We have in the past few weeks implemented root-finding algorithms to create images as the first class project. The Bisection method, the Regula Falsi method, Newton’s method, and Secant method are all suggested to be used to supply roots to a ray-casting program. The project not only confirms the importance of considering algorithm differences and errors associated with different algorithms, but also has shown the applicability of root-finding methods in image rendering.

It becomes clear that this project scenario is inherently capable of being extended, especially when we have most recently learned the algorithms for differentiating and integrating functions. In particular, ray tracing is one of the algorithms employed in global illumination.¹ Images rendered using global illumination algorithms often appear more photorealistic than images rendered using only direct illumination algorithms.² The raytracer could be extended to further illumination of these images and more. For instance, differentiating techniques as the central difference operator and Richardson’s extrapolation can be used for light illumination application, while Riemann sums, the composite trapezoid rule, and Romberg’s algorithm to integrate functions.

There are two sets of interesting images displayed as sample results for a computer science lab experiment metalball raycaster from a link from Ohio State University ⁴. First, the site displays visually comparable smiling faces rendered by using different derivative approximation rendering methods, it would be interesting to take a look at the same smiling face produced using the methods: Analytical, Central Differences, Richardson’s Extrapolation with 1, 2, and 3 iterations respectively:

http://www.cse.ohio-state.edu/~meyerb/report3/⁴

As one scroll down, you will also see the distinctive images of teddy bears allow us to see the different effects x-rays have on metaballs, when using Riemann sum, contributed by different rendering methods: composite trapezoid approximation, and Romberg’s Extrapolation approximation respectively. The pictures with comments are pretty self-explanatory. Wish you find them fun. =)

Finally, as a practical example of a new method to render images, please refer to the reference link from Freepatentsonline.com ³ for detailed descriptions of a patent method of simulating the appearance of smoke in an electronically generated video image. Many people in this class might come up with patent methods as well. =)

References:
1) http://en.wikipedia.org/wiki/Ray_tracing
2) http://en.wikipedia.org/wiki/Global_illumination
3) http://www.freepatentsonline.com/6184857.html
4) http://www.cse.ohio-state.edu/~meyerb/report3/