Wavelets on the sphere

Exact reconstruction with directional wavelets on the sphere

Y. Wiaux, J. D. McEwen, P. Vandergheynst, O. Blanc
Institute of Electrical Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Astrophysics Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
E-mail: yves.wiaux@epfl.chE-mail: mcewen@mrao.cam.ac.ukE-mail: pierre.vandergheynst@epfl.ch
July 14, 2019
Abstract

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999) and Wiaux et al. (2005). The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation is firstly identified. A scale discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background (CMB) data, such as the imprint of topological defects, in particular cosmic strings, and for their reconstruction after separation from the other signal components.

keywords:
methods: data analysis, techniques: image processing, cosmology: cosmic microwave background
pagerange: Exact reconstruction with directional wavelets on the sphereReferencespubyear: 2007

1 Introduction

Very generically, the scale-space analysis of a signal with wavelets on a given manifold defines wavelet coefficients which characterize the signal around each point of the manifold and at various scales (Mallat, 1998; Antoine et al., 2004; Antoine & Vandergheynst, 2007). Wavelet techniques find numerous applications in astrophysics (Starck et al., 2006a). It commonly concerns the analysis of data distributed on the real line of time, or images on the plane. But other experiments also acquire data in all directions of the sky. This is notably the case of observations of the cosmic microwave background (CMB) radiation, such as the current Wilkinson Microwave Anisotropy Probe (WMAP) satellite experiment, or the forthcoming Planck Surveyor satellite experiment. Sky surveys such as the NRAO Very Large Array Sky Survey (NVSS) also map data on a large fraction the celestial sphere. The scale-space analysis of such data sets requires wavelet techniques on the sphere. Various wavelet formalisms have been proposed to date (Holschneider, 1996; Freeden & Windheuser, 1996; Freeden et al., 1998; Antoine & Vandergheynst, 1998, 1999; Narcowich et al., 2005; McEwen et al., 2006). The formalism originated by Antoine & Vandergheynst (1999) in a group-theoretic context triggered various developments (Antoine et al., 2002; Demanet & Vandergheynst, 2003; Bogdanova et al., 2005), and was reconsidered in a more practical context by Wiaux et al. (2005). This approach notably found a recent and very interesting application in the analysis of the CMB data, as reviewed by McEwen et al. (2007b).

In particular, the denoising or the deconvolution of data represents a large field of application of wavelet techniques. Experimental data sets are indeed always affected by various noise sources, notably related to the instrumentation. The data can also be blurred by experimental beams associated with the instrumentation. Signals detected can also originate from different physical sources, which need to be separated. In that component separation perspective, each component can in turn be technically understood as a signal, while the other components are seen as noise. As an example, observed CMB data represent a superposition of the CMB signal itself with instrumental noise and foreground emissions, blurred by the experimental beam at each detection frequency. The denoising and the deconvolution of the signal, and the separation of its astrophysical components is of major interest for astrophysics and cosmology.

Signals with features defined at specific positions and scales typically have a sparse expansion in terms of wavelets. For such signals, denoising and deconvolution algorithms are generically much more efficient when applied to the wavelet coefficients (Mallat, 1998; Daubechies, 2004). However this requires a scheme allowing the exact reconstruction of the signals analyzed from their wavelet coefficients. Moreover, localized characteristics can be elongated, in which case directional wavelets are essential. The identification and the reconstruction of localized directional features in CMB data represents a very interesting application of such a framework. It typically concerns the imprint of topological defects such as textures or cosmic strings (Kaiser & Stebbins, 1984; Turok & Spergel, 1990; Vilenkin & Shellard, 1994; Hindmarsh & Kibble, 1995). This application can be recast in a component separation approach where all continuous and typically Gaussian emissions are seen as noise, in contrast with localized directional features. Let us also emphasize that such a framework can have many applications well beyond astrophysics, from geophysics to biomedical imaging, or computer vision.

At present, the simultaneous combination of the properties of exact reconstruction and directionality is lacking in the existing wavelet formalisms on the sphere. It has only been considered for the wavelet analysis of signals on the plane (Simoncelli et al., 1992; Vandergheynst & Gobbers, 2002). The primary aim of the present work resides in the development of a new scale discretized wavelet formalism for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. As a by-product, a new continuous wavelet formalism is also obtained, which allows the analysis of signals with a new family of wavelets relative to existing formalisms. But the continuous range of scales required for the analysis prevents exact reconstruction in practice, for which the scale discretized wavelet formalism proposed is essential.

