The geometric manipulations we have in mind are applied to data in the form of photographic images, and we carry out the manipulations using MATLAB or perhaps another programming language. We require that the software can open an image file as an array of numbers that can be manipulated. This initial step is complicated by the fact that there are several "modes" of images and several common file types. In order to get quickly to my main focus, I will assume that the reader has an image in RGB mode that is saved as a JPEG file. (RGB means "red, green, and blue" and indicates that the image is a standard color image. JPEG stands for Joint Photographic Experts Group and is a standard image compression routine.) Most images that come from the internet or from a digital camera are apt to be in that form already. And, if not, then many graphics applications that you might use to crop a photo or to adjust the brightness and contrast also allow you to put the image into RGB mode and to save it as a JPEG file.
An RGB image can be thought of as a rectangular array of "elements", one for each pixel of the image, where each element is a triple of numbers representing the red, green, and blue intensities of the pixel. The numbers used in each triple are typically required to be in the interval , with representing black and white. However, just to complicate matters a little more, it seems to be a common convention that the numbers are stored as 8-bit integers from 0 to 255. These integers are ultimately divided by 255 to yield values in .
The JPEG compression routine has little impact in this discussion -- once the image is read into the computer's memory as an array, the array size is the same whether or not there had been any compression in the stored file. Thus, a shot from a 3 megapixel digital camera may yield a 1536 by 2048 array, which is saved as a compressed JPEG file of only 1 megabyte. But when the file is opened in MATLAB it becomes, once again, an array of the original size. By the way, a supercomputer may be needed to work with image files of that size in MATLAB -- let's experiment first with arrays that are just a few hundred pixels on a side.