INTRODUCTION

With the increasing computing power and memory storage capabilities of computers, it has become feasible to analyze 3-D data as a volume. Among the most simple transforms extended to 3-D are the separable wavelet transform (decimated, undecimated, or any other kind) and the discrete cosine transform (DCT), because these are separable transforms and thus the extension is straightforward. The DCT is mainly used in video compression, but has also been used in denoising (Rusanovskyy and Egiazarian 2005). As for the 3-D wavelets, they have already been used in denoising applications in many domains (Selesnick and Li 2003; Dima et al. 1999; Chen and Ning 2004).

However these separable transforms lack the directional nature that has facilitated the success of 2-D transforms such as curvelets. Consequently, a lot of effort has been made in recent years to build sparse 3-D data representations that better represent geometrical features contained in the data. The 3-D beamlet transform (Donoho and Levi 2002) and the 3-D ridgelet transform (Starck et al. 2005a) were, respectively, designed for 1-D and 2-D feature detection.Video denoising using the ridgelet transform was proposed in Carre et al. (2003). These transforms were combined with 3-D wavelets to build BeamCurvelets and RidCurvelets (Woiselle et al. 2010), which are extensions of the first-generation curvelets (Starck et al. 2002). Whereas most 3-D transforms are adapted to plate-like features, the BeamCurvelet transform is adapted to filaments of different scales and different orientations. Another extension of the curvelets to 3-D is the 3-D fast curvelet transform (Ying et al. 2005),which consists in of paving the Fourier domain with angular wedges in dyadic concentric squares, using the parabolic scaling law to fix the number of angles depending on the scale; it has atoms designed for representing surfaces in 3-D. The Surflet transform (Chandrasekaran et al. 2004) – a d-dimensional extension of the 2-D wedgelets (Donoho 1999; Romberg et al. 2002) – has been studied for compression purposes (Chandrasekaran et al. 2009). Surflets are an adaptive transform estimating each cube of a quad-tree decomposition of the data by two regions of constant value separated by a polynomial surface. Another possible representation uses the Surfacelets developed by Lu and Do (2005). It relies on the combination of a Laplacian pyramid and a d-dimensional directional filter bank.