The remainder of this paper is organized as follows. In Section 2, we present an existing scheme for the definition of a continuous wavelet formalism on the sphere from a generic dilation operation. We consider directional and axisymmetric wavelets and discuss the cases of the stereographic and harmonic dilations. In Section 3, we propose a new kernel dilation. A corresponding family of factorized steerable functions with compact harmonic support is identified. We show that localization and directionality properties of such functions can be controlled through kernel dilation. In Section 4, we derive a new continuous wavelet formalism from the kernel dilation with continuous scales. We then derive a new scale discretized wavelet formalism that allows the exact reconstruction of band-limited signals in practice. We design explicitly an example wavelet. We finally recast the scale discretized wavelet formalism in an invertible filter bank approach. In Section 5, we describe an exact algorithm accounting for the multi-resolution properties of the formalism. The memory and computation time requirements are discussed and an implementation is tested. In Section 6, we discuss the application of the formalism to the detection of cosmic strings through the denoising of full-sky CMB data. We finally conclude in Section 7.

2 Wavelets from a generic dilation

In this section we discuss an existing scheme for the definition of a continuous wavelet formalism on the sphere from a generic dilation operation. We consider directional and axisymmetric wavelets explicitly. We finally discuss in detail the stereographic and harmonic dilations.

2.1 Directional wavelets

In the continuous framework developed by Antoine & Vandergheynst (1999) and Wiaux et al. (2005), the wavelet analysis of a signal on the sphere, i.e. the unit sphere , defines wavelet coefficients through the correlation of the signal with dilated versions of a local analysis function. Theoretically, the signal can be recovered explicitly from its wavelet coefficients provided that the local analysis functions satisfies some admissibility condition, raising it to the rank of a wavelet.

The real and harmonic structures of are summarized concisely as follows. We consider a three-dimensional Cartesian coordinate system centered on the sphere, and where the direction identifies the North pole. Any point on the sphere is identified by its corresponding spherical coordinates , where stands for the co-latitude, or polar angle, and for the longitude, or azimuthal angle. Let the continuous signal and the local analysis function be square-integrable functions on the sphere: , with the invariant measure . The spherical harmonics form an orthonormal basis for the decomposition of square-integrable functions. They are explicitly given in a factorized form in terms of the associated Legendre polynomials and the complex exponentials as

(1)

with , , and (Abramowitz & Stegun, 1965; Varshalovich et al., 1989). The index represents an overall frequency on the sphere. The absolute value represents the frequency associated with the azimuthal variable . Any function is thus uniquely given as a linear combination of scalar spherical harmonics: . This combination defines the inverse spherical harmonic transform on . The corresponding spherical harmonic coefficients are given by the projection , with , where the bracket generically denotes the scalar product for . This projection defines the direct spherical harmonic transform on .

Continuous affine transformations such as translations, rotations, and dilations are applied to the analysis function. The continuous translations by and rotations by are defined by the three Euler angles defining an element of the group of rotations in three dimensions . The operator in for the translation of amplitude of a function reads as

(2)

where is defined by the three-dimensional rotation matrices and , acting on the Cartesian coordinates associated with . The rotation operator in for the rotation of the function around itself, by an angle , is given as

(3)

where also follows from the action of the three-dimensional rotation matrix on the Cartesian coordinates associated with . The operator incorporating both the translations and rotations simply reads as and , with . The continuous dilations affect by definition the continuous scale of the function. The notion of scale may a priori be defined both in real or in harmonic space on . In the remainder of the present subsection we simply denote the dilated function as , where stands for a continuous dilation factor. We explicitly discuss two possible definitions of dilations in Subsections 2.3 and 2.4.

The analysis of the signal with an analysis function defines wavelet coefficients through the directional correlation of with the dilated functions , i.e. the scalar products

(4)

At each scale , the wavelet coefficients therefore identify a square-integrable function on the rotation group in three dimensions SO(3). They characterize the signal around each point , and in each orientation . This defines the scale-space nature of the wavelet decomposition on the sphere.

