Transformation is the art of calculating a bunch of numbers from another bunch of numbers.

    Why might one bother doing that? Why is that an interesting thing to do? These are legitimate questions. I will address them by way of an example.

    Suppose that we are given a digital picture. A digital picture is nothing more than a bunch of numbers. These numbers are used by a computer to decide what colors to paint on a screen such that the picture becomes visible.

    A typical picture that we see on our screen, say in the Visual Arts section of Kahany, is made up of about half-a-million numbers. Suppose that we wish to write down that picture as a bunch of numbers on a sheet of paper. Not only are half-a-million numbers too many to write, they will not even fit on one sheet of paper.

    If you are as lazy as I am, you might let your personal djinn out of his bottle and say: Look here, I wish to write no more than 10 numbers. None of these numbers should have more than two digits. I want to send these numbers to my friend in Timbuktoo, and I want her to use them to build up a similar sort of picture on her computer screen. What should those ten numbers be?

    You have just asked your djinn to perform a transformation.

    People given to mathematics, physics, and engineering use transforms every day. They use transforms that go by the names of Fourier, Laplace, Legendre, wavelet, and so on.

    They use them to study abstract mathematical spaces. They use them to solve partial differential equations. And they use them to "compress" images, video, and audio --- which is exactly what I do for a living. It is especially true of electrical engineering that one cannot go very far without some understanding of transformation.

    The main body of the article that follows this prologue must of necessity use mathematical notation. It is accessible only to people who have some understanding of set theory and calculus. Other than that it is my intention to provide the simplest possible explanation of the process of transformation.

    To go to the main body of the article, please click here (204 Kilobytes).

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