Digital Signal Processing Reference
In-Depth Information
Section 10.3 shows how a fast orthogonal Haar wavelet transform on the sphere
can be built using HEALPix. In Section 10.5, we present an isotropic wavelet trans-
form on the sphere that has similar properties as the starlet transform and there-
fore should be very useful for data denoising and deconvolution. This algorithm
is directly derived from the fast Fourier transform (FFT)-based wavelet transform
proposed by Starck et al. (1994) for aperture synthesis image restoration (see Sec-
tion 3.7.2) and is relatively close to the Freeden and Schneider (1998) method,
except that it features the same straightforward reconstruction as does the starlet
transform algorithm (i.e., the sum of the scales reproduces the original data). This
wavelet transform can also be easily extended to a pyramidal wavelet transform on
the sphere (PWTS), allowing us to reduce the redundancy, a possibility that may be
very important for larger data sets. In Section 10.6, we show how this new pyramidal
transform can be used to derive a curvelet transform on the sphere. Section 10.7 de-
scribes how these new transforms can be used for denoising, component separation,
and inpainting.
Sections 10.8.1 and 10.8.2 present how these new tools can help us analyze data
in two real applications in physics and in cosmology. Finally, guided numerical ex-
periments, together with a toolbox dedicated to multiscale transforms on the sphere
(MR/S), are described.
10.2 DATA ON THE SPHERE
Various pixelization schemes for data on the sphere exist in the literature.
These include the equidistant coordinate partition (ECP), the icosahedron method
(Tegmark 1996), the quad cube (White and Stemwedel 1992), IGLOO (Crittenden
2000), HEALPix (Gorski et al. 2005), hierarchical triangular mesh (HTM) (Kunszt
et al. 2001), and Gauss-Legendre sky pixelization (GLESP) (Doroshkevich et al.
2005). Important properties to decide which is best for a given application include
the number of pixels and their size; fast computation of the spherical harmonics
transform; equal surface area for all pixels; pixel shape regularity; separability of
variables with respect to latitude and longitude; availability of efficient software li-
braries, including parallel implementation; and so on. Each of these properties has
advantages and drawbacks. In this chapter, we use the HEALPix representation,
which has several useful properties.
10.2.1 HEALPix
The HEALPix representation (Gorski et al. 2005)
1
is a curvilinear hierarchical par-
tition of the sphere into quadrilateral pixels of exactly equal area but with vary-
ing shape. The base resolution divides the sphere into 12 quadrilateral faces of
equal area placed on three rings around the poles and equator. Each face is subse-
quently divided into
N
side
pixels following a quadrilateral multiscale tree structure
(see Fig. 10.1). The pixel centers are located on isolatitude rings, and pixels from
the same ring are equispaced in azimuth. This is critical for computational speed
of all operations involving the evaluation of the spherical harmonics coefficients,
1
See http://healpix.jpl.nasa.gov.