The real and harmonic structures of the rotation group in three dimensions SO(3) are summarized concisely as follows. As discussed, any rotation on SO(3) is given in terms of the three Euler angles , with , and . Let be a square-integrable function on SO(3): , with the invariant measure . The Wigner -functions are the matrix elements of the irreducible unitary representations of weight of the group in . By the Peter-Weyl theorem on compact groups, the matrix elements also form an orthogonal basis in . They are explicitly given in a factorized form in terms of the real Wigner -functions and the complex exponentials, and , as

(5)

with , , and (Varshalovich et al., 1989; Brink & Satchler, 1993). Again, represents an overall frequency on SO(3), and and the frequencies associated with the variables and , respectively. Any function is thus uniquely given as a linear combination of Wigner -functions: . This combination defines the inverse Wigner -function transform on SO(3). The corresponding Wigner -function coefficients are given by the projection . This projection defines the direct Wigner -function transform on SO(3).

At each scale, the direct Wigner -function transform of the wavelet coefficients is given as the pointwise product of the spherical harmonic coefficients of the signal and the wavelet:

(6)

Indeed, the orthonormality of scalar spherical harmonics implies the Plancherel relation for , and the action of the operator on reads in terms of its spherical harmonic coefficients as

The reconstruction of a signal from its wavelet coefficients with an analysis function is given as

(7)

In this relation, the scale integration measure is part of the definition of the dilation operation itself (see Subsections 2.3 and 2.4). The operator in is defined by its action on the spherical harmonic coefficients of a function : . The reconstruction formula holds if and only if the analysis function satisfy the following admissibility condition for all :

(8)

In this case the analysis function is by definition raised to the rank of a wavelet. From relation (8), the admissibility condition intuitively requires that the whole wavelet family {}, for , covers each frequency index with a finite and non-zero amplitude, hence preserving the signal information at each frequency. Notice that a direct connection exists between the generic relations (7) and (8) for the signal reconstruction, and the theory of frames on the sphere (Bogdanova et al., 2005).

We generally consider band-limited signals. Any function is said to be band-limited with band limit , for any , if for all with . Any function is said to be band-limited with band limit , for any , if for all with . From relation (6), if the signal or the wavelet are band-limited on , then the wavelet coefficients are automatically band-limited on SO(3), with the same band limit .

Let us already notice that the reconstruction is ensured theoretically from relation (7), through the integration on the continuous parameter for translations and rotations of the wavelet, and on the continuous dilation factor . But in practice, the reconstruction would require the definition of exact quadrature rules for the numerical integrations. Exact quadrature rules for integration of band-limited signals on exist on equi-angular (Driscoll & Healy, 1994) and Gauss-Legendre (Doroshkevich et al., 2005a, b) pixelizations of . HEALPix pixelizations (Górski et al., 2005)111http://healpix.jpl.nasa.gov/ of on provide approximate quadrature rules which can also be made very precise thanks to an iteration process. Pixelizations may for instance be defined on by combining pixelizations on with an equi-angular sampling of . Corresponding quadrature rules can be made exact on the pixelizations based on equi-angular and Gauss-Legendre pixelizations on , while those based on HEALPix pixelizations are approximate. This extension basically relies on the separation of the integration variables (Maslen & Rockmore, 1997a, b; Kostelec & Rockmore, 2003) from relation (5).

However, exact quadrature rules do not exist for the integration over scales . In practice, this prevents an exact reconstruction of the signal analyzed. A scheme allowing an exact reconstruction requires a discretization of the dilation factor. A scale discretized wavelet formalism is proposed in Section 4, thanks to a specific choice of dilation, and through an integration of the dilation factor by slices in relation (7).

2.2 Axisymmetric wavelets

Any general function explicitly dependent on the azimuthal angle , is said to be directional: . By opposition, any function independent of the azimuthal angle is said to be zonal, or axisymmetric: . It only exhibits non-zero spherical harmonic coefficients for : .

In this particular case, the directional correlation of a signal with reduces to a standard correlation obviously independent of the rotation angle (Wiaux et al., 2006). The analysis of with an axisymmetric analysis function defines wavelet coefficients through the standard correlation of with the dilated functions , i.e. the scalar products

(9)

At each scale , the wavelet coefficients identify a square-integrable function on rather than on SO(3). The spherical harmonic transform of the wavelet coefficients is still given as the pointwise product of the spherical harmonic coefficients of the signal and the wavelet:

(10)

This relation simply follows from relation (6) and the equality .

