The reader might want to jump right into the book, but I decided to give a guided tour (which one may leave and rejoin at will of course) through the chapters, to whet the reader's taste.

Antoine opens in Chapter 1 with a brief survey of the basic properties of wavelet transforms, both continuous (CWT) and discrete (DWT). In the latter case one learns about the intuitively very appealing concept of multiresolution analysis. Section 1.4 looks ahead to the two- and more-dimensional versions, and summarily brings out connections with well known symmetry groups of physics, and the theory of coherent states.

In the second chapter, also by Antoine, the 2-D wavelet transform is treated. Here the characterization as mathematical microscope must be further qualified, because it misses the new and important property of orientability of the 2-D wavelets, which the 1-D case lacks. A real-world microscope is not more sensitive in one direction than in another one, it is ‘isotropic’. But the mathematical microscope as embodied in 2-D wavelets has an extra feature: these wavelets can be designed in such a way that they are directionally selective. Apart from dilation and translation, one can now also rotate the wavelet, which makes possible a sensitive detection of oriented features of a signal (a 2- D image). In many texts the 2-D case is still limited to the DWT, and the wavelets are usually formed by taking tensor products of 1-D wavelets in the x and y-direction, thereby giving preference to horizontal, vertical and diagonal features in the plane.