Des perspectives nombreuses et variées s’inscrivent dans la continuation de ce travail. Notre méthode d’estimation spectrale à partir des coefficients de needlet a été testée sur des simulations intégrant une grande partie de la complexité du modèle physique. La prochaine étape est naturellement de l’appli- quer aux données disponibles issues des observations. Il sera possible d’utiliser simultanément l’en- semble des données publiques, ce qui devrait permettre de préciser les valeurs du spectre de puissance actuellement estimées. En dehors d’un important travail technique de préparation des données, cette application nécessitera d’un point de vue méthodologique la prise en compte des multiples petites zones de données corrompues qui criblent le ciel (voir la figure 1.12). Ces zones (qui correspondent physiquement à des sources ponctuelles d’émission lumineuse) pourraient être traitées exactement comme nous l’avons fait dans le cas de la séparation de sources (chapitre 8).
Concernant toujours les applications, l’utilisation de needlets optimalement localisées est envisa- geable pour d’autres analyses statistiques du CMB : séparation des différentes sources (autres que le CMB), tests de gaussianité et d’isotropie. Leur utilisation dans d’autres domaines faisant intervenir l’analyse de champs sphériques (comme la géophysique) pourrait aussi s’avérer fructueuse.
Du point de vue théorique, les résultats obtenus demandent à être approfondis et généralisés : au- delà de la convergence, il est nécessaire – et probablement peu coûteux en s’appuyant les résultats de Baldi et al. [151] – d’obtenir distributions asymptotiques et vitesses de convergence (toujours au sens inhabituel de la définition 4.1) des estimateurs proposés. Il serait intéressant en outre de les comparer sous cet angle aux autres estimateurs utilisés en pratique (par exemple ceux présentés dans la section 3.2), et d’étudier leur efficacité asymptotique.
1Logiciel libre semblable à Matlab. http://www.gnu.org/software/octave 2Hierarchical Equal Area isoLatitude Pixelization. http://healpix.jpl.nasa.gov 3Gauss-Legendre Sky Pixelization. http://www.glesp.nbi.dk
4Spherical Harmonics Tools. http://www.ipgp.jussieu.fr/~wieczor/SHTOOLS/SHTOOLS.html 5http://www.apc.univ-paris7.fr/APC_CS/Recherche/Adamis/Outreach/spherelab-fr.php 6nside≤ 512 avec HEALPix, permettant le calcul des coefficients de Fourier pour ℓ ≤ 1500
Jusqu’à présent, on a supposé le bruit indépendant de point à point, et le niveau de bruit σ(ξ) connu. Une généralisation intéressante serait de s’affranchir de ces hypothèses : modéliser la dépendance du bruit dans un modèle discret ou continu, et en estimer le niveau à partir des observations.
S’agissant des recherches à venir dans le domaine de l’analyse sur la sphère, deux aspects paraissent importants. Le premier est une comparaison plus systématique des différentes constructions d’on- delettes sphériques. Cela permettrait peut-être de trouver des conditions plus générales assurant les qualités de frame et de localisation. Par exemple, Geller et Mayeli [21] ont récemment construit des needlets “presque ajustées” (au sens des frames) à décroissance exponentielle. Le second est la géné- ralisation des needlets à des champs aléatoires sphériques de matrices symétriques (au lieu de fonc- tions ou champs aléatoires à valeurs réelles). De telles champs matriciels correspondent en effet à l’observation de la polarisation du Fond diffus cosmologique, qui est l’une des principales avancées observationnelles attendues dans les prochaines années.
Deuxième partie
Chapitre 5
Practical wavelet design on the sphere
Ce chapitre reproduit l’article du même nom, F. Guilloux, G. Faÿ, J.-F. Cardoso, Applied and Computational Harmonic
Analysis (à paraître)∗. Seules certaines notations ont été modifiées pour la cohérence du manuscript.
Abstract. This paper considers the design of isotropic analysis functions on the sphere which are per- fectly limited in the spectral domain and optimally localized in the spatial domain. This is motivated by the need of localized analysis tools in domains where the data is lying on the sphere, e.g. the science of the Cosmic Microwave Background. Our construction is derived from the localized frames introduced by Narcowich, Petrushev et Ward [33]. The analysis frames are optimized for given appli- cations and compared numerically using various criteria.