The reconstruction of from its wavelet coefficients reads as:

(11)

for any scale integration measure , and with the operator in defined by: . The reconstruction formula holds if and only if the analysis function satisfies the following admissibility condition for all :

(12)

2.3 Stereographic dilation

In the original set up proposed by Antoine & Vandergheynst (1999) the stereographic dilation of functions is considered, which is explicitly defined in real space on . The stereographic dilation operator on , for a continuous dilation factor , is defined in terms of the inverse of the corresponding stereographic dilation on points in . It reads as

(13)

with . The dilated point is given by with the linear relation . The dilation operator therefore maps the sphere without its South pole on itself: . This dilation operator is uniquely defined by the requirement of the following natural properties. The dilation of points on must be a radial (i.e. only affecting the radial variable independently of , and leaving invariant) and conformal (i.e. preserving the measure of angles in the tangent plane at each point) diffeomorphism (i.e. a continuously differentiable bijection). The normalization by in (13) is uniquely determined by the requirement that the dilation of functions in be a unitary operator (i.e. preserving the scalar product in , and specifically the norm of functions). Notice that the stereographic dilation operation is supported by a group structure for the composition law of the corresponding operator . A group homomorphism also holds with the operation of multiplication by on .

In this setting, the effect of the dilation on the spherical harmonic coefficients of the dilated function is not easily tractable analytically. Consequently, the admissibility condition (8) is difficult to check in practice. On the contrary, wavelets on the plane are well-known, and may be easily constructed, as the corresponding admissibility condition reduces to a zero mean condition for a function both integrable and square-integrable. In that context, a correspondence principle was proved (Wiaux et al., 2005), stating that the inverse stereographic projection of a wavelet on the plane leads to a wavelet on the sphere. This correspondence principle notably requires the definition of a scale integration measure identical to the measure used on the plane: . Notice that this measure naturally appears in the original group-theoretic context (Antoine & Vandergheynst, 1999).

2.4 Harmonic dilation

Another possible definition of the dilation of functions may be considered, which is explicitly defined in harmonic space on . It was proposed in previous developments relative to the definition of a wavelet formalism on the sphere (Holschneider, 1996; McEwen et al., 2006). The harmonic dilation is defined directly on through a sequence of prescriptions rather than in terms of the application of an simple operator. Firstly, an arbitrary prescription must be chosen to define a set of generating functions of a continuous variable , for each . These functions are identified to the spherical harmonic coefficients of through: for , and . Secondly, the variable is dilated linearly, , just as would be the norm of the Fourier frequency on the plane. For a continuous dilation factor , the spherical harmonic coefficients of the dilated function are defined by:

(14)

In the corresponding continuous wavelet formalism, the analysis function must satisfy the following form of the admissibility condition (8). On the one hand , which corresponds to the requirement that has a zero mean on the sphere:

(15)

This zero mean is of course preserved through harmonic dilation. As the zero frequency is not supported by the wavelets, only signals with zero mean can be analyzed in this formalism (see relation (6)). Let us remark that wavelets on the sphere dilated through the stereographic dilation do not necessarily have a zero mean. On the other hand, the scale integration measure can arbitrarily be chosen as . This leads to a simple expression of the remaining constraints for as

(16)

The left-hand side inequality implies for at least one of the first two generating functions: . In other words, either or must be non-zero on a set of non-zero measure on . The right-hand side inequality implies for all generating functions: . Hence, the generating functions must satisfy (this condition encompasses the zero mean condition (15) in the form ) and tend to zero when . With this choice of scale integration measure, the constraints summarize to the requirement that each generating function satisfies a condition very similar to the wavelet admissibility condition for an axisymmetric wavelet on the plane (Antoine et al., 2004)222The exact wavelet admissibility condition on the plane reduces to a zero mean condition for functions that are both integrable and square-integrable. defined by a Fourier transform identical to . Consequently, the wavelet admissibility condition (16) can be checked in practice and wavelets associated with the harmonic dilation can be designed easily.

For continuous axisymmetric wavelets, a unique generating function of a continuous variable is required. The admissibility condition (12) reduces to the following expression. The analysis function must have a zero mean and only allows the analysis of signals with zero mean. A unique additional condition holds independently of :

(17)

This condition actually encompasses the zero mean condition in the form , and also requires that the generating function must tend to zero when . The coefficients entering the reconstruction formula (11) read as , for .

2.5 Discussion

On the one hand, the harmonic dilation lacks some of the important properties which hold under stereographic dilation. As the harmonic dilation does not act on points, the question of the corresponding properties of a radial and conformal diffeomorphism make no sense. The harmonic dilation of functions is not either a unitary procedure. It does not preserve the scalar product in , or specifically the norm of functions. This is due to the requirement of definition of generating functions for any function to be dilated. A group structure for the composition of harmonic dilations holds only if successive dilations of a function are defined through linear dilation of the variable of a unique generating function for each . The same condition applies for the existence of a corresponding homomorphism structure with the operation of multiplication by on .

