The intention for Scalable Function Graphics (SFG) is to provide a means of storing photographic images as scalable mathematical models. Unlike raster graphics, which store images as an array of discreet pixel values, an SFG effectively stores an image as a complex, bivariate, vector-valued, continuous function. The exact mathematical function is determined by a machine learning algorithm during a typically time-consuming conversion process.

 

The input values to this function are decimal x,y co-ordinates, and the three outputs correspond to the red, green, and blue values of a precise point at that location. Before converting into an SFG the original image will have consisted of a fixed number of points (or pixels). Through the conversion process the learning algorithm steadily manipulates constants of an extremely long mathematical function such that the output of the function will directly match up with the RGB values of each pixel in the original image (so long as the inputs to the function were the co-ordinate values of the respective pixel).

 

Once this has been achieved the resulting function can be fed new co-ordinate values that weren't defined in the original image (co-ordinates in-between the original pixels). The resulting outputs act as effective estimates, allowing the image to be scaled up dramatically (as long as it is stored as an SFG) with the learnt mathematical function filling in the gaps.

 

A comparison can be drawn between Scalable Function Graphics and Scalable Vector Graphics (SVG). SVGs also store images as a scalable mathematical model, but with an important difference. SVGs store values that directly represent colours, heights, widths and other attributes of a fixed number of geometric shapes. As mathematically defined objects these shapes maintain the same characteristics at any scale. Unfortunately, however, this mechanism only remains useful for text, and relatively simple illustrations such as logos. Photographic images are far too complex to be stored using simple geometric objects. This is where Scalable Function Graphics are intended to be useful.

 

 

For machine learning algorithms to work, such as genetic algorithms, relatively small learning steps need to be taken which gradually build up. However using discontinuous functions meant that even small alterations in the function could result in dramatic changes to the resulting image. This obstructs subtle but beneficial changes from accumulating effectively. Additionally, almost any alteration to any constituent function, operator or variable (as opposed to constants) often resulted in relatively drastic effects. For example, changing 100+100 to 100-100, or max(0,x) to min(0,x), or y to x. Changes to mathematical constants on the other hand can be controllably subtle, so long as they are not contained within discontinuous functions (for example changing 50 to 50.00001).

 

Fortunately, limiting the learning process to the manipulation of constants also allows for a more efficient learning algorithm to be used called 'backpropogation'. Significant modifications have, however, needed to be made to the standard backpropogation algorithm to make it fit for the purpose of Scalable Function Graphic conversion. Additionally, to solve the problem within a tractable time frame the original image needs to be divided into a series of smaller tiles. A different mathematical function can then be trained to model each tile, with each one overlapping with neighbouring tiles slightly to ensure that the scaled up versions fit together seamlessly once the learning process is complete. Tiling is necessary as the ability of even an extremely large continuous function to model the complexity of an entire photographic image appears to diminish exponentially the larger the original image becomes.

 

For comparison here are the results of four different scaling techniques (including a Scalable Function Graphic) applied to a 20 by 20 tile within the wiki commons image, 'Feathered Dusk'. by Jessie Eastland. From left to right, 'nearest neighbour algorithm', 'bicubic algorithm', 'fractal algorithm', 'SFG(sharp edge mode)'. If you would like to see the length and complexity of the vector-valued function required for the SFG to produce such a high quality rendering (for just this one tile) click here.