TABLE OF CONTENTS Chapter 1 has given us a brief overview of the basic facets of the CWT in the simpler one-dimensional (1-D) case, including its relationship with the various discrete approaches and a glimpse of some applications. Now it is time to enter the proper subject of the book, namely, the two-dimensional (2-D) wavelet analysis.
In 1-D, the CWT (1.8) amounts to projecting the signal onto the wavelet ? b,a, obtained by translation and dilation of the mother wavelet ?. Thus the transform is fully determined by these elementary operations of the line. Accordingly, in order to derive the CWT in 2-D, a good starting point is to consider first the elementary operations we want to apply to our signals. Actually, as we will see later (Chapter 6), this point of view allows one to extend the CWT to much more general situations, such as wavelets in higher dimensions, wavelets on the sphere, time-dependent wavelets, etc.
By an image, we mean a two-dimensional signal of finite energy, represented by a complex-valued function defined on the real plane 
2 and square integrable, i.e., a function s ? L 2( 2 , d 2 
):
(sometimes it is useful to take s integrable as well). In practice, a black and white image will be represented by a bounded non-negative function:
the discrete values of s( ) corresponding to the level of gray of each pixel. However it is useful...