Moreover, the harmonic dilation is explicitly defined in harmonic space. The evolution in real space of localization and directionality properties of functions on the sphere through harmonic dilation is therefore not known analytically. However, in the Euclidean limit where a function is localized on a small portion of the sphere, this portion is assimilated to the tangent plane, and the stereographic and harmonic dilations both identify with the standard dilation in the plane. For each , the overall frequency index identifies with the continuous variable , corresponding to the norm of the Fourier frequency on the plane (Holschneider, 1996). So in particular, the evolution of localization properties of functions through harmonic dilation is at least controlled in the Euclidean limit.

On the other hand, the very simple action of the harmonic dilation in harmonic space also exhibits several advantages relative to the stereographic dilation. Notably, the harmonic dilation ensures that the band limit of a wavelet and of the corresponding wavelet coefficients, is reduced by a factor . Such a multi-resolution property is essential in reducing the memory and computation time requirements for the wavelet analysis of signals. It does not hold under stereographic dilation. Moreover, as already emphasized, a scheme allowing an exact reconstruction of signals from their wavelet coefficients requires a discretization of the dilation factor. The definition of a scale discretized wavelet formalism through an integration of the dilation factor by slices in the continuous wavelet formalism turns out to be very natural with a dilation defined in harmonic space, but not with the stereographic dilation. Indeed, one would like the dilation operation acting on scale discretized functions after the integration of the dilation factor by slices to be the same as the original dilation operation. It will become obvious that this property holds for a dilation defined in harmonic space, but not for the stereographic dilation.

In conclusion, no obvious definition of dilation is imposed for the development of a wavelet formalism on the sphere. But considering our aim for a scale discretized wavelet formalism, as well as the essential criterion of defining a formalism with multi-resolution properties, we will focus on a scale discretized wavelet formalism from a dilation defined in harmonic space. However, for any dilation defined in harmonic space, the evolution of the localization and directionality properties of functions in real space through dilation needs to be understood and controlled. In that regard, we amend the harmonic dilation (14) and define a kernel dilation to be applied on functions which are said to be factorized steerable functions with compact harmonic support. Moreover, the kernel dilation will also render the transition between the continuous and scale discretized formalism much simpler and more transparent than what the harmonic dilation can provide.

3 Kernel dilation

In this section we define the kernel dilation on factorized functions in harmonic space on the sphere. We consider in particular factorized steerable functions with compact harmonic support. We also study the localization and directionality properties in real space for such functions, as well as the controlled evolution of these properties through kernel dilation.

3.1 Factorized functions and kernel dilation

A function can be defined to be a factorized function in harmonic space if it can be written in the form:

(18)

for , and . The positive real kernel is a generating function of a continuous variable , initially evaluated on integer values . The directionality coefficients , for , and , define the directional split of the function. In particular, for a real function , they bear the same symmetry relation as the spherical harmonic coefficients themselves: . Without loss of generality one can impose

(19)

for the values of for which is non-zero for at least one value of . Hence localization properties of a function , such as a measure of dispersion of angular distances around its central position as weighted by the function values, are governed by the kernel and to a lesser extent by the directional split. Indeed, the power contained in the function at each allowed value of is fixed by the kernel only. The norm of reads as , where the sum runs over the values of for which is non-zero for at least one value of . However, the directional split is essential in defining the directionality properties measuring the behaviour of the function with the azimuthal variable , because of it bears the entire dependence of the spherical harmonic coefficients of the function in the index .

The kernel dilation applied to a factorized function (18) is simply defined by application of the harmonic dilation (14) to the kernel only. The directionality of the dilated function is defined through the same directional split as the original function. For a continuous dilation factor , the dilated function therefore reads as:

(20)

Let us emphasize that the directionality coefficients are not affected by dilations, on the contrary of what the complete action of the harmonic dilation (14) would imply. The kernel and harmonic dilations strictly identify with one another when applied to factorized axisymmetric functions , for which the directional split takes the trivial values for .

3.2 Compact harmonic support

Any function can be said to have a compact harmonic support in the interval , for any and any real value , if

(21)

where denotes the largest integer value below . Notice that the compactness of the harmonic support of can be defined as the ratio of the band limit to the width of its support interval.

