Discrete Wavelet Transformations: An Elementary Approach with Applications

CHAPTER 5

CONVOLUTION AND FILTERS

We are interested in processing signals. These signals might come in the form of images or digital sound files. The reason we process signals is so that we can perform tasks such as data compression, image enhancement and audio denoising. A fundamental tool for processing signals is convolution.

A convolution product of two sequences of numbers (for the time being we will think of our signals as bi-infinite sequences) h and x results in a new sequence y. Although we can give a formal definition of convolution and we will see that we can convolve any two sequences h and x we desire, it is practical to think of x as input data and h as a processor or filter of the input data x.

In this chapter we define the convolution operator * and become familiar with it. We then develop some basic properties of convolution and next, study the interplay between the convolution with filter h and the Fourier series H () for h. We continue by defining two special types of filters and how they can combine to produce a filter bank. We conclude the chapter by illustrating how to represent convolution as a matrix and discuss the idea of inverting or deconvolving the filter process.

Note: All sequences and filters are denoted using bold lowercase letters.

5.1  CONVOLUTION

In Chapter 2 we reviewed some basic ideas involving vectors. Vector addition and subtraction are ...