Mathematical Structures for Computer Graphics

Chapter 1Basics

It is rather amazing that a finite rectangular array of colored dots (called pixels as an abbreviation of picture elements) is sufficient to display the nearly limitless collection of images we recognize as realistically or symbolically representing portions of our world. The power of combinatorics helps us to explain the situation (millions of possible colors for each pixel in the large display array), but we can hardly conceive of all the images we have already seen let alone those that are yet to be seen. From this reductionist viewpoint, the whole idea of computer graphics is to set the right pixels to the right color. Easier said than done. Yes, a plain red square is easy, but one that looks like it is made of bricks is tougher, and one that includes a human face taxes the best of known algorithms.

Of course, the computer graphics enterprise includes any and all manipulations of images. We can start from scratch and produce a photo-realistic image of a new airliner or perhaps construct a landscape design complete with a variety of plants. Maybe the challenge is to translate CAT (computerized axial tomography) scan data into an image of the brain or correct the color balance in a photo being readied for publication. To bring some order to the very long list of possibilities, it is helpful to consider two main categories: either we are generating images, or we are processing existing images. Both require mathematical tools, but the first category encompasses ...