For a factorized function of the form (18), the compact harmonic support in the interval is ensured by the choice of a kernel with compact support in the interval :

(22)

The compactness of the harmonic support of can simply be estimated from the compact support of the kernel as . One has when , and when . Typical values would be corresponding to a compactness , or leading to a higher compactness .

By a kernel dilation with a dilation factor in (20), the compact support of the dilated kernel is defined in the interval . The compact harmonic support of the dilated function itself is thus defined in the corresponding interval , where denotes the smallest integer value above . In particular, the compactness of the harmonic support of a function remains invariant through a kernel dilation.

3.3 Steerable functions

The notion of steerability was first introduced on the plane (Freeman & Adelson, 1991; Simoncelli et al., 1992), and more recently defined on the sphere (Wiaux et al., 2005). By definition, is steerable if any rotation of the function around itself may be expressed as a linear combination of a finite number of basis functions :

(23)

The square-integrable functions on the circle , with , and , are called interpolation weights. Intuitively, steerable functions have a non-zero angular width in the azimuthal angle , which renders them sensitive to a range of directions and enables them to satisfy the steerability relation. This non-zero angular width naturally corresponds to an azimuthal band limit in the frequency index associated with the azimuthal variable :

(24)

It can actually be shown that the property of steerability (23) is equivalent to the existence of an azimuthal band limit (24).

On the one hand, if a function is steerable with basis functions, then the number of values of for which has a non-zero value for at least one value of is lower or equal to : . This was firstly established for functions on the plane (Freeman & Adelson, 1991), and the proof is absolutely identical on the sphere. As a consequence, the function has some azimuthal band limit , with .

On the other hand, if a function has an azimuthal band limit , then it is steerable, and the number of basis functions can be reduced at least to . This second part of the equivalence can be proved by explicitly deriving a steerability relation for band-limited functions with an azimuthal band limit . Any band-limited function can in particular be steered using rotated versions as basis functions, and interpolation weights given by simple translations by of a unique square-integrable function on the circle :

(25)

for specific rotation angles with . One may choose equally spaced rotation angles as , with . The function is then defined by the Fourier coefficients for and otherwise. Notice that the angles and the structure of the function are independent of the explicit non-zero values .

Typically, if has a non-zero value for at least one value of for all with , then and the function is optimally steered by these angles and the function described. On the contrary, when values of , with , exist for which for all values of , then and one might want to reduce the number of basis functions. Depending on the distribution of the values of for which has a non-zero value for at least one value of , the number of basis functions required to steer the band-limited function may indeed be optimized to its smallest possible value . This optimization is notably reachable for functions with specific distributions of the values of , corresponding to particular symmetries in real space. For example, a function is even or odd through rotation around itself by if and only if has non-zero values only for, respectively, even or odd values of . This property notably implies that the central position of the function identifies with the North pole, in the sense that its modulus is then always even through rotation around itself by . The combination of an azimuthal band limit with that symmetry reads as:

(26)

with

(27)

In this particular case, and one may choose equally spaced rotation angles as , with , and steer the function through relation (25). The function is defined by the Fourier coefficients for and otherwise.

In summary, the property of steerability is indeed equivalent to the existence of an azimuthal band limit in . For a factorized function of the form (18), steerability constraints such as (24) and (26) are ensured by the directionality coefficients , independently of the kernel. Consequently, any relation of steerability remains unchanged through a kernel dilation (20), which by definition only affects the kernel.

3.4 Localization control

Let us consider the Euclidean limit where a function is localized on a small portion of the sphere which can be assimilated to the tangent plane. As discussed, the harmonic dilation (14) identifies with the standard dilation in the plane in that limit (Holschneider, 1996). Hence, for factorized steerable functions with compact harmonic support, the kernel dilation (20) certainly shares the same property if it identifies with the harmonic dilation itself in the limit . This is ensured by considering functions with directionality coefficients which become independent of in the limit . Consequently, the evolution of localization properties of functions through kernel dilation is also controlled in the Euclidean limit. But a much more important localization property holds for the kernel dilation at any frequency range for factorized functions with compact harmonic support.

A typical localization property of a function is a measure of dispersion of angular distances around its central position, as weighted by the function values. The corresponding measure in harmonic space is defined by the dispersion of the values of around their central position, as weighted by the values of the spherical harmonic coefficients , for each value of . It is well-known that the smaller the dispersion in real space, the larger the dispersion in harmonic space. An optimal Dirac delta distribution on the sphere exhibits an infinite series in of spherical harmonic coefficients: . On the contrary a spherical harmonic , completely non-localized in real space on , by definition exhibits a unique frequency .

