Discrete Wavelet Transformations: An Elementary Approach with Applications

CHAPTER 6

THE HAAR WAVELET TRANSFORMATION

We are now ready to start building wavelet transformations! We have most of the tools we need. We begin with a very simple lowpass filter and an associated highpass filter. Since both can be represented by matrices, we start there. We combine these matrices and after some more straightforward modifications we essentially have our transformation. What’s more, the steps we take are utilized in subsequent chapters — we just start with a different lowpass and highpass filter pair(s).

Hopefully, you have an idea of what obstacles we must overcome to build a nice transformation to use in applications. We must deal with the invertibility issue as well as the fact that on a computer, there are no infinite-length sequences — we must think about truncating the matrix we build. Ultimately, it is our goal to produce a transformation that is very useful in applications — toward this end, we insist that our transformation will effectively decompose the input data.

We motivate our basic construction in the next section. In Section 6.2 we fine-tune the transformation so that it is even more useful in applications. Our work in this chapter is to construct a matrix that we can apply to a finite-length sequence or vector. In Section 6.3 we learn that it is very easy to generalize the ideas of the previous two sections to build a two-dimensional transformation. The immediate application here is to apply a two-dimensional transformation to images. We conclude ...