Introduction

One of the most basic operations on signals in any domain, including 2-sphere, is linear filtering or convolution. Yet, unlike conventional time-domain signals, for signals on the 2-sphere this is not consistently well defined and there exist competing definitions.

So the first aim of this chapter is to study existing definitions of convolution on 2-sphere in the literature and identify their properties, advantages and shortcomings. We determine the relationship between various definitions and show that two seemingly different definitions are essentially the same (which stems from the azimuthally symmetric or isotropic convolution property inherent in those definitions) (Kennedy et al., 2011). Our framework reveals that none of the existing formulations are natural extensions of convolution in time domain. For example, they are not commutative. Changing the role of signal and filter would change the outcome of convolution. Moreover, in one definition, the domain of convolution output does not remain on the 2-sphere.

Recognizing that a well-posed general definition for convolution on 2-sphere which is anisotropic and commutative is more difficult to formulate and that a true parallel with Euclidean convolution may not exist, in the second part of this chapter we use the power of abstract techniques and the tools we have studied to show that a commutative anisotropic convolution on the 2-sphere can indeed be simply constructed. We discuss additional properties of the proposed convolution, especially in the spectral domain, and present some clarifying examples.

We begin with the familiar case of convolution in time domain or convolution on the real line, which shall guide our subsequent developments for convolution on the 2-sphere.

This is our book on the theory of Hilbert spaces, its methods and usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. There are many fine books on Hilbert spaces and our intention is not to generate another book to stick on the pile or to be used to level a desk. So from the onset, we have sought to synthesize the book with special goals in mind.

The needs and concerns of researchers in engineering differ from those of the pure sciences. It is difficult to put the finger on what distinguishes the engineering approach that we take. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a foundational concept that links deeply with other areas of mathematics or is part of a long-standing human intellectual endeavor — not that engineering, in comparison, concerns less intellectual pursuits. As an example, Carleson in 1966 proved a conjecture by Luzin in 1915 concerning the almost-everywhere convergence of Fourier series of continuous functions. Carleson's theorem, as it is called, has its roots in the questions Fourier asked himself, in French presumably, about the nature of convergence of the series named after him. As a result it is important for mathematics, but less clear for engineers.

Introduction

The processing of signals whose domain is the 2-sphere or unit sphere1 has been an ongoing area of research in the past few decades and is becoming increasingly more active. Such signals are widely used in geodesy and planetary studies (Simons et al., 1997; Wieczorek and Simons, 2005; Simons et al., 2006; Audet, 2011). In many cases of interest flat Euclidean modeling of planetary and heavenly data does not work. Planetary curvature should be taken into account especially for small heavenly bodies such as the Earth, Venus, Mars, and the Moon (Wieczorek, 2007). Other applications, for the processing of signals on the 2-sphere, include the study of cosmic microwave background in cosmology (Wiaux et al., 2005; Starck et al., 2006; Spergel et al., 2007), 3D beamforming/sensing (Simons et al., 2006; Górski et al., 2005; Armitage and Wandelt, 2004; Ng, 2005; Wandelt and Górski, 2001; Rafaely, 2004; Wiaux et al., 2006), computer graphics and computer vision (Brechbühler et al., 1995; Schröder and Sweldens, 2000; Han et al., 2007), electromagnetic inverse problems (Colton and Kress, 1998), brain cortical surface analysis in medical imaging (Yu et al., 2007; Yeo et al., 2008), and channel modeling for wireless communication systems (Pollock et al., 2003; Abhayapala et al., 2003). This type of processing exhibits important differences from the processing of signals on Euclidean domains—such as time-based signals whose domain is the real line R, or 2D or 3D signals and images, whose domain is multi-dimensional, but still Euclidean.

Introduction to Hilbert spaces

The basic idea

Hilbert spaces are the means by which the “ordinary experience of Euclidean concepts can be extended meaningfully into the idealized constructions of more complex abstract mathematics” (Bernkopf, 2008).

If our global plan is to abstract Euclidean concepts to more general mathematical constructions, then we better think of what it is in Euclidean space that is so desirable in the first place. An answer is geometry — in geometry one talks about points, lines, distances and angles, and these are familiar objects that our brains are well-adept to recognize and easily manipulate. Through imagery we use pictures to visualize solutions to problems posed in geometry. We may still follow Descartes and use algebra to furnish a proof, but typically through spatial reasoning we either make the breakthrough or see the solution to a problem as being plausible. Contrary to any preconception you may have, Hilbert spaces are about making obtuse problems have obvious answers when viewed using geometrical concepts.

The elements of Euclidean geometry such as points, distance and angle between points are abstracted in Hilbert spaces so that we can treat sets of objects such as functions in the same manner as we do points (and vectors) in 3D space. Hilbert spaces encapsulate the powerful idea that in many regards abstract objects such as functions can be treated just like vectors.

To others, less fond of mathematics, Hilbert spaces also encapsulate the logical extension of real and complex analysis to a wider sphere of suffering.