TABLE OF CONTENTS The mathematical concept of a transform is a powerful tool that is employed in many areas and can also serve as an approach to image compression. Section 5.1 discusses this concept in general, as well as the Fourier transform. An image can be compressed by transforming its pixels (which are correlated) to a representation where they are decorrelated. Compression is achieved if the new values are smaller, on average, than the original ones. Lossy compression can be achieved by quantizing the transformed values. The decoder inputs the transformed values from the compressed stream and reconstructs the (precise or approximate) original data by applying the inverse transform. The transforms discussed in this section are orthogonal. Section 5.6.1 discusses subband transforms.
The term decorrelated means that the transformed values are independent of one another. As a result, they can be encoded independently, which makes it simpler to construct a statistical model. An image can be compressed if its representation has redundancy. The redundancy in images stems from pixel correlation. If we transform the image to a representation where the pixels are decorrelated, we have eliminated the redundancy and the image has been fully compressed.
We start with a simple example, where we scan an image in raster order and group pairs of adjacent pixels. Because the pixels are correlated, the two pixels ( x,y) of a pair normally have similar values. We now consider each pair of pixels a point in two-dimensional space, and we...
