI agent the other a world-renowned mathematician, who use a combination of brawn and smarts to solve crimes. In an episodde from the first season (Episode Information, Information about the Math of Numb3ers) some bad-guys make some counterfeit money and the only evidence the cops have is a low-quality video. The Don (the FBI brother) asks Charlie (the math brother) for some help. Turns out (very conveniently for Don) that Charlie happens to have a new technique for improving the quality of video. His method uses extrapolation of the colors from each pixel to “guess whats between the dots” and turn a crummy 7-11 video into HD-greatness (well, not quite, but you get the idea). So basically he is taking a bunch of points, fitting a function to them, and then using the function to evaluate values at other points he doesn’t have information about (is any of this starting to sound familiar, or is spring break too close).

While it’s cool to see math on TV (at least if your a power-nerd like me), the math behind image processing has been a very hot area of research lately and plays an important role in many places. The ray-casting that we did in the first project dwells in the realm of pretty pictures and animations. However, hard-core numerical computing techniques are used to solve the more difficult problem of image processing. Like in the TV show, one of the areas of research is enhancing pictures. This technique uses information from the existing pixels (like color, hue, brightness, etc.) to try and generate more pixels by guessing what they should look like. In another application, a researcher may be trying to take a picture of one side of someone’s face and use extrapolation techniques about human facial structure (ie how far your eyes and ears are apart) to generate an image of the person’s entire face. It’s not hard to imagine how this type of analysis may be useful to law enforcment agencies.