Sparse Image and Signal Processing: Multiscale Geometric Analysis on the Sphere

INTRODUCTION Many wavelet transforms on the sphere have been proposed in past years. Using the lifting scheme, Schroder and Sweldens (1995) developed an orthogonal Haar wavelet transform on any surface, which can be directly applied on the sphere. Its interest is, however, relatively limited because of the poor properties of the Haar function and the problems inherent to orthogonal transforms. More interestingly, many papers have presented new continuous wavelet transforms (Antoine 1999; Tenorio et al. 1999; Cayon et al. 2001; Holschneider 1996). These works have been extended to directional wavelet transforms (Antoine et al. 2002; McEwen et al. 2005). All these continuous wavelet decompositions are useful for data analysis but cannot be used for restoration purposes because of the lack of an inverse transform. Freeden and Windheuser (1997) and Freeden and Schneider (1998) proposed the first redundant wavelet transform, based on the spherical harmonics transform, which presents an inverse transform. Starck et al. (2006) proposed an invertible isotropic undecimated wavelet transform (IUWT) on the sphere, also based on spherical harmonics, which has the same property as the starlet transform; that is, the sum of the wavelet scales reproduces the original image. A similar wavelet construction (Marinucci et al. 2008; Fay and Guilloux 2008; Fay et al. 2008) used the so-called needlet filters. Wiaux et al. (2008) also proposed an algorithm that permits the reconstruction of an image from its steerable wavelet transform.