In particular, we need to understand the evolution of this localization property of a factorized steerable function with compact harmonic support through the kernel dilation (20). Let us consider for simplicity a factorized axisymmetric function with compact harmonic support, for which the kernel and harmonic dilations identify with one another. For an initial compact harmonic support in the interval , the kernel dilation by a factor modifies the interval to . Hence in harmonic space the width of the harmonic support interval is multiplied by . This also measures the evolution of the dispersion in harmonic space. In real space, one can intuitively consider that the corresponding dispersion of the values of the angular distance around the North pole (which is the central position of any axisymmetric function) is multiplied by . This intuition is actually only exact in the Euclidean limit , reached when . But a weaker property holds though, on a wide class of axisymmetric functions on the sphere, in particular on factorized axisymmetric functions with compact harmonic support. It takes the form of the following upper bound through the kernel dilation by of such an axisymmetric function at a given angular distance from the North pole:

(28)

for any integer and for some constant depending on and (Narcowich et al., 2005). The ratio of the bounds at the North pole and at any fixed angular distance simply reads as . When increases, this ratio gets closer to unity and the bound is less constraining, enabling a larger dispersion of the values of the angular distance around the North pole. When decreases, the ratio increases and the bound is more constraining, hence imposing a smaller dispersion of the values of the angular distance around the North pole. This ensures a good behaviour in real space for the kernel dilation, when applied to factorized axisymmetric functions with compact harmonic support.

In summary, the dispersion of angular distances around the central position of a function defines a localization property. We have shown that the evolution of the localization of factorized axisymmetric functions with compact harmonic support through kernel dilation is controlled by the bound (28). For completeness, the corresponding bound should be analyzed for the kernel dilation of factorized steerable functions with compact harmonic support, but this goes beyond the scope of the present work. The verification of more detailed localization properties in real space for a function designed from its spherical harmonic coefficients requires a numerical evaluation of sampled values of that function.

3.5 Directionality control

Let us consider a typical directionality property of a function , such as measured by its auto-correlation function. The auto-correlation function of is defined as the scalar product between two rotated versions of the function around itself by angles . This auto-correlation only depends on the difference of the rotation angles and is therefore considered in the space of square-integrable functions on the circle : . The peakedness of the auto-correlation function in can be considered as a measure of the directionality of the function: the more peaked the auto-correlation, the more directional the function (Wiaux et al., 2005). From the Plancherel relation for , and the expression for the action of the operator on , one gets

(29)

The value of the auto-correlation function at obviously defines the square of the norm of the function : .

For a factorized function (18), the auto-correlation function is strongly related to the directional split. Let us also recall that in the case of a steerable function defined through (25), the interpolation weights depend on the values of for which the spherical harmonic coefficients have non-zero values and on the rotation angles , but not on the values themselves. This leaves enough freedom to design a suitable auto-correlation function and thus control the directionality of the function. Let us also consider a compact harmonic support (21) in the interval . We analyze the particular case where the directionality coefficients are independent of for ,

(30)

and where is lower or equal to the lowest integer value above the lower bound of the compact harmonic support interval, i.e. . In that limit, the auto-correlation reads as

(31)

In other words, the square of the complex norm of the directionality coefficients identifies with the Fourier coefficients of in . Notice that a better directionality of a steerable function, as measured by its auto-correlation function, is inevitably associated with a larger band limit , and with a larger number of values of for which has a non-zero value for at least one value of . Indeed, on the circle as on the plane or the sphere, the smaller the dispersion of in real space, the larger the dispersion of in harmonic space. Consequently, a better directionality of a steerable function requires an increased number of basis functions.

We need to understand the evolution of this directionality property of a factorized steerable function with compact harmonic support through the kernel dilation (20). The correlation function of two dilated versions and by factors and in is defined through the scalar product . We consider again the case where the directionality coefficients are independent of for and where the azimuthal band limit for steerability is lower than the lower bound of the compact harmonic support of each of the two dilated versions of : and . In that limit, the correlation reads as

(32)

and appears to be simply proportional to the auto-correlation function of . For , this result states that the auto-correlations of and are proportional. The control of directionality of through the auto-correlation function is therefore preserved through kernel dilation. For , this result essentially ensures that the kernel dilation does not introduce any unexpected distortion in the shape of the function in real space on .

