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0) Map of Auburn
This is Math Dept. Team Alpha, checking in from the Loveliest Village on the Plains: Auburn, AL.
1) Toomer's Corner being rolled
Here at Auburn, we love our traditions. And one of our most unique traditions, is the rolling of Toomer's Corner.
2) Toomer's Drugs exterior
According to AuburnTigers.com, the tradition of rolling Toomer's Corner is said to have begun when Toomer's Drugs had the only telegraph in the city.
3) Toomer's Drugs interior
After Auburn University would win a football game on the road, employees of our local drug store would throw the ticker tape from the telegraph onto the power lines. At some point down the line, Auburn fans joined in by throwing toilet paper, and the rest is history.
4) Toomer's Corner feed website
Teams at the Auburn, AL Hackathon competition were prompted with several suggestions, one of which was to take advantage of the live feed of Toomer's Corner provided by the City of Auburn. What started as a joke amongst our team of mathematicians quickly turned serious: using a little bit of calculus, we can programmatically identify when Auburn's famous corner is littered with celebratory TP.
5) Level curves and gradient vectors
A quick Cal 3 refresher: given a function of two variables, the gradient vector field aways points in the direction of greatest change. As a result, a gradient vector will always be perpendicular to the corresponding level curve of the function.
6) Edge-detected Toomer's Corner, no paper
The function we wish to consider is simple: using the well-known Canny edge detection algorithm, we may isolate the edges found in a photo of Toomer's corner.
7) Edge-detected Toomer's Corner, no paper, with gradient vectors
In this image, the corner has not been rolled yet, so the edges and normal gradient vectors are not oriented in a common direction.
8) Edge-detected Toomer's Corner, with paper
However, once Auburn fans get to rolling the corner...
9) Edge-detected Toomer's Corner, with paper, with gradient vectors
... due to gravity, the vertical strips of toilet paper introduce numerous vertial edges, and therefore horizontal gradient vectors. By summing up the number of almost-horizontal gradient vectors, our application can say with a high degree of confidence that Toomer's Corner is being rolled. Our application does the obvious thing: it sends out a tweet to @IsToomersRolled, and updates our website IsToomersCornerBeingRolledRightNow.com.
10) Product page screenshot
Despite the silliness of our concept, our application was met with a high level of excitement from the Auburn community, including the judges of our regional competition and local press. Following our selection to move on to the global competition, we put the app into production using Amazon Web Service's EC2 cloud servers, where it watches the Toomers Corner webcam 24/7 for that next rolling. We look forward to serving the Auburn community this fall as football begins, but we're also excited to celebrate non-football victories which result in the Auburn Family rolling Toomer's Corner as well.
11) Toomer's Corner feed website
Of course, as this tradition is certainly unique to the Auburn community, we struggled to figure out how to generalize our application for global use. While there certainly are some other specific applications, we came to ask ourselves: "Is there another mathematical tool we can use to recognize arbitrary unusual activity from a video feed?"
12) Line of least squares regression
Another quick math lesson. Given a collection of datapoints in a two-dimensional space, the line of least squares regression may be used to identify where further typical data points are expected to be positioned.
13) Principle Component Analysis
This technique may be generalized to three dimensions, and higher, known as principle component analysis, or PCA. And that's a good thing, because a 400-by-300 pixel photograph is essentially a 120,000-dimensional vector. By using footage of a typical day at Toomer's corner, we may use this technique to identify a hyperplane of least squares regression, capturing what you could call "average" states of the web cam feed.
15) Photoshopped image of Toomer's Corner with dinosaurs
Using this hyperplane, we're able to identify any... not-so-typical scenes, by simply computing the Euclidean distance of a photo from the hyperplane of typical photos.
16) Photoshopped image of Toomer's Corner with police cars and barricades
PCA is used in facial recognition techniques, but we see it being used in reverse here: by registering when a photo of an intersection does not look familar to the algorithm, drivers could be alerted to find an alternate route to avoid the unusual activity being observed. While this application is still a prototype, our team is excited as this approach can recognize arbitrary strange activity for any video feed, rather than the specific criteria we search for in our production Toomer's Corner analysis.
17) Picture of team or website
Thanks for considering our project for the Smart City Hackathon! We're honored to have had this opporunity to use mathematics and technology to improve our local and global communities. Be sure to check out our project READMEs on GitHub to learn more about each component of our applications. And to our Auburn Family: Waar Eagle, Hey!