## Extrapolatable Function Graphics

Here the wikipedia commons photograph 'Feathered Dusk' taken by Jessie Eastland, has been converted into a mathematical function using a machine learning algoirthm. Through an iterative process the algorithm has developed a bivariate, vector valued, mathematical function, the outputs of which correspond to the RGB values of each pixel in the image (as long as the two inputs are the x,y co-ordinates of the respective pixel).

With the entire image modelled as a single mathematical function, feeding this function x,y co-ordinates that exist outside of the original image's borders produces interesting results. The top right image shows the rendered results of one such mathematical function. The white corner markers identify where the original borders of the image existed. The pixel co-ordinates that exits outside of these markers did not exist in the original image yet the function is still capable of rendering relatively plausable results. During the learning process, heavy restrictions were intentionally placed on the detail that the function would be able to model (by limiting the size and complexity of the function).

The next image down shows a larger, more complex mathematical function, modelled on the same image. With less restrictions on size and complexity the algorithm has learnt to model finer details of the original image but to the detriment of a plausable rendering beyond the image borders. The more capable it becomes at exactly replicating the image, the less capable it becomes as generalising outside of the image borders based on what it has learnt.

The slider below shows the results from a string of attempts to solve this problem, mainly by using multiple mathematical functions with each one trained to pick up on different features within the same image. The results are interesting, but largely unsuccessful.

Also see Scalable Function Graphics.