Pierre Vandergheynst

2-D Cauchy wavelets and symmetries in images

We recall the definition and basic properties of 2-D directional wavelets. We then focus on a particular class of wavelets, namely the Cauchy wavelets introduced by Antoine et al. (1996), and give several examples of its use in detecting and handling symmetries in images. We also compare these results with those obtained using the well-known 2-D Morlet wavelet. These directional wavelets can be used to evaluate the global symmetry of an object or to measure its intrinsic orientation.

Two-Dimensional Wavelets and their Relatives: Applications of the 2-D CWT. II: physical applications

In the previous chapter, we have discussed a number of applications of the 2-D CWT that belong essentially to the realm of image processing. Besides these, however, there are plenty of applications to genuine physical problems, in such diverse fields as astrophysics, geophysics, fluid dynamics or fractal analysis. Here the CWT appears as a new analysis tool, that often proves more efficient than traditional methods, which in fact rarely go beyond standard Fourier analysis. We will review some of these applications in the present chapter, without pretention of exhaustivity, of course. Our treatment will often be sketchy, but we have tried to provide always full references to the original papers. Astronomy and astrophysics Wavelets and astronomical images Astronomical imaging has distinct characteristics. First, the Universe has a marked hierarchical structure, almost fractal. Nearby stars, galaxies, quasars, galaxy clusters and superclusters have very different sizes and live at very different distances, which means that the scale variable is essential and a multiscale analysis is in order, instead of the usual Fourier methods. This suggests wavelet analysis. Now, the main problem is that of detecting particular features, relations, groupings, etc., in images, which leads us to prefer the continuous WT over the discrete WT. Finally, there is in general no privileged direction, nor particular oriented features, in astrophysical images.