Sommaire
Introduction . . . 81 5.1 Needlets frames . . . 82 5.1.1 Notations and background . . . 82 5.1.2 Needlet frames . . . 83 5.1.3 Our setting . . . 86 5.1.4 Practical computation of needlet coefficients . . . 87 5.2 Design of optimally localized wavelets . . . 87 5.2.1 Examples. . . 88 5.2.2 L2-concentration and variations . . . 90 5.2.3 Statistical criterion for optimal analysis with missing data . . . 93 5.3 Examples, numerical results . . . 96 5.3.1 Comparison of filters for various criteria . . . 96 5.3.2 Robustness of needlets coefficients . . . 97 5.3.3 Some MISE-optimal filters for axisymmetric weight functions . . . 97 5.4 Conclusion . . . 100 5.5 Proofs . . . 101
Introduction
Data defined on the sphere are found in numerous application domains. During the past decade, localized analysis for spherical data has motivated many developments, whether it be in cosmo- logy [28, 133], geophysics [184, 51, 15, and the references therein], biomedical engineering [see e.g. 191, and the references therein], computer vision, hydrodynamics, etc. When the data are avai- lable on the whole sphere, they can be analyzed by projection onto the spherical harmonics, with the spherical harmonic transform (SHT) providing the spherical equivalent of the Fourier Series on the circle. But in many practical situations, data are defined or available only on a fraction of the sphere. For example, cosmologists try to give sharp estimates of the cosmic microwave background (CMB) or its power spectrum but strong localized foreground emissions superimpose to the CMB making it unreliable to use the full sphere for accurate spectral estimation. Even when data are available over the whole sphere, spatially localized tools are invaluable to process non stationary fields or stationary fields corrupted by non-stationary noise. In such situations, spherical harmonics are not adequate due to their lack of spatial localization. A mathematically elegant solution of proven practical efficiency is provided by multiscale and wavelet theories. These are well developed for functions defined on Euclidean spaces ; some practical aspects of their extension to the sphere is the focus of this paper. Adaptation to the sphere of the “wavelet” transform (in the broad sense of filtering by spatially and spectrally localized functions) was introduced a dozen years ago [39, 46, 7, 34, 38, 20]. Since then, Antoine and Vandergheynst [5] showed that any Continuous Wavelet Transform (CWT) on the sphere can be viewed locally as a regular CWT on the tangent plane, thanks to the stereographic correspon- dence between the sphere and the plane [5, 50]. A discretized version of this CWT has been presented by [6], leading to wavelet frames. This approach has already been followed in astrophysics for the analysis of the CMB [see e.g. 142, 28]. The stereographic projection on the tangent plane helps to identify the similitude group on the sphere and derive the CWT in a group-theoretical manner but in practice, the whole analysis is performed directly on the sphere. In the above-mentioned references, the spherical wavelets that are given as examples or are used in the real data analysis are defined on the tangent plane and transformed (once for all) to the sphere by the stereographic projection. For instance, the Mexican hat wavelet family is constructed from the Gaussian density function on the R2-plane ; there is no deep rationale for using it as a spherical function. Moreover, these wavelets are defined in the spatial domain and have infinite support in the frequency domain (which must be truncated in practice).
In the present work, we follow and extend the approach of Narcowich, Petrushev and Ward [32, 33] and their construction of “needlets”. A similar construction can be found in Starck et al. [134]. The needlet transform has important distinctive properties. Firstly it is intrinsically spherical. No inter- mediate tangent plane is needed to define it. Secondly, it does not depend on the particular spherical pixelization (sampling) chosen to describe the data. Thirdly, although the needlets still have an ex- cellent spatial localization, they have a finite spectral support adjustable at will. They are axisymme- tric (which is convenient when dealing with statistically isotropic random fields) and thus the needlet coefficients are easily computed in the spherical harmonic (Fourier) domain. Filtering is obtained by multiplication of the spherical harmonic coefficients by well chosen window functions (which is equivalent to a convolution in spatial domain). Needlets are well defined in theory and the statistical properties of their coefficients have already been established for isotropic Gaussian fields by Baldi et al. [151]. However, the performance of a needlet-based analysis depends on the particular shape of the needlet.
We are concerned with the development of a flexible spectral analysis on the sphere which remains practical at high resolution. For instance the P space mission1of the European Spatial Agency
will produce CMB maps as large as 50 mega-pixel with reliable multipoles up to order ℓ ≃ 4000. This paper focuses on the design issue, namely the optimization of the window functions which define the needlet transform. We consider only band-limited needlets. This choice is motivated by applica- tions in high-precision cosmology. Indeed, the angular power spectrum of the CMB varies over a large dynamic range (power-law decay) so that using band-limited wavelets guarantees the absence of spectral leakage from the very large low-frequency components to the much weaker high-frequency components.
Strict simultaneous confinement of a needlet in space and frequency is, of course, impossible. One of the most famous expression of this fact is the Heisenberg uncertainty principle which has been extended to the sphere [34, 22]. Recently Fernandez [13] constructed band-limited wavelets which minimize the uncertainty product. However, paraphrasing Slepian [42], this criterion which is relevant to quantum mechanics may not be as meaningful as a simple Lpcriterion in our context.
In this contribution, we investigate the design of band-limited spectral window functions in two dif- ferent directions : 1) By requesting the best spatial localization of associated needlets, in an energy- sense (L2) which is easily solved. This is an application of the work of [41] which adapted to the
sphere the problem solved by Slepian [187] on the real line, giving rise to the well known prolate spheroidal wave functions (PSWF). 2) By the following statistical problem : given some region of the sphere in which the data is missing (or has to be discarded), we seek to minimize, in the needlet ana- lysis, the error introduced by the missing data. More criteria and applications to cosmological science will be given in a future work.
The paper is organised as follows. In Section 5.1, we expose the general construction of needlets. In Section 5.2, we define and optimize the two above-mentionned criteria (geometrical and statisti- cal), yielding needlets which are optimal in some well-defined sense. Their efficiency is illustrated in Section 5.3 with numerical simulations following the model of a masked observation of the CMB. Conclusions are given in Section 5.4 and proofs are postponed to Section 5.5.