INTRODUCTION

The ridgelet and curvelet transforms generalize the wavelet transform. Firstly they incorporate angular alignment information, and then in addition length of the alignment is covered. As with all of these transforms, multiple scales are supported. The motivation for these transforms is to build up an image from edge-related building blocks. Furthermore, as in previous chapters, the efficiency of computing these transforms is an important practical aspect.

In this chapter we consider the ridgelet transform and a number of algorithms for its implementation. Then we proceed to the curvelet transform and algorithms for it.

BACKGROUND AND EXAMPLE

Wavelets rely on a dictionary of roughly isotropic elements occurring at all scales and locations. They do not describe well highly anisotropic elements, and contain only a fixed number of directional elements, independent of scale. Despite the fact that they have had wide impact in image processing, they fail to efficiently represent objects with highly anisotropic elements such as lines or curvilinear structures (e.g. edges). The reason is that wavelets are nongeometrical and do not exploit the regularity of the edge curve. Following this reasoning,new constructions have been proposed such as ridgelets (Candès and Donoho 1999) and curvelets (Candès and Donoho 2001, 2002; Starck et al. 2002).

Ridgelets and curvelets are special members of the family of multiscale orientation-selective transforms, which have recently led to a flurry of research activity in the field of computational and applied harmonic analysis. Many other constructions belonging to this family have been investigated, and go by the name of contourlets (Do and Vetterli 2003b), directionlets (Velisavljevic et al. 2006), platelets (Willett and Nowak 2003), bandlets (Le Pennec and Mallat 2005; Peyre and Mallat 2007), grouplets (Mallat 2009), shearlets (Labate et al. 2005), dual-tree complex wavelet transform (Kingsbury 1998; Selesnick et al. 2005) (see Section 3.4), and other complex directional wavelet transforms (Fernandes et al. 2003; van Spaendonck et al. 2003; Fernandes et al. 2004).

The ridgelet and the curvelet (Candès and Donoho 1999, 2002) transforms were developed as an answer to the weakness of the separable wavelet transform in sparsely representing what appears to be simple building-block atoms in an image, that is, lines, curves and edges. Curvelets and ridgelets take the form of basis elements which exhibit high directional sensitivity and are highly anisotropic (Donoho and Duncan 2000; Candès and Donoho 2002; Starck et al. 2002).