Most large celestial objects are found to be approximately spherical in shape, e.g. moons, planets and stars. This is because the cause and the dominating force in cosmic structure formation is gravity: a force which pulls mass in all directions equally. Another sphere is the celestial sphere – the heavens above us upon which we observe astronomical observations. This means that data collected on planetary surfaces or observed on in the sky live natively on the sphere.
Hence, as we consider a wider-and-bigger picture (or technically speaking, large field-of-view observations or a whole-sky/whole-Earth survey, where Euclidean limit no longer holds), it is essential to account for this underlying geometry of the data for accurate data analysis; the most effective analysis techniques for signals that are naturally defined on the sphere are those that respect this spherical geometry.
Moving on from the global geometry of the data, one may want to probe different signal content depending on the problems at hand. An example is to extract filamentary structures as seen in Fig. 1, a whole-sky map of the polarised interstellar dust emission obtained by ESA’s Planck satellite. These structures are localised, scale-varying, directional, and exhibit geometrical features and often serve as a key diagnostic in astrophysics, helping us to identify and track down physical processes and properties of cosmic media involved in the “life journey” of the cosmic messenger (photon in this case). The next questions are then how do we extract them? And can we do it all in one go? That is, can we simultaneously extract both position and scale information, as well as directional and geometric information of the signals in a self-consistent and an efficient manner? Curvelets, a special type of wavelets designed for such a purpose may serve as the tool. They were firstly introduced by Candes and Donoho in 1999, and by design satisfy a parabolic scaling relation (width ≈ length²) to efficiently represent highly anisotropic signal content. To deal with spherical data sets as mentioned above, we want curvelets defined on the sphere.
During the first-year of my PhD, I worked with Dr Boris Leistedt, Dr Thomas Kitching and Dr Jason McEwen on developing second-generation (spin) curvelets on the sphere. Our second-generation curvelets have many desirable properties that are lacking in first-generation constructions: they live directly on the sphere, exhibit the parabolic scaling relation typical to curvelets, are well localised in both spatial and harmonic domains, support exact analysis and synthesis (i.e. capture all of the information content of the signal without loss) of both scalar signals (e.g. intensity) and spin signals (e.g. spin-1 for vector fields, spin-2 for polarisation signals and cosmic shear), and are free of blocking artefacts (since it is not reliant on a specific pixelisation such as HEALPix). Fig. 2 and 3 show examples of the constructed curvelets. An interesting point to notice is how the geometric feature of curvelets, afforded by the satisfaction of their characteristic (width ≈ length²) relation, makes them highly anisotropic and directional sensitive, thereby providing an efficient representation of linear and curvilinear structures (e.g. edges) within signals. Fig. 4 shows an example application where we decomposed the global image of Jupiter (obtained using the Hubble Space Telescope, by the NASA’s Goddard Space Flight Center, the Jet Propulsion Laboratory) using scalar curvelets and scaling functions. Curvelets probe the high-frequency content of the signal and the scaling function probe the low-frequency content, i.e. the approximate information, of the signal not probed by curvelets. Notice how curvelets pick up various localised directional features on the surface of Jupiter.
Useful but not only limited to the applications in astrophysics and cosmology, our constructed curvelet transform is a general tool that is applicable to data acquired on or mapped onto the sphere, and a powerful tool to detect elongated structures. It may therefore find wide applications in various disciplines e.g. remote-sensing analysis, medical imaging and computer vision. Fig. 5 illustrates the effectiveness of how curvelets are highly sensitive to edges in the spherical image of a natural scene, the Uffizi Gallery in Florence (image data from Paul Debevec, 1998). Such an advantage can be further exploited in image compression and processing, e.g. denoising and inpainting.
The best analysis tool to adopt always depends on the problem to solve. Now added to the toolbox we have curvelets, which are useful for efficient representation of elongated structures in signals of arbitrary spin acquired on the sphere. Spherically-cool (i.e. cool in every direction)!
More details of this work can be found in our paper recently published in the IEEE Trans. Sig. Proc. “Second-Generation Curvelets on the Sphere”, and available on arxiv:1511.05578. The code for our fast and exact curvelet transform is made publicly available at s2let.org. A poster on this work can also be found here.
By Jennifer Chan, a PhD student at the lab working on large-scale astrophysics and cosmology, with a focus on investigating how large-scale magnetic fields evolve and how cosmic reionisation proceeds in a magnetised Universe.