iSAP

Data on the sphere

For denoising and deconvolution, wavelets have alsodemonstrated how powerful they are for discriminating signal from noise. In this chapter, the Section 2.2 overviews the problem of pixelization on the sphere and introduces the two solutions used by MRS package: the HEALPix pixelization scheme 2.2.1 and Gauss-Legendre Sky Pixelization (GLESP) 2.2.2.The Section 2.3 introduces the spherical harmonics transform which could be considered as an extension of Fourier’s transform to the sphere. Section 2.4 is a short introduction to multiscale methods on the sphere and to their applications. These methods are fully presented in Chapter 3 and algorithm on the sphere bases on them are given in Chapter 4 for data restoration, in Chapter 5 for sparse component analysis and in Chapter 6 for blind source separation. The Chapter 7 is dedicated to statistics on the sphere which includes the detection of non-gaussianities. Sections 5.2.3 and 5.3.2 present how these new tools can help us to analyze data in two real applications, in physics and in cosmology.

2.4 Multiscale methods on the sphere

• Many wavelet transforms on the sphere have been proposed in the past years. Using the lifting scheme (Schr¨oder and Sweldens 1995) developed an orthogonal Haar wavelet trans-form 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; Cay´on et al. 2001a; 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 in-vertible isotropic undecimated wavelet transform (UWT) on the sphere, also based on spherical harmonics, which has the same property as the isotropic undecimated wavelet transform, i.e. the sum of the wavelet scales reproduces the original image. A similar wavelet construction (Marinucci et al. 2008; Fa¨y and Guilloux 2008; Fay¨ et al. 2008) used the so-called needlet filters. (Wiaux et al. 2008) also proposed an algorithm which per-mits to reconstruct an image from its steerable wavelet transform. Since reconstruction algorithms are available, these new tools can be used for many applications such as de-noising, deconvolution, component separation (Moudden et al. 2005; Bobin et al. 2008; Delabrouille et al. 2008) or inpainting (Abrial et al. 2007; Abrial et al. 2008).

• In this area, further insight will come from the analysis of full-sky data mapped to the sphere thus requiring the development of a curvelet transform on the sphere. The MRS package offers an implementation of ridgelet and curvelet transforms for spherical maps.Those implementations are derived as extensions of the digital ridgelet and curvelet transforms described in (Starck et al. 2002a). The implemented undecimated isotropic wavelet transform on the sphere and the specific geometry of the Healpix sampling grid are important components of the present implementation of curvelets on the sphere.

5.3 Inpainting on the sphere

5.3.1 Algorithm

• Inpainting algo-rithms strive to interpolate through the gaps in the image relying on the available pixels, the continuation of edges, the periodicity of textures, etc.

• To make the link between building sparse representations and inpainting, consider the effect of a rectangular gap on the set of Fourier coefficients of a monochromatic sinewave : because of the non-locality of the Fourier basis functions it takes a large number of coefficients to account for the gap, which is known as the Gibbs effect.

• Following (Elad et al. 2005), an inpainting algorithm on the sphere is readily built from the Morphological Component Analysis on the sphere described in the previous section.

mrs-alm-inpainting -v input mask output