In addition to the auto-correlation function, symmetry properties may also be imposed on the spherical harmonic coefficients , which translate into simple directionality properties in real space for . Firstly, in the framework of a wavelet analysis on the sphere, one generally imposes the symmetry relation in order to restrict to real analysis functions: . Secondly, the constraint that has non-zero values only for even or odd values , for any azimuthal band limit , implies that the function is respectively even or odd under a rotation around itself by : . Thirdly, for such functions, the additional constraint that the spherical harmonic coefficients are real for even values of , and purely imaginary for odd values of , implies that the function is respectively even or odd under a change of sign on : . These symmetries are defined up to a rotation of the function around itself by any angle , which amounts to a multiplication of the spherical harmonic coefficients by a complex phase . The three properties discussed are obviously preserved through kernel dilation of factorized functions. They indeed only concern the directionality coefficients , which are not affected by the kernel dilation.

In summary, the auto-correlation function and additional symmetries define directionality properties of a function. We have shown that the directionality properties studied are essentially preserved through kernel dilation of factorized steerable functions with compact harmonic support. Again, the verification of more precise directionality properties in real space for a function designed from its spherical harmonic coefficients unavoidably requires a numerical evaluation of sampled values of that function.

4 Wavelets from kernel dilation

In this section we begin with the derivation of a new continuous wavelet formalism from the kernel dilation with continuous scales, and for factorized steerable wavelets with compact harmonic support. We then derive the scale discretized wavelet formalism from the continuous wavelet formalism. The transition is performed through an integration of the dilation factor by slices. We emphasize the practical accessibility of an exact reconstruction of band-limited signals from a finite number of analysis scales. We also illustrate these developments through the explicit design of an example scale discretized wavelet. We finally recast the scale discretized wavelet formalism developed in a generic invertible filter bank perspective.

4.1 Continuous wavelets

We simply consider the continuous wavelet formalism exposed in Section 2, and particularize it to the kernel dilation defined in Section 3. Hence the scales of analysis are still continuous. The translations by and proper rotations by of the wavelets are still defined through the continuous three-dimensional rotations from relation (2) and (3).

For application of the kernel dilation, we consider continuous factorized steerable functions with compact harmonic support:

(33)

for a continuous kernel defined by a positive real function and a directional split defined by the directionality coefficients . The compact harmonic support of the wavelet in the interval is ensured by a kernel with compact support in the interval , with a compactness :

(34)

The steerability of a wavelet with an azimuthal band limit in ensured by the directional split:

(35)

with

(36)

for all . Continuous axisymmetric wavelets with compact harmonic support are simply obtained by the trivial directional split with for all .

The analysis of a signal with the analysis function gives the wavelet coefficients at each continuous scale , around each point , and in each orientation , through the directional correlation (4). The reconstruction of from its wavelet coefficients results from relation (7). The zero mean condition (15) for the admissibility of implies . One can also set arbitrarily . The admissibility condition (16) summarizes to:

(37)

which actually also encompasses the zero mean condition. The coefficients entering the reconstruction formula are for . In other words, the kernel must formally be identified with the Fourier transform of an axisymmetric wavelet on the plane.

Notice that for a factorized wavelet , the directional correlation defining the analysis of a signal may also be understood as a double correlation, by the kernel and the directional split successively. The standard correlation (9) of the signal and the axisymmetric wavelets defined by the kernel of , provides intermediate wavelet coefficients on at each scale . The spherical harmonic transform of these coefficients reads as:

(38)

At each scale , the directional correlation of the intermediate signal obtained at that scale and a directional wavelet defined by the directional split of provides the final wavelet coefficients on SO(3):

(39)

This reasoning obviously holds independently of the steerability or compact harmonic support properties of .

In conclusion, the definition of the kernel dilation provides a new continuous wavelet formalism, where scales, translations, and proper rotations of the wavelets are all continuous. As the previously developed continuous wavelet formalism based on the stereographic dilation, it finds application in the identification of local directional features of signals on the sphere. The wavelets defined bear new properties of compact harmonic support and steerability, which are preserved through kernel dilation. These properties can give a new insight for the analysis of local directional features. However, as already discussed the continuous scales required for the analysis prevent in practice the exact reconstruction of the signals analyzed from their wavelet coefficients.

4.2 Scale discretized wavelets

Scale discretized wavelets can simply be obtained from continuous wavelets through an integration by slices of the dilation factor . Through this transition procedure, scale discretized wavelets remain factorized steerable functions with compact harmonic support, and are dilated through kernel dilation.

We consider the analysis of a signal