Signal analysis on the sphere

Data are measured on the surface of a sphere in fields as diverse as computer graphics, computer vision, geophysics, planetary science, molecular biology, acoustics, and astrophysics, to name only a few. As soon as observations are made over directions, the resulting data naturally live on the sphere. However, the majority of informatics and signal processing techniques developed to date are restricted to Euclidean space. These informatics techniques have proved exceptionally useful in many areas of engineering and physics; however, they cannot at present be applied to the large variety of data-sets defined on the sphere. To realise the benefits of informatics techniques on spherical data-sets, we will extend Euclidean informatics techniques to the sphere, focusing on three areas of fundamental theoretical and practical importance: namely, sampling theory, wavelet transforms, and techniques to solve inverse problems on the sphere.

The Nyquist-Shannon sampling theory is a seminal result in information theory, describing how to capture all of the information content of a band-limited signal from a finite number of samples. From an information theoretic perspective, the number of samples required to capture the information content of a signal is the fundamental property of a sampling theorem. Sampling theory on the sphere is less mature than in Euclidean space. Very recently McEwen developed a new sampling theorem on the sphere that reduces the spherical Nyquist rate by a factor of two compared to the previous canonical sampling theorem developed by Driscoll & Healy in 1994. We will extend this result to the space of three-dimensional rotations defined by the rotation group SO(3), often parameterised by the Euler angles. This will reduce Nyquist sampling of signals defined on the rotation group by a factor of two. Furthermore, we will develop fast and exact algorithms to compute the Fourier transform of signals defined on the rotation group, the so-called Wigner transform.

Wavelets are a powerful signal analysis tool due to their ability to localise signal content in scale and position simultaneously. McEwen recently constructed exact wavelet transforms on the sphere to perform a directional analysis of scalar functions defined on the sphere. At present a wavelet transform capable of performing a directional analysis of spin signals on the sphere, such as polarised light, does not exist. We will construct such a wavelet framework and will develop fast and exact algorithms, based on our fast Wigner transforms, to apply this wavelet transform to big spherical data-sets.

A sampling theorem and sparse decompositions like those afforded by a wavelet transform are the building blocks of the revolutionary new paradigm of compressive sensing. In compressive sensing, the sparsity of natural signals (in an efficient representation) is exploited to recover a signal from fewer measurements than typical by solving an inverse problem. Encouraged by this theory, sparse regularisation techniques to solve inverse problems have recently found widespread application and shown considerable promise. We will develop a generic, flexible and coherent framework for solving inverse problems on the sphere by promoting sparsity, exploiting our novel sampling theory and wavelet transforms described above.

Our research programme targets areas designated by EPSRC as priorities for growth due to their strategic economic and societal benefits to the UK. Specifically, we focus on digital signal processing, which has implications for the digital economy. In the era of big-data, teasing useful information out of big data-sets is of paramount importance in many industries. We address precisely this challenge, focusing on spherical data-sets that are found in a wide variety of applications.

Diffusion magnetic resonance imaging (dMRI) is a non-invasive medical imaging modality commonly used to probe the structure of white matter in the brain. However, clinical use of diffusion MRI is limited due to very long acquisition and processing times. In each voxel of the brain, a spherical deconvolution problem must be solved. The efficient sampling schemes and sparse reconstruction methods that we will develop to solve inverse problems will be directly applicable to improving both the acquisition and processing times, respectively, helping to render clinical use of diffusion MRI feasible. Indeed, McEwen has taken the first steps towards studying the application of his techniques to diffusion MRI already. Opening diffusion MRI up to clinical use will provide wide-ranging societal benefits for healthcare, particularly for the treatment of patients with neurological disorders such as strokes.

Computer graphics and vision techniques underly a number of large industries in the UK, for example film and computer gaming, which combined contribute well over a billion pounds to the UK economy. Many data-sets in computer graphics and vision are defined on the sphere, for example global environmental illumination maps. Acting as a consultant to Geomerics Ltd (http://www.geomerics.com), McEwen applied his wavelet methods on the sphere to the forward problem of fast rendering. We plan to explore the application of our methods to the inverse rendering problem of recovering the reflectance properties of objects, which is a deconvolution problem defined on the sphere. Object reflectance properties form an important input of all rendering algorithms, which are widely applied in the generation of special effects in film and in synthetic rending in games, where photo-realism depends heavily on an accurate characterisation of the properties of real objects. Such techniques would have a significant impact across a number of large industries where computer graphics and vision techniques are instrumental.

The techniques that we will develop will not only have an impact in medical imaging and computer graphics, but they will be of use in a wide variety of industries where data are defined on the sphere. For example, in acoustics, molecular biology and many others. We will ensure that our techniques are well-known in these fields by disseminating our results at high-profile international meetings and conferences.

We will develop techniques to tease meaningful information out of big data-sets and in the process will learn many lessons and new skills related to the analysis of big-data. The UK hosts a burgeoning information economy, where big-data challenges are playing a major role in shaping new industries and revolutionising current ones. We will share with these industries our experience in tackling the big-data challenge and the novel solutions that we will develop to address this challenge. Our research will therefore help to develop expertise in big-data, which will have positive spin-off implications for the wider UK economy. In addition to the targeted dissemination of our research to the big-data community, we will perform regular public engagement activities to raise the awareness and highlight the implications not only of our own research, but also of the wider challenge of developing novel techniques to analyse big-data.