Digital Signal Processing Reference
In-Depth Information
10
Multiscale Geometric Analysis
on the Sphere
10.1 INTRODUCTION
Many wavelet transforms on the sphere have been proposed in past years. Using
the lifting scheme, Schr oder 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 trans-
forms (Antoine 1999; Tenorio et al. 1999; Cay on 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 Schnei-
der (1998) proposed the first redundant wavelet transform, based on the spherical
harmonics transform, which presents an inverse transform. Starck et al. (2006) pro-
posed an invertible isotropic undecimated wavelet transform (IUWT) on the sphere,
also based on spherical harmonics, which has the same property as the starlet trans-
form; that is, the sum of the wavelet scales reproduces the original image. A similar
wavelet construction (Marinucci et al. 2008; Fa y and Guilloux 2008; Fa y 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. Since
reconstruction algorithms are available, these new tools can be used for many ap-
plications such as denoising, deconvolution, component separation (Moudden et al.
2005; Bobin et al. 2008a; Delabrouille et al. 2008), and inpainting (Abrial et al. 2007;
Abrial et al. 2008).
Extensions to the sphere of two-dimensional (2-D) geometric multiscale decom-
positions, such as the ridgelet transform and the curvelet transform, were presented
by Starck et al. (2006).
The goal of this chapter is to overview these multiscale transforms on the
sphere. Section 10.2 overviews the hierarchical equal area isolatitude pixelization
(HEALPix) of a sphere pixelization scheme and the spherical harmonics transform.
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